Groupe International de recherche University of Udine

Developing Form

alThinking in P

hysics

Groupe International de recherche sur l’enseignement de la Physique

University of UdineInterdepartmental Centre

for Research in Education (CIRD)

editors Marisa Michelini

Marina Cobal

Developing Formal Thinking in Physics

First International Girep seminar 2001Selected contributions

Editorial boardManfred Euler, President of Girep, Department of Physics Education, IPN, Kiel, GermanyMarisa Michelini, Vicepresident of Girep, CIRD, University of Udine, ItalyIan Lawrence, Vicepresident of Girep, Department of Education, University of Birmingham, UKSeta Oblack, Secretary of Girep, Board of Education, Ljubljana, SloveniaLorenzo Santi, Physics Department, University of Udine, ItalyChristian Ucke, Physics Department E20, Techn. Universität München, Germany

EditorsMarisa MicheliniMarina Cobal

© 2002 Forum, Editrice Universitaria Udinese srlVia Palladio 8, I - 33100 Udine, Italy

All rights reserved. No part of this publication may be translated, reproduced, stored in a retrieval systemor transmitted in any form or by other any means, electronic, mechanical, photocopying, recording orotherwise, without prior permission of the publisher.

Printed in Italy - Lithostampa, Pasian di Prato (UD) - December 2002ISBN:

Developing Formal Thinking in PhysicsSelected contributionsof the First Girep seminar, 2-6 September 2001, Udine, Italy

First International Girep seminar 2001Selected contributions

Groupe International de Recherche sur l’Enseignement de la Physique

University of UdineInterdepartmental Centre

for Research in Education (CIRD)

editors Marisa Michelini

Marina Cobal

Developing Formal Thinking in Physics

Table of contentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1. Background Aspects

IMAGERY AND FORMAL THINKING: APPROACHES TO INSIGHT AND UNDERSTANDING IN PHYSICS EDUCATION,M. Euler, Leibniz Institute for Science Education, University of Kiel (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

PHYSICS CURRICULUM REFORM: HOW CAN WE DO IT?,G. Fuller, Physics Department and Astronomy, University of Nebraska-Lincoln (USA) . . . . . . . . . . . . . . . . . . . . . 27

REAL-TIME APPROACHES IN THE DEVELOPMENT OF FORMAL THINKING IN PHYSICS,E. Sassi, Physics Department Science, University “Federico II” of Napoli (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

DIFFERENCES BETWEEN THE USE OF MATHEMATICAL ENTITIES IN MATHEMATICS AND PHYSICS AND THE CONSEQUENCES FOR AN INTEGRATED LEARNING ENVIRONMENT,T. Ellermejer and A. Heck, AMSTEL Institute, University of Amsterdam (Netherlands). . . . . . . . . . . . . . . . . . . . . 52

AN EPISTEMOLOGICAL FRAMEWORK FOR LABWORK IN EXPERIMENTAL SCIENCES,M. Vicentini, Physics Department, University “La Sapienza”, Roma (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

IS FORMAL THINKING HELPFUL IN EVERYDAY SITUATIONS?,S. Oblak, Faculty of Education, University of Ljubljana (Slovenia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

THE FORMAL REASONING OF QUANTUM MECHANICS: CAN WE MAKE IT CONCRETE? SHOULD WE?D. Zollmann, Kansas State University (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

JUMPING TOYS: A TOPIC FOR INTERPLAY BETWEEN THEORY AND EXPERIMENTS,C. Ucke, Physics Department E20, Technical University of Munich (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2. Special Aspects

A. INTERPLAY OF THEORY AND EXPERIMENT,De Ambrosis, Physics Department, University of Pavia (Italy)G. Rinaudo, Department of Experimental Physics, University of Torino (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

EXPERIMENTS AND REFLECTIVE LEARNING,M. Bandiera and M. Vicentini, Physics Department, University “La Sapienza”, Roma (Italy) . . . . . . . . . . . . . . . . 109

GALILEI’S EXPERIMENT ON INCLINED PLANE,A. De Ambrosis, Physics Department, University of Pavia (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

WATER COOLING: HOW TO BUILD A PHYSICAL MODEL FOR AN EVERY DAY LIFE EXPERIMENT,A. Sconza, Physics Department, University of Padova (Italy). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

LEARNING PHYSICS VIA MODEL CONSTRUCTION,R.M. Sperandeo-Mineo, Department of Physical and Astronomical Sciences, University of Palermo (Italy) . . . . 117

THE DEVELOPMENT OF FORMAL REASONING,I. Lawrence, School of Education, University of Birmingham (UK),P. Guidoni, Department of Physical Sciences, University of Napoli (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

TOYS FOR LEARNING PHYSICS,C. Ucke, Physics Department E20, Technical University, Munich (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

NEW TECHNOLOGY AND COMPUTER IN PHYSICS LEARNING,L. Rogers, School of Education, University of Leicester (UK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

TEXTBOOKS AS AN IMAGE OF PHYLOSOPHY OF TEACHING,Z. Golab-Meyer, Physics Department, Jagellonian University (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3. Topical Aspects

3.1 Laboratory and Theory

USING EXPERIMENTAL LABORATORIES TO TEACH FORMAL PHYSICS,S. Kocijancic and C. O’Sullivan, Faculty of Education, University of Ljubljana, Slovenia . . . . . . . . . . . . . . . . . . . . 129

PHYSICS: FACING THE PRESENT TO FOSTER THE FUTURE,T. Lobato and M. Saraiva-Neves, Escola Secundaria de Fonseca Benevides, Lisbon (Portugal) . . . . . . . . . . . . . . . 132

TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR,S. Lasič, G. Planinšič, Faculty of Mathematics and Physics, University of Ljubljana (Slovenia),G. Torzo, Physics Department, University of Padova (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

ELECTROSTATIC MOTOR,N. Miklavčič, Srednja Pomorska Sola Portoroz (Slovenja) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

HOLOGRAPHY: A PROJECT-TYPE APPROACH FOR CONTEXTUALIZED TEACHING OF OPTICS,P. Pombo and J. Pinto, Physics Department, University of Aveiro (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

EXPERIMENT AND THEORISATION: AN APPLICATION OF THE HYDROSTATIC EQUATION ANDARCHIMEDES THEOREM,L. Santos and M. Talaia, Physics Department, University of Aveiro (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.2 Problem Solving

PROBLEMS IN THE PHYSICS OLYMPIADS,G. Cavaggioni, A.I.F. Committee, responsible for the Physics Olympiads in Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

PROBLEM SOLVING ACTIVITIES IN TRAINING THE ITALIAN CONTESTANTS OF THE PHYSICS OLYMPIADS IN A SUMMER SCHOOL,D.L. Censi, Italian Physics Olympiads National Group of the Associazione per l’ Insegnamento della Fisica (AIF, Italy) 151

HOW TO START A PROBLEM,F. Minosso, Board of Olympiads (AIF, Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

DOES PHYSICS FORMAL KNOWLEDGE REALLY HELP STUDENTS IN DEALING WITH REAL-WORLD PHYSICS PROBLEMS?N. Grimellini Tomasini and O. Levrini, Physics Department, University of Bologna (Italy). . . . . . . . . . . . . . . . . . . 164

INFLUENCE OF NARRATIVE STATEMENTS OF PHYSICS PROBLEMS ON THEIR COMPREHENSION,E. Llonch, M. Massa, P. Sanchez and E. Petrone, Facultad de Cs. Exactas, Ingeniería y Agrimensura,Universidad Nacional de Rosario (Argentina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

DIFFICULTIES ON INFERENCIAL PROCESS. A STUDY OF THERMODINAMIC PROBLEMS,M. Massa, M. Yanitelli and S. Cabanellas, Facultad de Cs. Exactas, Ingeniería y Agrimensura,Universidad Nacional de Rosario (Argentina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

MATHEMATICS OF DIMENSIONAL ANALYSIS AND PROBLEM SOLVING IN PHYSICS,D. Pescetti, Physics Department, University of Genova (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

DEVELOPING THINKING IN PHYSICS THROUGH PROBLEM SOLVING,S. Sawicka-Wilgusiak, Faculty of Pedagogy, Warsaw University (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.3 Modelling

MATHEMATICAL FORMAL MODELS FOR THE LEARNING OF PHYSICS: THE ROLE OF AN HISTORICALEXAMPLE,G. T. Bagni, Department of Mathematics, University “La Sapienza”, Roma (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . 188

MODELLING PHYSICAL PROCESSES: THE EXAMPLE OF A MAGNET GLIDER,M. D’Anna, Liceo Cantonale Locarno (Switzerland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

MODELS IN SCIENCE AS METAPHORS OF THE WORLD AND OF THE HUMAN BODY,F. de Stefano, Liceo Scientifico “G. Marinelli”, Udine (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??

6

MODELLING PHENOMENA IN VARIOUS EXPERIENTIAL FIELDS: THE FRAMEWORK OF NEGATIVE AND POSITIVE FEEDBACK SYSTEMSC. Fazio, A. Giangalanti, G. Tarantino, I.P. “Enrico Medi”, Palermo (Italy), R.M. Sperandeo-Mineo,Department of Physical and Astronomical Sciences, Univesity of Palermo (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

THE LEARNING OF MODELING: A SCIENTISTS’VISION,S. M. Islas, Departamento de Formación Exactas-Universidad Nacional del Centro de la Provincia de Buenos Aires, Campus Universitario, Tandil (Argentina),M.A. Pesa, Departamento de Física, Facultad de Ciencias Exactas y Tecnología, Universidad Nacional de Tucumán (Argentina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

MODELLING WITHOUT NUMBERS,I. Lawrence, School of Education, University of Birmingham (UK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

LEARNING DATA ANALYSIS,D. Moreno, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico (Mexico),G. del Valle, Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana,Azcapotzalc (Mexico) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

MODELS-THE BASICS OF PHYSICAL THINKING: CONCLUSIONS FOR MULTIMEDIA,T. Romanovskis, Institut für Experimentalphysik, Universität Hamburg (Germany), on leave from Faculty of Physics and Mathematics, Latvia University, Riga (Latvia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

MODELS, MENTAL IMAGES AND LANGUAGE IN SCIENTIFIC THINKING,J.A. Smit, Potchefstroomse Universiteit vir CHO, Potchefstroom (South Africa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

LEARNING PHYSICS VIA MODEL CONSTRUCTION,R.M. Sperandeo-Mineo, Department of Physical and Astronomical Sciences, Univesity of Palermo (Italy) . . . . . 224

3.4 Hands-on/Toys

FROM PLAYING WITH TOYS TO MEASUREMENTS,M. Bertoncelj and A. Gostinčar Blagotinšek, Faculty of Education, University of Ljubljana (Slovenia) . . . . . . . . 224

HOW SCIENCE CENTERS AND MUSEUMS CAN SERVE THE FORMAL LEARNING IN THE SCHOOLS,P. Cerreta, ScienzaViva,Associazione per la divulgazione scientifica e tecnologica, Sezione AIF di Calitri (Italy). . . . . . 228

TOYS IN MOTION,A. Gostinčar Blagotinšek, Faculty of Education, University of Ljubljana (Slovenia) . . . . . . . . . . . . . . . . . . . . . . . . 233

PHYSICS IS MIGHTY AS IT IS EASY,K. Papp, A. Nagy, M. Molnar and J. Bohus, University of Szeged, Department of Experimental Physics (Hungary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

TEACHING MECHANICS AND BIO-MECHANICS,P.B. Pascolo, University of Udine and Bioengineering Department of CISM (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . 240

SIMPLE EXPERIMENTS HELP IN GAINING A BETTER UNDERSTANDING OF PHYSICS CONCEPTS,G. Planinšič, Faculty of Mathematics and Physics, University of Ljubljana (Slovenia),M. Kos, Ustanova Hiša exsperimentov, Slovenian Hands-on science centre, Ljubljana (Slovenia) . . . . . . . . . . . . . . 244

LEARNING BY PLAYING OR PLAYING BY LEARNING,N. Razpet, The National Education Institute, Ljubljana (Slovenia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

DEVELOPING FORMAL THINKING THROUGH TOYS AND EVERYDAY OBJECTS FOR THE FORMATION OF FUTURE PRIMARY SCHOOL TEACHERS,D. Allasia, V. Montel, G. Rinaudo, Department of Experimental Physics of the University of Torino (Italy) . . . . . 254

SOME PHYSICS TEACHING MITHS, C.H. Worner,Instituto de Fisica, Universidad Católica de Valparaíso (Chile) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

3.5 First Steps in Formalization

BUILDING FIRST EMPIRICAL CONCEPTS AND KNOWLEDGE BY ONESELF AND OTHERS, IS THE WAY TO FORMAL THINKING FOR YOUNGSTERS,C. Balpe, Institut Universitaire de Formation de Maîtres d’Aquitanie (France) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

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INTRODUCING THERMAL PHENOMENA QUANTITIES,D. Ferbar, Pedagoska Fakulteta Ljubljana (Slovenia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

WHAT FRACTION OF PUPILS REALLY REACH THE STAGE OF FORMAL THINKER IN PHYSICS?,R. Krsnik, P. Pećina, M. Planinić and A. Sušac,, Physics Department, PMF, University of Zagreb (Croatia),I. Buljan, Primary School “Zaprude”, Zagreb (Croatia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

COGNITIVE LABS IN AN INFORMAL CONTEXT TO DEVELOP FORMAL THINKING,A.Stefanel, C. Moschetta, M. Michelini, Research Unit in Physics Education, University of Udine (Italy) . . . . . . 276

MAKING PHYSICS FASCINATING TO…ALL!?,G. Zini, Dipartimento di Fisica, Università di Ferrara (Italy),A.Turricchia, Aula Didattica Planetario, Comune di Bologna (Italy), L. Bernacchio, Osservatorio Astronomico,Padova (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

3.6 Strategies: Methods and ToolsSWITCHING FROM EVERYDAY FACTS TO SCIENTIFIC THINKING,P. León, Univesidad “Simón Bolívar” (Venezuela), M. Castells, Universitat de Barcelona, Catalonia (Spain) . . . . 288

ENGLISH AS A MEDIUM OF INSTRUCTIONS IN SCIENCE TEACHING,C. Haagen-Schützenhöfer and L. Mathelitsch, Institute for Theoretical Physics, University of Graz ( Austria) . . 293

DEVELOPMENT OF FORMAL THINKING ON KINEMATICAL ASPECTS OF MOTION FROM CHILDREN KNOWLEDGE TO EARLY MATHEMATISATION,M. Gagliardi, N. Grimellini Tomasini, B. Pecori, Physics Department, University of Bologna (Italy) . . . . . . . . . . . 296

WHY DO WE RUN WHEN WE WANT TO MOVE FASTER?,E. Reichel, Bundesgymnasium und Bundesrealgymnasium, Graz (Austria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

TIME TRAVEL – MORE THAN A PHYSICAL CONCEPT?,T. Tajmel and L. Mathelitsch, Institute of Theoretical Physics, University of Graz (Austria). . . . . . . . . . . . . . . . . . 306

THINKING ON VECTORS AND FORMAL DESCRIPTION OF THE LIGHT POLARIZATION FOR A NEW EDUCATIONAL APPROACH,M. Cobal and M. Michelini, Physics Department, University of Udine (Italy),F. Corni, Physics Department, University of Modena and Reggio Emilia (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

HIGHER ORDER THINKING IN OHYSICS EDUCATION (HOT-PHYSICS),J.D. Holbech and P.V. Thomsen, Centre for Studies in Science Education, University of Aarhus (Denmark) . . . . 320

COMPREHENSION AND TEST RESULTS AFTER INTRODUCTION OF WORKSHOP PHYSICS,T. Lundstrom, M.D. Lyberg and A. Svensson, School of Mathematics and System Eng., Department of Physics, Växjö University (Sweden). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

3.7 Mathematisation

ON HOW TO BEST INTRODUCE THE CONCEPT OF DIFFERENTIAL IN PHYSICS,R. Lopez-Gay and J. Martinez-Torregrosa, Dpto. de Didáctica General y Didácticas Específicas,Alacant (Spain),A. Gras-Martí, Dpto. de Física Aplicada, Universitat d’Alacant (Spain),G. Torregrosa, Dpto. de Análisis Matemático y Didáctica de las Matemáticas, Alacant (Spain) . . . . . . . . . . . . . . . . 329

TEACHING QUANTUM THEORY,H. Grassmann, Physics Department, University of Udine (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

ANALYSING DIFFERENT FORMS OF PRESENTING NEWTON’S LAWS EMPHASIZING THE RELATED CONCEPTS,J.L. Jiménez, Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa (México),I. Campos, Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México (México),G. Del Valle, Área de Física Atómica y Molecular Aplicada, Universidad Autónoma Metropolitana-Azcapotzalco (México) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

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HOW TO “AVOID” WORK. UNDERSTANDING THE WAYS IN WHICH PHYSICS USES MATHEMATICS TO RECOGNIZE THE CONSTANT OF MOTION OF MECHANICAL ENERGY,M. Michelini and G.L. Michelutti, Physics Department, University of Udine (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . 340

ONE DIMENSIONAL QUANTUM SYSTEMS AND SQUEEZED STATES,C.A. Vargas, Área de Física, Departamento de Ciencias Básicas, UAM-A, Azcapotzalco (México),A. Zúñiga-Segundo, Departamento de Física, Escuela Superior de Física y Matemáticas-IPN, Zacatenc (México) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

3.8 Software Packages-Multimedia

GRAPHS AS BRIDGES BETWEEN MATHEMATICAL DESCRIPTION AND EXPERIMENTAL DATA,L. Rogers, School of Education, University of Leicester (UK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

THE USING OF MULTIMEDIA COURSEWARE FOR COLLEGE PHYSICS RELATIVITY TEACHING AT HARBIN NORMAL UNIVERSITY,Z. Changbin, S. Guilian and M. Hongchen, Physics Department, Harbin Normal University (China) . . . . . . . . . . 361

DEVELOPING STUDENTS COMPETENCES BY MEANS OF SIMULATION: USE OF A RESEARCH TOOL FOR UNDERSTANDING ION-MATTER INTERACTIONS IN SOLID STATE PHYSICS,F. Corni, Physics Department, University of Modena and Reggio Emilia (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

MULTIMEDIA PROGRAM-ELECTRIC CURRENT,L. Koníček, E. Mechlová, Department of Physics, Faculty of Science, University of Ostrava (Czech republic). . . 375

DO YOU HEAR THE SEA FROM A SHELL?,I. Verovnik, National Education Institute of Slovenia, Ljubljana (Slovenia),L. Mathelitsch, Institut für Theoretische Physik, Universität Graz (Austria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

3.9 Text Books

DISCUSSING THE PROBLEMS IN TEXTBOOKS,C. Escudero and M. García, Physics Departmen., Fac. de Ingeniería, Universidad Nacional de San Juan (Argentina),S. González, Physics and Chemistry Department, Facultad de Filosofía, Humanidades y Artes, Universidad Nacional de San Juan (Argentina)M. Massa, Physics and Chemistry Department, Facultad de Ciencias Exactas, Ingeniería y Agrimensura,Universidad Nacional de Rosario (Argentina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

DO PRIMARY AND SECONDARY TEXTBOOKS CONTRIBUTE TO SCIENTIFIC REASONING?,M. Massa and H. D’Amico, Physics and Chemistry Department, Facultad de Ciencias Exactas,Ingeniería y Agrimensura, Universidad Nacional de Rosario (Argentina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

3.10 Teacher Training

FORMALIZING THERMAL PHENOMENA FOR 3-6 YEARS-OLDS: ACTION-RESEARCH IN A TEACHER-TRAINING ACTIVITY,L. Benciolini, Dipartimento di Georisorse e Territorio, University of Udine (Italy),M. Michelini, Physics Department, University of Udine (Italy), A. Odorico, Facoltà della Formazione,University of Udine (Italy). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

THE DIDACTIC LABORATORY AS A PLACE TO EXPERIMENT MODELS FOR THE INTERDISCIPLINARYRESEARCH,M. Fasano, University of Basilicata (Italy), F. Casella, IRRE, Basilicata (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

A MODERN TEACHING FOR MODERN PHYSICS IN PRE-SERVICE TEACHERS TRAINING,M. Giliberti, Physics Department, University of Milano and INFN Milano (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . 400

TEACHING ENERGY IN HIGH SCHOOL: CRITICAL ANALYSIS AND PROPOSALSJ. Ll. Domènech, D. Gil-Pérez, Universitat de València, Spain,A. Gras-Martí, J. Martínez-Torregrosa, Universitat d’Alacant, Spain,G. Guisasola, Euskal Herriko Unibertsitatea, Spain, J. Salinas, Universidad Nacional de Tucumán, Argentina . . 406

9Developing Formal Thinking in Physics

OBSTACLES TO THE DEVELOPMENT OF CONCEPTUAL UNDERSTANDING IN OBSERVATIONAL ASTRONOMY: THE CASE OF SPATIAL REASONING DIFFICULTIES ENCOUNTERED BY PRE-SERVICE TEACHERS,Ch. Nicolau and C.P. Costantinou, Learning in Physics Group, University of Cyprus (Cyprus) . . . . . . . . . . . . . . . 410

A STUDY OF THE COMPETENCE AND PROFESSIONAL DEVELOPMENT OF SCIENCE/PHYSICS TEACHERS IN ANTALYA PROVINCE OF TURKEY, I.S.Ustuner, Akdeniz University, Antalya (Turkey),Y. Ersoy, Middle East Tech. University, Ankara (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

4. The Seminar

4.1 First International Girep Seminar

DEVELOPING FORMAL THINKING IN PHYSICSUniversity of Udine, 2-6 September 2001, M. Cobal, L. Santi, Physics Department of the University of Udine (Italy) . . . 427

4.2 Welcome of the Rector, F. Honsell, University of Udine (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

4.3 Introduction to the Seminar, M. Michelini, CIRD, University of Udine (Italy) . . . . . . . . . . . . . . . . . . . . . 433

4.4 The organization of the Seminar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

4.5 Structure of the Seminar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

10

Introduction

11

This book includes a selection of the contributions presented at the First International GirepSeminar on Developing Formal Thinking in Physics. The selection has been done by a wide groupof referees: each paper has been examined by two or more referees chosen in between the expertsof the field.The First International Girep Seminar has represented a completely new way of meeting toexchange researches, experiences and connected activities at an international level. Participantshave been selected in a number to realize such a working condition that the discussions on thevarious topics could be favoured with respect to the presentation of those works already prepared.The large majority of the time during this Seminar has been dedicated to thematic discussions inspecific Workshops. Some of the invited talks – which appear in Chapter 1 of this volume – offeran overview of the problematic involved with the Seminar’s topic.Chapter 2 contains the short reports from each Workshop responsible. These reports include theoutcomes of the discussions carried out in the Workshops, which had as a starting point the paperspresented by the participants.Chapter 3 collects the selected papers of the participants to the Seminar. These papers areorganized by topics according to the main problems identified for the development of the formalthinking in physics.Finally, Chapter 4 recalls the characteristics of the Seminar.

Chapter 3 is surely the wider one and shows some of the more important studies related to thedevelopment of the formal thinking in physics. Therefore, can be useful to give here somecomments on the single paragraphs.

In Laboratory and Theory some interesting experimental proposals are presented: advanced, forexample related to holography, as well as elementary, for example an application of theArchimedes’s principle. They offer indications on how to set up and carry out the experimentalwork in order to favour the formal reflection.The Problem solving for the development of the formal thinking is illustrated as a methodology ofapproach but also as a modality to overcome the disconnected view that students seem to havebetween the “common sense” vision of the world and the scientific one learned in the school.The way to implement this activity and the presence of some cases from specific physics fields offerus a rich source of examples that comes in parallel with an analysis of the difficulties that can beencountered.Modelling activities as a methodological tool for physics are proposed in a more profound way,with contributions of historical examples, with examples in various fields of physics, with proposalsrelated to the experimental work, with indications for the qualitative activities and withcontributions which make use of the communication, information and multimedia technologies.The informal education and the importance of an active role of the learner, realized as well bymeans of games and toys, constitute a motivational proposal for all ages: contributions for theprimary school and examples of didactic on kinematics for engineering studies, are some of theexperiences offered, together with a reflection based on the learning by playing and on thelearning from everyday life.First step in formalization offers proposals of didactical activities and cognitive analyses, which area challenge to the basic formation in physics starting from the primary school, for the scientificformation of all the citizens, as a complement to the group of works on toys.

Strategies: Methods and Tools is a paragraph related to the cognitive link with everyday life andrelated to the understanding of the role of formal tools, as vectors, and/or of formalized contexts,as the kinematics one.Mathematization treats specifically the problem of the role of mathematics and of the link withmathematics in various fields for the didactic of physics.It is discussed here the conceptual role of the formalism in quantum mechanics and the meaningof the Newton’s laws, as well as how the formalism can lead to the recognition of the constants ofmotion.Software Packages and Multimedia are representatives of the ways in which the ICTs can either bea bridge between the experimental results and the mathematical description of the phenomena, orcan allow to perform measurements with sensors connected to the computer, or can offer usefulsimulation to analyse predictions or can constitute the courseware contest.The last paragraph of Chapter 3 is devoted to the teacher training. Some proposals are presentedto activate formalization processes or thoughts on how to train and educate teachers. The goal isfor the teachers to be able to start a deep process to get the formal tools in physics, knowing howto deal with the meanings.

The choice of publishing a thematic book has been driven by the need of specialization thatemerged in the studies of didactic and physics education research and in the teachers training paths.We worked in a rigorous way and with dedication to offer an useful tool. We are sorry for thepossible errors that have been done, and we wish that this could be useful to teachers andcolleagues who work in the physics education research and in the teacher education fields.

The Editors:Marisa Michelini and Marina Cobal

12 Introduction

1. Background Aspects

IMAGERY AND FORMAL THINKING: APPROACHES TO INSIGHT ANDUNDERSTANDING IN PHYSICS EDUCATION

Manfred Euler, Leibniz-Institute for Science Education (IPN), University of Kiel, Germany

This lecture is dedicated to the memory of the late GIREP president Karl Luchner.Physik ist überall (Physics is everywhere)[1]

K. Luchner

1. Deficits in science education: the quest for understanding and insightThis is the first GIREP seminar held between the biannual meetings. In focusing on the develop-ment of formal thinking the present conference addresses a problem, which is of vital importancefor the future of physics education. Since long, we can observe a declining interest in physics. Thistrend is ubiquitous, and many reasons can be found to account for this phenomenon. In close corre-spondence with the motivational deficits the cognitive outcomes of physics teaching are far fromwhat is expected.An adequate unfolding of formal thinking is undoubtedly one of the crucial prob-lems of learning and understanding physics.The development of formal thinking can be seen as the acquisition of networks which assignmeaning to symbolic elements and which allow students to navigate in the world of physics [2].Formal thinking, creating symbolic descriptions that model structures and processes in the realworld, is at the heart of the methodology of physics.As formal thinking obviously is not the normalmode of mental activity, it is also at the heart of the difficulties of learning physics. Thus, findingbetter ways of promoting formal thinking and of moving from the concrete to the abstract is essen-tial for restructuring the ways we teach and learn physics.It is useful to consider the subject of the present seminar in the context of international efforts tomonitor scientific literacy. The status and the effectiveness of science education in many countriesare not considered satisfactory and sufficient to master the challenges of the future. We are amidsta transformation process from the post industrial to the knowledge society. There is the generalfear that scientific literacy and the public awareness of science do not comply satisfactorily withthe needs of a global knowledge societ [3].The OECD Programme for International Student Assessment (PISA) has addressed somequestions which are considered vital in view of the rapid global changes [4]. Are students wellprepared to meet the challenges of the future? Are they able to analyse, reason and communicatetheir ideas effectively? Do they have the capacity to continue learning throughout life? The term“literacy” is used in a metaphoric way to describe a broad conception of knowledge and skills forlife, which are broken down to various processes, including, among many other aspects, the abilityto apply knowledge from science in more or less authentic real world situations. Due to thecomprehensive approach of assessing student performance and of collecting ample contextinformation, PISA provides the empirical framework for a better understanding of the causes andpossible consequences of observed skill shortages.Among other findings, PISA confirms the results of earlier studies like TIMSS (Third InternationalMathematics and Science Study [5,6]). With respect to the consequences, I can speak only for mycountry: The TIMSS results have shattered the long-held beliefs about the high standards ofphysics education. Science education is more or less efficient only with respect to imparting theknowledge of facts. However, broad deficits exist on the level of more demanding scienceprocesses, e.g. applying knowledge to new situations. Broadly speaking, physics teaching focuses onconveying factual knowledge (“know what”). Approaching the “know how” and the “know why”

poses big problems. Physics education fallsshort of attaining more challenging goalslike flexible application of knowledge innew contexts, and of fostering insight andunderstanding. Formal thinking, as such, isnot addressed in these studies but theirresults clearly point out deficits in applyingwhat has been learned. This deficit is closelyrelated to formal thinking, as learners areunable to abstract sufficiently and transferknowledge from one context of experienceto a related domain. However, internationalcomparisons do show that there are specificdifferences between countries in attainingthese more demanding goals.Under what conditions can one expectphysics education to promote insight andunderstanding? This article approaches theelusive phenomenon of insight from variousperspectives and discusses its specificmeaning for cognitive processes by usingexamples from physics and mathematics.Intuitive pictures, mental transformations ofimagery and reflections about theseprocesses are considered important for learning and for doing physics as well. In this context,images represent the (more or less) concrete symbolic substrate from which formal thinkingemerges. All learning theories agree on learning being an active process. Active transformationprocesses of mental imagery are necessary to build up more and more abstract representations.How can we link the mental images and their transformation processes with concrete experiencesin a meaningful way?On a more general level the question of the relation between instruction and autonomousconstruction is touched upon. Insight and understanding result from genuine individual mentalconstruction processes. From an external perspective these processes are difficult to investigate asthey are accessible only indirectly and incompletely. Is it possible to stimulate processes of insightand to support meaningful learning by suitable instructional means? We are far from being able tooffer a patent solution but the present article intends to focus on these issues and to initiate broaderdiscussions.

2. The elusive nature of insight: is it something special? By insight one generally means the ability of immediately knowing and of seeing clearly by mentalactivities which are not fully transparent.This mode of thought is felt in sharp contrast to rational andanalytical reasoning. In the context of physics and mathematics, insight means the ability to “see” thesolution even without running logically through all the steps of problem solving. In insights, theessence of phenomena or processes is intuitively grasped or linked with other phenomena withoutfully decoding the path of argumentation. There is a close connection to creative thinking, when newsolutions to problems are generated. Insight describes the transition process of our minds from thestate of ignorance to the state of clarity about the solution to the problem [7].There are many metaphors for insight; the “sudden beginning to see the light” is one, the “flash ofgenius” is another. In a letter to a friend the famous mathematician Gauss describes the process inthe following way: Just like lightning strikes, the puzzle is solved. I myself would not have been able toprove the guiding line between what I already knew ... and how it was possible to prove it finally [8]. A

14 Background Aspects

Fig. 1. Insights in insight: How to circumvent self referentialcircles?

common characteristic in almost all of the reports about sudden insight in mathematics and thesciences is – in addition to the affective moment – its visual character. Insight depends on picturesthat entangle concrete experience and visual imagination with abstract, symbolic elements.An overview of the current state of psychological research may be found in the volume entitled TheNature of Insight [9], which also gives some useful references for insights relevant to physics. Aseveryone will know from more or less intensive personal introspective experience, processes ofinsight are associated with the following characteristics [10]:• suddenness: A discontinuity or an abrupt transition from ambiguity to understanding the prob-

lem and its solution is perceived instead of a gradual steady transition process.• spontaneity: Insight cannot be forced, but rather happens intrinsically and spontaneously without

an apparent agent.• unexpectedness: Insights can emerge even when one is not consciously involved in the problem.• correctness: Often, insights are accompanied by an immediate feeling of correctness, although a

full logical justification is missing.• satisfaction: A deep feeling of satisfaction may accompany or even precede the moment of

insight.Dissolving the tension of hitherto unsolved cognitive conflicts may result in exclaiming a loudEureka!, as we know from folklore reports about Archimedes and his bathtub insight on buoyancy.A more quiet everyday form of satisfaction is the Aha! experience. In any case we must notunderestimate the strong affective element of the moment of insight, which drives our mentalactivity and finally rewards tedious thoughts and hard labour.Generally, one finds two basic assumptions about the character of insight which express two con-tradictory points of view [11]:• Insight is not a special process. It is rather considered as a smooth extension of common percep-

tion, learning and thinking processes. This point of view emphasises the continuity, the cumula-tivity and the associativity of the process: The way, how a new problem is solved only depends onprior knowledge. One produces new solutions by combining what one already knows. Thus,insight is nothing more than a chain of associations of elements available to the somehowprepared mind. It is doubtful whether it is possible to solve new problems according to a kind of“flash of genius” independent of previous experience.

• Insight is a special process. It can be understood as an acceleration or - in the extreme - even asa circumvention of the normal, conscious chain of arguments. Insight is considered an uncon-scious process, restructuring knowledge in such a way that new connections emerge. Insights areseen in close analogy to visual perception processes. Similar to the emerging of visual perceptionsfrom optical input signals by means of an active structuring process, knowledge is restructured increative thought processes. This can take place in various ways, e.g. by completing missinginformation, by reformulating the problem, by eliminating thought blockades, by separating anobject from its functional context, or by transforming it to an analogous problem for which thesolution is already known.

While the first point of view developed out of traditional psychological research, the secondapproach came from the field of Gestalt psychology. It was scorned for a long time for its presumedlack of methodological strictness and theoretical clarity. This point of view, it was argued, is based onanecdotic evidence and not on facts that can be proved by experiments. Emotionally, however, it isvery attractive as it describes characteristics of insight that everyone can perceive introspectively. Canone’s own intuition (and the intuition of many great spirits of the scene) be so wrong?From the perspective of modeling cognitive processes within the framework of neural networks,the differences in the two points of view become irrelevant. Interestingly enough, many features ofthe phenomenology of Gestalt perception can be modelled by neural networks [12]. Instead of anexclusive “either - or” it seems to be better to view the two perspectives as the extremes of anapproach to insight which allow a continuum of possible intermediate forms depending on thecomplexity of the problem and the level of expertise the problem solver has.


3. Metaphors and models for insight: metamorphoses of internal images To resolve the conflict between the “business as usual” standpoint and the view of insight beingsomething special, we discuss optical pattern recognition processes. They can serve as a metaphoror even an analogy of higher cognitive processes. This analogy suggests that there is nothing specialabout insight processes. They are compatible with conventional sensory, learning and thinkingprocesses. At the level of perception analogous processes are continually taking place almostunnoticed. Nevertheless, we as our own internal observers perceive them as having the char-acteristics described in the second point of view.This is one of the oddities of the internal and external view of complex systems, and our head isundoubtfully such a complex system. The internal observer’s view and the descriptions of the“participant” can be incompatible with those of the “detached observer”, a fact that is just asimportant for physics as for cognitive science [13]. The view from within the system compared tothe view from the outside, the endoperspective and the exoperspective, are conflicting. Thisincompatibility expresses more than the normal observer dependency of descriptions of reality, inwhich the results of measurements may depend on the individual frame of reference. In the lattercase, it is possible to construct smooth transformations that relate consistently the outcomes of anexperiment in one system to the descriptions of reality given in another system.The problem of internal observers is more deeply rooted and cannot be reduced to smoothtransformations from one perspective to the other. It is closely connected to the emergencephenomenon, where new entities are generated by a spontaneous re-ordering process, as soon asthe (open) systems are complex enough. In physics, these processes occur in dynamical systems farfrom equilibrium (cf. par. 7).Two perception experiments are presented that elucidate by analogy how this puzzle of conflictingdiscreteness and continuity might be solved. On the one hand, one can maintain the belief gainedfrom the view of the internal observer that insight is a discontinuous process. On the other hand,in spite of the emergent discontinuity, the underlying processes can be regarded as fully continuous,building upon what one already knows. In order to accept this unifying view, one has to change theperspective and compare the view from within the system with the description from the outside. Ingeneral, such a switch from the internal view to the reflective meta-perspectives is also a decisivestep moving from learning facts to conceptual understanding.

Experiment 1: Gestalt-Perception - spontaneous and induced switching processesHave a look at Fig. 2 on the left (Necker Cube). You clearly perceive a cube in two possible spa-tial configurations. Although the drawing is one dimensional and unique, the percept is spatial andbistable.The actions of the observer’s conscious mind appear to animate the image. In case that youhave no prior experience with this figure, the switching between the two arrangements of cubes is


Fig. 2. The Necker cube. Bistability in visual perception can be perceived as aspontaneous process, but it can also be stimulated externally or induced voluntarilyby guiding the eye movements

completely spontaneous. This subjective impression of spontaneity is fully correct as psycho-physical experiments confirm! When the switching sequence is registered (by pushing a button,whenever the percept snaps into another configuration), a time sequence of events results that willpass all the tests for randomness. The switching sequence corresponds to a Poisson-process which,for instance, also shows up in spontaneous emission of light quanta from excited states or inradioactive decay, both phenomena being physical archetypes for spontaneous processes [14]. Theobserved spontaneous switching processes from one percept to the other exhibits manycharacteristics of insight that were described earlier. Snap! The insight shows up spontaneously andthe subject appears to have no possibility of controlling the process.From the external view, however, the processes are transparent and they can be controlled (theo-retically at least). In fact, an “expert” with enough background knowledge can voluntarily inducethe transition from the one state to the other. To such an expert, the spontaneity vanishes. A simpletrick suffices to turn a novice into an expert for switching Necker cubes. A change in the directionof gaze indicated by the two dots in Fig. 2 (right) helps to break the symmetry of the figure.Focussing on one of the two dots will induce the switch. One can even move the tip of a pencilperiodically back and forth along the lines, and ask the subject, to follow the tip. In this case, thesubject will perceive the switching in synchrony with the periodic motion. This little experimentturns the spontaneous switching into an induced transition process, that can even be guidedexternally.Transferred to the insight-experience, we can conclude that, whenever a sufficient level of knowl-edge is available, the insight-feeling is turned into nothing special. There is “business as usual”, andthe feeling of “not being able to do anything about it” disappears. The mystery of insight becomesdemystified in so far as a fully regular and continual association with other mental or bodily activi-ties can be produced. In the present experiment these activities (eye movements) are rather trivial.The next demonstration focuses onthe crucial role of prior knowledge.

Experiment 2: When do Picturessay More Than a 1000 Words?Do you recognise the two rathercoarse grained faces in Fig. 3? A hint:Squint your eyes and have a blurrylook or look from a distance - thathelps! Optical pattern recognition -like insight - requires at least incertain phases oversimplifying andgeneralising. After a short “incuba-tion period” the insight should comeabruptly. All the faces shown arethose of physicists. Insight should come quickly in the case of the right-hand picture, since theperson is well known and almost everyone has seen this specific picture. Additionally, when youknow that this person once stuck his tongue out at obtrusive photographers, then the task is an easyone. Previously acquired knowledge and the access to available information determine how quicklya new problem can be solved.Recognising the face on the left is more difficult; it is practically impossible! This person does notexist in the classical sense. It is a superposition of the faces of two physicists, Feynman and Gell-Mann.As the prior knowledge is not available, no insight can arise. Insight without some form priorknowledge is apparently not possible according to this visual analogy.A reflection of the processes that generate meaning in these cases give some hints about the condi-tions under which pictures work effectively. The proverb a picture says more than a thousand wordsis incorrect in the naive, literal version. There is a plethora of potential information in pictures (just


Fig. 3. Pictures of famous physicists. Who is who?

think of the many kilobytes or even megabytes that uncompressed images use in computermemory). The effective information that is actually involved in making the image “happen” andactivating internal rearrangement processes that finally switch to the percept is, on the other hand,quite limited. In the example of the Einstein picture about 200 pixels with 3 bits each (23 shades ofgrey), i.e. about 600 bits - far less than the information contained in 1000 meaningful words - sufficeto describe the person and even what is happening in the scene.Both experiments tell us something about dynamical processes in visual perceptions that shed lighton the function of mental imagery. The theoretical biologist von Bertalanffy remarked The formsof life are not, they happen [15]. Along the same line on could reformulate the statement withrespect to images and insight: Images are not, images happen. Images are no static objects. Theycorrespond to dynamic processes and are linked with prior experience. Images are models that canbe run mentally.When do images occur? The first experiment shows, that they occur spontaneously- but nevertheless they can be triggered or stimulated by suitable arrangements. The secondexperiment demonstrates the “happening”, that is the linking process with prior experience. Bothexperiments can be used as metaphors for insights.Although deep insights and visual pattern recognition differ considerably by the time scale of theresponse and the complexity of the percepts it is not unreasonable to assume close links. If this isacceptable, one cannot circumvent the conclusion that insights (like visual perceptions) can be fos-tered and stimulated to some extent - a conclusion that has far reaching implications for teachingand learning. In science teaching, images are often considered from a passive perspective, showingthings and situations statically. We have to do more in teaching and learning to make imageshappen. Fostering insightful learning processes requires a delicate balance of instruction (providingsuitable learning arrangements) and construction (giving ample space to explore and reflect ownapproaches).Deficiencies in meaningful mental imagery can be considered one of the causes for the poorachievements of students in the “know how” tasks reported earlier. The following selection of ex-amples is guided by the assumption that intuitive pictures, their metamorphoses, or, to be more spe-cific, their active transformation processes, and the conscious reflection of these transformationsare crucial for learning and for doing physics.

4. No insight without internal images: imagery and scientific imaginationWhat role do insights and images play in learning? Certainly their role will not be very differentfrom research processes. Thus a short glance at the latter will be helpful. Although research in sci-ence and mathematics is generally considered as being especially analytic, logical and rational, thecreative processes in these fields are inspired by imagery and intuition. Much evidence for this canbe found in the biographies and correspondence of well known researchers, although such reportshave to be read with a certain amount of critical distance.An especially notable example is W. Pauli, who is generally considered one of the most rational andcritical minds in physics [16]. In spite of his rational attitude he has always tried to account for whathe called “underground physics” giving way for creative mental processes. The Pauli-Jungcorrespondence documents his effort to understand the role of unconscious processes in creativescientific thought and demonstrates the meaning the attached to symbols and symbolism, light aswell as dark symbolism [17,18]. Pauli’s reports show the difficulty of reconstructing the mostlyunconscious creative processes rationally, due to the interweaving of dreams, concrete pictures,abstract ideas, symbols (scientific as well as religious) and highly emotional personal experiences.There is no insight without internal images! This is how the theoretical chemist Primas saw creativeprocesses in the field of mathematics and the sciences [19].As he emphasises, the pictorial intuitionis, however, only one part of the creative process.What is intuitively seen must be corroborated andcritically questioned by rational reconstruction, otherwise intuition is nothing more than shallowfantasies. An adequate interplay between intuition and rational reconstruction is crucial not onlyfor doing physics but also for learning physics.


The evidence that images promote insight and understanding in science (and mathematics as well)is overwhelming. So we have to ask, why only little effort is made in physics education to accountfor the role of imagery for a better understanding of physics. Apparently, visual approaches havebeen banned by a formalistic tradition. The relations between the concrete and the abstract are notsufficiently balanced. While the formalistic approach tries to prevent misconcepts based on falseintuitions it often falls short of giving the abstract symbols and formal operations a concrete mean-ing which we need to operate successfully with these concepts.We must reconcile ourselves and our students with the fact, that we need concrete images and sym-bols of physical entities, even though physical reality is not fully imaginable in terms of everydayreality. We are forced to use abstractions, but there remains some important residual concretenessin the abstract symbols. There are physical as well as psychological grounds for these “elements ofreality” in the abstract formalism.Although many textbooks use images on a superficial level, only few encourage visualisation andeven fewer reflect the role of imagery and imagination. The Feynman lectures are a notable excep-tion. In a chapter on the solution of Maxwell’s equations in free space Feynman lines out the de-mand on scientific imagination in the context of classical field theory. He conveys a vivid picture ofhis own struggle for a visualisation of the invisible and the untouchable that mixes abstract symbolswith concrete actions and perceptions. He confesses that I have... no picture of this electromagneticfield that is in any sense accurate. ....When I start describing the magnetic field moving through space,I speak of the E- and B-fields and wave my arms and you may imagine that I can see them. I’ll tellyou what I see. I see some kind of vague shadowy, wiggling lines – here and there is E and B writtenon them somehow, and perhaps some of the lines have arrows on them – an arrow here or there whichdisappears, when I look too closely at it. .... I have a terrible confusion between the symbols I use todescribe the objects and the objects themselves [20].Retreating to the purely mathematical view is not a solution either. On the one hand, any attemptto make the electromagnetic field fully “touchable” is bound to fail. On the other hand, the electro-magnetic field is more than merely an abstract and arbitrary mental construction. Although ourminds are unable to conceive an adequate picture of the electromagnetic field, like, for instance amechanical model that we can grasp, the instruments that we build according to the laws of physicscan “grasp” the field in a predictable way.An analysis of Feynman’s struggle for appropriate visualisation points out some reasons, why it isso difficult to convey images of physical processes. On the psychological side there is the absoluteprivacy of mental imagery. Images are one’s own individual constructions that depend on prior ex-perience. Even though two persons see the same picture, the meaning they attach to it may be verydifferent. Images cannot simply be “implanted”. Images are often highly resistant to teaching andchanges of mental imagery is a long process, as overwhelming evidence from research onconceptual change and reasoning in physics shows [21].On the physical side, meaningful images are everything but arbitrary inventions. They must be con-sistent with the part of reality that we try to model. This requires sufficient complexity of ourimages to map the essentials of the target domain on the structural and functional level. Such arequirement still leaves ample space for concrete, engineering type of models, for instance modelsused in graphical model building that are based upon state and rate variables. These models can beviewed as having some concrete underlying quasi mechanical or fluid-dynamical machinery. On amore abstract level, however, our images must be in accord with general principles of physics likesymmetry and invariance. Ultimately, such general principles will “kill” all concrete images of someunderlying machinery and will leave us alone with abstract mathematical structures.Somehow, the role of mental imagery in the modelling process of physics can be compared to aladder. We climb up the ladder from concrete pictures to increasingly formal and abstractrepresentations. Sometimes we even forget about the ladder, that we used to climb up. From timeto time it is good to step down the ladder, reflect the situation, and link the formal and abstractideas of physics with the concrete experiences, from which they emerged. Therefore, a glimpse on


concrete imagery for problem solving at basic levels of mathematics and physics is helpful to reflectthe role of imagery for formal thinking.

5. Linking the concrete with the abstract: examples for image driven problem solvingThe potential of concrete images for problem solving and the richness of individual images andstrategies can be demonstrated by the problem in Fig 4. The item is taken from the exposition ofthe PISA framework [22]. Two arrangements of T-shirts and drinks are shown and the problem isto find out the respective prices. Iexposed students and teachers withthe item at various occasions.Usually, they start writing down twoequations and begin to solve themformally. They have to be instructedto solve the problem mentally withoutwriting down equations.After this hintmost people are fascinated by thevariety of successful methods ofsolving the problem and of “seeing”the solution.One way of solving the problem isthe following. Because of thesymmetry of the above arrange-ment, one knows the price of one T-shirt and one drink.This amount canbe taken off mentally from the lower arrangement. From the remaining parts on both sides oneknows the price for two drinks and the problem is solved. This procedure combines visual elements(symmetry) with mental arithmetic and algebra (substitution, carrying out the same operation onboth sides of the equation).There exist other solutions which are even more “visual”. Some students have argued in the wayshown in Fig. 5, which comes very close to a “proof without words”. Both images are extrapolatedas a logical sequence (both on the side of the objects and the respective prices). This approach is avery convincing example for thecreative role of visual schemata fornon-routine problem solving. Thistype of problem and theencouragement of the non-formalyet highly insightful way of problemsolving is very far away from thestandard repertoire of textbookproblems, however.Another example for insight refersto a problem that elementary schoolchildren are able to solvespontaneously. The parallelogrampuzzle, investigated by Wertheimeris a classical example frompsychological research of insight[23] and the underlying processesare basic to modelling in physics onvarious levels of abstraction.Children explore how the area of arectangle can be measured by


Fig. 4. How much is a T-shirt, how much is a drink? Explain how you found the solution.

Fig. 5. A visual solution of the problem

counting the number of small squares, thatcover the rectangle. After the childrenhave understood the method, they areconfronted with the problem of measuringthe area of a parallelogram with the samewidth as the rectangle. Of course, theoriginal method fails as the oblique sidesdo not fit the squares (Fig. 6).Wertheimer gives a very detailed and vividaccount of the case of a little girl (age 5.5years!). Confronted with the parallelogramproblem she says, “I certainly don’t knowhow to do that.” Then after a moment ofsilence: “This is no good here,” pointing tothe region at the left end;“and no good here,”pointing to the region at the right. “It’stroublesome, here and here.” Hesitatinglyshe said: “I could make it right here . . . but .. . “Suddenly she cried out, “May I have apair of scissors? What is bad there is justwhat is needed here. It fits”. She took thescissors, cut the figure vertically, and placedthe left end at the right ([23], p. 49). Otherchildren bend the parallelogram to a ringso that the oblique parts fit together andfind the solution that way.This problem solving task demonstratesthat visual imagination is connected withconcrete experience (cutting out areas,putting them together). Yet there is muchmore involved that goes beyond mechanically carrying out actions. Some children are able toanticipate their actions, finding ways to solve the problem without ever having seen the solutionbefore. They go beyond their past experience and create something new.The underlying visually driven problem solving processes reach out deeply into a type of mentalmodelling which is relevant both in physics and in mathematics. In an abstract sense the little girl’sinsight is an early example for seeing invariants and constructing conserved quantities by suitablemental transformation processes (which have to be grounded by concrete experience).The require-ments on mental modelling are closely related to modelling processes in physics, creating a deeperinsight into the concept of torque and of angular momentum. For instance, in order to understandKepler’s second law one must find out why in any central force field planets sweep equal areas in equaltimes. As Newton’s geometrical derivation of Kepler’s second law shows, this requires the trans-formation of the areas of parallelograms (respectively of triangles) [24, 25].The little girl’s insight opens up a broad road towards mental modelling and a deeper understandingof heavenly and earth-bound motion. This line of thought shows that great ideas and achievements inthe history of science are not very different from everyday insights that even children can develop atan early age.As a consequence, we must make our teachers sensible for the potentials of visual literacyand foster cognitive development along these lines at a much earlier age.The potential of images is not sufficiently exploited in physics and mathematics teaching. Imagesdo have an important function not only as and external medium of representation but also as aninternal medium of problem solving. In general, teachers do only very little to challenge the mentalimagery of pupils. They do not support systematically multiple ways of problem solving, in which


Fig. 6. The parallelogram problem. Children who know how tomeasure the area of a rectangle are asked to measure the areaof a parallelogram

images and their reflections would quite naturally play a stronger role.This can be seen from video-analyses of physics lessons [26]. The education of future physics teachers has to address theseproblems adequately. At present, however, the formal way of physics teaching at universities is, inmy view, one of the biggest obstacles for making future teachers susceptible to more appropriateapproaches to formal thinking in schools. It is quite natural to copy bad examples!

6. From concrete actions to general principles: insights require reflectionsFig. 7 shows a mechanical balance at equilibrium. Similar to the T-shirt item shown in Fig. 4, thisproblem can be solved more effectively by using concrete pictorial operations rather than by solvingan equation. Removing mentally one brick from each side of the scales is a good strategy and a goodindicator for insight, although the argumentation - in comparison to the formal algebraic solution -“only” takes place at a concrete level.The example demonstrates the close interplay between the pictorial, concrete, physical access andthe symbolic, formal mathematical one.The physicalobject “balance” is a possible realisation of themathematical object “equation”. Algebraicoperations at the abstract level correspond tophysical processes at a concrete level. The equi-librium is not disturbed when the same processestake place on both sides of the physical ormathematical object. Our mental images and theabstract mathematical symbols that we use are“embodied” in such a way that they are based onconcrete experience about physical systems.It is interesting to note that the physical system“balance” corresponds to a basic mathematicalmetaphor giving birth to algebra, an abstract subjectwithin mathematics.Algebra comes from the Arabicword “al jabr”, the meaning of which is connected with balance and compensation [27]. As we allknow that mathematical formalism develops a life of its own, many students (and teachers too)often forget about this correspondence. In his criticism of the pure formalist approaches tomathematics, the famous Russian mathematician Arnold opens an article on teaching mathematicswith the statement Mathematics is a part of physics and he goes on saying Mathematics is the partof physics where experiments are cheap [28]. Penrose, a master in visualizing abstract ideas,promotes an evolutionary view of our mathematical abilities, which he considers an incidental fea-ture, a by-product of evolution [29]: For our remote ancestors, a specific ability to do sophisticatedmathematics can hardly have been a selective advantage, but a general ability to understand couldwell have. An important feature of understanding is mental model making, including reflectionsabout models and the modeling process. Models are executable mental images, mapping essentialfeatures of reality and allowing to anticipate the future. They can be run at practically no cost andtheir biological relevance for survival is evident.To some, Arnold’s criticism may sound too strong but in my personal view it is important to keepin mind that the mathematical enterprise is based primarily upon physical experience and gains itspotential in a social context (communicate an externalize ideas). Our conceptual system (includingthe formal mathematical system) is embodied. Most abstract ideas in the domain of science andmathematics arise via conceptual metaphors. This is a mechanism that projects concrete, embodied(that is, sensory-motor) reasoning to abstract reasoning. Thus, many abstract inferences can beconsidered as extensions or projections of sensory-motor inferences. Such a line of thought aboutformalism has been developed from linguistic approaches [30]. It may sound heretic to a formalist,but for learning and doing physics it is important to keep in mind that formal thinking has a humanface and builds upon concrete experience.


Fig. 7. The scale is in equilibrium. On the left scaleis a 1kg piece and half brick. On the right scale isa full brick. What is the mass of the full brick?

Nevertheless, the abstraction processes (as the development of physics shows) transcend the con-crete experience on which they are based. How is that possible? There is a creative element inmaking models. Going from the level of concrete experience to the level of generalisations (or,more formally, to the level of axioms) always involves a creative jump.This is very clearly expressedin Einstein’s view on the modelling cycle. His EJASE-scheme connects the level of experience (E)with the level of axioms (A) and the conclusions (S), formally derived from the axioms, by a crea-tive process depicted by J for jump [31] for a more detailed exposition). In many pedagogicaldiscussions on the modelling cycle this creative element is completely ignored! It is important to reflect the transition from concrete experience to abstraction in detail. Again, wecan use a simple example, closely related to the scale problem and transformations of mechanicalequilibrium. Fig. 8a shows a visual approach to the lever principle which is attributed to Archimedes[32]. We start with a symmetric configuration. We accept that it is in balance due to the symmetryprinciple. With a few transformations, that add additional loads and shift them, leaving the systemin equilibrium, we arrive at an asymmetric configuration: One unit of load at triple distance fromthe center is in balance with three units of load at one unit of distance. The sequence of picturesspeaks by itself. It acts quasi like “proof without words”. But nevertheless it is not a proof. Why? There is a circularity in the argumentation. Some of the transformations (shifting the loads by oneunit in one direction and compensating the effect by an equal shift in the opposite direction) takefor granted the principle that is intended to be proved by the whole procedure! The same holds forthe demonstration of the center of mass (Fig. 8b). In rigid body mechanics, there are three princi-ples that are closely interrelated: The lever principle, the superposition principle of torques and thecenter of mass - principle. They are part of experience that we take over to the system of axioms.They cannot be proved. But taking one of these principles for granted, the others can be derived.There is a certain freedom of constructing the system of axioms that requires a creative jump fromexperience to theory. Making this transition explicit is one of the most difficult stumbling blocks inmodel making and formal thinking.

7. Metamorphoses of complex open systems: metaphors for insight?In our attempts to foster understanding and insightful learning of physics we have to put more


Fig. 8. Insight into thelever principle (a) andthe center of massprinciple (b).

emphasis on the nature of human understanding and the concrete “embodied” base of abstractionand formalisation. Although understanding has the appearance of being a simple and common-sense quality, it is impossible to define it. In his popular writings on atomic physics and humanknowledge Bohr noted that an analysis of the very concept of explanation would, naturally, beginand end with a renunciation as to explaining our own conscious activity [33]. While our experiencestarts with concrete concepts, the development of physics has led to the insight that physical realityis not comprehensible in terms of concrete touchable systems. Questions, that have arisen in theearly days of quantum physics are still on the agenda today and will persist as we cannot circum-vent the abstractness of physics. The way to abstract and formal thinking is unavoidable, but whatcan we do not to lose too many students on that way?Making physics more meaningful to the inquiring mind is important. This includes, among manyother aspects, the question, what kind of physics we have to learn, in order to make the “physics inour heads” more tangible. The embodied mind, our mental imagery and intuition are basedconsiderably on mechanical experience. In accordance with the concrete nature of experience wehave restricted the present discussion to examples that are more or less mechanical in nature.Mechanical systems can be grasped, they are “visual” and “comprehensible”. Promoting mechani-cal imagery and its critical reflection is highly promising. It is surprising how far we can get by usingmechanically inspired imagery. Rather simple mechanical systems of our everyday world likedriven oscillators or coupled clocks behave in surprisingly complex ways [3] and the experimentsdescribed there). These mechanical systems serve as models for complex dynamical processes ofopen systems far from equilibrium. New properties emerge, like adaptive behavior and perceptionlike qualities, that we usually refer to the mental realm. The more we understand the complexbehavior of matter far from equilibrium, the more we also learn about the workings of our mind.

Along this line of thought we can even try to modelthe metamorphoses of internal images. The phe-nomenon of bistability that is found in manycomplex systems can serve as a model for mentallyswitching bistable percepts like the Necker cube inFig. 9. The spontaneous switching corresponds to athermally driven process, giving rise to a Poissonsequence of events that compares to thespontaneous switching of the cube.This model evenallows predictions for the transition fromspontaneous to the induced switching mode.When the eyes follow a periodically moving dot asthe experiment in Fig. 2 shows, the switchingprocess locks to the periodical signal in spite of thespontaneous noise. The switching becomessynchronized with the external drive signal. Thereis a close connection with the phenomenon ofstochastic resonance that is ubiquitous in nonlineardynamics. Noise plays a highly constructive role inthat phenomenon on many levels of biologicalinformation processing and in neural computation[34].These models of complex phenomenon emanatefrom “mechanical” intuition. Nevertheless, they

lead to concepts that are no longer mechanistic in the strict sense. One hardly recognizes the “me-chanical substrate” they originated from. They even supply models for “physics in the head”, e.g.for the metamorphoses of pictures or for other complex dynamic processes in our brains. Thesesystems can be modelled from first principles by field-theoretical approaches and show, to a certain


Fig. 9. Transformations of inner landscapes.A model for mentally switching bistable figures.

extent, universal behavior. In my personal view the universality of dynamical processes far fromequilibrium is one important factor why our brain can represent relevant aspects of reality.Considering mental images through the looking glass of physics creates new insights on theworking of our brains, whose collective effects over the centuries finally set up the program ofphysics. Is the theoretical frame of today’s physics sufficient to model internal observers?How far can such imagery guide our intuition? Although concrete pictures are necessary for ourunderstanding, physical reality transcends concrete imagery. Behind all our attempts to create con-crete pictures “lurks” the paradox. Our visual imagination is powerful enough even to handleparadoxical situations. Fig. 10 shows a strange picture of reality, which is also a metaphor for gaininginsight into the modelling method of physics. We make categories and models for certain aspects ofreality that appear consistent as long as they are viewed separately.The two shelves with the objectson them might correspond to the particle or the field approach in physics. However, when we tryto put both systems together (as it happened in the development of quantum theory), we arrive ata paradox (the impossible shelf).Such impossible pictures link physics with arts. Combining physics and arts is one importantsubject. Physics as the art of model making is another one.The real challenge, however, is to conveyhow much our naive conceptions of reality are challenged by the program of physics! Theperception of physics in the future will depend critically on how we can convey to a broader publicthat Physics is everywhere, how the late GIREP president Karl Luchner put it in his book. There ismuch to be done to make physics education more attractive and effective!

References[1] K. Luchner, Physik ist überall, München, (1991).[2] M. Michelini, Introduction to the First GIREP Seminar, Udine, (2001).[3] M. Euler, Physics and Physics Education Beyond 2000:Views, Issues and Visions. In, R. Pinto, S. Surrinach (Eds.),

Physics Teacher Education Beyond 2000, Proc. Int. GIREP Conference Barcelona 2000, Paris, (2001).[4] Knowledge and Skills for Life. First Results from the OECD Programme for International Student Assessment

(PISA) 2000, Ed.: OECD, Paris, (2001).


Fig. 10. Images ofphysics: from everydayobjects to strangerealities.

[5] A.E. Beaton et al, Mathematic Achievement in the Middle School Years: IEA’s Third Int. Mathematics and Sci-ence Study (TIMSS), Chestnut Hill, (1996).

[6] A.E. Beaton et al., Science Achievement in the Middle School Years: IEA’s Third Int. Mathematics and Sci-ence Study (TIMSS), Chestnut Hill, (1996).

[7] R.E. Mayer, Thinking, Problem Solving, Cognition, New York, (1992).[8] C.F. Gauss, Brief an Olbers, Werke, Vol. X.1, Leipzig (1917), 24.[9] R.J. Sternberg, J.E. Davidson (Hrsg), The Nature of Insight, Cambridge, Mass., (1995).[10] C. M. Seifert, D. E. Meyer, N. Davidson, A. L. Patalano, I. Yaniv, Demystification of Cognitive Insight:

Opportunistic Assimilation and the Prepared-Mind Perspective, in [9], 65.[11] R. E. Mayer, The Search for Insight: Grappling with Gestalt Psychology’s Unanswered Questions, in [9], 3.[12] M. Spitzer, Geist im Netz: Modelle für Lernen, Denken und Handeln, Heidelberg, (1996).[13] H. Atmanspacher, G. Dalenoort (Hrsg.), Inside Versus Outside: Endo- and Exo-Concepts of Observation and

Knowledge in Physics, Philosophy and Cognitive Science, Springer Series in Synergetics, Berlin, (1994).[14] F. Moss, D. Pierson, D. O’Gorman, Stochastic Resonance: Tutorial and Update, Intern. Journal of Bifurcation

and Chaos, 4, (1994), 1383[15] L. von Bertalanffy, Theoretische Biologie, Francke, Bern, (1951).[16] W. Pauli, Physik und Erkenntnistheorie, Braunschweig, (1984).[17] C. A. Meier (Hrsg), Wolfgang Pauli und C.G. Jung: Ein Briefwechsel 1932-1958, Berlin, (1992).[18] H. Atmanspacher, H. Primas, E. Wertenschlag-Birkhäuser (Hrsg.), Der Pauli-Jung-Dialog und seine Bedeutung

für die moderne Wissenschaft, Berlin, (1995).[19] H. Primas, Es gibt keine Einsicht ohne innere Bilder, GAIA 1 (1992), 311[20] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, II, Ch.

20-3, (1963).[21] See articles on students’ knowledge and learning in: A. Tiberghien, E.L. Jossem, J. Barojas (Eds.), Connecting

Research in Physics Education with Teacher Education, ICPE-Book, 1997; available for download athttp://physics.ohio-state.edu/~jossem/ICPE/BOOKS.html

[22] Measuring Student Knowledge and Skills: A new framework for Assessment, OECD, (1999).[23] M. Wertheimer, Productive Thinking, Harper & Row, New York, (1959).[24] J. Fauvel, R. Flood, M. Shortland, R. Wilson (Eds.), Let Newton be! A New Perspective on his Life and his

Works, Oxford Univ. Press, Oxford, (1992).[25] D.L. Goodstein, J.R. Goodstein, Feynman’s Lost Lecture, Norton, New York, (1996).[26] T. Seidel, M. Prenzel, R. Duit, M. Euler, H. Geiser, L. Hoffmann, M. Lehrke, C. Müller, R. Rimmele, Lehr-

Lernskripts im Physikunterricht und damit verbundene Bedingungen für individuelle Lernprozesse,Unterrichtswissenschaft, 30, (2002), 52

[27] A.K. Dewdney ,A Mathematical Mystery Tour, Wiley, New York, (1999).[28] V. I. Arnold ,On Teaching Mathematics, Russian Math. Surveys 53, 1 (1998), 229[29] R. Penrose, Shadows of the Mind, Oxford Univ. Press, Oxford, (1994).[30] G. Lakoff, Nunez R.E, Where Mathematics Comes From, Basic Books, New York, (2000).[31] G. Holton, Thematische Analyse der Wissenschaft, Suhrkamp, Frankfurt, (1981).[32] K. Simonyi, Kulturgeschichte der Physik, Harri Deutsch, Frankfurt, (1990).[33] N. Bohr, Atomic Physics and Human Knowledge, Wiley, New York, (1958).[34] E. Simonotto et al, Phys. Rev. Letters, 78, (1997), (1186).


PHYSICS CURRICULUM REFORM: HOW CAN WE DO IT?

Robert G. Fuller, Department of Physics and Astronomy, University of Nebraska -Lincoln, USA

The physics community in the United States of American is facing a crisis. This crisis has beendescribed in presentations and papers at professional meetings in various ways. Let me introduceyou to the crisis faced by the physics community by discussing three different published papers thatpresent views of this crisis.First is the paper by David Goodstein, Provost of the California Institute of Technology and aphysicist who co-authored The Mechanical Universe television series and textbook. According toProvost Goodstein “The Big Crunch” occurred in the 1970s which was the end of 100 years ofexponential growth of science in the USA. No longer was increase financial support and nationalinterest in science guaranteed. Our educational institutions are poorly adapted to deal with adifferent future.In the USA science education has produced the Paradox of Scientific Elites and ScientificIlliterates. We have a small cadre of exceptional scientists and a broad population of scientificilliterate people. The model for science education in the USA has been described as a leakypipeline. This is the wrong. Mining and Sorting is a better metaphor for science education in theUSA. The purpose of the education enterprise has been to sort out the student unworthy of aprofessional degree in science. It reaches its culmination in graduate school. Research professorsobtain external funding, independent of the needs of the institution, to run their researchprograms. In the steady state each professor needs to turn out ONE Professor for thenext generation. In the golden era of physics, research professors would turn out one PhD per year!The profession of teaching physics in the USA today has only two purposes:to turn out physicists and to act as a gate keeperWe must turn the problem around. Physics has tremendous assets. Physics is a vast body of humanknowledge. In some ways, physics is the central triumph of human intelligence. Physics has pavedthe way of civilization to our victory over mystery and ignorance. The methods of inquiry andanalysis used in physics that have produced that body of knowledge. The reasoning patterns usedand developed in the study of physics are the keystone of scientific reasoning. Hence, anundergraduate physics major program must become the essence of a liberal education for the 21stcentury. Unfortunately, everything about the way we teach physics today is useless for this visionand I do not know the first step in that direction.How do we teach physics for all citizens rather than just a scientific elite? I believe the key toteaching anything is to remember what it was like not to understand the thing. Provost Goodsteinends his paper by pointing us in a direction for physics education for the next generation [1].The next paper I want to discuss is a paper by Sheila Tobias. Ms. Tobias is not a physicist. She wastrained in the humanities and first became noted for her famous book on mathematics anxiety.Sheila Tobias first wrote Overcoming Math Anxiety in 1978. In her updated version, published byW. W. Norton in 1994, she enlarges on her analysis of the attitude and approach variables thatinterfere with students’ performance in college-level mathematics. In her paper published in theAmerican Journal of Physics in 2000, Ms. Tobias raises several questions about various aspects ofphysics education in the USA.Many of the most prestigious secondary schools in the USA, offer advanced placement (AP)physics courses. These AP courses are a second year of physics intended for an elite corps ofsecondary school students. How useful is AP physics?There is a national movement in the USA towards standardized testing and in-class examinations.Yet examinations can constrain educational innovation.There are a variety of national rankings of physics departments. How do we figure teaching into adepartment’s rank? In fact, if you look at the descriptions of the interests of physics facultymembers at major universities you will almost find none who express a professional interest inphysics education!

We must start a national movement to require high school physics for entrance into college. Thismeans that we must develop courses in “Science for all”...not just an option for some.Physics departments must cultivate their “clients.” They need to establish permanent liaisons withthe engineering and life science communities who require their students to take physics courses.Universities need to revisiting the issue of class size. Is class size a meaningful arena for change?Finally, Ms. Tobias urged physics departments to examine and transform the physics major forundergraduates. It needs to become, not just a path for physics elites who intend to go to graduateschool in physics, the physics major must become attractive for students with undecided careergoals [2].The third paper I want to discuss is the paper by Professors Ruth Howes and Robert Hilborn, bothformer presidents of the American Association of Physics Teachers (AAPT). Their paper was alsopublished in the American Journal of Physics in 2000.Professors Howes and Hilbom assert that Physics departments may not have changed much in thepast few decades but the educational, but the scientific and social environments in the USA havechanged considerably.Physics has expanded and spun off numerous subfields.The educated public views the frontiers of science as in the life sciences and physics is no longerwhere the action is.The educational environment has changed. The students are more diverse.Client disciplines have begun to consider teaching introductory physics themselves.The number of undergraduate physics majors has sunk to below pre-Sputnik levels while the totalnumber of undergraduates has doubled.Four principles need to guide our response:1 A wide spectrum of physicists recognize the need for change, but many still do not.2. The fundamental unit of change is the department.3. An undergraduate physics program is more than just the curriculum.4. Every physics department is different.The new environment is unlikely to return to its state of 30-some years ago. It will probably takesustained efforts on many fronts before we see substantial results [3]. Taken in toto these threepapers are an urgent plea of major reforms in the physics curricula used in the USA. I want tosuggest a direction for such curriculum reform efforts by looking back at the work of a famousphysicist and physics educator and try to draw from his work guidelines for national physicscurriculum reform efforts.

Basing Physics Curriculum Reform On the Second Career of Robert KarplusLet me begin by telling you about the first career of Robert Karplus. He was born in Vienna,Austria, in 1927. His family moved to the USA when he was 10 years old. His first career was intheoretical and experimental physics. He obtained a double degree from Harvard University inphysics and chemistry in 1945 and one year later got a masters degree in chemistry, also fromHarvard University. He completed his Ph. D. in chemical physics at Harvard in 1948. His thesisresearch included both experimental and theoretical work on microwaves for Professor E. BrightWilson, Jr. He moved from Harvard to the Institute for Advanced Studies, directed by J. RobertOppenheimer, at Princeton University in 1948. He married Elizabeth Fraizer in December of 1948.He began to work in quantum electrodynamics (QED). In 1950 Karplus and Kroll published thefirst detailed calculations of a physics observable based on QED [4].In 1950 Dr. Karplus returned to Harvard University where he served as an assistant professor ofphysics from 1950 until 1954. In 1954 he moved to the University of California, Berkeley where hewas an Associate Professor of physics from 1954 until 1958 when he was promoted to full professor.From 1948 to 1962 he published 50 research papers in physics, mostly in QED, but also on the Halleffect and Van Allen radiation. He was the senior or only author of the first 19 papers. He publishedwith 32 different scientists, including 2 Nobel prize winners. More than 90% of his co-authors are


now fellows of the American PhysicalSociety. Professor Karplus made his firstvisit to his daughter’s elementary schoolclass in 1959-60. He probably did anelectrostatics demonstration with aWindhurst machine1. Some thinghappened to his intellectual curiosity inthose visits to his daughter’s class and hebecame more and more interested in thekind and quality of science being taughtto children in elementary schools in theUSA. He joined in an elementary schoolproject with some other University ofCalifornia Berkeley faculty in 1959. Hepublished his first education paper withJ. M. Atkin in 1962, “Discovery orInvention?” in The Science Teacherperiodical [5].Karplus and Herb Thier started the Science Curriculum Improvement Study(SCIS) in 1961 withfinancial support from the National Science Foundation. Over the next several years they and theirco-workers developed a complete K-6 science curriculum, (for children ages 5 through 11) the SCIScurriculum, that is still in use today.Robert Karplus was president of the AAPT in 1977 and he received the Oersted Medal in 1980. Hesuffered a cardiac arrest while jogging in June of 1982 which ended his professional career . He diedin 1990.As a part of faculty development leave in 1999, I collected a sample of his publications in scienceeducation and based on those works I want to lift up for your consideration the enduringcontributions the work of Robert Karplus has made to science education [6].

Robert Karplus’s enduring contributions to scienceeducation:1) He took Piaget’s work seriously. Robert Karplus was one ofthe first educators in the USA to see the relevance of the workof Jean Piaget to curriculum development. Based on his studyof Piaget’s work he came to believe that new knowledge mustbe constructed by the mind of the learner and not simplytransmitted by the teacher. In this sense, then Karplus was oneof the first constructivists.Therefore, I think a start toward understanding the impact ofthe work of Robert Karplus in science education is to do abrief review of the life and work of Jean Piaget.Professor Jean Piaget (1895-1980) lived and worked most ofhis life in Geneva, Switzerland. His work can be described bythree different periods. His first period (1922-29) began inBinet’s laboratory in Paris, France. He began his style of semi-clinical interviews, placing simple apparatus in front ofchildren and asking them to explore and explain its behavior.He discovered and described children’s philosophies, such as


1 He and Betty were the parents of seven children bom between 1950 and 1962, three daughters and four sons.

their belief that the sun followed them, a property of children’s thinking that he called egocentrism.The second period of the work of Piaget (1929-40) was when he studied his own three children. Hetraced the origins of a child’s spontaneous mental growth and realted it to infant behavior. Hepostulated a variety of conservation reasoning patterns whereby the child comes to believe thatcertain properties of a system remain the same even though the appearance of the system maychange dramatically, as in the game of Peek-a-Boo.In the third period of his work (1940-80) Piaget concentrated on the development of logicalthought in children and adolescents. He observed how a child constructs one’s world. A child’smind is not a passive mirror. At first a child can reason about things but not about propositions.Based on his own observations, Piaget developed his concept of the stages of cognitive develop assummarized in the table below:

Logical KnowledgeStages of Cognitive Development

(Jean Piaget)

Stage Characteristics Approximate AgeRange (Years)

Sensory-Motor Pre-verbal Reasoning birth -2

Pre-operational No cause-and-effectUses verbal symbols,simple classifications,but lacks conservation reasoning 1-8

Concrete Operational Reasoning is logical,But concrete ratherthan abstract 8 - ?

Formal Operational Hypothetical- deductive reasoning 11 - ?

According to Piaget, cognitive development explains learning. Development occurs by four mainfactors:

MaturationExperience - the effect of the physical environment on the mental structures of intelligenceSocial, or education, transmissionEquilibration or Self-regulation-the fundamental one

Professor Karplus wrote his summary of major ideas of Piaget’s work for educators as a part of theWorkshop on Physics Teaching and the Development of Reasoning that he developed for theAAPT.Major Ideas from Piaget’s Work (by Karplus, 1975)1. Piaget’s theory describes two major stages of logical, operational reasoning in human intellectualdevelopment, the stage of concrete reasoning and the stage of formal reasoning. Earlier stagesidentifiable in the behavior of young children may be called pre-operational.2. Each of these two major stages is characterized by certain reasoning patterns, used by individualsto classify observations, interpret data, draw conclusions, and make predictions.

Characteristics of concrete and formal reasoningConcrete ReasoningIndividuals-(a) Need reference to familiar actions, objects, and observable properties.


(b) Use classification, conservation, and seriation reasoning patterns in relation to concrete itemsa) above. Have limited and intuitive understanding of formal reasoning patterns.

(c)Need step-by-step instructions in a lengthy procedure.(d) Are not aware of their own reasoning, or inconsistencies among various statements they make,

or contradictions with other known facts.

Formal ReasoningIndividuals-(a) Can reason with concepts, relationships, abstract properties, axioms, and theories; use symbols

to express ideas.(b) Apply classification, conservation, seriation, combinatorial, proportional, probabilistic,

correlational, and controlling variables reasoning in abstract items (a) above.(c) Can plan a lengthy procedure given certain overall goals and resources.(d) Are aware and critical of their own reasoning, actively seek checks on the validity of their

conclusions by appealing to other known information.

3. The formal stage is an idealization in that most persons after age twelve use formal reasoningpatterns under some conditions and concrete reasoning patterns under others. The latter is likelyto occur whenever the subject matter is unfamiliar, as is the case for a student beginning work in anew area. The former is likely to be the case for an experienced worker in the field.4. The process of self-regulation plays a vital role when an individual advances from the use ofconcrete reasoning patterns to the use of formal reasoning patterns. Self-regulation begins withone’s awareness that the concrete reasoning patterns are inadequate. It proceeds through directexperience with phenomena supplemented by the introduction of related organizing principles andmajor concepts.5. A person who uses only concrete reasoning patterns is likely to proceed through self-regulationin a new subject much more slowly than a person who reasons formally in connection with otherstudies. The latter individual benefits from the possibility of transferring formal reasoning patternsto the new area, especially if the new and old are closely related as is the case with mathematicsand physics.6. Some students who are required to learn formal-level material in a subject in which they so far haveonly used concrete reasoning may go through self-regulation spontaneously. Other students, with lessexperience or self-awareness, are not likely to experience the necessary self-regulation; instead, theywill memorize certain prominent words, phrases, formulas, and procedures, but will apply these withlittle understanding unless the teaching program takes their specific needs into account.7. Tests should be designed to evaluate the students’ reasoning and also help them engage in self-regulation.8. The Learning Cycle can be an effective strategy in classes where some students display concretereasoning patterns and some formal reasoning patterns [7].

Robert Karplus’s enduring contributions to science education (continued):2) Professor Karplus focused on student reasoning. Based on his understanding of the work ofPiaget, he developed a series of paper and pencil tasks that he could use with students to reveal thereasoning patterns they used. He published this work in a number of articles about reasoningbeyond elementary school. The first puzzle he used by the Mr. Short-Mr. Tall Puzzle [8], as follows:

The Mr. Short - Mr. Tall Puzzle The figure below is called Mr. Short.We used large round buttons laid side-by-side.To measure Mr.Short’s height, starting from the floor between his feet and going to the top of his head. His heightwas four buttons. Then we took a similar figure called Mr. Tall, and measured it in the same waywith the same bottom. Mr. Tall was six buttons high.


Professor Karplus collected a number of student responses to the Mr. Short/Mr. Tall task and thenhe developed a classification system for these responses. Finally, he organized the classificationsystem into a sequence of reasoning patterns based on the stages of cognitive developmentpostulated by Piaget.

Student Responses to the Mr. Short/Mr. Tall Task:Category N: no explanation or statement, “I can’t explain.”Category I (intuition): An explanation referring to estimates, guesses, appearances, or extraneousfactors without using the data.Examples of the predictions and related explanations are:

9 1/2-The big man just looks that much bigger than the little man.9 -Because that’s what I think it is.9 1/2-You use a ratio factor, but I don’t remember how, so I guessed10-A guess.

Category IC (intuitive computation) :The subject makes use of data haphazardly and in an illogicalway. Examples are:

16-By multiplying.10-I figured out that if he had bigger paper clips, you add 6 with the other 4 and you have 10.12 1/2-Half of 12 is 6, so you’d take the paper dips and measure him twice

because he’s longer than Mr. Short, so naturally it would be 121/2.10 3/4- Since it took 4 biggies for Mr. Short and 6 1/4 smallies, there is 2 1/4 difference, and it

tool, 6 biggies for Mr. Tall, 2 more than for Mr. Short, so I added 2 1/4 + 2 1/4 together and got 4 1/2and added that to 6 1/4 and got 10 3/4 for an answer.Category A (addition) : An explanation using all of the data, but applying, the difference ratherthan the ratio at measurements. Examples are:

8-The little man was 4 of his and 6 of mine so I added 2.8 1/2 -When you did it, the large man was 2 big paper clips bigger than your small man. So mine

must be 8-if 6 smallies equal4 biggies, then 6 biggies must equal 8 smallies.

Category S (scaling) :The subject makes a change of scale when he predicts. He does not relate thisoperation to the scale inherent in the data, thereby failing to see the whole problem. He expressesa tentative attitude toward his estimate. Examples are:


1. Measure the height of Mr. Shortusing paper clips in a chain provided.The height is

2. Predict the height of Mr. Tall ifhe were measured with the same paperclips.-

3. Explain how you figured out yourprediction. (You may use diagrams,words or calculations. Please explainyour steps carefully.)

12-The large man is two times bigger than the little man; the little man is 6, so I think it is 12.12-I think, it is twelve because in biggies it is six, and small ones are about half that size, and so

I thinl, it is about 12.Category AS (addition and scaling) : The subject focuses on the excess height of Mr. Tall, but scalesup the excess number of jumbo paper clips by a factor of two to compensate for the size difference;An example is:

10-1 think, two smllies are as big as one biggie, so I added four smallies for the two extraCategory P (proportional reasoning): The subject uses proportionality and makes clear how theratio is derived from the measurements on the two figures. He may or may not use the word “ratio.”Examples are:

9 -There is a mathematical problem in 4 big, and 6 big, 4 is 2/3 of 6, so it should be 9.9 -6 ÷ 4 = 1 1/2, 6 x 1 1/2 = 9.9 -The ratio of the biggies is 2:3, so you figure the small paper clips would also have the ratio 2:3.9.75 -It was a direct ratio and proportion, small to large, small to large.9 -Set up a ratio, 6/4= x/69 6/4 =x/6, x = 9

The results Professor Karplus obtained for students from ages 9 (4th grade) to 17(12 th grade) areshown in the bar graph below:


Another task that Professor Karplus developed by the island puzzle [9], see below:

The Islands puzzleThe puzzle is about Islands A, B, C, and D in the ocean. People have been traveling among theseislands by boat for many years, but recently an air-line started in business. Carefully read the cluesabout Possible plane trips at present. The trips may be direct or include stops and plane changeson an island. When a trip is Possible, it can be made in either direction between the islands. youmay make notes or marks on the map to help use the clues.

First Clue: People can go by plane between Islands C and D.Second Clue: People cannot go by plane between Islands A and B, even indirectly.Use these two clues to answer Question 1. Do not read the next clue yet.Question 1: Can people go by plane between Island B and D?

Yes_________ No _________ Can’t tell from the two cluesPlease explain your answer.

Third Clue (do not change your answer to Question I now!): People can go by plane between IslandB and D.Use all three clues to answer Question 2 and 3.Question 2: Can people go by plane between Island B and C?

Yes_________ No _________ Can’t tell from the two cluesPlease explain your answer.

Question 3 :Can people go by plane between Islands A and C?Yes_________ No _________ Can’t tell from the two cluesPlease explain your answer..

Island Puzzle ResponsesCategory N: no explanation or statement “I can’t explain.”Category I (pre-logical): an explanation which makes no reference to the clues and/or introducesnew information. Subcategories are the mere repetition of the answer to be explained (to #2, “Yes,because there are flights”), appeal to the diagram itself (to #2, “Yes, because it is the diagonal” orto #3, “Yes, because it is close”), and fanciful stories (to #1, No, because there is a strong air pocketthat no one can survive” or to #2, “No, because the plane can run out of gas and go down in thewater”)Category Ila (transition to concrete models) : direct appeal to or repetition of clues (#1, “No,because you did not say so” or to “#1 “Can’t tell because you didn’t say”). Since all three questionsrequire inferences, a direct appeal to the clues does not provide a logical justification.Category Ilb (concrete models) : the clues are used to construct models which are then used tomake the predictions. The most common model provides for the presence or absence of airportfacilities on an island, according to whether flights were or were not said to reach it (to #l, “Can’ttell, because Bean Island has an airport, but Bird Island might or might not have an airport”; to #2,“Yes, because there must be an airport on Bird Island, so the people from Fish lsland can get there”;to#3, “No, Snail must be the one with no airport, so people from Fish Island can’t get there”). Thismodel-based approach, when correctly used, leads to correct answers to all three questions in theproblem. It assumes information not given in the clues, however, and cannot be generalized to solvesimilar puzzles with different data.Category IIIa (transition to abstract logic): logical explanation to question 2, that Bird Island can


The puzzle is about Islands A, B, C, and D inthe ocean. People have been traveling amongthese islands by boat for many years, butrecently an air-line started in business.Carefully read the clues about Possible planetrips at present. The trips may be direct orinclude stops and plane changes on an island.When a trip is Possible, it can be made ineither direction between the islands. you maymake notes or marks on the map to help usethe clues.

certainly be reached from Fish Island by way of a stop at Bean Island (to #2, “Yes, Fish to Bean toBird”). Since the logical inference from the two positive statements (clues I and 3) needed forquestion 2 is easier, in our view, than the use of the negative statement (clue 2), question 2 does notmake maximum demand on the subject’s reasoning ability.We have therefore classified the logical answer here as being transitional to the abstract stage,rather than representing attainment of the abstract stage.Category IIIb (abstract logic): logical explanations to questions 1 and 3 (to #1, Can’t tell becausethere is no information linking either Bean or Fish Island with Bird Island”; to #3, “No, because aflight between Fish and Snail would make possible a route between Bird and Snail via Bean andFish; this contradicts the second clue”).Karplus collected responses to the Island Puzzle from students from ages 10 to 17 as well as frommembers of the National Science Teachers Association and the AAPT. The results of hisinvestigations are shown in the following bar graph.


Based on the work of Karplus we wondered about the reasoning patterns typically used by collegestudents in the USA. Furthermore, we decided that proportional reasoning is a pattern that isessential for understanding college level mathematics and physics.So we solicited the help of many faculty colleagues to collect data about the reasoning of a largenumber of college students doing a variety of proportional reasoning tasks.We give three examples of the kinds of proportional reasoning tasks we developed [10] to use withcollege students2.

The Wahoo PuzzleThe state of Ohio is converting all of their highway distance road signs to a dual English-metricsystem. Shown below is an example of a sign that you might see as you drive towards Cleveland,Ohio.

2 With the help of other faculty members we collected student responses to proportional reasoning tasks from more thateight thousand college students in the USA. We developed five categories of responses, ala Karplus, to these tasks.

CLEVELAND

94 MILES

152 KILOMETERS

Assume that the state of Nebraska also converts its road signs to the same system. As you drivetowards Wahoo, Nebraska, you might see the following sign.

WAHOO

_______ MILES

380 KILOMETERS

Are you able to compute the number to put in the blank shown on the sign above from the datayou are given on this page? Yes_______ No _______Explain your answer-If you can compute the number to put in the blank, please do so. Writeit in the blank above and explain in words how you calculated your result.Show your work below:

The Recipe PuzzleA recipe for pumpkin pie requires that milk be added along with otheringredients. This modern recipe gives both the old English and the newmetric equivalents as shown below.

Recipe #1 Pumpkin Pie

Add21 teaspoons of milkor99 milliliters of milk

Another metric recipe calls for a similar ingredient but does not give theEnglish unit equivalent.

Recipe #2 Chocolate Cake

Add_______teaspoons of milkor231 milliliters of milk

Are you able to compute the number to fill in the blank shown in Recipe #2 ?Yes_______ No _______Explain your answer-If you can compute the number, please do so. Write your answer in the blank above, and show andexplain how you did the calculation below.


The Shadows PuzzleWalking back to my room afterclass yesterday afternoon, Inoticed my six-foot frame cast ashadow eight feet long. A rathersmall tree next to the sidewalkcast a shadow eighteen feet long.My best guess of the height of thetree would be Please explain thereasoning you used to find youranswer

Proportional Reasoning Responses1. Intuitive: No response or aguess with little evidence of reasoning.Examples: Can’t tell.I’m not good at numbers.2. Additive: Adds or subtracts to obtain an answer.Example (Shadows): 8 is to 6 as 18 is to 16.3. Ratio attempt: Attempts a ratio but fails for reasons other than arithmetic: wrong ratio, can’tsolve for x, etc.Example (Recipe): May try to find how many times 99 goes into 231, what remainder will exist,then somehow try to convert the 33 ml remainder into teaspoons.4. Ratio formula: Uses proportional reasoning to set up an equation and then solve for unknown.Example (Shadows): 6/8 = x/18 so x = (6/8) 18 or 13 feet.5. Conversion: Introduces a new quantity as a conversion factor then multiplies or divides. Example(Shadows): The height is 6/8 or 75% of the shadow so the tree is 0.75 x 18 = 13.5 feet high.You will notice on the following bar graph of our results that a significant fraction, about fortypercent, of typical USA college students do NOT systematically use proportional reasoning.Before we can begin to develop a new physics curriculum, we must address a more fundamentalquestion. How do we foster thedevelopment of more advancedreasoning by college students?Let us turn our attention back to thework of Robert Karplus as we attemptto answer that question.

Robert Karplus’s enduring contributionsto science education (continued):3) Karplus developed the learningcycle instructional strategy [11]. Hebelieved this strategy was most likelyto encourage students to go throughthe process of self-regulation anddevelop more advanced reasoningprocesses.Classroom activities may play a centralrole in the improvement of studentreasoning. A classroom instructionalstrategy based upon the work of Piaget


and Karplus is called the Learning Cycle. The entire learning cycle consists of three phases that arecalled exploration, and application.During exploration the students learn through their own more or less spontaneous reactions to anew situation. In this phase, they explore new materials or ideas with minimal guidance orexpectation of specific achievements. Their patterns of reasoning may be inadequate to cope withthe new data, and they may begin self-regulation.During the invention phase, a new concept is defined or a new principle invented to expand thestudents’ knowledge, skills, or reasoning. This step should always follow exploration and relate tothe exploration activities. It will thereby assist in your students’ self-regulation. Do encourageindividual students to “invent” part or all of a new idea for themselves, before you present it to theclass.During the last phase of the learning cycle, application, students find new uses for the concepts orskills they have invented earlier. The application phase provides additional time and experiencesfor self-regulation to take place. It also gives you the opportunity to introduce the new conceptrepeatedly to help students whose conceptual re-organization proceeds more slowly than average,or who did not adequately relate your original explanation to their experiences. Individualconferences with these students to identify their difficulties are especially helpful.A wide variety of evaluation projects were able to demonstrate the efficacy of this instructionalstrategy.

Robert Karplus’s enduring contributions to science education (continued):4) Professor Karplus developed a scientific process of curriculum development.It seems so logical now. The process that Karplus developed has become widely used by peoplewho are unaware of his work. He brought the feedback loop of the scientific method to the taskof curriculum development. Field testing materials with students was a keystone of his process.The process that he used is to develop the materials, then field test them in a wide variety ofclassrooms and then use the feedback from the classroom activities to revise the lessons andfield test them again and revise them again, etc. In such a process, only the very best lessonssurvived [12].5) Dr. Karplus realized that the central figure in the school learning experience of science is theteacher. Elementary school teachers had to become comfortable with hands-on science activities inorder for his SCIS program to succeed. Karplus emphasized teacher development. He created aseries of teacher workshops on the theme of science teaching and the development of reasoning[13].In addition to the workshops, Professor Karplus and his co-worked created a series of movies thatshow how students react to a variety of reasoning tasks, many of them taken from Piaget’s work.The film of most use to college faculty is his film on formal reasoning patterns. It shows thebehavior of students from 10 to 17 years of age as they approach tasks requiring combinatorialreasoning, proportional reasoning, separation and control of variables and multiplicativecompensation [14].6) Karplus emphasized that science for everyone, not just an elite few. As a result of his convictionabout this, they tried SCIS lessons on students of all kinds. The modern day movement of sciencefor all students can look back and find its roots in the work of Robert Karplus.7) Finally, Karplus loved the act of discovery. He never tired of discovering things for himself andhe wanted children to know the joy of discovery. “Don’t tell me, let me find out.” is the title of thefilm that they made about the SCIS curriculum.In conclusion, then, it seems clear to me that important curriculum work in physics and othersciences will find its strength in the enduring contributions that Robert Karplus has made toscience education. A successful physics curriculum for the 21st century will be one in whichstudents in physics classes are encouraged to have their own wonderful ideas [15].


References[1] D. Goodstein, “Now Boardng: The Flight from Physics”, Amer. J. Phys., 67(3), (1999), 183-6.[2] S. Tobias, “From innovation to change: Forging a physics education reform agenda for the 21st century”, Amer.

J. Phys., 68(2), (2000), 103.[3] R. Howes and R. Hilborn, “Winds of Change”, Amer. J. Phys., 68(5), (2000), 401-2.[4] R. Karplus et al., Fourth Order Corrections in QED to the Magnetic Moment of the Electron, Phys. Rev. 77,

(1950), 536-49.[5] R. Karplus and J.M. Atkin, “Discovery or Invention?”, The Science Teacher 29(5), (1952), 45-47.[6] “A Love of Discovery: Science Education - the Second Career of Robert Karplus”, R.G. Fuller (Ed), Kluwer

Academic/Plenum Publishers, “, (2002).[7] R. Karplus et. al., Physics teaching and the development of reasoning workshop. American Association of

Physics Teachers, (1975).[8] R. Karplus and R. W. Peterson, Intellectual development beyond elementary school ii: ratio, a survey, School

Science and Mathematics, 70(9), (1990), 813-820.[9] E. F. Karplus and R. Karplus, Intellectual Development Beyond Elementary School 1: Deductive Logic, School

Science and Mathematics, 70(5), (1970), 39@-406.[10] M.C. Thornton and R.G. Fuller, how do college students solve proportion problems? J. Res. Sci. Teach., 18,

(1981), 335.[11] R.G. Fuller, R. Karplus, and A.E. Lawson, Can physics develop reasoning?, Physics Today, 30 (2), (1977), 23;

R. Karplus, Science Teaching and the Development of Reasoning, Journal of Research in Science Teaching, 14(2), (1977),169-175.

[12] R. Karplus, Strategies in Curriculum Development-The SCIS project, Strategies for Curriculum Development,Jon Schaffarzick and David H. Hampton (Eds), McCutchan Publishing Corp., Berkeley, CA, (1975), 69-88.

[13] R. Karplus, A. E. Lawson, W. T. Wollman, M. Appel, R. Bernoff, A. Howe, J. J. Rusch and F. Sullivan. Workshopon Science Teaching and the Development of Reasoning, Berkeley, CA: Lawrence Hall of Science, March (1976)(trial edition), Final edition, (1977)]

[14] R. Karplus and R. Peterson, Formal Reasoning Patterns (a film illustrating Jean Piaget’s developmental theoryof intellectual development), San Francisco: Davidson Films, (1978).

[15] [Eleanor R. Duckworth, The having of wonderful ideas & other essays on teaching & learning, 2nd ed, TeachersCollege Press, Teachers College, Columbia University, (1996).



REAL-TIME APPROACHES IN THE DEVELOPMENT OF FORMAL THINKING INPHYSICS

Elena Sassi, Department of Physical Science, University of Naples, Italy, EU project STTISScience Teacher Training in an Information Society, Italy

1. IntroductionThere is a widely acknowledged consensus, both in the educational research and in the schoolcommunities, about the very relevant role of Formal Thinking in building a well sound knowledgein Science Education, and expecially in Physics Education. A piece of evidence is the decision ofhaving this Seminar and the many contributions presented.In school practice several learning difficulties are very likely related to insufficient awareness of therelationship between experiment and theory, and/or phenomenology and its formal description.For the sake of brevity no detailed discussion of Formal Thinking will be addressed here; thepresentation of this Seminar has offered many hints and a broad meaning has been proposed inthe presentation of WorkShop1 (De Ambrosis & Rinaudo, 2001) .To frame the content of this paper it may be useful to briefly recall some features of the currentsituation.A first one is the general consensus nowadays acknowledged by science education researchersabout the urgent need of connecting more intensely research results with the fields of teachers’education, class-practice and work of practitioners. (Osborne & Millar, 2001; Viennot 2000, 2001).Another one is the growing interest in Pedagogical Content Knowledge or PCK which, since thelast ’80s, calls attention on the awareness that, in order to reach a teaching practice of high quality,the indispensable knowledge of disciplinary content needs to be transformed into a subject matterknowledge for teaching.A main focus is that at least these types of knowledge are key components of PCK: representationsof content suitable for teaching; teaching strategies coherent with these representations; students’common-sense knowledge and learning difficulties (De Jong, 2001).A third one is that educational research is more and more demanded to demonstrate that itsresults are valid and useful contributions to teachers and school practice. In the the presidentialaddress of 2001 ESERA Conference, Robert Millar (Millar 2001) pointed out that “in manycountries, research in education (including science education) is under increasing pressure todemonstrate that it is useful – to teachers and to other such as policy-makers – and can help themto do their jobs more effectively”. A growing effort is being put in developing research basedteachers’ education programs, for instance in the EU Project STTIS (cfr. in the following) (Colinet al, 2001; Monroy et al. 2001;; Pintò et al., 2001; Stylaniadou et al., 2001).This paper presents some viewpoints about the main contributions that Real-Time approaches cangive to the development of Formal Thinking (FT) in basic physics education.The results underlying these viewpoints come from three sources: research studies of the physicseducation group I am working with, which has been involved, since the early ‘80s, in studying andusing Real-Time approaches; the EU three years project STTIS (Science Teacher Training in aninformation Society) which very recently has been completed1 and literature.Section 1 indicates very briefly some aspects of the development of Formal Thinking in physicsand science education and some main difficulties encountered in this process.Section 2 discusses the main contributions that Real-Time Experiment and Images can give toFormal Thinking.In Section 3 presents comments about the current use of RTEI in ordinary class practice and themain transforming trends of its rationale.Section 4 concludes with a possible project aimed at facilitating RTEI adoption.

1 Public Documents of the STTIS project can be found at http://www.blues.uab.es/~idmc42/archive/index.html

2. Learning/teaching problems related to development of formal thinkingIt is commonly acknowledged that the development of Formal Thinking is a complex process,involving the acquisition of multiple types of knowledge and the integration of skills belonging todifferent areas. A schematic, non exhaustive representation is shown in Fig.1; it depicts two largeareas which do play important roles in the process and which have been labelled “Models” and“Experiments” for the sake of brevity.


Fig.1: Some components of the development of Formal Thinking

To develop a sound FT in physics the knowledge of the language and meaning of Mathematics isa crucial element. The relationship between Physics and Mathematics has been defined as a“constitutive” one (Levy Leblond 1991) and the students need to be helped to become aware ofthe fact that Mathematics is the language needed by Physics to articulate its description of that partof natural reality which has been and is studied scientifically. Other crucial elements are theknowledge of the basic principles of physics and the awareness of the values and beliefs that thephysics community has developed along the development of the discipline. This framework isrepresented in the schema as a kind of net that is a background element of the process.The two blobs Models and Experiments are linked through many activities, as for example thoseindicated between the two arrows. These activities aim at acquiring capabilities about the linksamongst phenomenological observations, experimental results, data of different types and theirpossible formal representations, such as: - data fitting, exploring/using models already made,building one’s own models. A well sound FT in physics implies the capability of moving back andforth phenomenology and its formal descriptions and explanations.The blob “Models” refers to all those learning/teaching activities that help to become aware of thevarious types of models (qualitative, quantitative, mathematical, etc..); of the difference betweennatural reality and models as mental representations of some of its aspects; of the various levels offormalisation involved in modelling (Sperandeo 2001). Above “Models” are indicated somespecific activities related to: awareness of the limits of models; capability of prevision ofexperiments and extrapolated trends; verification of models’ validity; comparison amongst models;criteria for choosing an appropriate model, etc…The blob “Experiments” indicates those activities related to lab-work, observation and explorationof phenomenology that are typical of an experimental discipline as physics. Below “Experiments”some key aspects participating to the development of FTare summarised; from right to left somesignificant areas a learner goes through in becoming capable of FT. The identification of

significant versus not significant variables and the issues of approximation and errors, unavoidablein every measure and experienced in laboratory work with real apparatuses, are contents to befamiliar with in order to be able to think in formal terms.They are followed, in the schema, by threeother components of the complex conceptual chain leading to FT: - to distinguish between trends ofexperimental data and specific details (the first inform on the global behaviour of the evolving physicssystem; the second allow to study local aspects); - to explore the evolution of the studied phenomenonaccording a variational approach which changes one condition at a time and allows to answerquestions like “What changes or remain constant if this or that varies?”; - to practice a “from Realphenomena to Ideal cases” rationale that helps becoming aware of the many steps needed to proceedfrom the features of complex, real phenomena to the formal representation of ideal cases in terms ofphysics laws. The last three aspects are labelled with a star to indicate that they can be experiencedmore easily (or almost uniquely) in Real-Time approaches (cfr. in the following).Very many learning and teaching difficulties have been researched (cfr. Bibliography by Pfundt andDuit, 1994) according to age level of students, type and level of addressed physics content, teachingstrategy and teacher’s beliefs and convictions (Hewson & Hewson, 1988;Tobin, 1988; Briscoe, 1991;Bell and Pearson, 1992; Crawley & Salyer, 1995; Couchouron et al., 1996; Hirn, 1998; Van denAkker, 1998; Van Driel & Verloop, 1999). Most of the comments presented here refer to problemsencountered in secondary education, at age range about 13-17. Several of them influence thedevelopment of FT or are kinked to difficulties related to reasoning in formal terms.A list of common problems in the development of FT contains many items and is not exhaustive;here may be useful to recall some amongst the relevant ones.Both experienced teachers and educational researchers do know that there are common difficultiesin linking formal representations of physics phenomena with the naive and interpretativedescriptions students give using ordinary language, intuitive images, spontaneous metaphors, etc..,based on their robust common-sense knowledge. The “translation” processes needed to relate suchdifferent languages with forms of FT are not easy nor immediate for young students, specially whenabstract representations are involved, as mathematical expressions and/or graphs. Unfortunately,very often there is insufficient emphasis on helping the students to become aware of the linksbetween physics phenomenology and mathematical model describing and representing it;this weakfocus may easily produce obstacles in developing those aspects of FT needed to build a long lasting,well integrated physics knowledge,Difficulties are also encountered by students in becoming familiar with the various levels offormalisation required, from regularities that can be inferred from phenomenological observation,to formal rules representing such regularities, to mathematical models that represent and predicttrends/features, to synthetic physics laws of ideal cases referring to abstract situations.Quite often students have problems in acquiring the capability of understanding and usingmultiple-representations of the same data and of choosing the most suitable one according with theobjective aimed at. An example case is that of graphic representations where the same informationcan be represented in different modalities emphasising different aspect of the same phenomenon.Last but not least, several learning/teaching obstacles are found in the interplay of experimentaland modelling activities; the latter still being not enough practised (STTIS 2000a). The result isoften missing a “royal way” to help the development of FT.

3. Formal thinking and real-time experiment and imagesThe Real-Time approaches are based on experiments with sensors driven by computers, on-linedisplay of collected data and use of images of resulting graphs, (the main “language” forrepresenting the measures); for the sake of brevity, the acronym RTEI (Real-Time Experiment andImages) will be used here in order to avoid specific reference to any of the available systems.RTEI has been first proposed in the early ‘80s and many user friendly systems are available2; it uses


2 Information on Real-Time systems can be found at:www.vernier.com; www.pasco.com;http://www.cma.science.uva.nl/english/index.html

a mature technology and therefore it is not a novelty, its innovative educational value is supportedby many research studies and experiences. (Sokoloff and Thornton, 1997; Workshop Physics, 2000;Ellermeijer and Heck, 2002; Sassi 1995, 19996, 1997, 2001a). Nowadays many researches do focuson its value as a cognitive tool other than a powerful technological system and propose its use inschool practice. Very briefly, its major areas of impact on the quality of learning/teaching processesdeal with the integration of diverse types of knowledge, as for instance perceptual, common-sense,abstract and experimental knowledge; the re-structuring of the contents to be taught, both usuallyaddressed ones and new ones; the implementation of innovative strategies for addressing robustlearning/teaching difficulties that impair the understanding of basic physics, as for instance the“from Real phenomena to Ideal cases” rationale; the possibility of easy practice of the effectivePEC (Prediction Experiment Comparison) learning cycle which usually is not much practised,given the long time required by its iteration with traditional lab-work.Before discussing some aspects of the contributions RTEI can give to Formal Thinking, in the spiritof the scientific rationale of this Seminar, the many links of this paper with other products of theSeminar have to be explicitly indicated. Three Workshops have addressed themes deeply related towhat is discussed here. In Workshop 1 “Interplay of Theory and Experiments” (De Ambrosis, 2002)two of the proposed questions: “Computer based laboratory: how can it favour the formalisationprocess?” and “Discovery experiments: how easy is it to prepare them and to relate them withtheory?” relate much to RTEI. Workshop 2 “Learning Physics via Model Construction” has(Sperandeo, 2001) looked at the integration between laboratory work and use of modellingsoftware and has discussed how to bridge qualitative, informal description of data trends and theirrepresentation through mathematical functions. The study of everyday, complex phenomena, wellknown to students in terms of their perceptual and naive knowledge, is made possible and facilitateby RTEI; this helps the development and the interpretation of explanatory models of these familiarphenomena. Finally, RTEI is obviously a not trivial theme of Workshop 6 “New technologies andcomputers in physics learning” (Rogers, 2002).Many reasons justify and support the use of RTEI to foster the development of FT in physics.Amongst the main ones are the following.First of all is the fact that phenomenological exploration and experiments are an indispensable,ground basis in the path aiming at understanding the role of formalisation in physics and atacquiring capabilities in FT. Amongst the different types of experimental activities, RTEI plays avery important role, for its unique feature of displaying the results of the measurement while thestudied phenomenon happens and evolves and for the many measurable physical quantities giventhe available variety of sensors.Second is the bridge between common-sense knowledge interpretations of phenomena and someof their possible abstract representations, and amongst these the one through Cartesian graphs.Since the “natural” RTEI language is that of graphs showing the evolution in time of the measuredvariables, this bridge is facilitated by the very fact that the graph is built and displayed while thephenomenon happens and evolves.Third, user-friendly software tools for data fitting are nowadays present in all available RTEIsystems (and sometimes also a modelling environment is provided); therefore modelling activitiesare facilitated and fostered. Expecially in the case of phenomena well known to students in termsof their naïve knowledge, the data fitting phase is a useful first step toward the development andthe interpretation of explanatory models.Since RTEI is an approach and not a proposal to teach a specific subject, it is transversal withrespect to specific disciplinary contents, even though its implementation in class practice calls for aspecific content to be addressed. An adoption of RTEI coherent with its rationale implies acomplex process usually resulting in not minor re-design of teaching and deep changes in schoolpractice. This shows usually through becoming able: of implementing RTEI approaches in variouscontexts; of realising learning situations focused on both phenomenological and formal aspects; ofcentring on learning difficulties to be addressed.


Several peculiar features of RTEI do contribute to foster the development of FT in physics;according to research and experience at least the following ones play an important role.* In developing FT it is crucial to identify which variables are significant for the description of thephenomenon being studied; RTEI allows and facilitates a variational analysis through a rapidrepetition of experiments under different conditions. Changes that do (and do not) influenceexperiment’s results can be easily identified through activities of the kind “What happens if ..?”,worthwhile both from cognitive and motivational viewpoints.The sensors used in RTEI usually collect tens of measures per second, this allows and facilitates theobservation of very many details in the time evolution of the studied phenomena. A centralcapability in the process of developing FT is the awareness about which details should be describedby the formal representations aimed at. Given the richness of the measured data and theconsequent details, it’s necessary to choose which aspects to take into account and which todisregard, in order to proceed toward modelling activities. What becomes “visible” in a real-timeapproach fosters and supports this choice; the development of explanatory models of the studiedphenomena is therefore facilitated. Specially in the case of complex, everyday phenomena, wellknown to students in terms of common sense knowledge, a noticeable resource to build on, in thedevelopment of FT, is the comparison between predictions about their features (as for instancetrends in their time evolution, co-relation amongst significant variables, shapes of graphs, etc..) andthe result of experiments.The goal of making concrete important aspects of FT is facilitated by RTEI also through the easyimplementations of the PEC learning cycle (Prediction Experiment Comparison), a well known,not much practised learning tool, about whose educational usefulness there is wide consensus. PEChelps in acquiring high level cognitive capabilities, as expressing one’s own ideas and reasoningstrategies; analysing and modelling experimental data; comparing predictions with results; all threephases of the cycle contribute significantly to develop FT and to be able of its concrete applications.The phase Prediction allows to elicit students’ ideas which, in some cases, might conflict withacknowledged physics. Moreover, according to the language used, possible problems aboutrelationships amongst verbal expressions, graphs, schema, diagrams, formulas, etc.. may appear andbe addressed. All these elements play important roles in the construction of formal reasoning. Thephase “Experiment”, other than helping in acquiring operative skills in assembling and optimisingexperiment set-ups and in using the system software, allows to address possible confusion betweenphenomenology and explanatory models. The distinction between facts and models is essential inthe development of FT; a recent document of the Nuffield Foundation calls attention on the factthat :”Students tend to believe that theoretical models emerge directly from data, and that allfeatures of a theoretical model correspond directly to features in the real world. Students often failto recognise the conjectural and tentative nature of many scientific explanations, and that scientificexplanations are often expressed in terms of theoretical entities which are not ‘there to be seen’ inthe data.” (Teaching about Science, http://www.nuffieldfoundation.org/aboutscience/).In the phase “Comparison” the students’ reasoning strategies and their formal expressions playimportant roles. If there are non trivial discrepancies amongst meaning of predictions, their formalrepresentations and results of experiments and/or models, it is needed to identify plausiblefactors/reasons for them and decide what has to be changed in the next run of the cycle. On theother hand, if prevision and results agree, it is appropriate to consolidate this coherence. Theiteration of PEC cycle is a powerful opportunity, not only to address conflicts between previsionsand results (and proceed in the development of FT), but also to become aware of the fact that theexistence of various (disciplinary correct) viewpoints is a resource to be exploited, in order tobecome capable of choosing the most suitable solution (and its formal representation) for a givenproblem. One aspect of formal reasoning is the capability of identifying the most convenient (interms of global effort) path(s) to solve a problem.* Another noticeable resource to build on, in the development of FT, is an educational rationale,made possible by RTEI, that have been tested at length in our research; for the sake of brevity, it


is labelled as “from Real to Ideal” rationale.The conceptual chain proposed starts from explorationof real, complex (often everyday) phenomena and proceed toward models of ideal cases. Theintermediate steps of the chain are: to identify and recognise regularities in the evolution of thestudied phenomenon; to co-relate diverse abstract representations (mainly graphical ones) of theseregularities; to infer rules from them and express these rules in formal terms; to choose the effectsto be disregarded and consequently perform “more clean” experiments; to model the experimentresults; to abstract to the ideal case and its formal physics law.In traditional teaching this approach is seldom practised, almost always the presentation of theideal case is the starting point and lab –work is intended and proposed as a verification of somealready studied features of ideal situations. When exposed to such an approach young studentsoften encounter learning difficulties; on one hand it requires a non trivial capability of abstraction,on the other it may reinforce the rather common idea/interpretation that physics laws are “objects”of the natural reality and not constructions of human minds aimed at representing some aspectsof the physical world.* Another important component of the FT development process is that of modelling activities,which, again, is easily supported and fostered by RTEI approaches. Fig. 2 shows a schematicrepresentation of the main relationships between RTEI and modelling and of the range of activitiesthat may help to connect physics phenomena and their mathematical, formal descriptions

Fig. 2: Schematic relationships between RTEI and Modelling

On the left, for the indicated types of activities the horizontal dimension suggests the developmentof abstraction capabilities, while the vertical one recalls a path going from qualitative toquantitative analysis. The first type of activities is related with the identification of qualitativetrends in the observed phenomenology which can be expressed in words and with order relationsas “when x increases (decreases or is constant), y increases or ….”. The second type addresses therepresentation of the same data in diverse forms and the identification of co-relation amongst suchrepresentations. Both aspects help in developing an important aspect of formal reasoning; namelystudents often find difficulties in establishing such co-relation since a particular representation is


perceived as the “right” one, as for instance position versus time in the kinematics description ofmotion. The third type of activities is related with the fit of experimental data that quite often isthe first type of modelling activity, since data fitting aims at finding simple mathematical functionstaking into account the most important aspects of data trends. All available RTEI systems havefriendly data fitting tools , students are therefore facilitated in focusing their attention on the fitmeaning , the role of different elements of the mathematical function, the distinction betweenpredominant effects and secondary ones. In brief they are helped to reflect about the descriptivepower of the fit/model. The forth type of activities aims at becoming capable of extrapolating thebehaviour of the studied phenomenon in variables’ ranges different from the measured ones;consequently new experiments can be planned and performed and the fit/model can be checked.Last but not least, the fifth link indicates those activities that help students become aware ofcapabilities and limits of data fitting procedures.The right part of the figure schematises a range of modelling activities that help interpreting FTalso as a continuous bridge linking Physics and Mathematics. Rather common learning difficultiesare related to issues as: - the same mathematical model describes different physics phenomena,according to the meaning given to the model variables; - the capability of interpreting proposedmodels is other with respect to that of building a model for the case in study. The latter needs thatsome idea about aspects/nature of the phenomenon comes from outside the data or adds to theiranalysis. The RTEI systems now available often offer also modelling environments, so activitiesaimed at acquiring the above mentioned capabilities are facilitated.

3.1 An example of a “Real to Ideal” pathFor the sake of clarity, an example of a “from Real to Ideal” path is briefly described. It deals with“Inversions of motion” and has been proposed in teacher’s education programs and in ordinaryclass-work in secondary school basic physics courses. It is part of KINFOR - Teaching and Learningabout Kinematics and Force (SECIF 2001a), a web-supported set of resources developed in theframework of SECIF, a recently completed in physics education project, funded by Italian Ministryof Education and Research (SECIF 2001b).This path aims at two main objectives: to address/overcome common learning difficulties related tounderstanding motion inversions; to introduce impulsive forces, as those that intervene in hits, inorder to make plausible and clarify some aspects of the third Dynamics principle. The experimentsproposed deal with motions well known by students in terms of common-sense knowledge asregular walks, ball bouncing, cart going up/down ramps. Both motion and force sensors are used.The motion inversions addressed are of two kinds: - “natural” ones as those occurring when themoving system inverts its motion without any interaction with another systems so that the actingforces are the same than before the inversion (ex: a cart moves up a ramp, reaches the maximumheight and moves down, a ball launched in air, at the top starts coming down, etc… ); - “induced”ones as those when on the moving system intervenes a force that was not acting before theinversion (ex: a cart goes down a ramp, hits an obstacle at the bottom and moves up; a falling ballhits the floor and bounces up; etc…)The first type of experiments proposed explore regular walks of students who move away andtoward the motion sensor, the inversion is decided by the walker who changes direction as fast aspossible, according to his/her type of reflexes. By analysing position and velocity versus timegraphs, students are helped to identify as indicator of motion inversion a maximum (minimum) ins(t) and a zero in v(t). Moreover, emphasis is put on the fact that the iconic indicator of motioninversion in a v(t) graph is not a step like mathematical function but a smooth transition throughzero, the smoothness being related to the inversion duration. An important component in thedevelopment of FT is the awareness that instantaneous changes are forbidden in physicsphenomena and therefore some formal representations (as step discontinuity) are extrapolation toabstract models.


The second type of experiments, where the reflexes effect has been eliminated, deals with a cartmoving on an horizontal smooth track, it hits an obstacle at the track end and moves back. Theanalysis of the kinematics graphs is aimed at: comparing with the motion inversions in walks;discussing the effects of the hit; introducing impulsive forces and eliciting intuitive ideas aboutthem. Useful clarifications of common difficulties related to such themes come from experimentsabout the collision of two carts with force probes mounted on them; the probes touch when thecarts collide and analysis of the force versus time graphs allow to address the symmetry of theinteraction and offers elements of plausibility for the “Action-Reaction” 3rd principle.The third type of experiments deals with a cart moving up/down a ramp: when launched up itreaches a maximum height and moves down to the bottom where it hits an obstacle; it goes up againand so on. In this motion the two types of inversion are present; the graphs’ analysis allow toaddress aspects of formal representation of motion inversions as for instance how they appeardiversely in s(t); the influence on the representation of a change in the coordinate system (sensorat top or bottom of ramp) and other aspects relevant in formal reasoning.The forth type of experiments deals with a multi-bouncing of an elastic ball falling on the floor andthe comparison of its motion with the cart going up/down the ramp.The last type of experiments studies the “natural” inversion of motion of an spring- mass systemthat oscillates vertically. The identification of similarities and differences with the previouslyanalysed inversions allows to recognise rules valid for any motion inversion and proceed to theirformalisation.

4. RTEI and its use in current class practiceDespite the many didactic advantages of real-time approaches, also from the viewpoint of thedevelopment of FT and the many years since RTEI first proposal, the “naturalisation” process isslow(STTIS 2001a); that is RTEI adoption is not yet kind of a default choice made by teachers,when appropriate. While very many physics teachers are interested in adopting RTEI, its use inordinary class-practice is not common as it would have be expected according to research results.The STTIS project studies have addressed how RTEI and others innovations are being used byteachers; a “transformative” viewpoint has been adopted, i.e. to look at teachers as transformers ofthe didactic intentions and/or of the rationale of the proposed innovations (STTIS 1998). One ofthe results is that several transforming trends have been detected in the case of RTEI; while someteachers have resonated with the proposed rationale, making enriching transformations, many haveinterpreted it in reductive ways and dissonant transformations have resulted.A component of teacher training, not yet much acknowledged, is the analysis of the difficulties teachersencounter when adopting innovative approaches and/or teaching proposals of specific contents. Thiscomponent is rather important since it addresses the possibility of changing ineffective teaching patternsand of starting new valuable ones; its implementation may proceed through a two-steps procedure: -recognition of difficulties that fellow teachers have found in adopting RTEI; - experimentation ofstrategies/materials facilitating RTEI adoption and its integration in class practice.Some of the observed transforming trends made by teachers using RTEI should be taken seriouslyinto account in those teachers’ education programs that aim at increasing the capabilities to guidestudents in the development of Formal Thinking.These transforming trends are briefly discussed here in order to suggest aspects to focus on in atraining program;, for the sake of brevity, only the main dissonant transformations which relatemore to the development of FT are presented.They can be summarised in four groups which for brevity are labelled as: PEC cycle, Variationalapproach and Global versus Local viewpoints, “from Real to Ideal” rationale, and Iconicdifficulties. The first three are reductive interpretations of RTEI rationale while the forth is spreadall over the use of RTEI and, very likely, is linked with an insufficient awareness of problemsstudents have in reading and interpreting Real -Time graphs. These transforming trends are due to


several factors, such as: teachers’ ideas about teaching/learning processes; insufficient focus ondevelopment of formal reasoning; possible uncertainties in disciplinary knowledge; capability ofintegrating RTEI with other class activities; experience with computers and lab-work; objectivecircumstances; etc..* PEC cycle: it has been observed that both the practice of reasoned prediction before experimentsand of critic comparison is scarcely adopted. Similarly, not much emphasis is given to the iterationof the cycle, when not minor discrepancies results in the comparison phase. Both attitudes canimpair the learning of important aspects of formal reasoning.* Variational approach and Global versus Local viewpoints: insufficient resonance with theseaspects of the proposed RTEI rationale has been observed. The scarce focus on the powerfulstrategy “What happens if …?” that makes observable the effects of changes in the experiment set-up and other conditions weakens the awareness of central features of FT, such as what is invariantin the time evolution of the studied phenomenon. Similarly, the research results indicate a loweffort in helping students to become capable of using both a global and a local viewpoint (andswitching between them) in analysing the Real-Time graphs. Often the distinctions between thesetwo viewpoints, that is relevant for the development of a well sound formal reasoning, has beenoverlooked; the local details revealed by the many data have been disregarded without anyjustification and the experimental results have been forced to resemble ideal trends, as if they wereof a poor quality, as if only mathematically regular graphs were worthwhile to analyse. With suchan attitude there is the risk that the process for reaching a formal descriptions of data isoversimplified and that some important steps in the abstraction process are missed.* “from Real to Ideal” approach: it has been observed that often this rationale has been reductivelyinterpreted, with changes or overturning of the proposed conceptual chain, omission orundervaluing of crucial phases or changes in the order of proposed tasks. In some cases, the idealcase has been taken as starting point, despite the acknowledgment, on a declarative level, aboutthe didactic value of the proposed approach. Other steps significant for the development of formalreasoning were not much practised, as for instance the “searching for regularities and rules” andthe “modelling” phases that have been omitted or undervalued. These transformations may berelated to specific viewpoints about teaching strategies and/or to difficulties in appropriation of anapproach which implies to move away from the usual ones .* Iconic difficulties: a global low awareness of students’ difficulties in reading and interpretingimages of Real-Time graphs has been observed. This transforming trend has appeared in manyways teachers have used images of graphs; for instance: uncertainties in clarifying what system’sartefacts are and their difference with respect to relevant local details ; weak focus on the co-relation amongst two (or more) graphs in the same image; over-focus on the shape of the graphthat shadows the meaning of the variables on axis; weak effort to improve the readability of theused images and to teach how to optimise a graph; etc.. . A scarce awareness of teachers’ aboutstudents iconic obstacles may, very likely, results in impairing the development of FT; namely thecapability of reading/interpreting graphs in such process plays an important role.These results about transforming trends made by teachers in ordinary class use of RTEI stronglysuggest to reflect on them, also from the viewpoint of the development of FT; not only to improveteachers’ training about RTEI, but also to support the guidance teachers offer to students inacquiring capabilities in formal reasoning.

5. A possible project for speeding RTEI “naturalisation”The results and comments presented above suggest that it is now appropriate to suggest guidelinesand to produce materials in order to facilitate the adoption of RTEI in current class practice, alsofrom the viewpoint of fostering and supporting the development of Formal Thinking, since earlyphases of basic science education. While there is an increasing emphasis and consensus on teachingas a research-based profession, there are still difficulties in the process of constructing a culture in


which communities of researchers and teachers work together to create a shared body ofknowledge. This effort should address many types of resources, a not exhaustive list contains:resources for teachers’ professional growth and education; research-based materials for classactivities and self-learning; research findings; environments for connecting research andeducational practices; environments for sharing experiences/problems; joint projects betweenteachers, students and innovators; suggestions for policy-makers.This body of knowledge should bemade easily available, in a format easily accessed by a vast audience; in the current situation thediffusion of ICT and the interest for its solution suggest a RTEI web portal.The main added value of a project about such a portal comes from the objective of synergeticintegration of different viewpoints. The didactic perspective, together with the cognitive,pedagogical and technological one, should be integrated in a global vision containing also theviewpoints of class practice, teachers education, student self-learning, life-long learning of thecitizen; etc. All these perspectives are very significant in physics and science education; up to nowthey are not available in an integrated format aimed at a wide public.To design such an integration of viewpoints it is necessary to reflect on the accumulatedexperiences/expertise and to come up with an innovative structure for the portal. Several researchgroups have worked with RTEI approaches since long time and there is a large amount of resultssuitable for contributing to such an integrated format.The envisaged audience is potentially a large one, namely the knowledge of RTEI rationale, itseducational potentialities and related experiences can be interesting and beneficial to variousgroup of people.Teachers may improve and innovate traditional teaching strategies expecially whose results are notpositive. Their self education process may also receive substantial trigger and benefit.Teachers’ trainers may find both criteria and materials to be directly used in training sessions andhints to design personalised activities.Students may explore and use the available materials and be helped in their learning process andin developing ideas/hints for their contribution to class activitiesResearchers in science education may find another easy channel to exchange the results of theirstudies. This sharing helps novices entering RTEI research, through an easy availability of alreadyaccumulated knowledge which may minimise the risk of “re-inventing the umbrella”. It also helpsexperienced researchers by offering, for instance, suggestions for both rapid synergeticcombinations of results and future study.Schools may offer the most significant RTEI didactic experiences made by their teachers andclasses in order to share them. This process is beneficial for at least these reasons: a) submergedresources of good quality become available to a wide audience; b) teachers’ practical research isrecognised and acknowledged, thus favouring increasing of self-esteem; c) isolation of teachersdecreases; d) another internal trigger enters the complex process of school practice innovation.Policy-makers in Science Education may find suggested criteria and guidelines; their task isfacilitated by a contact with research results and school experiences.Scientific Associations (as GIREP, ESERA, National Teachers Groups, etc..) may find suggestionsand occasions that help to reach some of their objectives.EU funding Agencies may take advantage, in deciding their support, by the availability of this vast,continuously updated body of knowledge.As a project launched in an European context and given the “de facto” current situation it seemsappropriate to have an English version of the portal. Because of the aimed audience (cf. below), itis necessary also to implement versions in National languages, in order to make possible andfacilitate the use of the portal in various EU countries.In conclusion, to trigger the start of the project the following preliminary suggestions can beconsidered.The project of an “RTEI portal” implies an international collaboration aimed at making available


a first core of web-supported, various types of resources. A non exhaustive list contains: researchbased results, example school experiences, proposals for class activities, materials for teachers’education, information on the available RTEI systems, commented summary of emblematicexperiments, links between experimental and modelling activities, suggestions for supportingeducational innovation; ….An international group, under the aegis of a scientific association, for instance GIREP, couldproduce a preliminary identification of basic features of this core in terms of:– global structure and content;– conceptual design of the portal features and maintenance;– scheme of content implementation and maintenance;– scheme of technical maintenance; etc...The design and implementation of this first core obviously

needs to allow and support easy future expansion.If the trigger of such a project might be numbered amongst the effects of this Seminar onDevelopment of Formal Thinking in Physics, it will be a useful contribution.

ReferencesBell B., Pearson, J. “ Better learning”. International Journal of Science Education, 14 (3) (1992), 349-361Briscoe C., The dynamic interaction among beliefs, role metaphor and teaching practices. A case study of teacher

change. Science Education 75 (2), (1991), 185-99Colin P., Chauvet F., Rebmann G. and Viennot L., Designing research based strategies and tools for teacher training,

Reading images in optics. Proceedings of ESERA Conference, Thessaloniki, August 2001Couchouron M., Viennot, L. and Courdille, J.M. Les habitudes des enseignants et les intentions didactiques des

nouveaux programmes d’électricité en classe de quatrième, Didaskalia, n° 8, (1996), 81-96Crawley, F.E., Salyer, B.A., Origin of Life Science.Teachers’ Beliefs Underlying Curriculum Reform in Texas, Science

Education, 79, (6), (1995), 611-635De Ambrosis A., Rinaudo G., Interplay of Theory and Experiment, in “Developing Formal Thinking In Physics”, (M.

Michelini, M. Cobal Eds), Forum, Udine (2002), 108-112.De Jong O., Exploring science teachers’ pedagogical content knowledge. Proceedings of ESERA Conference,

Thessaloniki, August 2001, VOL I, 7-9, (2001)Ellermeijer T. and Heck A., Differences between the use of mathematical entities in mathematics and physics and

the consequences for an integrated learning environment, in “Developing Formal Thinking In Physics”, (M.Michelini, M. Cobal, Eds), Forum, Udine (2002), 56-75.

Hewson, W.P. & Hewson M.G., An appropriate conception of teaching science: a view from studies on sciencelearning. Science Education 72 (5), (1988), 597-614

Hirn C. Transformations d’intentions didactiques par les enseignants: le cas de l’optique élémentaire en classe deQuatrième. Unpublished thesis, Université Denis Diderot (Paris 7), (1998)

Levy Leblond J.M., Mathématique de Mathématiciens and mathématique des physiciens. Revue du palais de ladécouverte, 40, (1991)

Millar R. (2001). What can we reasonably expect of research in science education?. In Proceedings of ESERAConference, Thessaloniki, August 2001, Presidential address

Monroy G., Sassi E., Testa I., Lombardi S. Teachers’ adoption of Real-Time approaches: transforming trends andresearch-based training materials. In Proceedings of ESERA Conference, Thessaloniki, August 2001, Vol I, (2001),156-158

Osborne J. and Millar R. (eds), (2001, Beyond 2000 - Science education for the future. http://www.kcl.ac.uk/depsta/education/be2000/index.html

Rogers L., New Technology and computers in physics Learning, in “Developing Formal Thinking in Physics”, Udine(2001), (M. Michelini, M. Cobal, Eds), Forum, Udine (2002), 129-130.

Pintó R., Gutierrez R., Couso D., Teaching about energy conservation and degradation. using research into teachers’transformations to design material for teacher training. In Proceedings of ESERA Conference, Thessaloniki,August (2001)

Pfundt, and Duit R., Students’, Alternative Framework and Science Education. IPN Kiel, Germany, (1994)Sassi E., Esperimenti in tempo reale e didattica della fisica.TD Tecnologie Didattiche, No. 7, (1995), 46-55, (in Italian)Sassi E., Addressing some Common Learning-Teaching Difficulties in basic physics courses through computer based

activities. Proceedings of the XVI GIREP-ICPE International Conference, Ljubljan, (1996), 162-179Sassi E., Computer-based laboratory to address learning/teaching difficulties in basic physics. In Proceedings ESERA

Summer School on Theory and Methodology of Research in Science Education, Barcelona, (1997), 55-64.Sassi E. and Olimpo G., Teacher Training and Information Technology Proceedings of the conference. “The Role of


Experimental Work and Information and Communication Technologies in Science Learning”, Lisboa. CDROMForum Educaçao, (1999)

Sassi E., (2001a)Computer supported lab-work in physics education: advantages and problems, - “Proceedings of theInternational Conference Physics Teacher Education Beyond 2000” R. Pintò & S. Surinach (eds). CD ProductionCalidos. ISBN: 84-699-4416-9 Barcelona

Sassi E. (2001b). Le difficoltà di apprendimento in fisica ed il laboratorio in Tempo Reale. Atti del convegno TED2001, Tecnologie Didattiche e Scuola. A cura di D. Persico, http://www.itd.ge.cnr.it/TED/ATTI.htm, 80-83

Sassi E. (2001c). La formación de profesores y los enfoques del trabajo experimental en tiempo-real, el caso de laFísica. VI International Congress Of Research Into The Teaching Of Science: Challenges For Science Teaching InThe XXI Century, Barcelona, September 2001

SECIF Project (2001) (Spiegare e capire in Fisica, Coord. P. Guidoni). E. Sassi, G. Monroy, I. Testa, S. Lombardi.Insegnare ed apprendere la cinematica e le forze con KIN-FOR.

http://www.na.infn.it/gener/did/kinfor/secif/index.htm.Sokoloff D., and Thornton R., Using Interactive Lecture Demonstrations to Create an Active Learning

Environment. Phys. Teach. 35, (10), (1997), 340-345Sperandeo-Mineo R.M., (2001): “I.MO.PHY. (Introduction to MOdelling in PHYsics-Education): A Netcourse

Supporting Teachers in Implementing Tools and Teaching Strategies” in (R.Pinto, S.Surinach Eds) Physics TeacherEducation beyond 2000: Selected Papers of 2000-GIREP Conference-Elsevier-Paris,135-140

SECIF Project, (2001a), (Spiegare e capire in Fisica, Coord. P. Guidoni). E. Sassi, G. Monroy, I. Testa, S. Lombardi.Insegnare ed apprendere la cinematica e le forze con KIN-FOR.

http://www.na.infn.it/gener/did/kinfor/secif/index.htm.SECIF Project (2001b) (Spiegare e capire in Fisica, Coord. P. Guidoni). Website: http://pctidifi.mi.infn.it/SeCIFStylianidou F., Boohan R. Ogborn J. (2001) Computer modelling and simulation in science lessons: using research

into teachers’ transformations to inform training. In Proceedings of ESERA Conference, Thessaloniki, August2001

STTIS (Science Teacher Training In an Information Society) Project Reports: Outline and Justification of ResearchMethodology: Work Packages WP1, WP2 and WP3, RW0, (1998); The nature of use by science teachers ofinformatic tools. RW1, (2000a); Investigation on the difficulties in teaching and learning graphic representationsand on their use in science classrooms. RW2, (2000b); Investigation on teacher transformations whenimplementing teaching strategies. RW3, (1999); Teachers Implementing Innovations: Transformation Trends. RW4,(2001a); Teacher training materials favouring the take-up of innovations. TR WP5, (2001b)

Tobin K., Improving science teaching practice. International Journal of Science Education 10, (1988), 5475-484Van den Akker J. The science curriculum: between ideals and outcomes. In Fraser, B. J. and Tobin, K. J. (eds),

International Handbook of Science Education, Kluwer, London , (1998), 421-447Van Driel, J. H., & Verloop, NTeachers’ knowledge of models and modeling in science, International Journal of

Science Education, 21 (11), (1999), 1141-1153Viennot, L., 2000, Anticipating Teachers’ Reactions to Innovative Sequences: Examples in Optics. In R. Pinto & S.

Surinach (Eds.), Physics Teacher Education Beyond 2000, Paris: Elsevier, 65-72Viennot L. , (2001), Relating research in didactics and actual teaching practice: impact and virtues of critical details.

In Proceedings of ESERA Conference, Thessaloniki, August 2001Workshop Physics (2000), Principal Mentor: Priscilla Laws, Department of Physics and Astronomy, Dickinson

College. http://physics.dickinson.edu/



DIFFERENCES BETWEEN THE USE OF MATHEMATICAL ENTITIES INMATHEMATICS AND PHYSICS AND THE CONSEQUENCES FOR ANINTEGRATED LEARNING ENVIRONMENT

Ton Ellermeijer, André Heck, AMSTEL Institute, University of Amsterdam, Amsterdam,Netherlands

1. BackgroundIn The Netherlands, like in a few other countries, two fields of application of computers in scienceeducation have been dominant: for data-acquisition and data processing, and for modelling.Experience with data-acquisition started around 1980, as well by enthusiastic teachers, as well bythe Physics Education group of the University of Amsterdam (now part of AMSTEL Institute).The interest in modelling has been triggered by the work of Jon Ogborn (DMS).Soon the demand for an integration of these tools arose. Even in 1985 at an initiative of theUniversity of Amsterdam group, a small group with Philip Harris, London University, and DutchTelecom defined a concept for an integrated tool for Science and Technology education(Ellermeijer et. al, 1988). Unfortunately this initiative was not granted by EC, but it did have amajor influence on further developments in The Netherlands. Due to the fact for the Dutch schoolsystem IBM-PC’s (or compatibles) became standard, we could start to develop for more powerfulcomputers. In 1988 the first integrated software with support for an interface for data logging andpowerful analysis tools including modelling was realized (IP-Coach 2).Over the years other valuable tools have been added to the environment, like for Control activitiesand for analysing digital video. The Coach 4 environment, still a DOS-software package, becamede-facto standard in Dutch schools, together with the hardware for data logging. Because of thesesituation publishers, curriculum committees, teacher-training institutes could focus on thisenvironment, and schools and teachers received a consistent and strong support. Because of itsversatile character it is applied not only in Physics, but also in Chemistry, Biology and Technology.The present environment, Coach 5, internationally achieved a breakthrough. This environment isbased on a new concept, made possible by the Windows environment. In addition to all the toolsfor Science and Technology education teachers/developers have powerful authoring tools toprepare activities for their students. They can select and prepare texts, graphs, videos andmeasurement settings and choose the right level according to age and skills of their students. Inthis way, Coach 5 can be adapted for pupils/students ranging from age 10 up to undergraduatestudents.Up to date information on the Coach environment can be obtained on the website:http://www.cma.science.uva.nl/english. A demonstration version can be downloaded from thissite. A more detailed description of Coach 5 can be found in (Mioduszewska & Ellermeijer,2001).

2. Demands for a learning environment for science and technologyThe development of the Coach environment has been driven by a few key starting-points.The choice for powerful tools, like data-acquisition, processing, modelling, was motivated becausewe like that what we teach at schools should reflect the way scientist work at present. And since the60’s computers have become more and more dominant in all areas of scientific research and alsoin industry.Since the 70’s many science educators are in favour of more realistic science education. With thiswe mean the treatment of science concepts in a realistic, everyday life context. The powerful toolshave proven themselves as very helpful to bridge the gap between physics concepts and real lifecontext.From the implementation point of view an environment that teachers and students can usefrequently and throughout the curriculum has many benefits. It means for them that it makes senseto become familiar with the environment, and a learning path from simple use up to the use inadvanced projects can be developed. The integration of tools enables them to move data to other

parts of the environment and compare for instance in the same graph data obtained frommeasurements with data from modelling.On the other hand, such a versatile tool might become complicated. It has to be possible to adaptthe complexity and the features needed to the level of the student. In the Coach 5 environment anauthor has the facility to create tailor-made activities.The development of the Coach environment is an ongoing research project since 1986 driven byclassroom experience, educational research and technology. A multi-disciplinary team of scienceeducators, curriculum developers, teachers, software and hardware specialists is responsible for thedevelopment and for the new implementations. Many smaller improvements are based onexperience from teachers, from inservice course, and from authors of activities (publishers, teachers,curriculum developers).More dramatic changes are most of times supported and guided by intensive scientific research, forinstance in the framework of PhD studies.The extensions planned for Mathematics are an exampleof this and in the next parts of this contribution we report about it. Another important futuredevelopment will address the connectivity with Internet. Internet will allow for collaborative workof students, for remote laboratories and remote control and for easy exchange of data and lessons.Also the hardware tools for students will become more flexible and powerful. Think of dataloggersthat can communicate with powerful computers applying wireless techniques.The challenge will beto use these new technologies for the benefit of science education and to keep our teachersinformed and prepared.

3. ICT-rich mathematics activities The first step in the redesign of Coach towards an integrated environment is to investigate whatmathematics can be dealt with already, what kind of problems one encounters in doing so, andwhat needs can be identified. After all, the present software environment offers manymathematical tools: the use of tables, diagrams, data analysis, computer models, amongst otherthings. Here, we shall briefly report about recent try-outs of mathematics activities and presentsome findings about Coach for mathematics. More details can be found elsewhere (Heck, 2000;Heck & Holleman 2001a, 2001b).

Modelling human growthThe recent Dutch human growth study (Wit,1998) has been used to create learning materialfor pupils in their first year of the second stage ofpre-university education (age 15-16 yr.), whohave no experience with practical investigationtasks, and who have not worked with thesoftware environment before. Main objectivesare to let pupils• work with real data and with diagrams that

are actually used in health care;• experience how much useful information can

actually be obtained from diagrams;• see that the change of a quantity is often as

important and interesting as the quantityitself;

• practice ICT-skills;• carry out practical work in which they can

apply much of their mathematical knowledge.A mathematical highlight is the ICP-model that models mean height for age within millimetres. Itis used in medical literature and yet consists of mathematical submodels that are studied at school,viz., exponential growth, quadratic growth, and logistic growth.


Fig. 1: Manual function fit of childhood component.

By seeing various regression models in a menu, pupils feel free to try any function type in the tool,for example, to fit growth data by a modified rational function or by a modified exponential curve.When asked a simple function-fit, most pupils interpret this as a fit with a straight line; a quadraticfit is not ‘simple’ for them. They have difficulty in analysing a curve in parts or at least they do notdo this independently. And the software does not support them much in this respect: all data areused in regression; one cannot restrict the function fit to some range of the data. This would havebeen useful in the study of the growth during infancy and puberty. The lack of possibilities torestrict operations to parts of data also holds for other data processing and analysis tools in Coach.In the current version, you can only copy a column in a table, discard unwanted data manually, andapply processing or analysis tools to this adjusted copy.

4. Investigating bridges and hanging chainsThe main objectives in this practical assignment for pre-university pupils are to let them• work with real data collected from digital images;• apply mathematical models to

investigate shapes;• practice ICT-skills, in particular use

tools to collect data from video clipsand images;

• carry out practical work in whichthey can apply much of theirmathematical knowledge.

The main task for the pupils, who arenovice users of Coach, is to get familiarwith the video tool and to use it toinvestigate the mathematical shape ofbridges and hanging chains.Let us describe briefly how the imagemeasure-ment activity shown in thescreen dump to the right takes place.First of all, the activity allows collection


Let us discuss some findings out of classroom trials about the Coach tools to graph and analyse growth data. For this, it is important to realise that regression is not a subject in the Dutch mathematics curriculum. How does it work in this activity? During childhood (from 3 years of age to the onset of puberty), a simple quadratic function fits growth during this period very well: 22

222 ctbtaH ++= . Pupils are expected to search for a parabola that on the one hand fits well

the height between 3 and 10 years of age, and that on the other hand reaches its maximum at the age of 20 years, when height growth usually stops. This curve fitting process is supported in the software by a manual function fit, as shown in the screen dump. The formula of a parabola,

cbxaxy ++= 2 , has been selected as function type; the selected column corresponds with the mean height of Dutch boys. The icon of the pin on the screen dump is such that the approximation has been fixed at that location. By dragging another point of the parabola with the mouse one can now re-scale the graph. When the fixed pin is released by double clicking, then one can translate the parabola, i.e., independently change the parameter c. This quadratic model is one of the regression models that can be described by a formula of type µ ++= )( xfy , where f is a simple mathematical function. The good news is that pupils have little difficulty in using this manual function fit. On the other hand, we notice that some important mathematical regression models are missing, e.g., the cubic polynomial model and the (modified) logistic model ( ) deay cbx ++= 1 . This is a pity, the more so because these models are principally of the same form that is handled by the manual function fit.

Fig. 2: Analysis of the Zeeburger Bridge.

of position data from the digitalimage. It is possible to place the originof the coordinate system at anydesired position and to rotate the axes,if necessary. One chooses the correctscale by matching a ruler with anobject in the image of known distance.coordinates are gathered by clickingon the location of points of interest.Data can be plotted and used forfurther analy-sis. In the lower-leftwindow of Figure 2, the regressiontool has been used to find thequadratic function that fits the databest. In the lower-right window, yousee the collected points once more,together with the data plot of thedifference quotients (dy/dx) ofconsecutive points (plotted with respect to a second vertical axis). The difference quotients lieapproximately on a straight line. The best line fit can again be found with the regression tool. Thethird column of the table in the upper-right window shows as well clearly the pattern for thedifference quotients. It follows that the shape of the arch is a well described by a parabola. But,with the regression tool one could easily discover that the sinusoidal model y = a sin(bx + c) + dworks almost as good. More is needed for good understanding!Figure 3 shows a screen dump of the activity in which the mathematical shape of a hanging chainis investigated. By collecting positions on the digital image and by trying a quadratic curve fit onthe measured data, a pupil quickly finds out that the form of the chain is not a parabola (as Galileierroneously claimed). At once, the simple question “How does a chain hang?” becomes ameaningful and challenging problem.In the activity, we let the pupils study a related, but simpler problem: “How does a chain with fiveobjects of equal weight symmetrically attached hang under gravity?” The case of weights at equalhorizontal distan-ces is investigated first. Measurements in the digital images (see Figure 4) reveal


Fig. 4: Measuring properties in the image

Fig. 3: Analysis of a hanging chain

that the slopes of the right segments of the chain have a fixed ratio, viz., 1:3:5. Measuring in otherimages would convince pupils that this does not depend on the length of the segments or how farthe suspension points are apart from each other. It turns out that one always has the following fixedratio of positive slopes: 1:3:5:7:9:… Basic physics can explain this: equilibrium of forces holds ateach point of application where a weight is attached. This simple observation about slopes allowscomputation of the shape of the system and it explains that the points of application are necessarilyon a parabola under the given circumstances. In case of weights at equal distances along the chain,one still has the above ration of slopes and one can follow in the footsteps of Huygens to concludethat the points cannot be on a parabola.What do we learn from trials of this assignment in classroom? On the one hand, pupils like this kindof practical work and they have little difficulty in learning how to handle the video and imagemeasurement tool. On the otherhand, we observe that the pupilsare not much used inmathematics lessons to make bythemselves choices in coordinatesystems, origin, scaling, to applysymmetry, and so on. And this isan essential part of a video andimage measurement activity.The only technical problems thatpupils encounter in using Coachin this task actually all have to dowith the fact that the software atpresent only has a data video toolbuilt-in and that still images mustbe treated as video clips withidentical frames.This makes it, forexample, cumbersome to add or insert more measurements afterwards. Once you have carried outmeasurements, you cannot sort the data later on and at the same time maintain the immediate linkbetween table entries and measured points on the digital image. Some tools are missing, too: forexample, you cannot measure the length of a curve or the area of some region of the image.In the pupils’ reports, the presence of units of length for slopes indicates that some pupils confuseslope and increase of a quantity. Maybe this is caused by the difference between mathematics, inwhich tangents are dimensionless, and science, where slope is treated as a quantity. Anotherinteresting difference between the use of diagrams in mathematics and science pops up in theclassroom experiment when pupils are making graphs invisible in a plot (by checking visibilityboxes of quantities). To their surprise, some pupils get weird diagrams with no coordinate system,one axis only, or no labels near the axes. They are thinking of a graph as a representation of afunction, i.e., as a representation of a single object, so that it suffices to work with one variable. Thisis common in mathematics. In science however, a graph represents a relation between quantities.Then one must work with at least two variables. The envisioned integrated learning environmentmust somehow link up with both graphical concepts.

Iterative processesMathematical functions are in Coach described by lists of numerical values. This has to do with thescience background of physical quantities, between which a functional relationship exists. Values ofquantities can be obtained in various ways:• via a real measurement with sensors• via measuring a video clip or digital • manually filling out a table


Figure 5. Animation of a cobweb diagram.

• importing results from a file• by a formula• via a computer program Let us have a closer look at the last possibility. Figure 5 shows a cobweb diagram of iterations ofthe function f (x) = 3.3(1 – x) x. The program to make an animation is shown. A pupil only needsto enter the initial value, the number of computational steps, or the function definition to doinvestigations.The first three lines of the program code illustrate that the definition of a function in thesoftware differs from the usual mathematical notation. Another remarkable point is that althoughCoach is list-oriented with respect to variables in graphs and tables, the system only allows the storageof a single value at a time in a computer program. The programming code of this example and othermathematical computer tasks would be a lot easier if working with lists was allowed, too. Some othersoftware limitations, which come out from trials of Coach for meaningful mathematics, are:• Quite some mathematical notions, e.g., functions, equations, and recurrence relations, cannot be

introduced and used in the standard mathematical way.• A calculation window for doing numerical and symbolic computations in the same style as a

calculator is missing. At present, you can only write computer programs for numerical purposes.• Regression and simple descriptive statistics of tabular data are the only tools for doing statistics.

Other statistical notions such as box plot, histogram, statistical distribution, and hypothesis testare not supported.

• Combination of diagrams originating from different sources, say combining graphs of functions,parametric curves and measured data, is cumbersome in Coach

• Sometimes the user of Coach is confronted in his/her activities with inconveniences that revealthe science roots of the software. For example, all tables in an activity have the same number ofrows, viz., the number of measured data in an computer experiment. In mathematicalinvestigations however, it is not unusual to have tabular representations with various table sizesat the same time.

Many of these limitations are not fundamental and can be or have been dealt with by the developers;they only show that the present version of Coach is not yet optimal for doing mathematics. The morecrucial limitations all have to do with the issue of what kind of mathematical objects are supported bythe software. Two things must be kept in mind: Firstly, we quote (Yerushalmy, 1999): “The tool IS thedesign”. Coach as a science tool has been designed for collecting, processing and analysing data, andfor working with computer model. These design intentions for example underpin the choice ofvariables as references to samples of values, either in tabular or in graphical form. In a computermeasurement, one connects a column of a table with a measured quantity; derived quantities arecreated by menu commands or by formulas in terms of columns and/or connected quantities. So, thephysical set-up of an experiment, where sensors have been connected to specific channels of ameasurement panel, determines strongly the mathematical processing and analysis of data. Anapproach that is more detached from the physical set-up, instead identifies measured quantities bysymbols, and gives the notion of variable a more central role in activities, seems to be a more fruitfulapproach towards an integrated math and science learning environment.Secondly, Yerushalmy writes: “The tool reflects curricular agenda”. In this respect, the Coachenvironment can be described as a set of tools to explore natural phenomena. It is an activity-basedsystem for students to carry out practical investigation tasks and to do research work at their level:a student can use the computer environment to collect, process, and analyse data, and to reportabout his/her work. Depending on the subject, a student can make a choice of tools. We wish toextend this view of the use of a computer learning environment to the field of mathematics. So, theintegrated learning environment must provide tools to study mathematical phenomena, methods,and techniques. For example, a natural choice would be a definition-window, a graph-window, atable-window, and a calculation-window to have access to mathematical methods, at least at thesame level of functionality as a (symbolic) calculator provides.Anyway, the development of the integrated learning environment is not only technology-driven,


but can also be seen as an attempt to answer the question “What are the characteristics ofinvestigations that can lead to good math and science problems for students and how cancomputers help?”. (Clements, 2000) lists the following characteristics for mathematics:• are meaningful to students;• stimulate curiosity about a mathematical or nonmathematical domain, not just answer;• engage knowledge that students already have, about mathematics or about the world, but

challenges students to devise solutions;• invite students to make decisions;• lead to mathematical theories about (a) how the real world works or (b) how mathematical

relationships work;• open discussion to multiple ideas and participants; there is not a single correct response or only

one thing to say;• are amenable to continuing investigation, and generation of new problems and question.Replace the word “mathematics” by “physics”, and you get a sensible list of characteristics of goodphysics activities. The second part of the above question, “how can computers help with a problem-centered approach”, is from software developer’s point of view the most interesting one. To bebetter prepared for the creation of an integrated math and science learning environment, thedevelopment team must have idea of how several concepts such as the notion of variable, function,table, and graph are used in the various disciplines. As far as the mathematics component of thelearning environment is concerned, one must have a clear view on what requirements physics andother sciences make for the mathematics and how this links up with the requirements frommathematics as a discipline itself. In the next three sections we shall restrict ourselves to adiscussion of the concepts of variable, function, and graph.

5. The meaning of variable is variable in mathematics In mathematics, the concept of variable has several meanings. See (Schoenfeld & Arcavi, 1988) orthe following examples from school mathematics, taken from (van Etten, 1980).


12: +xxf a wlA ×= 7=+ ba

)3)(3(92 += aaa Nx 33 +=+ xx

n is a divisor of 24 xx =2 823 +=+ xx

p is a prime number yx = rA 2=

abba +=+ 92 =a QS

1sin3cos =+ xx 9<x222 zyx =+

k is parallel to the x-axis P lies on l

Even if letters are used for numbers only, different roles of letters in the algebraic context can be distinguished (Kücheman, 1981; Usiskin, 1988). It may be • an indeterminate, in statements like )3)(3(92 += aaa .

• an unknown, in equations such as 7=+ ba .• a known number like .• a variable (generalised) number, e.g., in Nx , in declaring p a prime number, and in

differences like )()1( afaf + .

• a computable number like A in the formula rA 2= .• a placeholder, e.g., in function definitions 12: +xxf a or 12)( += xxf .

• a parameter, e.g., as a label in the function definition xpxf p =)( to distinguish several cases.

• an abbreviation like 3,2,1=V .The multiple meanings of the term ‘variable’ make it hard for secondary school pupils to understand this concept. So, educators are constantly searching for ways to familiarise students with


however, is much used to applying the same algebraic symbolism for many purposes: 2xy = may stand for an equation, a function definition, an abbreviation of the expression 2x , as well as for the process of computing the value of y from the value of x .The equal sign is anyway an intriguing symbol in mathematics. Consider the following two statements:

a) For every real number x,23

32

2

1

1

12 ++

+=

++

+ xx

x

xx.

b) For every real number x,2

1

1

1

23

322 +

++

=++

+

xxxx

x .

In a formal sense, these statements are equivalent. But it cannot be denied that the first statement is simply about computing the sum of two rational expressions, whereas the second one represents a partial fraction decomposition. In the first statement the equal sign is not read as “is formally equivalent to”, but as “yields”: it is about addition. The second statement can be interpreted as “the

fraction23

322 ++

+

xx

x yields2

1

1

1

++

+ xxwhen it is decomposed.” Here, the expression refers to a

process of simplification. Again, the expression is not just a mathematical object with the structure of an equation, but it has a process aspect, too.With respect to learning algebra, much attention has been given in educational research to the dual nature of mathematical entities, which have a procedural and a structural or conceptual aspect, and to the way pupils can obtain such versatile understanding. In early algebra, a mathematical expression is often introduced as a means to describe a process of computing. But gradually, pupils are acquainted with the idea that an expression can also be viewed as a result of a computational

i b h i l bj i hi h b i l d

variables (e.g., see Kieran, 1997; Graham & Thomas, 2000) and to make the transition from arithmetic to algebra, or generalised arithmetic as it is commonly called, easier for them. The main strategy is to treat variables as primitive terms that are best learned by practice. If one cannot define the concept of variable rigorously, then maybe one can better show how and for what reason variables can be used. Basic idea of this approach is that pupils will learn from the examples and the exercises and that they will gradually sense the meanings of variable. Another obstacle in the transition from arithmetic to algebra is that mathematical meaning is often determined by context rather than by formal rules and notation. For example, what does the symbolism )( exa + mean? Which of the following meanings would you choose?

• a generalized number )( exa +× . By the way, does the symbol e stand for the base of the natural logarithm?

• the function a applied to ex + (or do you care that a, used as a function, is usually not in italics). • a function in x with parameters a and e.• a function in two or more indeterminates. • the instruction a applied to the argument ex + .Because of your training and experience, you probably answered that you could not make a choice without knowing the mathematical context or the wording used about the expression. However, for a secondary school pupil it takes time and practice to get used to the fact that a variable actually gets meaning in mathematics through its use (as indeterminate, as unknown, as parameter, etc.), through its domain of values, and through the context in which it is used.

By the word ‘context’ in the last sentence we also mean the context of ‘doing school mathematics’, which has its own mathematical conventions. For example, the word ‘formula’ has a special meaning in school mathematics and the role of the letters in the formula 2xy = is not the

same as in the equation 02 =xy . The words ‘formula’ and ‘equation’ are used to distinguish between the case of a functional relationship between the isolated variable y that depends on the other variable and the case of a more general relationship between unknowns. For pupils it is important to make a clear distinction between these different notions. A mathematician or scientist,


Variables as placeholders mostly occur in function definitions: for example, the

definitions 22),( yxyxf += and 22),( babaf += both define one and the same function, viz., the

norm in two-dimensional Euclidean space. One also refers to these placeholders as dummy variables: they do not indicate objects anymore, but rather the locations for replacements with certain kind of objects. If other variables are present in a function definition, e.g., in

cxbxaxf ++= 2)( , they are distinguished from the dummy variables and they are called parameters. At first sight this is an easy distinction, but the use of parameters is in practice more complicated and more difficult to master (e.g., see Furinghetti & Paola, 1994; Drijvers & van Herwaarden, 2000). One of the greatest obstacles for a pupil to be able to handle a parameter is that (s)he must see the structure of the formula, for instance see that x2 and 12 +x are both examples of the linear expression bxa + . 2. as a polyvalent name, i.e., a name for an object that can take a multitude of values. If n is a divisor of 6, the letter stands for any of the numbers 1, 2, 3, and 6. In the statement that we have a real number x such that 022 =+ xx , the letter x refers to a number that is yet unknown, but can be computed, and that has the property that the sum of this number, its square, and 2 is equal to 0. Without knowing its exact value, one can deduce that for this number holds 232 xxx += . Solving the equation means finding the x for which the statement is true. A priori x is indeterminate, a posteriori x can take two values. 3. as a variable object, i.e., a symbol for an object with varying value. In mathematics, the object to be thought of can be a number whose value may change like in the

process. An expression becomes a mathematical object on its own, which can be manipulated. In (Gray & Tall, 1994), the authors introduce the word “procept”, which is a contamination of process and concept, for the combination of symbol, process, and concept, to make clear that a mathematical object never completely looses its process nature. For example, in the expression

12 +x the + is not only a symbol that defines the object ‘the sum of x2 and 1’, but it also refers to the process of ‘add 1 to the product of 2 and x’. Another example: nn alim represents both the process of ‘tending to a limit’ and the concept of the ‘value of the limit’. The interested reader is referred to (Tall et al, 2001) for further discussion of the process/concept duality in algebra and its impact on learning to think mathematically. The alternative conceptions are also referred to as “process and object” (Sfard, 1991), where the word “reification” is used for the objectivation of a process, and as “procedure and structure” (Kieran, 1992).Gradually pupils learn to see variables as a replacement not only for numbers, but also for expressions. They learn and practice the various ways in which algebraic expressions can be manipulated: combining literal terms, replacing subexpressions, factoring, completing the square in a quadratic polynomial, rationalising the denominator, subtracting the same term from both sides of an equation, solving systems of linear equations, and equivalence testing are examples of activities that are present in all mathematics curricula. In many research studies (e.g., see Kieran, 1989; Sfard at el, 1994; MacGregor and Stacey, 1997; Stacey and MacGregor, 2000; Tall et al, 2001) is reported that understanding generalised arithmetic is not easy: literal symbols are like numerals and words, yet they are different, and pupils have to learn to deal with these differences. For many a pupil,

21

21 +x relates to a straight line, but the equivalent expression 2)1( +x is only considered as a rational expression, which is something completely different in their eyes. The previous examples illustrate that pupils must understand the underlying structure of algebra and become familiar with the dual character of algebraic expressions (operational/structural, process/object) to gain competence in mathematics. Let us return to the concept of variable. That it is almost impossible to rigorously define the term does not mean that one cannot classify the various appearances of variables in mathematics. The following three uses of variable are distinguished in (Freudenthal, 1983): 1. as a placeholder, which denotes the places in an expression where the same object is meant.


pendulum. Another example is the role of the initial height 0y , the initial velocity 0v , and the

release angle in the mathematical formula 22

100 )sin( tgtvyy += that gives the vertical position

of an object that is thrown away. Typically, one keeps values of all parameters except one fixed and draws graphs for several values of the free parameter in one picture, i.e., the parameter in its generalising role is used to distinguish several cases.Freudenthal mentions another occurrence of parameter, which does not seem to fit so well in the previous three roles of parameters at first sight, viz., the parameter representation of curves and surfaces. For example, the unit sphere in the x-y plane is parameterised by means of the arc length sfrom a fixed point as )sin(),cos( sysx == . The parameter s arises as dependent variable (arc length), dependent on the point of the curve. A posteriori it is used as independent variable in order to represent the curve. Thus, the parameter in the parameter representation of a curve is mainly used as a variable object.Besides, parameter representation of curves and surfaces, there exist many more applications of the concept of parameter in school mathematics. We mention three applications, in which software can help to visualize parameters and to study the effect of changing parameter values (e.g., see van der Giessen, 2001):• curve fitting: linear fit ( bxay += ), exponential fit ( xgyy 0= ), and sinusoidal fit

( dcxbay ++= )sin( ).

locutions “2n for n from 1 upward” and “an converges to 0 as n goes to infinity”. Of course, one can replace these locutions with n2 for Nn and 0lim =nn a , but then one looses the kinematic aspect of the variable. However, this kind of variable very often occurs in mathematical models that are constructed to study real-world phenomena. The object can be a physical quantity such as time, position, and temperature, or an economic quantity such as price, capital, and income. Clearly quantities, often depending on time, with varying values. A variable object may be related with others. One speaks of independent variables, whose values one is free to choose, and of dependent variables, whose values one can compute given the values of the independent variables. Many applications can be given: the position of a moving object depending on time, the room temperature depending on the time of the day, the temperature of a long rod as a function of place, and so on. The roles of independent and dependent variables are often not fixed during a computation. For example, studying the motion of a sprinter, one may on the one hand consider acceleration as a function of time, but on the other hand describe it as a function of the velocity of the sprinter. One of the big ideas in calculus, and in mathematics in general, is the freedom of choosing independent and dependent variables. The above distinction of three uses of variable can be applied to parameter as well: in the role of a placeholder, the parameter has one value at a time. For example, in the formula glT 2= for the

period of the mathematical pendulum, the letter g stands for the gravitational constant. You may study the motion of the pendulum on earth and as well as on moon, but always stands g for one value only. The given formula expresses the relationship between the period of the pendulum (T)and the length of the pendulum (l). So the letter l plays the role of a variable object. Rephrasing Freudenthal, one speaks about “the use of a secondary – as it were sleeping – independent variable, which, if need be, can be accounted for – as it were wakened up – for instance in order to get a system lT of periods by the variability of the parameter l.” Thirdly, as a polyvalent name, a

parameter allows you to write general formulas or to distinguish various cases: the label p in the function definition xpxf p =)( is used to distinguish several cases. Coming back to our example of

the pendulum: (angular frequency), and (phase) are used in the formula )sin( += tAu todescribe harmonic motion mathematically. and are according to their origin dependent variables (dependent on the pendulum and its position at a certain time), but according to their appearance independent variables and they serve to describe the general periodic motion of the


single function value are mixed up easily and apparently without much harm: a physical quantity is sometimes a function of time, in other occasions a finite sample of values measured at different times, and sometimes a function value at a certain fixed time. As concrete example we consider Boyle’s law constantVp = , for pressure p and volume V. The variables are in fact functions of time, viz., )(p:p tt a and )(V:V tt a . Boyle’s law says that the product of these two functions is a constant function, i.e., constant)(V)(p =× tt , at every time t. In an experiment, one verifies or rediscovers the relationship by measuring p and V under different conditions. Here, one compares samples of function values. But in a school problem like “suppose that dl10V = when bar2p = , how much is V when bar4p = ?”, p and V do not represent functions or samples of function values anymore, but they represent single function values, viz., the pressure and volume at a certain fixed time. In mathematics, this ambiguous use of notation for function, sample of function values, and function value rarely occurs. In mathematics it is often allowed to change names in an expression without changing the meaning: for example, ( ) 1, =yxyx is the same set of ordered pairs as ( ) 1, =baba . There only exist some

generally agreed conventions about the use of letters such as x and y in order that almost every mathematically trained person reads bxay += as a functional relationship between these two

• exploring transformations of a function: e.g., starting from a standard function like 2)( xxf =

investigate the behaviour of )(),(,)( axfxfaaxf ++ and )( xaf .

• Solving problems numerically or graphically: e.g., “find a value of the parameter a such that the graph of )sin( axy += passes through the point (0,1) or that the graph of xay = and its derivative are equal.”

6. Variables in physics

In (Vredenduin, 1979), the author discusses some of the differences in terminology and notational systems between physics and mathematics. Some of his essential differences have to do with the use of variables and have been incorporated in Table 1. Below we work out the details.

Mathematics Physics Generalized arithmetic with dimensionless variables is used.

Quantity arithmetic with its own rules and use of units is dominant.

Names of variables are free to choose and changing names in an expression does not change its meaning

A variable is often related with some physics concept and its name is an abbreviation of this notion.

Irrational numbers like 2 , and e areimportant; floating-point numbers are without accuracy: 1.0 = 1.000.

In measurements, only natural numbers and floating-point numbers with accuracy occur:1.0 1.000

There is a strong focus on special properties of functions, e.g., on asymptotic behaviour.

Properties and assumptions may rule out parts of mathematical interest.

Words like “big”, “small”, “negligible” have little meaning.

A small change of a quantity Q is also a quantity Q with its own arithmetic.

Table 1: Some essential differences between the use of variables in mathematics and physics.

In physics, a variable is most often used as a name for a quantity that can vary (often with respect to time) and that in many cases can be measured. Physical quantities can be magnitudes such as tem-perature, mass, and length, or vector quantities such as velocity, acceleration, and force. One striking feature of the symbolic writings in physics is that function, sample of function values, and


this formula. When manipulating the formulas in a real world context, a pupil better keeps the dimensions in mind to verify work. Physics teachers have good reasons to request from their pupils that they check the dimensions and the units of measurement in their answers. The focus of mathematical thinking may differ radically from the one in a study of a real world problem. For example, when one encounters in mathematical work a rational function of the form

bx

xa

+ the attention goes almost automatically to the singular behaviour as x approaches –b. Compare

this with the study of enzyme kinetics, where the Michaelis-Menten expression for the initial rate of

transformation of a substrate, S, by an enzyme, is [ ][ ]SK

SVv

m += max , where [ ]S is the concentration of S,

maxV is the maximum rate, and mK is constant called the Michaelis-Menten constant. Since the parameters and concentrations are positive, the singularity is never encountered and the common mathematical analysis is irrelevant. In physics, words like “big”, “small”, “relatively small”, “negligible” can be used while talking about quantities. A small change of a quantity Q is also given a name, such as Q , and one

variables with parameters a and b. This conventional use of letters makes a discussion about a mathematical problem easier, because it takes away the need to explain every time the notation in use. In physics, replacements of letters are almost always forbidden: replacing traditional names for energy E, mass m, and the velocity of light c in Einstein’s law 2cmE = will ruin it. The reason is simple: most variables in physics are not meaningless but deal with concepts in physics that have the property of quantity. For this reason, one often chooses the first character of the name of the concept as the name of the variable: F for force, m for mass, and a for acceleration in Newton’s law

amF = is one example of many. Physical quantities can often be measured, but the measured values are always natural numbers (in counting processes) or floating-point numbers, possibly with margins of error. Irrational numbers like 2 or do not occur in measurements. Nevertheless, physicists mostly compute with quantities as if they take real values. But they treat floating-point numbers differently than most mathematicians do. For many a mathematician, the number 1.23000 is the same as the number 1.23 without the trailing zeros. However, in numerical analysis and in physics, the notation 1.23000 implies that the number is known more accurately than 1.23. If 1.23000 is a measured value of a quantity, you know it is between 1.229995 and 1.230005. A value of a physical quantity actually consists of three parts, viz., the numerical value (a number), the precision (the number of significant decimals or the margins of error), and the unit that is used to measure the quantity. This makes quantity arithmetic more difficult to learn and to use than reference-free number arithmetic. The following example, taken from (van der Kooij 1999), illustrates this. Compare the given answers to the following problem:

“Peter and John walk in a straight line in the same direction from the same starting point, with the same speed of 2 m/s. Peter starts first at time t=0 and John 3 seconds later. Give the formula for the distance s walked by John after t seconds, for t > 3 sec.” Answer 1: )3(2= ts . Answer 2: 62= ts .

In generalized arithmetic, the two expressions are equivalent: a factored and expanded form. In quantity arithmetic, which takes dimensions into consideration, the first answer )3(2= ts

represents a time-approach of the problem via a timespeeddistance ×= formula and the second answer represents a distance-approach 62= ts via a distancedistancedistance = formula. So, thinking about dimensions and units of measurement makes clear that in real life problems not only the variables but also the numbers have contextual meanings. Especially the number 1 is tricky because it is always left out of expression: no one writes the formula tNN 20= of exponential growth as

)1/(0 2 tNN = to show that the doubling time is 1 time unit and that dimensions are actually correct in


inversely proportional to 1Q , and therefore the quantities 2Q and 11 Q are proportional, it may be wise to plot 2Q against 11 Q and see if it gives a straight line through the origin. This helps a secondary school pupil to learn that relationships between quantities do not go without saying in physics and that people to better understand or to predict relationships have invented the physical laws. Another purpose of a physics graph is that it allows easier discussion about the physics problem and that it provides a good means for presenting results to other interested persons.For science, one could certainly say that graphing is not a context-independent skill. Rather, competencies with respect to graph interpretation are highly contextual and are a function of the scientists’ familiarity with the phenomena to which a graph pertains and their understanding and familiarity with representation practices. (Roth & Bowen, 2001) take this as a starting-point for implications for teaching graphing in school mathematics and science settings. The authors take a sociocultural orientation toward graphing as practice (by professionals), instead of the cognitive-psychological perspective, in which problems and misconceptions in students’ reading and interpreting graphs are identified (e.g., Leinhardt et al, 1990). Their analysis of how professionals

manipulates it as any other variable, except that one often ignores higher order terms like ( )2Q to get a simpler model description. Going from calculus of small changes to infinitesimal change and calculus with differentials is then a natural step. The following example shows how it works. We look at the formula 2tas = for a moving body with s being the distance travelled as function of time t. Suppose that one is interested at the speed of the object during its fall. The change in distance travelled )( 1ts during a small time interval [ ]ttt +11, is given by

( ) ( )21

21

211 2)( tattatattats +=+= . A physicist will say that, when t is small, the term with

( )2t can be neglected. So, for the rate of change one has 11 2)( tatts . In the limit case of infinitesimal changes, one has 11 2)( tatdtds = . So, the speed at time t1 is equal to 12 ta . The mathematician, on the other hand, is not happy with the sentence “when t is small, you can neglect the term with ( )2t ”. He or she will say that tatatts += 11 2)( and that therefore

110 2)(lim tattst = . The same conclusion, but a different style of working. The above differences add a good deal to the understanding of the difficulties that pupils have in relating the mathematical methods and techniques that are used in physics to what is learned in mathematics lessons. In mathematics, they use variables mostly as placeholders and polyvalent names. Emphasis is on generalised pure arithmetic, with reference-free numbers, and on the concept of function defined as a special kind of correspondence between nonempty sets A and B which assigns to each element in A one and only one element in B, i.e., on the Dirichlet approach to the concept of function. In physics, the third kind of variable, viz., the variable object, comes into play. One is involved with functional relationships between varying quantities, in which one distin-guishes between dependent and independent variables. In working with variable objects and relationships between them one uses mainly the theory and practice of solving equations in known and unknown quantities and one uses the calculus of change.

7. Different contexts for graphing in mathematics and physics

Why do we ask pupils to make graphs? The answer to this question differs from discipline to discipline, but the reasons for using graphs are commonly divided into two classes: analysis and communication (Friel et al, 2001).For example, a physics teacher (Barton, 1998) may say that the graph is simply a means to an end: plotting graphs helps to interpret measured data. A diagram gives an overview of the measured data and from its shape one may get a clue about the possible relationship between the physical quantities in which one is interested. In order to better see or verify these relationships all kinds of scaling of graphs are at hand, such as (semi-)logarithmic and double-logarithmic plots. Derived quantities can be introduced to make the relationship clearer: for example, if a quantity 2Q is


in case the pupils has to do it by pencil and paper. In a graph coming from a real context, the axes represent quantities and the names of these quantities appear at the axes together with the chosen units. An arrow represents the positive direction of the axis, because this is not fixed anymore in applications. Frequently, physical quantities have by origin a limited range of possible values. Then these ranges determine the plot range of a graph and accordingly the origin can be different from (0,0). As was said before, scaling is a matter of choice and does not have to be linear: if it is more convenient to choose another one, say a logarithmic scale, to clarify a relationship between quantities, then one a free to do so.The quantities represented by the axes in a physics graph are not the only quantities presented by the diagram. Also the ratio between proportional quantities, the gradient of a function, and the area between the graph of the function and the horizontal axis represent quantities. For example, in a position-time graph, the gradient represents speed (or velocity if direction of motion is taken into account) and the area gives the distance travelled during some time. The ratio between mass and

read graphs shows that competent graphing practices are related to understanding of both the real world phenomena and the structure of signifying domain, familiarity with conventions underlying these two domains, and familiarity with translating between the two domains. To read a graph competently, one needs more than instruction on the mechanical aspect of producing graphs. One must also be familiar with representation-producing mechanisms, data-collection devices, feature-enhancing techniques, etc. In education, extensive interaction with phenomena and representational means seems to be a prerequisite for graph sense. And this is where ICT is expected to contribute to the learning process. In mathematics, drawing the graph of a function has not much to do with finding a relationship between quantities; in most cases, the function is already given by a formula or a table of function values, and has nothing to do with a real world context. One does not have to look for the mathematical object of study and one makes the graph mainly to present a single view on various properties of the mathematical object. From the graph of a function one can get an idea about the number and location of zeros, maxima, and minima, about the asymptotic behaviour of the function, about points of interest such as discontinuities and bending points, about increase and decrease, and so on. The shape of a curve is important issue in mathematics lessons. Calculus helps to make graphical observations more precise: e.g., computing the derivative and making a sign diagram of it gives precise information about change, and solving equations provides appropriate answers to questions about zero’s and extrema of the function and about specific function values. Comparison of graphs can be done to illustrate the effect of transformations of functions. Because a diagram is in the field of mathematics essentially a set of numeric 2-tuples, it is easy to combine graphs of different origin in one picture. For example, one can draw without difficulty in one picture a graph of the function 2xy = , the graph of the inverse xy = , the upper part of the unit circle via the parameter representation )sin(),cos( sysx == , and the straight line going through (0,0) and (1,1). In all cases, a mathematical graph is used to show various aspects of a function in a single picture and it is one of the means to solve mathematical problems. The graphs in mathematics and physics also differ from technical point of view. A pupil studying a Dutch mathematics textbook can easily recognize when (s)he is dealing with a mathematical graph or with a graph coming from an application in a real world context. In a mathematical graph, the coordinate system consists of two perpendicular number lines of the same kind intersecting at (0,0), and the horizontal and vertical axes are labelled by x and y, respectively. Unless otherwise defined, the domain and range of a function are infinite. Accordingly, a pupil is free to choose the linear scale of the number lines such that all interesting parts of the graph of a function are visible, but always 0 appears on the lines. The introduction of the graphing calculator, which allows non-linear scaling of axes and a plotting area that does not contain (0,0), has not brought much change in the mathematical graphs in the Dutch textbooks. Often textbook authors make a distinction between ‘plotting a graph of a function’, when the calculator is used, and ‘drawing the graph of a function’,


about a v-t diagram or an x-t diagram. This is opposite ordering of names used in mathematics. It might help pupils to better understand graphing if physics textbooks and teachers would regularly use the terms ‘ )(tv diagram’ and ‘ )(tx diagram’, and if mathematics textbooks and teachers would regularly use the terminology ‘graph of y as function of x’.The word ‘range’ has a different meaning in mathematics and physics. In the latter field, it refers to the values that a physical quantity can take. So, it is legitimate to talk about the range of the quantities at the horizontal and vertical axes of a graph. In mathematics, the words ‘domain’ and ‘range’ are used in connection with the notion of function and they specify for what values the function is in principle defined and to what set the function values belong.In many physics textbooks, the difference between steepness of a straight line and its slope is stressed. The reason lies in the use of dimensions: a quantity like speed is connected with the

volume of matter is a derived quantity, viz., the mass density. In a mathematical graph, the directional coefficient of a straight line is a dimensionless number as well as the slope of a graph in a given point, which only has a geometric interpretation as the limit of a difference quotient or the slope of the tangent line in that point.

We summarise our findings so far in Table 2.

Mathematics Physics Graph Represents a single object, viz.,

a function. Main purpose is to give a single view of various aspects of a given function.

Represents a relationship between two quantities. Main purpose is to explore or to present the relationship between quantities.

Axes Dimensionless numbers are represented. Scaling is by default linear.

Values of quantities are expressed in some unit. Scaling is a matter of choice and may be non-linear.

Origin (0,0) is the fixed position. Arbitrary position. Plotrange

In principle infinite Determined by the ranges of the quantities.

Slope/Gradient

Dimensionless number having a geometric interpretation only.

Represents the change of a quantity with respect to another and is on its turn a quantity with a unit.

Table 2: Different contexts of graphing in mathematics and physics.

Graphs in mathematics and physics do not only differ in their construction or purpose; also reading of graphs is different and this is reflected in the language that is commonly used. We give some examples. In mathematics, one speaks about the origin of a coordinate system and about the zero on the number line, in the sense of the point that represents the integer 0 with the property of the unit element in an additive group. In physics one does not speak about the origin of quantity values, but instead one uses the wording ‘zero’. For example, temperature can be expressed in various units depending on the zero that one chooses: 0 Kelvin refers to the lowest temperature that an object can have, whereas 0 °C refers to the temperature of melting ice. Another common example of the freedom to choose the zero of a quantity in a physics problem setting is the potential of a force field, say an electric or gravitational field. A mathematical graph represents a single object, the graph of a given function. Accordingly one speaks about ‘the graph of the function y’, ‘the graph of )(xy ’, or ‘the x-y graph’. A physics graph represents a relationship between two quantities and one generally speaks about ‘the graph of one quantity versus the other’. In this wording one uses the name of the quantity represented by the vertical axis first, especially when the horizontal axis represents time. For example, one speaks

8. Consequences for the design of an integrated learning environment.What can be learned from the discussion of mathematical notions in the last three sections?Certainly, analysing differences between the use of variables, functions, and graphs in mathematicsand physics is not enough when it comes to the design of an integrated learning environment. Onething to keep constantly in mind is that doing mathematics on a computer differs essentially fromtradition pencil-and-paper work. In (Heck, 2001), the author discusses the differences between theconcept of variable in mathematics and science and in computer algebra. He lists the followingproperties of computer algebra variables that make them different from variables in mathematics:• A computer algebra variable always points to a value, which can be almost anything.• Manipulation of computer algebra variables has its own rules, in which internal storage ofexpressions, automatic simplification, ordering of commands, and evaluation scheme play a role.• An expression can represent a mathematical object as well as describe a particular process to becarried out. Both notions are frequently used.• Some variables have special meaning distinct from standard mathematics.• Although modern computer algebra systems try to mimic mathematical notation as much aspossible, their users still have to translate to and from standard notation on many occasions.• In computer algebra, there is a strong focus on solving generic problems, i.e., special cases such asspecial values of parameters are not taken into account.Being aware of such differences between traditional work and computer-based work, what kind ofsuggestions and recommendations can we make to designers of educational software environments?First of all, a lesson from the past: many successful education software environments like dynamicgeometry systems have been primarily designed on the basis of what should be done in education insteadof on the basis of what can be done technically. In other words, the activities of students and teachers havebeen chosen as the basis to work from,not the knowledge that apparently must be incorporated. In short,we would like to advocate the following primary design goals:• to provide a useful, powerful, and scalable set of facilities for students and teachers;• with an easy to use and easy to learn, appealing user interface;• ideal for teaching and learning activities;• that runs on moderate and affordable equipment;• allows easy exchange with other software products such as spreadsheets, text processors,presentation tools;• supports easy network communications such as internet browsing and distance learning.


steepness of the tangent line at points in the x-t diagram. The steepness of a tangent line can be approximated by a difference quotient, i.e., the quotient of two differences of quantities. In mathematics, which uses dimensionless numbers, the slope of a tangent line or the coefficient of direction, as it is also called, is mainly interpreted geometrically as the arctangent of the directional angle of the line, but only in case equal horizontal and vertical scales are used. The slope has to do with angle in a geometrical picture and it is not a physical quantity.The above differences in the use of language for graphing, which is summarized in Table 3, may contribute to the difficulties that pupils have in using graphs similarly, but yet so differently in mathematics and physics.

Mathematics Physics Tangent line, slope. Gradient, steepness. Origin (of coordinate system). Zero (of a quantity). Domain and range (of a function). Range (of quantity values). Graph of )(xy , x-y graph. v-t diagram, x-t diagram. Set of 2-tuples (x,y) Plot of y vs. x

Table 3: Some different languages for graphs.

We envision a scenario of a teacher and students using a set of tools for the study of natural andmathematical phenomena. This set of tools is integrated in one open environment designed for theeducational setting. Such environment is open in the sense that it is• a flexible and customisable multi-purpose tool;• an environment for solving open problems that need definition, set-up, exploration, etc., i.e., athinking tool;• free of didactic context or principles, i.e., it less considered as a pedagogical tool and more asmathematical and scientific tool.Some technical recommendations are:• provide many useful representations of mathematical objects;• allow flexible notation of mathematical expressions and user-defined objects;• allow more than one way of creating and manipulating objects;• make tools customisable to users.• provide an easy to learn and easy to use programming language for writing computer model andextending the built-in library of user commands.Let us be a little bit more specific and technical about the above guideline for the development ofan integrated learning environment that allows the user to express easily mathematical andscientific thinking. Firstly, what kind of model for the calculus of variables and expressions do weenvision and what do we learn from our analysis of the use of variables? Whatever the answer is,the model should support the objects and operations from mathematical and scientific contexts ina natural way. In other words, the way that the software treats them should be consistent with theexpectations of a user. For a software system that is meant for education, this is even moreimportant, since the software representations will influence the formation of concepts.In general, one can say that the more structure the model has, the more support it can give. But astricter structure also implies less freedom and introduces the need of more conversions of onestructure type to another.Current mathematical software systems take different approaches towards this design choice. Matlabsupposes that everything is a matrix. Axiom uses strong typing, together with a system for typeinference based on mathematical categories. Derive, Maple and Mathematica have no strong typing ofobjects, and give the user an enormous freedom. Such a freedom has a price; the interested reader isreferred for a discussion about Derive to (Artigue, 1997), (Guin & Trouche, 1998), and (Drijvers & vanHerwaarden, 2000). A price to pay is in general that although a user may enter an expression that ismathematically unsound, it can very well be syntactically sound. In this case the user will only benotified by a runtime error, or will not be notified at all. Moreover, more structural commands areneeded in the language to make up for the lack of structure in the model.We are of opinion that one can follow here an intermediate route and choose for a strong typing, butof a very simple nature. Few atomic objects like numbers, booleans and strings are required, and theremust be a way of creating indexed lists of objects. In most contexts, the types of the objects will followimmediately from the context; one does not have to declare them beforehand. In this way, we hope andexpect to have found a proper balance between power in expressiveness and structural support.Variables are ubiquitous in science and mathematics. In general, one thinks of a variable assomething with a symbol as name and a value. The symbol may be used in computations andmathematical expressions as if it were a number. The value may be not determined, it may change,and it may vary. This description is still somewhat vague, and in textbooks the concept is often illdefined or not defined at all. We find Freudenthal’s classification of the various appearances of theconcept of variable very useful, even under the constraint that variables always need a finiterepresentation on a computer. The finite representation of the variable poses for the use of avariable as a placeholder or polyvalent name not a particular problem. It is a symbol without aparticular value, or with a value that can be changed. The meaning of the symbol depends on themathematical context: expressing statements, solving equations, defining functions, and so on. Butin the third type of use, i.e., the variable as variable object, the finiteness of any representation ona computer does impose restrictions. We see for an object with varying value two possibilities:


• a finite representation with a finite indexed listing of values;• a finite representation with an algorithm expressed in finitely many terms.It is the first possibility of a notion of variable with indices to express the variability that leads usto a type for variables: e.g., the type float[i,j] will mean that the variable has as values floating pointnumbers that depend on the indices i and j. So, such a variable has a symbol (name) and a typeassociated to it, possibly together with a value that can be a number, a list of numbers, a list of lists ofnumbers, etc.To keep track of the origin of the data, variables will have a source attribute that containsa reference to the instance (file, URL, sensor, GUI-element, etc.) that actually sets the values of thevariable. Although such a typing system allows structured support for the user and a short andtherefore powerful language, it is the purpose of the model to ensure that most users do not have toset the types explicitly. In fact, only advanced users will have to understand the typing system.The second possibility is about algorithms. This leads us at first to formulas: mathematicalexpressions containing arithmetic operations, function calls, variables and indexed lists. Formulasare in some way the easiest algorithms, but one will soon feel the need for other algorithms, e.g.,with repetitions and conditionals, or operations in which also the indices are involved.This is wherethe programming language comes in. Expressions can be formed in terms of variables, indexed listsand operations on those. A prototype for internal use by software developers at the AMSTELinstitute shows that this model for calculating with variables of different types is feasible.The second major theme of our discussion about consequences for the design of an integratedlearning environment is graphics and its role in mathematics education. First of all, there seems tobe general consensus about some starting-points of teaching and learning calculus:• equally important role of graph, table, formula, and context (certainly in the beginning).• use of multiple representations of the same mathematical object.• stress on relational understanding instead of instrumental understanding: the difference is in

understanding why and how. For example, drawing a graph of a function is of instrumentalnature, whereas curve fitting requires knowledge about properties of functions.

• use of real contexts for concept building and application.• importance of dynamics for deepening mathematical insight.Experimental research of recent years (e.g., Hennessy, 1999) supports the assumption that graphing

calculators can already stimulate• the use of realistic contexts and bring mathematical modeling and interpretation of results of

simulations within pupils’ reach;• the exploratory and dynamic approach of mathematics: think of pupils’ activities like

investigating and classifying parabola, and graphical techniques like zooming;• a more integrated view of mathematics: pupils can use multiple representations of mathematical

objects;• a more flexible problem solving behavior: new strategies like solving equations graphically are

possible.The question is not whether technology can contribute, but more how it can contribute and whatrequirements can be made for an integrated environment to support this view on calculuseducation. We limit ourselves to role of graph, table, formula, and situation.Graph and table are mathematical instruments to represent data in a clear structured way.They arein general more than an ordered set of point and numbers: they tell a story or represent a process.For example, in an interpretation of a velocity-time diagram qualitative issues such as change of agraph and asymptotic behavior are often more important than the quantitative aspects.To facilitatediscussion about a diagram and to make it possible to construct diagrams in the same format as onesees in textbooks, it is necessary that one can make diagrams more beautiful. Use of differentcolors, different line styles, various characters, annotations and legends, auxiliary points and lines,and so on, must be possible. Various zooming facilities such as automatic zooming or the use of a(dedicated) zoom box allow the user to focus on interesting parts of a diagram. To betterunderstand the role of a parameter in a mathematical model or to carry out a sensibility analysis, alot is gained when a range of parameter values is available via graphical user interface elements


like a slider bar. This immediate graphical response to changing a parameter is expected tocontribute to mathematical understanding. Construction of graphs and table should be easy andnatural. It must be easy to comply with a request like “give me a straight line”, “give me aparabola”, or more generally, “give me a graph of a function of this well-known type”, and providea graph together with handles for further adjustment. Good-quality freehand sketches of graphsmay be difficult to make, but standard techniques like the use of Bezier curves may already sufficefor making a sketch. Also, when a mathematical function f is defined in a standard way, say (f(x) =x2 – 3x + 1, dragging and dropping the symbol f into a diagram window may already produce thegraph of f for some default domain. Similarly, a definition like y: = x2 – 3x + 1 can be dragged anddropped into a diagram window to get the isolated variable y plotted against the independentvariable x. The same holds for dragging a symbol or definition into a table window. Graphs andcolumns in tables do not need to come from formulas exclusively. For example, to create anincrease diagram of some functional relationship or to filter some data it is enough to select someappropriate (menu) command. Certainly, in case you want to be able to carry out operations ongraphs of functions before the required algebraic manipulation skills have been learned andpracticed by students, this kind of graph manipulation without a formula is fruitful. A more generalrecommendation to software developers is to de-emphasize algebraic input via keyboard orpalette, but to allow other ways of creating mathematical objects. One of the behaviors associatedwith graph sense is the ability to understand the relationships among a table, a graph, and thefunction or data being analyzed. Many research studies (e.g., Kaput, 1992) report about the possiblecontribution of the use of multiple, linked representations of mathematical object to mathematicalthinking.The main rationale of linked representations is to give students the opportunity to exploreideas from a variety of directions. Let us have a brief look at what this means for learning theconcept of function. The matrix in Table 4, developed in (Janvier, 1978), gives an overview of thetransitions between table, graph, formula, and situation.In mathematics lessons, the most used transitions are t → g, g →_t, f → g, f → t, and f → f , wheret, g, f stand for table, graph and formula. From an integrated learning environment one may expectthat it makes such transitions simple to do and that is allows student to concentrate onmathematical and scientific issues, instead on technical details. After all, the technology must solveproblems for its user and not create additional problems. But one may expect more: transitionsinvolving situation are possible in video and digital image measurement. And manipulations ofgraphs can be dealt with in a dynamic way.


from →_to situation table graph formulasituation restructuring,

searchingrelevant data

deriving data fromthe situation, e.g.,via data collection

sketching ofgraph form verbaldescription

modelling,finding aformula from averbaldescription

table reading and interpreting

changing one tableinto another, e.g.,adding an increasecolumn

plotting orderedpairs of numbersin a grid

finding aformula fortabular data

graph interpretingcharac-teristicsof functions

reading offcoordinates

making one graphfrom another

curve fitting

formula formularecognition

computing sketching a curvefrom a formula

simplifying andre-writing aformula

Table 4: Transitions between situation, table, graph, and formula.

We give the above long and yet incomplete list of recommended facilities for graphing because they playa major role in the setting in which graphs are represented, used, or learned. They have much to do withthe following list of features that contribute to students’ success in graphing (Ainly, 2000):• The presentation of a complete image allows students to take a global view of the graph, rather

than focussing on separate components. So, zooming facilities are important.• The use of a number of similar graphs encourages students to focus on similarities and

differences between examples and sharpens their discrimination. The easier to graph, the moreexamples pupils can study.

• The ability to manipulate the graph and change its experience is closely related to the previouspoint. Re-representing the same data in different ways helps to emphasize features of the data.

• A familiar and/or meaningful context allows students to feel ownership of the data and to makesense of it. This will obviously be true when they have collected and recorded data themselves.

• A purposeful task in which the graph is used to solve a problem encourages active use of graphs,in which it is natural to work on reading the graph and on manipulating the graph to make itmore readable, rather than seeing it as an illustration.

9. ConclusionFrom an integrated learning environment for math and science one may expect that students andteachers can • use a rich variety of tools to express mathematical and scientific ideas in a concrete form;• explore ideas from various directions• be explicit about why they are using a particular tool and method.However, to implement these wishes is nontrivial. A complicating factor is that mathematicalconcepts are not always used the same in mathematics and science.

ReferencesAinly, J., Transparency in graphs and graphing tasks. An iterative design process. Journal of Mathematical Behavior ,

19, (2000), 365-384.Artigue M., Le logiciel ‘Derive’ comme révélateur de phénomènes didactiques lies a l’utilisation d’environnements

informatiques pour l’apprentissage. Educational Studies in Mathematics 33, No 2, (1997), 133-169.Barton R., Why do we ask pupils to plot graphs? Physics Education 33, No 6, (1998), 366-367.Clements D.H., From exercises and tasks to problems and projects. Unique contributions of computers to innovative

mathematics education. Journal of Mathematical Behavior, 19, No 1, (2000), 9-47.Drijvers P., & Herwaarden O. van.,. Instrumentation of ICT-tools: the case of algebra in a computer algebra

environment. International Journal of Computers in Mathematics Education, 7, No 4, (2000), 255-275.Etten, B. van, Variabelen in de schoolwiskunde. [Variables in school mathematics] Wiskrant, 23, (1980), 15-18.Ellermeijer, A.L.(ed.), STOLE, Scientific and Technical Open learning Environment. Proposal for the E.U.

programme DELTA. Internal report of the University of Amsterdam, (1988).Freudenthal, HDidactical phenomenology of mathematical structures. Reidel Publishing Company, (1983).Friel, S.N. Curcio, F.R. & Bright, G.W., Making sense of graphs: critical factors influencing comprehension and

instructional implications. Journal for Research in mathematics Education, 32, No 2, (2001), 124-158.Furinghetti, F. & Paola, D. Parameters, unknowns, and variables: a little difference? In Da Ponte, J.P. & Matos, J.F.

(eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education, vol 2.University of Lisbon, Lisbon, (1994).

Giessen, van der C., The visualisation of parameter. Proceedings of the 5th international conference on technology inmathematics education. University of Klagenfurt, Klagenfurt, (2001).

Graham, A. & Thomas, M., Building a versatile understanding of algebraic variables with a graphic calculator.Educational Studies in Mathematics, 41, No 3, (2000), 265-282.

Gray, E.M. & Tall, D.O., Duality, ambiguity and flexibility.A proceptual view of simple arithmetic. Journal for Researchin Mathematics Education, 26, No 2, (1994). 115-141.

Guin, D. and Trouche, L., The complex process of converting tools into mathematical instruments: the case ofcalculators. International Journal of Computers for Mathematical Learning, 3, No 3, (1998), 195-227.

Heck,A. Coach: β−tool in spe en nu al inzetbaar bij wiskunde. [Coach: math and science tool in the making and alreadyusable for mathematics] Nieuwe Wiskrant, 19, No 4, (2000), 40-46.

Heck A. & Holleman A., (2001a) Modelling human growth. Proceedings of the 5th international conference ontechnology in mathematics education. University of Klagenfurt, Klagenfurt.

Heck, A. & Holleman, A. (2001b) Investigating bridges and hanging chains. Proceedings of the 5th internationalconference on technology in mathematics education. University of Klagenfurt, Klagenfurt.

Heck A., Variables in computer algebra, mathematics, and science. International Journal of Computers in Mathematics


Education, 8 No 3, (2001), 195-221Hennessy S., The potential of portable technologies for supporting graphing investigations. British Journal of

Educational Technology, 30, No 1, (1999), 57-60Janvier C., The interpretation of complex cartesian graphs – studies and teaching experiments. Ph.D. Thesis. University

of Nottingham, (1978)Kaput, J.J., Technology and mathematics education. In Grouws, D.A. (ed.). Handbook of research on mathematics

teaching and learning. MacMillan, New York, (1992)Kieran C., The early learning of algebra: a structural perspective. In Wagner, S. et al (eds.). Research issues in the

learning and teaching of algebra. NCTM/Lawrence Erlbaum, Reston, VA., (1989)Kieran C., The learning and teaching of school algebra. In Grouws, D.A. (ed.). Handbook of research on mathematics

teaching and learning. MacMillan, New York, (1992)Kieran C., Mathematical concepts at the secondary school level: the learning and teaching of algebra and functions. In

Nunes, T. et al (eds.). Learning and teaching mathematics. Psychology Press, Hove, (1997)Kooij, van der H. (1999). Modelling and algebra: how ‘pure’ shall we be? Paper presented to Topic Study Group 8 at

the 9th international conference for the teaching of mathematical modelling and applications.Kücheman D.E., Algebra. In Hart, K.M. (ed.). Children’s understanding of mathematics: 11-16, John Murray, London,

(1981).Leinhardt G., Zaslasvky O. & Stein M.K., Functions, graphs, and graphing: tasks, learning, and teaching. Review of

Educational Research, 60, (1990), 1-64.MacGregor M., and Stacey K., Students’ understanding of algebraic notation: 11-15. Educational Studies in

Mathematics, 33, No 1, (1997), 1-19.Mioduszewska E., and A.L.Ellermeijer. Authoring Environment for Multimedia lessons. In: Proceedings of GIREP-

conference ‘Physics Teacher Education beyond 2000’, Barcelona, 2000. On CD-ROM, 2001Roth W.-M. & Bowen G.M., Professionals read graphs: a semiotic analysis. Journal for Research in mathematics

Education, 32, No 2, (2001), 159-194.Schoenfeld A.H. & Arcavi A., On the meaning of variable. Mathematics Teacher, 81, (1988), 420-427.Sfard A., On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of

the same coin. Educational Studies in Mathematics, 22, No 1, (1991), 1-36.Sfard A. & Linchevsky L.,The gains and pitfalls of reification - the case of algebra. Educational Studies in Mathematics,

26, (1994), 191-228.Stacey K. & MacGregor K., Learning the Algebraic Method of Solving Problems. Journal of Mathematical Behavior,

18, No 2, (2000), 149-167.Tall D.O., et al Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science,

Mathematics and Technology Education, 1 No 1, (2001), 81-104.Usiskin Z., (1988), Conceptions of school algebra and uses of variable. In Coxford, A.F. et al (eds.). The ideas of

algebra, K-12 . NCTM 1988 Yearbook. NCTM, Reston, VA.Vredenduin P., Terminologie in natuurkunde en wiskunde. [Terminology in physics and mathematics] Euclides, 55,

(1979), 81-94.Yerushalmy M., Making exploration visible: on software design and school algebra curriculum. International Journal

of Computers for Mathematical Learning, 4, (1999), 169-189.Wit J.M., (1998), De vierde landelijke groeistudie. [4th Dutch growth study] Boerhave Commissie, Leiden, (1998).

AN EPISTEMOLOGICAL FRAMEWORK FOR LABWORK IN EXPERIMENTALSCIENCES

Matilde Vicentini, Department of Physics University “La Sapienza”, Roma, Italy

1. Introduction: knowing science and knowing about scienceWhen talking about the role of labwork in the teaching of experimental sciences two possible aimsof didactic communication using laboratory activity are intertwined. The first aim concerns thelearning of science by the students: labwork activity is here viewed as a mean of guiding studentsin their understanding of the relationship of natural phenomena to the theoretical models that thescientific community today accepts as valid interpretations and explanations.The second aim concerns students’ learning “about science”: labwork activity is here viewed as themeans of guiding the students in a metareflection on the production and evolution of the contentsof science. This means, on one side, the procedural methodology of making observations, planningexperiments, developing theories and models, and comparing theoretical analysis with empiricaldata. But it also means, on another side, discussing the social aspects of science-in-the-making


(research group organisation and interactions, publication procedures, role of conferences and sym-posia, scientific debates) and the socio-political aspects related to the interaction of the scientificcommunity with the society at large (criteria and sources of financial support, popularisationefforts, ethical problems….).It is reasonable to recognise that the two aims require involvement at two different cognitive levels.In both cases however the teacher (who is supposed to know science) needs an epistemological fra-mework in which to place the planning of didactical activities involving labwork. The expertiserequired to build such a framework may be found in part in the work of philosophers, epistemolo-gists and sociologists and in part in the work of scientists. However the two parts focus on differentaspects of the problem of interest to science educators and teachers. Moreover one often has theimpression that science educators recognise the competence of philosophers, epistemologists andsociologists on the issue while scientists are often assumed not to be reliable in talking about “whatscience is” as they are so involved in “doing science” that they do not have the time nor the com-petence to argue about what they are doing (Vicentini 1999 a, b). This is partly true; in fact a nor-mal scientist is, generally, so involved in “doing science” (and publishing research papers) thatshe/he does not care to reflect on the placing of his/her work in an epistemological framework.However some outstanding scientists have written about science: Duhem and Poincaré are exam-ples from the past, De Gennes (1994), Jacob (1997), Deutsh (1997), Gellmann (1994), Cini (1994),Levy Leblond (1996) more recent examples. In the educational literature one seldom finds refe-rences to them. Of course epistemologists and philosophers have the competence as they have stu-died the work of the scientists in historical records but little has been done on “science in themaking”. As Latour (1998) says “there is a philosophy of science, but unfortunately there is no phi-losophy of research. There are many representations and clichés for grasping science and its myths;yet very little has been done to illuminate research”.Something is available on the theory-data-reality relationship, on the interactions among scientistsin a laboratory set up (Latour 1979, Giere 1988 are examples) and on the role of technology(Gallison 1997). However we must note that scientists do not publish the detailed history of theplanning and definition of an experiment with their reasons for choices in the technology and thedead ends of trial measurements.In their articles one finds the final apparatus, with the indication of measurement instruments and,perhaps, procedures and experimental results. Sometimes it may happen that - from the planningof an experiment - the design of a new instrument or a calibration procedure are produced.However these technical details are generally published in the appropriate technical journalswithout reference to the experiments that prompted their development.The research articles therefore present a tale of the experimental research not as it has actuallybeen developed but as it could have been developed in an ideal world where no accidental effectsinterfere with the story.In this paper I will present a possible framework for the reflection of teachers and science educa-tors. The framework is the merging of ideas taken from the work of both kinds of experts(Interchange 1997, J. Research in Science Teaching 1998, Science Education 1997/1999, Mattews1994) but also from my personal history as an experimental scientist in the field of physics.Therefore, physics will act, in this paper, as the primary referent for my metareflection: it is fromthis referential position that I will try to argue about the role of laboratory work in the experi-mental science.Since science may be considered a complex abstract “object”, it may be looked at from differentpoints of view. Let me then use the metaphor of the observation of a concrete three-dimensionalobject. When we look at an object from different points of view we obtain different images accor-ding to the perspective of observation. From one particular point of view we see the details of oneface of the object while the aspect of other faces is hidden. The same happens when we try to talk“about science”: one perspective permits us to focus some details while hiding others. Therefore Iwill present in the following three sections of the article three possible perspectives on the problem.Each perspective will focus some aspects involving laboratory work in experimental sciences. The


final section will then try to unify them. Conceptual maps, a communication tool suitable in thenegotiation of meanings, will be used as needed in all three perspectives.

2. First perspective: the organisation and development of scientific knowledgeFrom a constructivist perspective a scientists, like any other human being, is the one who organises


World ofphenomena/Laboratory

Socialcommunity/scientificcommunity

a scientist

Producesartifacts Exchanges

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that are

which havethe capacityof

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and eventually

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Demonstrationverification/falsification

Fig. 2

his/her personal knowledge through the informations received from the world of phenomena andfrom the scientific community.The conceptual map in fig.1 gives an illustration of the sources and procedures of this organisation ofknowledge. For a scientist the world of phenomena is made up of two different sources of information:the natural world and the laboratory world.The scientist, in his/her work, produces artefacts that enter

the laboratory world to furnish answers to his/her questioning. The interaction with the social com-munity privileges the debate with the experts of the disciplinary field - the scientific community. Thusthe scientist, in his/her life, goes through schemes of knowledge which enable him to cope with thechanging spectrum of information collected from experimental work and interaction with his peers.An analogous map may then be obtained to illustrate the development of scientific knowledge justby changing the role of organiser of knowledge from the individual scientist to the scientific com-munity.The discussion inside the community is such that it gives value only to the schemes of know-ledge that acquire inter-subjective agreement through confrontation with empirical findings in thelaboratory and among scientists in a critical debate.A closer look at the relation of the world of phenomena and the schemes of knowledge is illustra-ted in the conceptual map of fig.2 which may be used, by reading the map step by step, for a firstdelineation of the work of an experimental scientists.After the choice, in the world of events, of particular phenomena of interest, the scientific studybegins with a qualitative observation aimed at the identification of the parameters suitable for thedescription of the system and of its temporal evolution. This step also involves a decision aboutwhat may be considered as accessory to the phenomenon under consideration and, therefore, to beexcluded from scientific analysis. A model system may then be organised for study in a laboratoryset up. This model system (which may be called the “empirical referent” of the phenomena) is thesystem on which to perform quantitative observations through measurement procedures of thedescriptive parameters. Experimentation in the laboratory may lead to the definition of relationsamong the parameters which then constitute a set of empirical laws (or primary model) that pro-vide a phenomenological quantitative description of the phenomena (an answer to the question“how” does the phenomena appear). The primary model then has a predictive capacity for newexperiments or for the development of technological applications. However a scientist is also inte-rested in finding possible explanations for the occurrence of the phenomena. The search for ananswer to the question of “why” the phenomenological behaviour is as observed involves the intro-duction of entities outside of the experimental determination which have the role of explanatoryconcepts for a set of primary models.We may call the conceptual framework which furnishes an answer to the question “why” a “secondarymodel” or theory. The theories then justify the predictions emerging from the primary models whilealso envisaging the possibility of new predictions and new technological applications.Thus map in fig.2 gives us a perspective on the work of an experimental scientist which, although indi-cating that the “protagonists” of scientific inquiry (the real world of phenomena, the laboratory worldproducing the primary models, the ideal world of theories), does not really describe the interplayamong them. In fact the picture could be interpreted by the reader as an inductivist perspective onscientific methodology. It is, therefore, necessary to present the problem from a different perspective.

3. Second perspective: experience and experimentThe words “experimental science”, with the semantic root of the adjective, point to a specific rela-tion between the reality of facts and phenomena of the natural world and the theory and modelsdeveloped in the scientific disciplinary field.The semantic root may be found to be in common with the verb “experiencing” or “to have expe-rience”. However there are nuances of meaning in relation to specific characteristics of the activity:experiencing, having experiences are words connected with all kinds of activities in which a humanbeing receives information from the environment not necessarily as answers to questions that s/heis asking. One may experience something also when one is not willing to do so.Experimenting, doing experiments are words that correspond to the activities in which a humanbeing is searching for answers to specific questions. This then implies the planning of some sort ofapparatus for finding answers to the questions.Therefore the activities of “experiencing” and “experimenting” are different both in the strategiesthat define the procedural aspects (the first a spontaneous activity with no special focus, the seconda planned activity with a special aim) and in the reality toward which the actions are directed.


The reality of “experiencing” is a complex reality of the natural world, the reality of “experimen-ting” is the reality of objects in a laboratory. Thus the laboratory, while belonging to the naturalworld of the concrete objects and instruments that constitute its furniture, is also an artificial worldas the same objects have been built by mankind.It may be useful to focus in more detail on the differences between the natural world and the labo-ratory world.In the natural world objects and phenomena are interrelated in a more or less complex way. Todayalso many artefacts produced by applied technological research have become part of the environ-ment of everyday life. In the laboratory world these objects are mainly technological ones built forthe inquiry of some phenomena by abstracting them from the complexity of the real world or ofphenomena which do not appear in the natural world but are, in fact, produced in the laboratory.All these concrete objects must be known to the experimental scientist (or by the research groupwho is constituted by people of specific expertise working in co-operation). Other objects must alsobe known to the researchers: these are the abstract object of models and theories of the discipli-nary field and of experimental procedures.It thus appears that the laboratory world is the actual, direct referent of reality for the scientists whilethe natural world constitutes an ideal, indirect referent: the findings from the laboratory world mustmake sense also in the natural world. Of course there are differences among the various disciplinaryfields of the experimental sciences: (e.g. in physics or chemistry most findings come from the labora-tory while in geology or biology the role of the external environment is more pregnant and often con-stitutes the laboratory in which objects and phenomena may be studied directly).In the past when the laboratory world was more similar to the external world one could imagine ascientist who was mainly using induction procedures for the organization of schemes of knowled-ge. The increased debate, in this century, about the role of induction procedures in the experimen-tal work is also related to the role played by models and theories in the definition of the experi-mental apparatuses and measuring instruments.One may say that today theories and experimental tools are so interdependent that it is difficult toseparate them. To understand interdependence we may analyse the development of an experimen-tal research project in-the-making. This corresponds to a more detailed description of the step“experimentation” in the map in fig.2.Schematically an experimental research project develops over 4 phases of work.The first phase concerns the definition of the research problem to be solved experimentally. Whileconstrained by the status of knowledge at the time of the experiment, the capacity to finding mea-ningful research problems does depend on the creativity of the scientist. In some cases it may bemeaningful to verify empirically specific theoretical hypotheses or predictions. In other cases theexperimental exploration of new phenomenological fields may be meaningful. New technologiesmay focus on particular problems.It is not easy to develop a complete overview of the various situations that have happened in thehistory of science. “Great scientific experiments” have found their place in historical reports, but itis not easy to find descriptions of the “little scientific experiments” which are published in scienti-fic journals, which may be recognised as relevant for publication in a particular period but whosecontribution to the development of knowledge is not taken into consideration in retrospect.Once the problem of research has been defined we have the planning phase of the experiment.Now a careful analysis of the phenomenon is required with the definition of the variables to be con-trolled or measured and the invention of the technical know-how appropriate to the inquiry. Trialsand errors of specific parts and of the whole apparatus will be needed to evaluate the response, theexperimental accuracy, and the appropriate measuring instruments.Preliminary data will be collected and analysed for the tuning of the apparatus to the problem que-stions. Again it is a phase in which the creativity of the scientists plays an important role - also forthe resolution of technical problems.The next phase, when everything is ready for the data collection, becomes then a routine phase. Nolonger creativity but care and patience are required in following the rules of the experimental game.


The final phase of data analysis partly overlaps to the third. It directs eventual changes in the expe-rimental conditions and the decision about the conclusion of the experiment. Here, once againcreativity is very important.

4. Third perspective: the structure of scientific knowledgeIn a way by introducing two perspectives connected with the development of scientific knowledgeand the procedures of experimental work, I have not focused on the central aspect of the structu-re of scientific knowledge. Now we need to focus on this aspect.Today science is organised in different fields, each one characterised by a common shared set of theories,models, empirical laws - and methodologies of research. This common shared knowledge is, as we havealready pointed out, the knowledge obtained about that part of the world which pertains to the field ofresearch by inter-subjective agreement among different scientists who have the competence for compa-ring models and theories with natural phenomena, experimental outcomes, technological products.Each sub-field, therefore has its own specificity - and scientists of different fields may have diffi-culties in communication. However the interrelations between theories, models, experimentalresults and technological products cross over the different fields of the experimental sciences(Bandiera, 1998).A first aspect of the relation between the ideal world of Scientific Theories and the real world of thelaboratory where phenomena are studied experimentally is shown in fig.3. In some sense the new con-ceptual map may be considered a reorganisation of the map in fig.2 in which the apparent inductivismdisappears. Here the focus is on the difference between the reality of the world of phenomena and theidealness of the world of theories. The phenomena - described by empirical laws constitute the expla-nandum for models and theories. However the empirical laws may also be considered as possibleexplanations of some phenomenon, with a difference in kind with respect to theories: an empirical law“explains” only in the sense of including the phenomenon in a family of phenomena (if you like, anempirical law “explains” a fact by defining its “sameness” with other facts).A theory tries to explain itby the use of concepts and variables which are not defined at the level of empirical laws and which areinvoked as “the reason why” for the relations among measured quantities.As empirical laws unify different sets of experimental data and theories unify different sets ofempirical laws, one gains the obvious advantage of economy of representation and communicationin reaching a more and more unified scientific knowledge (the overarching framework that scien-tists aim to construct: examples are classical and quantum mechanics in physics, the theory of evo-lution in biology). In a given field at a given time the degree of possible unification is often knownto scientists. However, for the solution of well defined problems it is more practical to use know-ledge at a lower level of unification.There is another difference between theories and empirical laws. Empirical laws maintain theirvalidity in correlating experimental data (in the accuracy range defined by the measurements) evenif the theory invoked to explain them is changed or rejected.The changes in the ideal world of theories - which, together with experimental results, models andtechnological artefacts constitute the universe of knowledge shared by a scientific community atany given time, are depicted in fig.4. Here the focus is on the distinction between theoretical andexperimental research which both contribute to interrelating of inductive and deductive steps and- to the new definition of the universe of shared knowledge. It may be clear, from the picture, thatthe statement “data are theory laden” should be changed “any research project, either theoreticalor experimental, is laden by all the shared knowledge, at a given time, in the three aspects of theo-ries/models, experimental outcomes, technological artefacts”.In other words the statement should be completed by the complementary one “theories are dataladen”. Therefore any kind of scientific work, either in theoretical or in experimental research,requires the acquisition of competences in the knowledge shared by the community: theories,models, empirical data and laws.Theoretical research may be concerned with the development of aspects related to new or anoma-lous data, with the logical organisation of theoretical aspects and in some case also with the defini-tion of theories following some kind of intuition.


Any experimental research therefore must depart from analysis of the status of knowledge availa-ble at the time of the experiment concerning theoretical and technological aspects, in addition tothe empirical results that may be relevant for the planning of the experiment. The research mayconcern technological development, experimental verification/falsification of theoretical hypothe-sis, experimental exploration of fields opened by theoretical or technological developments, analy-sis of experimental data in search of empirical correlation/generalisations, analysis of experimentaldata in the light of new theoretical hypothesis, collection of observational information and corre-lation. Problems that may be resolved by scientific research are contextually defined at any given


There is a "Real" world

characterized

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existence/occurrence of

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described

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obtained by

correlating

measured quantities

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for finding ways of

describing or explaining

"which may "explain" a fact/event or phenomenby its

inclusion in a class of event/facts/phenomena

observed

but

do not say anything about facts

events/phenomena

not pertaining to the class

Scientific

concepts

objects/processes

parameters and

Fig. 3

time and change over the course of time along with the evolution of shared knowledge and withchanges in the social context.

5. The social aspect of scientific workI have tried to present an epistemological framework for the role of laboratory practice in the expe-rimental sciences through three different perspectives which, while sharing some common points,focus on different aspects. In all three perspectives experiments and theories are strongly correla-


The universe of shared knowledge

Theories/Models/Data, techology

guide the

development of

Ideas

which may focus

Problems

to be solved by

Theoretical research research

Experimental

Experimental outcomes

to be confronted

with which may stimulate

which

produces which may open the way to new

technology

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shared knowledge

models, data,

-theories may be rejected

or their domain of validity

may be restricted

-data may be reinterpreted

or their validity questioned

as for as their accuracy is

concerned

-technology if functioning

enters the real world of

facts/events/phenomena Fig. 4

of the universe of

about theories

technology

Fig. 4

ted and both contribute, on more or less equal grounds, to the definition of the knowledge sharedby a scientific community.One point remains to be discussed and it concerns the ways by which a community of scientists rea-ches agreement.The scientific community judges the validity of research (theoretical or experimental) with argu-ments that can be both theoretical and experimental. Its aim is to reach an “objective” judgement,devoid of personal biases or opinions; however a “subjective” component cannot be completelyavoided. Biases and systems of beliefs may be particularly relevant when experts have to makedecisions on the financial support of research projects.Instruments to be used in the confrontation of ideas are scientific reviews, conferences and symposia(where new ideas may be debated and eventually accepted or rejected), scientific societies (which maystimulate new lines of research), workshops (for in-depth discussion among experts of a particularfield).The terms of confrontation are, on one side, the natural world of phenomena (which includes theexperimental outcomes of laboratory practice) and on the other, the different points of view of diffe-rent scientists and the exercising of accurate criticism. Thus, the development of science requires bothan “inter-subjective” agreement on rules, criteria, models of explanation, background knowledge, etc.,as well as competition between different points of view - via scientific debates and criticism.Another comment concerns social interaction inside an experimental group. In the past, aside fromsome individual scientists who contributed to the development of theories while performing experi-ments, experimental work was organised in small groups of people who shared all the competencesrequired for the job.The small group organisation is still present in today science but often the collabo-ration in the group is obtained with a separation of competences according to personal expertise. Thisis particularly true in the called “big science” in which a group may include several hundred persons.The social abilities of an experimental scientist are therefore needed on two sides: the ability to co-operate in common and the ability to stand up to confrontation with peers.One final comment on the social aspects, which I have tried to present from a somewhat idealisedpoint of view, concerns the ethical aspects related to any kind of social interaction. I will not treatthe problem in detail but let me conclude with the simple consideration that scientists are normalmen and women who, while committed to the rules of the scientific game, have their personalsystem of beliefs and behavioural rules which sometimes may produce biases and even misconduct.

ReferencesBandiera M., Dupré F., Tarsitani C., Torracca E., Seré M.G., Vicentini M– “Teachers’ images of Science and Labwork

Working Paper 5” – Project Labwork in Science Education (European Commission), (1998).Cini M., Un paradiso perduto, Milano, Feltrinelli, (1994).De Gennes O.G, Les objects fragiles, Paris, Plan., (1994).Deutsch D, The fabric of reality, London, Penguin., (1997).Gallison P., Image and logic, The Univ. Of Chicago Press, Chicago and London, (1997).Gellmann M., The quark and the jaguar, New York, Freeman., (1994).Giere R.N, Explaining Science. A cognitive approach, Chicago, London, The Univ. Of Chicago Press, (1988).Interchange, Special Issue on History and Philosophy of Science and Science Education, vol.28, n.2-3, (1997).Jacob F., La souris, la mouche et l’homme”, Paris, Edition Odile Jacob, (1997).Journal of Research in Science Teaching, – Special Issue on Epistemological and Ontological Underpinnings in Science

Education, 35, n°2, (1998).Kelly K., The third culture, Science, 279, (1998), 992-3.Latour B., Woolgar S., –“ Laboratory life”, Beverly Hills, Calif. Sage, (1979).Latour B., From the world of science to the world of research?, Science, 280, (1998), 208-209.Levy-Leblond J.M., 1996a – “La pierre de touche”, Paris, Gallimard 1996b – “Aux contraires”, Paris, GallimardMatthews M.R, Science Teaching. The role of history and philosophy of Science, New York, Routledge., (1994)Science and Education,– Special issues about the nature of science and science education, 6 and.7, (1997,1998).Vicentini M., - The nature of science: A didactical issue, Invited paper at the Conference History and Philosophy of

Science for Education, Como, Italy, (1999).Vicentini M., What is it “doing science”?, Contrasting views of scientists, philosophers and science educators in:

Practical work in Science Education, (Ed. K. Nielsen and A.C. Paulsen), Royal Danish School of Education Studies,Copenhagen, (1999), 9-17

e-mail [email protected]



IS FORMAL THINKING HELPFUL IN EVERYDAY SITUATIONS?

Seta Oblak, Faculty of Education, University Ljubljana, Slovenia

Formal thinking in physics usually means mathematical language. In physics teaching, especially inupper secondary school, this is a frame of formulas filled with a limited series of more or lessartificial examples. For most students, it has nothing to do with everyday life, it is just schoolknowledge triggered by a certain sort of questions that can be put only in physics lessons.On the other hand, the aims in science, especially in physics, go up to a very high level: studentsshould not only observe and describe their observations, measure, record and interpret data, theyshould also draw conclusions, employ a variety of sources of information, make generalisations etc.These abilities would be useful in all kinds of situations and desirable for any profession.Yet even to observe and decribe the observations is not as self-evident as one would think. Lastyear, one of our teachers reported [1] that she gave her students (age 17-18) a home experiment:they should observe total reflection in a glass of water and describe their observations. A quarterof students just memorised what they heard and saw in school without doing anything on theirown. The majority reproduced the school experiment. The descriptions were superficial and evenincorrect and in discussion only, students realised that the experiment could not be performedfollowing their reports. Only two out of 62 students described their observations sistematically andwith their own words.In traditional physics teaching, expressing with words is not regarded as important. Mathematicalyformulated laws are much simpler and much more exact. It is not even clear how certain lawsshould be expressed in everyday words, since physics requires a special and very strict language.Question like “In what way is the momentum of a body affected by the resultant force acted onit?” cannot be put in real life. In 1969, Eric Rodgers in his Oersted Medal Address [2] proposedthat teachers should give students the formal wording of Newton’s laws and ask them for acolloquial explanation. He said that they would be surprised at the difference of their answers.An analysis of external examinations in Slovenia in 1998 has shown that students (age 19) whochoose physics for matura - about 15% - are successful in solving problems with mathematicaltools, yet they are hardy able to answer simple questions. For example, they determined the period,the gravitational force and the acceleration of an artificial satellite orbiting around the earth, yetthe answers to the last question: Why does the satellite not fall to the Earth? proved that there isa great void between formal knowledge and understanding of phenomena.The answers were analysed on 42% of all examination papers (500 out of 1196 who had chosenthis structured question). About a quarter of answers (27%) the examiners found more or lesssatisfying. Only a few answers were similar to Newton’s original explanation:• The satellite falls all the time when it revolves, yet it never reaches the Earth.Many answers used forces as explanation:• The attraction force of the Earth acts as a centripetal force and changes the direction of velocity

of the satellite all the time, that is why it moves in a circle. It does not fall on the Earth because thevelocity is just big enough that the satellite does not escape.

• The satellite does not fall to the Earth because the force with which the Earth acts on the satelliteis rectangular to its velocity which is big enough, and therefore this force changes only thedirection of the velocity.

Many other answers were expressed more loosely and did not show if students really understoodthe phenomenon. Yet they were taken as correct since an open question does not strictly definehow the answer must be.• Because the satellite has just the right velocity.• Beacuse the satellite revolves with a velocity between the 1st and 2nd cosmic velocity.If the answers were long, there was more chance to make mistakes:• The satellite must fulfill two conditions for not falling to the Earth. It has to have a big enough

orbiting velocity and it has to be high enough. It does not fall to the Earth because of attraction ofother celestial bodies and because of its velocity.

This is a “dead-mouse-answer”, according to Eric Rogers. The student has shown that he does notunderstand, and can get no marks for his answer.More than a quarter of answers (28%) explained the phenomenon with two equal and oppositeforces - gravitational and centrifugal (or “centripetal”). In these answers, students did notdistinguish between centripetal or centrifugal force. Their way of thinking can be seen clearly: fromthe mathematical formulation Fc = Fg, they deduced that forces were opposite and equal and thenet force was zero.• Because it revolves around the Earth, and therefore a centripetal or centrifugal force appears

which pulls the satellite away.• Because the radial force on the satellite is equal but opposite to the attraction force of the Earth

on the satellite.• Because the centrifugal force which pulls it away from the Earth is equal to the centripetal force

which pulls it towards the Earth.• The velocity is big enough that the gravitational and centrifugal force are equal and therefore the

satellite continues in its state of rest or of uniform speed.The last answer shows that the student was aware that in case of two equal and opposite forces, thenet force should be zero and the satellite should not change its velocity; yet he forgot aboutchanging the direction of velocity.About 10% of students gave undefined answers:• Because the attraction force of the Earth is too small at this distance.• Because the gravitational acceleration is too small.3% quoted Newton’s 3rd. law:• Earth acts on the satellite with attractive force, yet at the same time the satellite acts on the Earth

with equal and opposite force, so it cannot fall to the Earth.A little less then one quarter of answers (22%) were meaningless. It is perplexing to see howexpert language can be used without any understanding.• Because of the tangential acceleration which has the same direction as the velocity.• Because the sum of the vector of velocity and the gravitational force has horizontal direction and

does not point downwards.• Because the acceleration of the satellite is directed away from the Earth and gravitaion towards the

Earth. The satellite revolves with such a velocity that the accelerations cancel each other. We saythat it revolves on its orbit.

• Because the sum of the attractive force and the force which pushes the satellite is exactly equal tothe force which rotates the satellite around the Earth.

• Force is used for radial acceleration and not for approaching.• Because they keep it in space, and there it stays. The Earth is very far from the satellite, so it cannot

attract it with its gravitational force. Gravitational acceleration in the space is smaller than on theEarth.

• Because it has an inertial force which keeps it on the orbit.• Because of the distance it has from the Earth. If it were very close the Earth would simply attract

it and it would hit it – yet because of the distance it is saved from this fate.And 10% of students did not answer this question at all.From these answers, it is evident that the reverse process - from formal thinking back to everydaylife – has been neglected.That physics teaching does not give the desired results has been known for at least four decades.During that time, many attempts were made to change the didactic approach to teaching and shiftthe importance from formulas to understanding (PSSC, Nuffield, problem-centered learning,constructivist method, context-bound teaching etc). In Nuffield project [3], teams of interestedteachers and didactic experts prepared textbooks, teacher manuals, equipment, questions books


etc. Much equipment has been developed which is still in use today: big demonstration instruments,ticker-timer, air track, ripple tank, electronic tubes etc. The importance of experimental work doneby students was realized and kits for practical work were developed. In England, practical work haseven been made part of external examinations on A-level.Why the situation in these forty years has not changed? Obviously, the novelties do not reach thebroad population. After a time, when teachers who have iniciated the project stop working, theproject as a whole dies. Although new approaches and methods have proved to be successful, theyhave been applied in limited cases and by enthusiastic teachers only. In Slovenia, a modern physicscourse for the first year of upper secondary school was prepared by a group of university and uppersecondary school teachers, with very little mathematics, with good equipment for demonstrationand laboratory work, with teachers’ experimental guide, textbook, book of questions etc. Schoolswere equipped for demonstrations and practical work.Yet this course was abolished partly becauseof political reasons, but also because of dissatisfaction of traditional teachers and especially oftechnical faculties, shortage of qualified physics teachers, experimental work as an extra burden,unchanged curriculum in the next stages and especially unchanged examinations at the end ofupper secondary school etc.Traditional teaching is, for the majority of teachers and students, still the easiest way to the goalwhich is a certificate. There is a general agreement among the universities about the content ofphysics in upper secondary school which is almost impossible to break. The number of students hasgone up, and since teachers have to go through the syllabus, there is less and less time for discussionwhich is the only way to understanding. In external examinations, it is very difficult to evaluateanswers to open questions in a comparable way, and solving of mathematical problems prevails.The importance of questions for examining students’ knowledge was first realized by Nuffieldproject. When preparing the new course in 1966, examination questions were prepared alongsideso that they conveyed the right message: it is important to understand. Nuffield questions bookshave lost nothing of their actuality.In Slovenia, before external matura was introduced there were entrance exams on differentfaculties where questions were prepared without deeper consideration. A knowledge of facts,formulas and of solving mathematical problems was enough, and feedback of these exams onphysics teaching in gymnasium was bad. Now, examination board for matura is aware that its wayof evaluating knowledge is of crucial importance for physics teaching, and in examination papers,a collection of formulas is included so that students do not have to memorize. Also, open questionsare put to candidates although evaluating such answers is quite a problem. But since there are inaverage only about 2000 candidates, a group of about 30 teachers can do the marking in two days,working together and discussing current problems with chief examiner and members ofexamination board.On the other hand, matura in Slovenia has also bad effects: physics as choice subject for maturainfluences very much the teaching in the first three years since it is the only external evaluation ofteachers’ work. In gymnasium, classical physics is now overemphasized and modern physics is leftfor the fourth year and the 15% only.Apparently, a change in physics teaching is not a problem of physics didactics only, but also apsychological and sociological problem. In Slovenia, early science was much easier to implementbecause there was no tradition to be changed. In upper secondary school, mathematical and formalapproach has great tradition and changing it means changing teachers’ attitude.To promote understanding in physics teaching, discussion among teachers and students is needed.Qualitative and semiquantitative reasoning is important on upper stage also. Demonstrationexperiments and practical work have now a long tradition, yet for practical work too, discussion isnecessary if the students should profit from it. Otherwise, they just “measure some points and plotthem on the graph” (http://quark.physics.uwo.ca/~harwood/humor5.html)Today, there are enormous possibilities to make teaching more effective, interesting and up-to-date: computer-supported laboratory, interactive simulations, interactive textbooks, multimedia etc.


– it is hardly possible to use all the new materials in the prescribed number of lessons. So selectionis very important. Yet to select, one must be an expert in all fields.The most important condition for good physics teaching is good teacher education. In manycountries, during undergraduate study physics is promoted and didactis is not taken as important.After some years of work, teachers start to feel the need for more didactic knowledge and it is verygood if they have the possibility to study further. It is important that teachers follow thedevelopment of didactics throughout their professional carrier. This is not always the case, contraryto physics where it would be impossible to work without the knowledge of what is currently goingon in the world. Yet all this is of no help if there is a shortage of physics teachers in the country.State institutions can be of great help if advisers are stimulated to study and cooperate indevelopmental work. They have an overview of what is really taught, and access to teachers andschools all over the country. In England, HMI had an important role in Nuffield project. Trial andimplementation of new courses can best be organised by advisers who cooperate in the project. InNetherland, Institute for development of curricula (SLO) functions as an intermediate betweenuniversities and schools and an organiser of projects. In Slovenia, the Board of Educationcooperated with university in Tempus projects in early science for many years. Science and primaryschool advisers worked at developing seminar topics and later organised seminars in schools indifferent parts of the country. In this way, the novelties were implemented on big scale, and whenthe curricula were officially reformed, the new curriculum of primary science has already beenbrought to life in schools by interested teachers.In in-service training, to promote understanding in physics teaching, discussion is needed andadvantage and disadvantage of using everyday language has to be analysed. All this can best bedone in workshops for “shredding” examination questions prepared by participants.

References[1] Vida Kariž Merhar: Total reflection in a glass of water, Fizika v šoli 7, (2001).[2] E. Rogers: Examinations – powerful agents for good or ill in teaching, AJP, (1969).[3] Wonder and Delight, Essays in Science Education, Editors Brenda Jennison & Jon Ogborn, IOP, (1994).

THE FORMAL REASONING OF QUANTUM MECHANICS: CAN WE MAKE ITCONCRETE? SHOULD WE?

Dean Zollman, Kansas State University, USA

The teaching and learning of quantum mechanics is very frequently postponed until relatively latein a student’s academic career. In U.S. universities students typically receive a quick introductionto some aspects of one-dimensional quantum mechanics from the end of the second or beginningof the third year of their university studies and then do not study quantum mechanics in any depthuntil the fourth year. Thus, the major concepts which have driven much of the development ofphysics and of modern technology during the 20th century are delayed until the end of a physicist’sacademic career and are frequently not studied by other students at any time during their careers.One reason for this delay is the rather abstract nature of quantum mechanics itself. We can easilyargue that, for the way in which quantum mechanics is traditionally taught, students need to havegenerally developed their formal reasoning skills. For example, formal operations, in the Piagetiansense, include hypothetical and deductive reasoning, abstract thought, use of symbolicrepresentation, and the use of transformations. Quantum mechanics is a hypothetical system forunderstanding very small objects. It relies heavily on the use of symbolic representations anddeduction to apply quantum mechanics to a variety of situations. Symmetry arguments, andtherefore transformations, are a significant part of many presentations of quantum mechanics.Therefore overall, we can assume that the traditional mode in which quantum mechanics is taughtis very abstract and requires rather sophisticated formal operational procedures.


Significant research dating back to the 1970s has shown that many university students have not yetdeveloped formal operations (McKinnon & Renner 1971). In fact, the traditional way of teachingclassical physics is a significant mismatch for many of these concrete operational students. Thus, itis not surprising that many physicists conclude that quantum mechanics is not understandable bystudents who are not studying physics very carefully and are in their third or fourth year at auniversity. Many people have concluded that learning quantum mechanics at a lower level is notpossible and thus should not even be attempted (Arons 1990). They argue that the students willonly be able to memorize isolated facts and repeat things without true understanding. Thus, thestudents are better served if we spend all of our time on classical physics where concrete learningexperiences can more easily be constructed rather than attempting to teach them something thatthey could learn only with great difficulty, if at all.

1. Why teach quantum mechanics to non-physicists?The discussion about the abstract nature of the normal presentations of quantum mechanics seemsrather valid. The simplest response to these conclusions is to avoid teaching this topic at any butthe most advanced levels. However, some arguments favor attempting to find ways to teach thetopic to students who have not yet reached full formal operations. For example, quantummechanics was the most important development in 20th Century physics, and it has dominatedphysics and technology for well over a half a century. Thus, at the beginning of the 21st Century itis time to allow all interested people access to these ideas. Further, many experts predict thatwithin the next 10 years miniaturization of electronics will reach the quantum mechanics limit. Itwould be nice if people who are trying to take the next step – development or business –understood what that meant. Finally, many other very complex and abstract processes – the electionof an American President, for example – fill our lives. Perhaps an appreciation of quantum physicscan help us understand the role of measurement in these events.

2. Making quantum mechanics concreteOur group at Kansas State University has been convinced that we should make the teaching ofquantum mechanics more concrete than it normally is. We have worked to develop both thepedagogical style and the presentation of content so that students who are still developing their formaloperational skills can appreciate and understand some of the features of contemporary quantumphysics. For a pedagogical strategy we have adopted the basic Learning Cycle.The Learning Cycle wasdeveloped by Robert Karplus about 30 years ago and has been successfully used in almost all levels ofteaching (Karplus 1977, Karplus et al 1975, Zollman 1990). While many people have adapted orchanged the basic Learning Cycle, we find that the one that Karplus originally introduced works quitewell for our teaching situation. Each Learning Cycle begins with an Exploration where studentscomplete activities prior to the introduction of a new concept. These activities prepare them for theintroduction of new concepts which can explain their observations during the Exploration. Theconcept introduction provides the new principles on which the students will build and frequentlyincludes the development of models that can help explain the observations. Once the students have thenew concepts and models they complete an Application in which they apply the newly learnedinformation to situations that are similar but not identical to the ones they have already studied.With the addition of model building in some cases our Learning Cycle comes very close to theModeling Cycle that has been developed by Hestenes and his co-workers (Wells et al 1995). Wealso emphasize collaboration among students as they are learning.This cooperative effort is also animportant part of the Modeling Cycle.We have created Learning Cycles for a variety of different types of students. Our basic approach isthat all students, even those who are more advanced in their reasoning skills and academic careers,can profit from a more concrete or intuitive approach to the abstract ideas of quantum physics. Webegan by developing materials for secondary school students and those university students whowould complete a physics course but not study physics beyond one year. As these materials weredeveloped, they were used by faculty who were teaching higher level courses to physics students.


We then created a set of materials for that group and have recently expanded to include materialsspecifically aimed at medical students and physics students in the last year of their undergraduateuniversity careers. Each set of materials has a somewhat different approach and a different level ofmathematical sophistication. However, all of them follow a basic Learning Cycle and focus onconcrete visualization rather than abstract mathematical deduction.

3. Device orientation One way to make abstract ideas concrete is to connect the concept directly to something in thestudents’ experiences. In one way such a connection is easy for quantum physics. Almost everycontemporary technological device could not exist if a designer of that device did not have anunderstanding of quantum science. At the same time the connection between quantum science andsomething as ubiquitous as the television remote control is not immediately obvious. Thus, we havecombined hands-on experiences, visualizations, and traditional instruction to help the students seethese connections.Students should recognize these objects and see them in their everyday life. Light emitting diodes(LEDs), for example, are everywhere. Although many students do not know the name, they haveseen them in their computers, remote controls, etc. By examining the properties of LEDs thestudents learn that LEDs are different from other light sources. Then, with the help of computervisualizations they understand how the light emitting properties are related to the quantization ofenergy in atoms.We occasionally use devices that students may have heard about and may have seen pictures of, butthey have probably not encountered. The scanning tunneling microscope is the best example. Wedo not expect students to use a scanning tunneling microscope although it is possible for studentsto build one. But most students will not be able to build such a device. So, in this case, we use acombination of a simulation and an interactive program (Rebello et al 1997).We also use a variety of solid light sources. Infrared detector cards are a rather interesting example.They are a fairly recent development – at least fairly recent for inexpensive versions.TV repair peopleneed to know if a television remote control is emitting infrared. How can they do that? It is rathersimple if they have a video camera. The camera responds to IR and shows a bright spot where the IRis emitted. So, every TV repairperson needs a video camera, and he/she can find out whether there islight coming out of the remote control. But that is rather expensive. Another way to detect IR is withrattlesnakes, which are sensitive to infrared. So, every TV repairperson could have a rattlesnake. Butthat is rather expensive in a different way. However, one can buy a little card that responds to IR byemitting visible light. Thus, it absorbs low energy light and emits higher energy light.The Star Trek Transporter is also a quantum mechanical device. If one reads the Star Trek Users’Manual, one finds that the Transporter has a component called a Heisenberg Compensator(Sternbach 1991). When one of the writers for Star Trek was asked, “How does the HeisenbergCompensator work,” he responded,“Very well” (Time 1994). Because Werner Heisenberg is one ofthe founders of quantum science, we must assume that this Compensator is related to hisUncertainty Principle. In one of our units we ask students to address the fantasy device in termsof basic quantum mechanics principles. These and several other devices are introduced to students.In each case we show how the devices are related to quantum mechanics. Further, the studentslearn how the devices work at the atomic level.

4. Using visualization & model buildingIn the Visual Quantum Mechanics instructional materials we provide as concrete a description aspossible about how we know about atoms and how we use that knowledge to build models. One ofour learning units focuses on spectroscopy and its role as evidence for energy quantization. Thisunit begin with a study of the light emitting diode (LED). We can convince students that the LEDis related to contemporary physics because they can read statements such as A genuine White LightSuper Bright LED … utilizes an advanced Quantum Well technology…” (Electronics 2000) Afterobserving how different colors of LEDs respond to changes in voltage and observing the spectra


from both LEDs and gas spectral tubes, the students are ready to build an energy level model ofthe atom. A visualization program, Spectroscopy Lab Suite, provides a set of simulatedexperiments, similar to the ones that they have just done, that are coupled to building energymodels of atoms. The activities include the emission and absorption of light by gases, the emissionby solids – particularly LEDs, several types of lasers, and common emission processes such asfluorescence and phosphorescence (Rebello et al 1998).Students generally use the Gas Emission program after observing the spectra emitted by gas dischargetubes.The design of this component was motivated by the results from a preliminary field test.We foundthat students related the spectral lines for a gas to the discrete energy levels, rather than transitionsbetween these energy levels. The Emission module was created to alleviate this misconception.Studentscreate a trial spectrum for a gas by manipulating the energy level diagram of a gas, indicating thetransitions on it. They compare their trial spectrum with the real spectrum for the gas.The component screen for the Emission module (Figure 1) shows an array of simulated gas lampsand a power supply on the left. To create the feel of the real experiment that the students havealready completed, they must drag one of the gas lamps into the power supply. This action causesthe lamp to emit light and its spectrum appears at the top of the screen. There are five known gases

(hydrogen, helium, neon, lithiumand mercury) available to thestudent, and an unknown gas. Inthe case of the unknown gas, thestudent can change the spectrallines to create any hypotheticalspectrum.A scale which represents energy inthe atom is displayed on the lowerright side of the screen.The students’task is to manipulate energy levelsand transitions and reproduce thespectrum of the gas. This procedureaddresses our research aboutstudents’ understanding of atoms.Students can move the energy levelsand observe the correspondingchanges in the energy of the spectralline. To make the spectral line in thetrial spectrum coincide with one inthe real spectrum, the students mustcreate a transition between two

energy levels whose difference in energies is equal to the energy of the emitted light.By using this program students learn that the energy of the emitted light is equal to the change in energywithin the atom. More importantly they see that only certain discrete energy levels are needed toexplain the observed spectrum. From knowledge of energy conservation and the data presented bythe spectrum of a gas, students can discover that energy states in atoms are quantized. This criticaldiscovery of 20th Century physics follows from empirical results and an explanation in terms ofenergy – no knowledge of wave functions or the Bohr Atom is needed.The Gas Lamps Emission component does not enable students to determine the exact energylevels of a given gas, but rather construct a model based on energy differences. When thiscomponent is used in a classroom environment with students working in small groups, differentgroups of students may arrive at different energy levels within the models to explain the spectrumof the same gas. Rather than tell some students that they are wrong, a teacher can use this situationto discuss the nature of scientific models and limitations based on the models by available data. In


Figure 1. The emission program from Spectroscopy Lab Suite. Studentsdrag the light source on the right to the power supply. Then, they build anenergy level model of the atom to match the observed spectrum.

creating their models the students had available to them only the observed spectrum and theconservation of energy. With no further information they could create several different sets ofenergy levels which match the data. (See Figure 2.) By having students compare their results withothers in the class, they can begin to understand how more than one solution to a problem can be“right” when it is based on limited information. However, while they cannot create a completepicture of the atom, all students agree that discrete energy levels are necessary.

Figure 2. In one class students will frequently create two variations of the energy level model. Fig. 2a shows all of thetransitions beginning at the same initial state while Fig. 2b indicates that all transitions end on the same state.

Other modeling in Spectroscopy Lab Suite is similarly connected to experiments. For example,students create energy band and gap models for the observation that LEDs which emit differentcolors of light have different threshold voltages. They also interpret the behavior of electrons inconduction, valence and impurity bands to explain why a glow-in-the-dark toothbrush stopsglowing if it is placed in liquid nitrogen. In all cases the energy model building is connected directlyto an observation that the students can make.In building the instruction that led to these programs, we expected the students to be interactingwith each other and with the teacher. For example, the observation that different energy levels cangive the same result is effective because two groups of students obtain different but equally correctanswers. The teacher and the students’ peers are, thus, important to our teaching-learning process.

5. Conceptual approaches to wave functionsStudents with concrete reasoning skills can move beyond spectra and energy models of the atomto learning activities involving wave functions. Developing experiments with real equipment toexplore and apply wave functions is rather difficult. However, we can create visualizations whichhelp the students explore. For the students who are not science or engineering majors we avoid themathematics of quantum mechanics and rely heavily on visualization in which the studentsmanipulate variables, and the computer solves Schrödinger’s Equation. The students must theninterpret the results in terms of the conceptual knowledge.For both the secondary students and beginning physics students we begin the study of the wavenature of matter with an experimental observation – electrons can behave as waves. After thestudents have discussed how interference patterns indicate wave behavior and have observed theinterference of light, we turn their attention to electrons. They can observe a real experiment ifthe equipment is available, use video simulations (Kirstein 1999) or see pictures in books. Toinvestigate the wave nature of electrons further, the students use a simulation program whichenables them to control variables in electron, two-slit experiments. Using results such as thoseshown in Figure 3, the students can discover a qualitative relation between the wavelength of theelectron and its energy. They compare the changes in the pattern for changes in energy ofelectrons with similar changes when one observes the interference of light at different


(a) (b)

wavelength. They can easily conclude that the wavelength of electrons decreases as the energyincreases.An issue that students will sometimes raise is the relation between the particle’s charge and itswave behavior. Their reasoning is, “Diffraction is the spreading out of a wave. Like charges repel.In a beam the charge must cause the electrons to spread out.”We test this hypothesis by comparingthe diffraction pattern of simulated proton and neutron, two-slit experiments. The patterns areidentical, so charge must not be a factor.After a few more experiments, including a variation in mass, we introduce the deBroglie equation.We have not actually derived the equation experimentally but have given a feasibility argument forit. While this approach is not historically accurate, it seems to provide students with a somewhatmore concrete introduction to an abstract concept than stating deBroglie’s hypothesis and thenusing interference experiments to verify it.To connect the matter waves to probability we return to the experiment illustrated in Figure 3. Settingthe particle flux to a few per second the students watch the pattern develop.After a few particles havehit the screen, as shown on the left side of Figure 3, we have the students stop the “experiment.” Now,we ask them to predict where the next electron coming from our electron gun will appear on thescreen. The students very quickly fall into discussing the location in terms of probability. They canindicate some location where the electron will rather definitely not appear and several where it is verylikely to appear. However, they cannot give a definitive answer. Thus, we can introduce the wavefunction and its probabilistic interpretation based on the students’ experience with indeterminacy.With wave functions we emphasize conceptual understanding by having students manipulategraphic images in accordance with their knowledge. For example, we ask the students at all levelsto sketch wave functions qualitatively. Following procedures that appeared in French and Taylor(French & Taylor 1978), some sketching is done with paper and pencil. However, we find thatstudents can easily be very inexact with paper and pencil, and sometimes exactness is needed. So,we have created a program that does very little except that it allows the students to vary the wavefunction and match boundary conditions. Figure 4 shows a screen capture that would be created bystudents as they are preparing to study quantum tunneling.We have discovered some interesting ways in which the students use this program. First, if we tellthe students that the wave function is smooth, they will make it smooth to many derivatives. Theidea that two functions just stick together does not occur to them. Second, we use of the word“decreasing” for exponential decay. When we use “decay,” the students immediately think of


Figure 3. A simulated electroninterference experiment. As theenergy increases, the distancebetween minima decreases. Bycomparing this behavior with thatof light the students conclude thatthe wavelength decreases as theenergy increases.

radioactive decay. They interpret that to mean that the electrons are radioactively decaying in theregion where the total energy is less than the potential energy. So, we use the phrase “decreasingwave function.”The third and fourth year university physics students still use a basic Learning Cycle style approach.However, the Explorations require a little bit of formal operations. These students are still asked tomatch boundary conditions graphically. However, as shown in Figure 5, the Wave Function Sketcherprogram for these students includes the common mathematical language of physicists.In addition, the students are expected to work with both the wave function and its derivative whenthey are matching the boundary conditions.Using the Wave Function Sketcher for the advanced students provides an intuitive, and somewhatconcrete, approach to understanding the process of matching boundary conditions. After thestudents have completed these activities they are ready to use the mathematics involved inboundary value problems to complete the solutions for wave functions in various one-dimensionalsituations. While this type of Learning Cycle primarily focuses on formal operations, it doesprovide visualizations that are more concrete than typical mathematical symbols and, we hope,helps the students build their intuition about wave functions.

6. Does visual quantum mechanics work?The units have been used in secondary schools and in universities throughout the U.S. and in a fewother places. Actually, we do not know all the places that it is being used because, during the fieldtest phase, all the material was on the web and people download it. We have given materials topeople in Southeast Asia and throughout various parts of Europe as well as the U.S. Most of theoriginal units have now been translated into Hebrew. (Arieli 2001) Thus, the materials, except forthe new Advanced Visual Quantum Mechanics, have been thoroughly field-tested.Most of our reports, however, have come from the U.S.Approximately 175 different teachers in 160different schools have used the materials in classes and reported results back to us. Students’ attitudestoward these materials are very positive.They frequently make comments like,“I really like this betterthan our regular physics. Can we keep doing it?” (We don’t tell the instructors that.) Our staff hasobserved teachers using the materials in a variety of different schools. The students interact with thematerials and each other; and they seem to be learning. Most of the teachers also have positiveattitudes; a few do not. We certainly have the problem that many teachers in the U.S. do not have avery strong background in quantum mechanics. Even though we are approaching quantum mechanicsin a much different way than it is normally taught, some teachers still feel uncomfortable. Building theteachers’ confidence is very important. We are working on that aspect now by building a Web-basedcourse for secondary science teachers. (Connect to http://kzollman1.phys.ksu.edu.)Student learning was also rather good. During our observations of the teaching, we noticed that thehands-on component for both the real experiments and the visualizations was important. Someteachers decided that it was too much trouble to have the students work in a hands-on mode withall of these programs. So, they just demonstrated the programs to the students. In these cases


Figure 4. A screen capture from Wave Function Sketcher. This elementary version of the program is aimed at highschool students and non-science university students.

learning went down; attitudes went down; everything went down. Hands-on activities make adifference. Of course we should not be surprised because we built the material for the students touse; not for the teacher to talk about.With the second year physics students we have done some testing, but it has not been as extensive asfor the secondary students. These units are somewhat shorter and built to be in one to two-hour unitswithin a traditional “modern physics” course. We have found that the students’ attitudes are generallyvery positive toward this type of learning and material. However, occasionally a student would feel thathe or she was not getting all of the material that he or she would need for advanced level courses. Someof the first students with whom we tested the material are now taking a fourth-year quantummechanics course. We will be investigating with them how well the materials that they learned in ourcourse are serving them in the more advanced course. In terms of the student questions we found thatthese materials motivated students to ask very high-level conceptual questions. Rather than most ofthe questions about wave functions being concerned with the procedural efforts of manipulatingequations, the students were focused on what the wave function means and how it can be interpreted.We asked the students questions on examinations which were very similar to those that they might findin a higher level course. In general the performance on such questions was really quite good. So, overalleven though we have not tested the materials as carefully as the materials for lower level students, wefeel quite confident that the materials are teaching well and are providing conceptual understandingthrough concrete hands-on and visualized activities.We do not at this time have similar information about the materials for the fourth year students.


Figure 5. A screen capture from the advanced version of Wave Function Sketcher. This program is for theuniversity physics students.

These materials are very new and have yet to have a significant amount of classroom testsperformed on them.

7. ConclusionsConcerning the questions which we posed in the title of this talk, we believe that the VisualQuantum Mechanics project has shown that we can make quantum mechanics accessible tostudents who are at the concrete operational stage. Further, we believe that we must provide waysto allow students who are not formally operational to begin to understand some of the features ofthe most important scientific advances during the 20th century. Based on a large number of fieldtests and a rather careful evaluation of student attitudes and learning, we have concluded that theVisual Quantum Mechanics materials have been successful in teaching some abstract concepts tostudents who have limited science and mathematics background and who probably use concrete ortransitional operations. Our materials are also successful in teaching the conceptual ideas ofquantum mechanics to students who have stronger science and engineering backgrounds byproviding them with some concrete experiences. The combination of hands-on activities, pencil-and-paper exercises, and interactive computer visualizations seem to work well in a classroomenvironment where student-student and student-teacher interactions are taking place.Thus, we feelthat we have built a foundation for providing instruction in the most important aspects of 20th

Century physics to a broad range of 21st Century students.

AcknowledgmentsThe work described here has been supported primarily by the U.S. National Science Foundation.Additional funding has come from the U.S. Department of Education, the Eisenhower ProfessionalDevelopment Program and the Howard Hughes Medical Institute.The development of the originalVisual Quantum Mechanics teaching materials profited from significant work by N. Sanjay Rebello,Lawrence Escalada, and Michael Thoresen. Kirsten Hogg and Lei Bao were instrumental in thedevelopment of materials for second year physics students, while Waldemar Axmann is the primaryauthor and programmer for Advanced Visual Quantum Mechanics. Dr. Hogg has also been theprimary author of the Web-based materials while Kevin Zollman is the primary programmer forthe on-line course. Chandima Cumaranatunge programmed the visualizations described in thispaper. Rami Arieli has provided valuable feedback and suggestions for improvement while he hasbeen creating the Hebrew translation. We have worked with Manfred Euler, IPN – Kiel, andHartmut Wiesner, LMU – Munich, on some aspects of Visual Quantum Mechanics. We haveprofited greatly from input from undergraduate students and teachers at many other universitiesand high schools where the Visual Quantum Mechanics materials have been tested.

ReferencesArieli R., Visual Quantum Mechanics (Hebrew). Rehovot, Israel, Weizmann Institute of Science, (2001).Arons A., A Guide to Introductory Physics Teaching, New York, John Wiley & Sons, (1990).Dick Smith, Electronics, Flyer included with a white LED, Australia, (2000).French A,Taylor E., An Introduction to Quantum Physics, New York,W.W. Norton & Co., (1978).Karplus R., Science Teaching and the Development of Reasoning, Journal of Research in Science Teaching, 14, (1977), 169.Karplus R, Renner J, Fuller R, Collea F, Paldy L., Workshop on Physics Teaching and the Development of Reasoning.

Stony Brook: American Association of Physics Teachers, (1975).Kirstein J., Interaktive Bildschirmexperimente, Ph.D. thesis, Technical University, Berlin, (1999).McKinnon JW, Renner JW., Are colleges concerned about intellectual development?, American Journal of Physics,

39, (1971), 1047-52.Rebello NS, Cumaranatunge C, Escalada L, Zollman D., Simulating the spectra of light sources, Computers in

Phyisics, 12, (1998), 28-33.Rebello NS, Sushenko K, Zollman D., Learning the physics of the scanning tunnelling microscope using a computer

program, European Journal of Physics, 18, (1997), 456-61.Sternbach R., Star Trek : The Next Generation Technical Manual, New York, Pocket Books, (1991).Time, Reconfigure the Modulators! Time, (1994), 144.Wells M, Hestenes D, Swackhamer G., A Modeling Method for High School Physics Instruction, American Journal

of Physics, 63, (1995), 606-619.Zollman D., Learning Cycles in a Large Enrollment Class, The Physics Teacher, 28, (1990), 20-5.



JUMPING TOYS: A TOPIC FOR INTERPLAY BETWEEN THEORY ANDEXPERIMENTS

C. Ucke, Physics Department E20, Technical University of Munich (Germany)

I am going to talk about toys, especially about jumping toys – and animals.Probably all of you know this small toy. But to be sure, I will give everybodysuch a toy now. This is normally a dangerous idea during a talkbecause the members of the audience can play during the talk. But my idea is togive you information not only through hearing and seeing but also throughfeeling.The English trade name for the toy is ‚Springy Smiley Face‘.Everybody can take a sample or even two. There are about 200 toys.Furthermore, I would like to change the subtitle from “interplay between theoryand experiment” to “interplay between experiment and theory”. Theexperiment is for me the first and most important operation. Here is the contentof my talk. I hope to need not more than 35 minutes.

Probably all of you are familiar with this small animal, whether through your own experience or not.This animated picture comes from the Russian Zoological Institute in St. Petersburg, whichexplores all properties of the fleas (http://www.zin.ru/Animalia/Siphonaptera/index.htm)A flea jumps up to a height of about h = 0.5 m. It accelerates across a distance of about d = 2 mm.This leads to an acceleration of a = h·g/d = 0,5 m·g/0.002 m = 2500ms-2 = 250g (g = 10ms-2 = accelerationof gravity; uniform acceleration assumed). Since the jumping height of a flea is strongly influenced byair resistance and the acceleration is not uniform, it has, in reality, a greater initial acceleration. Thereare other animals with an even greater acceleration. Biologists have investigated very accurately thejumping mechanism of the flea. This is a topic which I cannot explain here.The jumps of fleas and other animals are difficult to measure and not very reproducible.This is a good reason for investigating a toy which gives better reproducible results.A man can only achieve up to 3g with a standing high jump.

jumping animals

© Zoological Institute in St. Petersburg/Russia


A small toy known as a jumping animal or pop-up makes some investigations easier and canilluminate the physics of jumping.The toy itself consists of a base, a spring, a suction cup and a head.You have to press the cup onto the base and thus load the spring. After some time the cup will loosenitself and the toy will jump up.Here you can see several different shapes of the toy. I started my own investigations with the toy on theleft. It was available in Germany one year ago but is out of production just now but will be available in afew months again.At the ‚Oktoberfest‘ in Munich I once got this strange item. The one on the rightside is what you have in your hands.You can build this toy easily by yourself.You need a compression spring with a length of about 6cmand a spring constant of about 500 Nm-1. In shops for household goods you can obtain simplesuction cups with a hook, as used in bathrooms or kitchens. The hook must be removed.The difference to the purchasable toy is that the base stays on the floor after the jump.

≈2m

m

height of the jump h ≈ 0.5m

acceleration distance d ≈ 2mm

acceleration a = h·g/d = 2500ms-2

≈ 250g

(uniform acceleration assumed; g = 10ms-2 = acceleration of gravity)

Pulex irritans(= human flea)

dhgaresults

advandghvFrom

/

22

=

==

jumping animals

man‘s acceleration ≤ 3g

purchasable items

simpleconstruction

jumping toys

ball pen

The use of appropiate ball pens is even simpler and gives almost the same properties. Probablymany of you remember experiments like this with ball pens in the school..If children have several of these toys then a lot of questions arise: Which toy will achieve thegreatest height? Which toy will start first?Children will start soon to discover more properties: What will happen if you remove the head? Andwhat, if you detach the base from the spring? Which height will the toy achieve jumping upside down?And you can ask them questions: How much force do you need to compress the spring? The headwill undergo an acceleration: When will the acceleration be at its maximum? How much time willthe starting process of the toy need? Which weight must be attached on the head of the toy so thatit will not jump at all? What will happen if you attach the base to the ground?And more questions ………I will now talk about experiments with this toy and not with the one you have because I made allexperiments with this toy.You can do these experiments on your own with your toy with other parameters.


jumping toys

Which toy will achieve the greatest height?

Which toy will start first?

What will happen if you remove the head?

And what if you detach the spring from the base?

Which height will the toy achieve jumping upside down?

Questions:

How much force do you need to compress the spring?

Which weight must be attached on to the head of the toy so that it will not

jump at all?

How much time will the starting process of the toy need?

………………………………..

Measured height of the jump h = 1.2m (±10%)

=> Epot = mgh = 0.0145kg·10ms-2·1.2m = 0.17J

Compressing on a – kitchen - scale

F ≈ 19N (= 1.9kg; ±10%); d ≈ 3.2cm (±10%)

Spring stiffness c = F/d ≈ 590Nm-1

=> Espring = 0.5·c·d2 = 0.5·590Nm-1·0.0322m2 = 0.30J

((=> h = Espringr/mg = 2.1m))

jumping toy

Simple experiments and calculations:

The first simple experiment is to measure the jumping height and calculate the potential energy.Another simple experiment is to compress the toy on to a – kitchen - scale and to measure theweight and the compression distance.From that you can calculate the spring stiffness and the energy stored in the spring.As you can see, there is a great difference, which I will explain later.The spring stiffness of the toy you have in your hands is about 400Nm-1.The spring stiffness of the ball pen springs is about 200 to 300 Nm-1.These values are comparable to that one I have here.On a very simple level it is possible to calculate with Newton's second law the initial accelerationof the head and, as you can see, it is remarkably high.When you let the toy jump upside-down the acceleration of the base is even incredibly high.I add some experiments with that toy you have have in your hands.The first simple experiment is to measure the jumping height.


a = F/m1* – g ≈ 1900m/s2 = 190g

(m1* = head + suction cup + 1/3 spring = 0.00984kg;

g = 10m/s2 = acceleration of gravity)

the mass of the spring cannot be disregarded

jumping toy

Initial acceleration of the base a = F/m3* – g ≈ 525g !!!

(m3* = base + 1/3 spring = 0.00369kg)

The asterisk always means that the mass of the spring has to be considered, but I amgoing to omit it in future to avoid overloading the formulas

Initial acceleration of the head

Measured height of the jump h = 0.3m (±20%)

Compressing on a – kitchen - scale

F ≈ 7.8N (= 0.78kg; ±10%); d ≈ 1.5cm (±10%)

Spring stiffness c = F/d ≈ 520Nm-1

jumping toy/Springy Smiley Face

Simple experiments and calculations:

Initial acceleration of the head

a = F/m1* – g ≈ 1700m/s2 = 170g

(m1* = head + suction cup + 1/3 spring = 0.00464kg)

Mass Head mHead = 4,510gMass Spring mSpring = 0,391gMass Base mBase = 1,174g


111

11 308 −− −≈−≈⎟

⎠

⎞⎜⎝

⎛−−= kmhms

c

gmd

m

cv

jumping toy

energyenergyenergy

kineticpotentialspring

vm

c

gmdgm

c

gmcd

c 21

111

2

12

222+⎟

⎠

⎞⎜⎝

⎛ −=⎟⎠

⎞⎜⎝

⎛−

Maximum velocity of the head

m3

m1

m3m3

m1

y

m1·g/c

d

0

jumping toy

Calculated height of the jump(conservation of linear momentum)

31311 )( vmmvm +=

mc

gmd

mmg

mc

g

vh 4.1

)(22

2

12

13

123

3 ≈⎥⎦

⎤⎢⎣

⎡−

+⋅

==

v3 = mean velocity of the whole toy after leaving the floor

Another simple experiment is to compress the toy on to a – kitchen - scale and to measure theweight and the compression distance.From that you can calculate the spring stiffness.The spring stiffness of the ball pen springs is about 200 to 300 Nm-1.These values are comparable to that one I have here.The spring is compressed, and at the time t0 = 0 the head starts with maximum acceleration. Thetime t1 is when the mass m1 (head + suction cup) achieves the position where the head is in theequilibrium situation.Equilibrium means the situation when the spring is not compressed and the head is in equilibriumwith the spring. The time t2 characterizes the position of the end of the spring without m1; it is theequilibrium position of the spring without m1.The head will attain its maximum velocity v1 at the time t1. Conservation of energy leads to thefollowing equation.If you calculate the velocity the result is v1 = 8ms-1 or about 30 km/h.This is not a high velocity whenyou compare it with the high value of the acceleration.The result is negative because the direction of the axis points downwards.To calculate the height of the jump you need the principle of conservation of momentum. t3 shouldbe when the bottom mass m3 leaves the floor. The head of the toy must pull the base (and partlythe spring) when leaving the floor.

Thus you get the height. The result is realistic when compared with themeasured height. A common mistake here is to use the entire spring energy for calculating theheight.This is not allowed because there are energy losses due to friction, and some energy is storedin the oscillation of the toy, as you will see later.The height as a function of the head's mass m1 is a complicated, nonlinear function which has amaximum. It is interesting to see that the design of the toy is almost optimized with regard to thejumping height.You can also roughly estimate the starting time.This formula is only valid for uniform acceleration!You need calculus to calculate the time exactly.8 milliseconds is too fast to take a normal video of this process. The time between two (half) videopictures is 20ms (PAL system).With the help of colleagues I made videos of the jump with a high-speed digital video camera, with1000 and 2000 pictures per second .The figure above shows some pictures taken from the video.


jumping toy

Estimation of the time from start until achieving maximum velocity (≈ leaving the floor)

mssa

st 2.80082.0

2=≈=

(s = 0.032m; a = 95g; this is half of the initial acceleration;uniform acceleration assumed)

Video

2000 pictures/second146 pictures = 73 ms

jumping toy

frames of a digital video with 1000 pictures per second

At 0ms the toy starts; at 7ms the head of the toy reaches its maximum velocity and the base leavesthe floor; at 9ms the spring is stretched to its maximum; at 16ms the spring is minimally stretched;at 23ms the spring is again maximally stretched. The pictures are not sharp because the higher thespeed of the digital video camera, the worse the resolution.The spring oscillates.This can not be observed with the naked eye because the oscillating frequencyis about 70Hz.The high speed video camera costs about 10000 US-Dollar.This is too expensive for a normal school.By video analysis, the position of the toy’s head is extracted and shown in the upper figure. Thisseems to be like a badly drawn straight line. But it contains a lot of information. If you analyze this,you get the bottom figure.It shows the velocity as a function of time. The maximum velocity of the head is about 7ms-1. Thisis in reasonable agreement with the previously calculated value of 8ms-1.


jumping toy

PositionExperimental results usingvideo data analysis programs(DIVA, Coach V, etc.)

0,00 0,01 0,02 0,03 0,040,00

0,05

0,10

0,15

0,20

Analysis jump3 (2000 pictures/s)

po

sit

ion

he

ad

[m

]

t ime [s]

0,00 0,01 0,02 0,03 0,040

2

4

6

8

velocitymean

= 72Hz0.0139s

originalsmoothed

time [s]

ve

loc

ity

[m

/s]Velocity

as derived from the upper diagram(using Origin)

jumping toy

0,00 0,01 0,02 0,03 0,040

2

4

6

8

velocitymean

= 72Hz0.0139s

originalsmoothed

time [s]

ve

loc

ity

[m

/s]

0 ,00 0,01 0,02 0,03 0,04-3000

-2000

-1000

0

1000

2000

originalsmoothed

ac

ce

lera

tio

n

[m/s

2]Acceleration

as derived from the upper diagram for the velocity

Velocity

Also you can see the oscillation time of the toy (0.0139s) here and calculate the frequency (72Hz).In the upper figure the velocity of the head is shown again as a function of time. The bottom figureis derived from that and shows the acceleration.The data are smoothed because the double derivation leads to great fluctuations. The measuredinitial acceleration of about 2000m/s2 (= 200g) is roughly equal to the calculated value of 190g.One has to be careful with smoothing. Especially the data near t = 0 are corrupted by this procedure.The coils of the spring that are pressed into the top of the suction cup can only move withconsiderable friction. Furthermore, there are unpredictable rotations and somersaults aroundseveral axes which are hard to predict.These cost energy, thus reducing the height.These influencesare almost impossible to measure quantitatively.Another level is the use of calculus. First year physics students can do this.You can write down the differential equation for the starting process. This is the well-known


Mechanical energy is lost in the suction cup.

Several coils of the spring are compressed into the suction cup;this part of the spring can decompress only with considerable friction.

Furthermore, there are unpredictable rotations andsomersaults which also need energy.

These losses are difficult to calculate quantitatively.

There is also a small fraction of the energy stored in the oscillation between head and base, which can be calculated.

jumping toy

Using calculus you can write down the differential equation for the starting process. This is the well-known equation for an oscillating mass hanging on a spring (harmonic oscillator), where damping is neglected:

The solution with the initial conditions and y(0) = -d is

cygmym −= 11 &&

0)0( =y&

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ −+= tm

c

c

gmd

c

gmty

1

11 cos)(

and

jumping toy

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−= t

m

c

c

gmd

m

cty

1

1

1

cos)(&&

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ −= tm

c

c

gmd

m

cty

1

1

1

sin)(&

equation for an oscillating masshanging on a spring (harmonicoscillator). Damping isdisregarded here but can also beintroduced.By derivation you get the velocityand the acceleration.If the base is attached to the floor,this equation means that the headoscillates with the amplitude of (d– m1g/c) and with the angular

frequency of

This is not the same oscillationfrequency as previouslycalculated between head andbase!

From the equation for the velocity the time for the starting process and the maximum velocity ofthe toy can be deduced.The function has ist maximum for this expression where the sinus has ist maximum.The velocity is, of course, the same as previously calculated.The American scientists Gerace, Dufresne & Leonard have investigated even more accurately theproblem of two masses connected by a spring. They call their design a springbok. They don’t takeinto account the mass of the spring.


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ −= tm

c

c

gmd

m

cty

1

1

1

sin)(&

jumping toy

you get the maximum velocity.)2.60062.0(2

1 mssc

mt ===

πFor

The time is not the same as previously calculated, when uniformacceleration was assumed.

The velocity itself is – of course - exactly the same as previously calculated without using calculus.

18/21Gerace/Dufresne/LeonardPhysics Teacher/Febr. 2001

This model assumes a damping force that is proportional to the relative velocity of the two springbok masses. This model does not attempt to take into account sliding friction. To obtain a "simple" expression for the time at which the springbok leaves the table, we assume that the damping coefficient is small.

1––––2π

√ c/m1 = 39Hz.

I do not expect that you can read or understand this. You can see that there can be involved muchmore mathematics.Dufresne, R.J. et al.: Springbok: The Physics Jumping, The Physics Teacher 39 (2001), 109-115.A completely another level is to model and simulate the toy on a computer.There are several programs which allow the simulation of mechanical

situations as, for example, the jumping toy.Here I have used the program ‚InteractivePhysics‘. It is possible to create and varyalmost all parameters of the toy as masses ofthe head and the base with the springconstant including damping. It is interestingto see that the results of the simulation arequantitatively very similar to the realexperiment, as you perhaps remember fromthe previously shown graphs.The simulation is very informative becauseyou can easily vary all the parameters.You can model easily your toy withInteractive Physics (cost of the programabout 200 USD)At the end let me express that there are notmany toys which have that manyadvantages.


jumping toyModelling and Simulation with Interactive Physics

There are not many toys which have that many advantages :

1) Cheap2) Interesting and motivating for children 3) Simple and transparent design4) Easy to build by yourself5) Interdisciplinary reflections6) Comparison experiment – theory7) Modelling and Simulation8) Different levels

But nothing is without disadvantage:

1) Certain danger2) Not always available

104Background Aspects

2. Special Aspects

INTERPLAY OF THEORY AND EXPERIMENT

Anna De Ambrosis, Physics Department, University of Pavia, ItalyGiuseppina Rinaudo, Department of Experimental Physics, University of Torino, Italy

1. OutlinesThe discussion was very lively, with several contributions from many members: the general trendwas to illustrate each argument with concrete examples, taken from the practice of everydayteaching in classrooms. The guidelines were essentially along the indications given in thepresentation of the workshop, that is:1) what is the meaning of “theory”, “experiment” and “interplay” in this context,2) the principal aims to be reached,3) types of experiments and their effectiveness in favoring the formalization process.The discussion however did not follow orderly the above sequence, but followed a rather randompath, in which, mainly starting from the discussion of a particular experiment, the aims of theexperiment were discussed and the meaning of “theory”, “experiment” and “interplay” in theparticular context were clarified.Essentially, in the discussion of each experiment, three basic questions, suggested by Matilde Vicentini,were systematically put forward: what? how? why? Only in the last session, after having examined alarge variety of experiments, the group went back, in a systematic way, to points 1) and 2).The experiments, which were presented and discussed, did not have, in general, a strikingoriginality, since most of them were well known to most of the participants. However the interestof their presentation was due to some particular aspect, generally not fully exploited but relevantfor the formalization process, and to the methods used or to the sequence followed, in thedidactical practice, to overcome well known student’s difficulties. It is just this panoramic view anddetailed analysis of many different experimental situations which resulted in the richness and,hopefully, in the usefulness of the workshop.In the following, we will first present the examples given of different types of experiments or ofexperimental situations and then report the outcomes of the general discussion.Examples of experiments or of experimental situationsIn order to guide the discussion, we agreed to analyze examples of the following types, which weconsidered to group most of the experimental activities done with or by the students:a) physics experiments in everyday lifeb) discovery experimentsc) demonstrative experimentsd) experiments to test a physical law e) computer based laboratoryf) historical experimentsAs said before, in each presentation we tried to stick to the three basic questions (what, how, why),the “why” being mainly focussed on the problem of the formalization. In some case it was difficultto classify rigidly the type of experiment, because it shared features of different types, for exampleit was “computer based” with peculiar aspects of “discovery” and related to a given “physical law”.

a) Physics experiments in everyday lifeThe positive aspects of experiments and phenomenology based on common objects or on eventsof everyday life are well known: the student is familiar with them, thus he has generally a positive

attitude towards objects that probably he likes and is interested to use more efficiently or towardsevents that he knows well and that probably he would like to understand why they happen in agiven way. Besides, he starts from his previous knowledge of how things work and what is their use,he has a natural language to describe them, taken from everyday life and made of words, used inthe common sense, which he understands.The negative aspects are also well known, the principal one is that a common object or a real-life-event is generally a complex thing, since the simplification, which is usually done to build a labexperiment, is lacking.However in many common objects and phenomenology this complexity can result in a greaterrichness, which helps rather than contrast the formalization process, at least in its first stages.Besides, the transition from the common language to the formal description is made easier by theassociation of real objects or events with the abstract concept.Examples were given (Pedro Pompo), at the level of university college students, on discovering themeaning of physical quantities such as power and energy consumption in domestic devices, as amicrowave oven.At the level of primary school pupils, many examples were mentioned (Giuseppina Rinaudo, formore details see the contribution presented to this conference, “Developing formal thinkingthrough toys and everyday objects for the formation of future primary school teachers” by AllasiaD., Montel V., Rinaudo G.) of simple toys which can help the formalization process. An example isa sling, used to throw a paper ball, where the abstract concept to be reached is that of force: with asling in his hand, the pupil can have a direct perception of abstract concepts related to the conceptof force, such as the fact that - the force is the expression of an interaction between two objects (the hand and the rubber of the sling),- the force has a direction,- there is a reaction force from the rubber of the sling towards his hand.

b) Discovery experimentsThe role of this kind of experiments is to focus the attention on some particular aspect, generallycurious, that one is led to try to understand and explain, through the experimental observation. Inthe search for an explanation, many different physics concepts are inevitably clarified, althoughgenerally only at a qualitative level.A nice experiment was suggested by Carlos Vargas on the “oscillating salted water solution”. Ahigh concentration solution of salted water is put in a vessel, which has a small hole in the bottomand which is put into a larger vessel containing pure water. Initially the hole is kept closed and thelevel of water is adjusted to be the same in the two vessels. Immediately after opening the hole,water flows from the inner vessel with salted water to the outer one, making the level in the innervessel to lower. However the flow reduces with time, until a flow starts in the opposite direction,i.e. from the outer to the inner vessel and the level in the inner vessel goes up. With time, the flowreduces and then it reverses again its direction and so on, with a periodic oscillation. What is the“discovery” and what the “formalization” in this experiment? To understand what happens andbuild a reasonable model, one has to review the meaning of quite a few physical quantities and oftheir relations, such as the relation between density, pressure, direction of flow, buoyancy, etc.

c) Demonstrative experimentsThey are used typically by the teacher to draw the attention to a particular phenomenology beforestarting the detailed discussion of an argument. Interesting examples, used in an university coursefor the formation of future teachers with an interactive methodology, were given by MatildeVicentini:- just put a jar containing water on a scale and put your finger in it: ask the students if they expect

that the scale indicates a change of the weight (concept of gravity force, buoyancy and forcetransmission in fluids);


- prepare four equal cylindrical cans by leaving one completely empty and filling the other threewith some powder material (sand, coffee...) in such a way that while one is completely full theother two are partially filled in a different way. The students are not told of the difference in thefilling. Put the cans on a curved guide where they oscillate with a damping rate which differsamong the four cans (two show a number of oscillations before stopping, two show a quickdamping). Ask the students the reasons for the difference. They usually attribute it to the weightbut assume a monotone increase of the damping rate with the weight. The discussion, after theverification that the damping rate is maximum for the intermediate weights, brings in theconcepts of friction, internal energy, entropy production, degrees of freedom and more aboutconnecting mechanics with thermodynamics.

Another use of demonstrative experiments was shown by Pedro Pompo to introduce the discussionon the meaning of “light ray” and of light propagation in straight direction. Put a layer of sugar onthe bottom of a transparent jar with flat sides, pour slowly water from the top trying not to disturbthe sugar and let the solution rest for at least half an hour. A concentration gradient will set in and,consequently, also a gradient of refraction index. If you shine a narrow beam of laser light throughthe solution in horizontal direction, perpendicular to the box side, just above the sugar layer, thebeam will bend towards the bottom of the jar; however the beam will continue practically in thesame direction if it passes through the higher levels of water, where the concentration is practicallyconstant. Besides the above concepts about light propagation in straight directions, thisdemonstrative experiment calls the attention to the meaning of the separation surface betweenlayers of different refraction indices in Snell’s law and of its perpendicular direction: inside thesolution there is no “separation surface”, but still the beam changes direction!

d) Experiments to test a physical law A few experiments of this type were presented. A preliminary question was put forward byMargarida Saraiva Neves and Lucilla Santos: should the “physical law” be known before going tothe lab to do the experiment? The answer was “it depends”. Lucilla suggested that the question isclosely linked to the sort of equipment used to “test the law”: A “black-box device” with on/offswitches only, gives very different results (as far as verification is involved) from an “open device”,were the students can actually see what is going on and interact with the equipment. An opendevice which allows manipulation and control of variables, contributes to test if a “law” iseventually wrong, because it helps students to realize if they are not getting the results expected,or if they are dealing with the wrong procedure, or variables.An example was presented by Grazia Zini. The experiment requires some guidance and some basicknowledge of a magnetic field, but it does not require necessarily to know the “physical law”. Thestudents are given a solenoid connected to a variable current source and a magnetic field sensorand they must discover, by trial and error, the spatial distribution of the field intensity, its directionalproperties and its dependence on the current intensity.A completely different example, in which the interplay between theory and experiment is muchtighter, was presented by Andrea Sconza. One might call this experiment a revised version of theclassical “hot tea experiment”, but the level at which it is done is quite higher: a can containing hotwater is left to cool down to room temperature and, while cooling, the temperature is measured asa function of time and recorded automatically on a computer (the experiment might indeed beconsidered a “computer based experiment”). To interpret in a satisfactory way all the details of thecooling down curve, the student has to recall all the physical laws he might have seen, including theusually neglected loss of energy by evaporation! A careful analysis shows in fact that thecontribution that one introduces in the first instance – the conduction through the walls - is notsufficient and also the introduction of the energy loss by radiation. The formalization in this caseconsists essentially in the ability of “reading”, in the trend of the experimental data, the expectedbehavior suggested by already known theoretical laws and by refining the process by going back totheoretical laws until the explanation is satisfactory.

106 2. Special Aspects

e) Computer based laboratoryAttention was drawn by Elena Sassi on real time experiments and on their main characteristics: thepossibility of collecting many data and of giving a graphical representation of a phenomenon. Forthese characteristics real time experiments can be used as cognitive tools. In particular they allowan approach which can be labeled as “from real and familiar phenomena to ideal models” ratherthan in the opposite direction as is almost always the case in the conventional textbook-basedapproach. For the sake of brevity this rationale, entitled “From Real to Ideal” suggests to start fromthe analysis of phenomena common in everyday life on which students already have experienceand ideas; to look for regularities and to describe them with rules; to propose a first model and tofind a formal description of it.. The paths implementing this rationale start from Real-Timeexperiments which explore real, complex facts well known to students in terms of common senseknowledge and elicit the relative naive ideas. They proceed to the identification ofphenomenological regularities which are transformed in rules, through more “clean” experiments,in which some secondary effects have been minimised (e.g. friction). They proceed further tomodelling these rules with simple mathematical functions. The final step is the abstraction towardthe ideal case/model representing the appropriate physics law. Real time experiments make itpossible an interplay of theory and experiment as a process which allows to pass from theobservation of complex phenomena described with common language to a formal representationof them. Other aspects of Real-Time approaches have been briefly discussed, referring also to thepresentation by E. Sassi in this Seminar about “Real –Time Approaches in the Development ofFormal Thinking in Physics”

f) Historical experimentsThe analysis of particular historical experiments can help students to focus their attention on theplanning of the experiment and to consider important aspects, such as the necessity to decide whatmeasurements to carry out and the difficulty of finding significant relations among collected data.One of Galilei’s experiments on inclined plane, the bell experiment, was mentioned as an example(Anna De Ambrosis).Galilei did not have a mechanical clock to measure time intervals, but he could detect equalintervals of time as part of an acoustic rhythm. Thus, if a ball falling on an inclined plane touchesmobile marks with bells, a series of sound is produced. The sounds can be produced at equalintervals with appropriate positions of the marks. This experiment introduces the idea thatdisplacements along an inclined plane are proportional to the square of corresponding timeintervals and gives the students the opportunity to differentiate the concept of position, velocityand acceleration which often are not understood to be distinct. Computer simulation of theexperiment allows students to plan individual experimentation and analysis.Pedro Pombo suggested other examples: the exam of the historical development of reflectionholography and of Gabor holography. Gabor’s historical experiment can be useful for a betterunderstanding of the characteristics of holographic principle and image formation and to and foran introduction of Fresnel lens concepts.

2. Final discussionThe final discussion focussed essentially on the meaning of the “interplay” between theory andexperiment. In all the presentations and contributions, it was clear that the words “theory” and“experiment” were interpreted in a broad meaning:- theory is any formal description, which brings to the definition of physics concepts, to their

representation and relations,- experiment is an experimental situation but also, in general, a relevant phenomenology which can

be observed, described and possibly measured.The question was then <<what do we mean with the word interplay?>>Tracing back through the examples given in the different presentations, we found the following


interpretations of the interplay between theory and experiment.Besides, there was also a general consent that it was useful to classify the different types ofinterplay because each type can have a different didactical application and can be useful indifferent circumstances.

1. “One way” interplay

It is the simplest and most common type of interplay, used in the normal practice of the laboratory:after the presentation of the theoretical laws in the theoretical lectures, the students go to the laband test the validity of the laws, or, vice versa, they first go to the lab and do the experiment as apreparation to the theoretical presentation. Is this kind of interplay useful for the formalizationprocess? It is useful if there is a tight link between the theoretical presentation and the experimentand if the link is used to reinforce the inference process, that is to learn to separate clearly what isreally inferred from the experimental evidence and what is accepted because it follows from thetheoretical law.

2. “Circular” interplay with “feedback”It is more powerful, but also more complex, than the previous type, because it requires that the time

of the theory is not separated from that of the experiment.

3. “Linear progress”Is typical of discovery experiments, in which the “theory” is discovered along with the experimentalfacts.

4. First stage of formalization It is the formalization done at the early stages, through words, drawings, graphics, before any theoryis even established.

5. Formalization by mathematical descriptionEllermeijer talk in the plenary session was very useful at this regard, because it presented both thepros and the cons of a formalization based on the mathematical description. In particular theexperiment is very important in understanding - which variables are “independent” and which are “dependent” (in mathematics the interest is all

focussed on the function which describes the relation between the variables, thus one can switch


theory experiment

from theory to experiment

theoryexperiment

from experiment to theory

theory experiment

from theory to experiment and back from experiment to theory and back

theoryexperiment

without difficulty from the function to its inverse, and, consequently, make the “independent”variable to become the “dependent” one),

- the limits of application of a particular law or theory (in mathematics the domain is establishedon the basis of the properties of the function which describes the relation between the variables,thus the domain to be explored can go easily from minus infinity to plus infinity or in regionswhere the variable has no physica meaning),

- the importance of the scale used, of the dimensional units, etc.

3. List of contributionsShort written contributions were asked to the participants to the workshop on the ideas that werepresented and discussed, if they were not already included in some presentation to this conference,because we considered that they could be useful to the readers.We include, below, the contributionsreceived up to now, we hope to receive others in the next days: if so they will be included in thefinal paper.“Experiments and reflective learning”, L. Bandiera and Matilde Vicentini“Galilei’s experiments on inclined plane”, Anna De Ambrosis“Water cooling: how to build a physical model for an every day life experiment”, Andrea Sconza

4. ParticipantsMost participants were presents to the four sessions, some took part only to one or two sessions.They were almost equally distributed between secondary school and university teachers, the latterbeing generally involved in the formation of future primary or secondary school teachers or intraining on-service teachers.

L. Austrilino (Brasil), A. De Ambrosis (Italy), E. De Masi (Italy), M. Giliberti (Italy), T. Lobato (Portugal), M. Lyberg(Sweden), D. Moreno (Mexico), H.G.B. Olyacee (Iran), P, Pecina (Croatia), D. Pescetti (Italy), P. Pombo (Portugal), G.Rinaudo (Italy), L. Santos (Portugal), M. Saraiva-Neves (Portugal), E. Sassi (Italy), A. Sconza (Italy), I. Ustuner (Turkey),C. Vargas (Mexico), M. Vicentini (Italy), G. Zini (Italy).

EXPERIMENTS AND REFLECTIVE LEARNING

M. Bandiera, M. Vicentini, Department of Physics, Univ. “La Sapienza”, Roma, Italy

1. IntroductionScience education is currently witnessing a change from a teaching approach aimed at the learningof facts and theories to one aimed at raising, confronting and solving problematic issues.Accordingly a major change would be required precisely in the presentation of experimentalactivities in the general context of university science courses (lectures and laboratories).In any case, direct contact with experimentation is obviously necessary in order both to familiarisestudents with the phenomenological aspects related to models and theories as well as to raise issuesconcerning the nature of science.The same prospective would require that the experiment be presented in a problematic way so thatthe students are encouraged – forced – to think.Discussion is an essential part of this didactic approach and, therefore, group work activities wouldbe planned.In the context of University courses specifically aimed at initial science teacher training, thelaboratory activities – required in the curriculum – acquire increased value as an opportunity:- for joining didactics to didactic research and for exploring from one year to the next, encounter

by encounter, the students’ knowledge and competence (without wasting time with constant,periodic tests);


- for placing on a lower level overused technical/methodological training and simple physicalcontact with objects and events in favour of the setting up of and training in productivemodalities for recalling theoretical knowledge and pulling out data from the sameobjects/events;

- for taking into account not only educational objectives but also the multiplicity of intelligences,of cognitive approaches and of basic disciplinary competences.

The present investigation considers two ways of carrying out lab activities: the “demo-show” whenthe teacher, interacting with a large group of students, variously alternates actions related to anexperimental procedure, interventions as facilitator, and provocation in order to enliven thediscussion; the “lab-party” where small groups of students (from 4 to 6) sit at a “set” table and,taking advantage of a pre-structured form, follow an itinerary of experimental and reasoning steps.The “demo-show” proved to be profitable for the prospective secondary school science teachers,who have degrees in a scientific discipline; the “lab-party” to the prospective primary schoolteachers who, coming from different secondary schools, have in common a generalisedinexperience in science.Both strategies require an appropriate choice of experiments which, in any case, must include thepossibility of asking questions, thus activating and illuminating the students’ knowledge and mentalrepresentations.The experiments we plan to present focus on ice-cubes melted over conducting and insulatingsupports, containers with various quantities of fillings, rolled down a sloped plane, and humanrespiratory and pulse frequency modifications as a result of muscular stress.They have in common the characteristic of being related to some known phenomenology of everyday life. Therefore, when asked to explain or predict the experimental occurrences and outcomes,the students tend to adopt intuitive or semi-intuitive explanations/predictions that totally orpartially conflict with the behaviour of the phenomenon when it is watched or performed. Such acontrast stimulates the discussion and the reflection on conceptual aspects.In sec.2 we illustrate an experiment carried out in the two methodologies. Other examples of“demo-show” experiments are illustrated in sec.3 while sec.4 reports other “lab-party” experiments.

2. The ice cubes experimentsThe “demo-show” starts with the request of a prediction of the difference in the time needed formelting two ice-cubes, one placed on a conducting sheet and the other on an insulating one. Thedimensions of the sheets exceed that of one face of the ice-cube. While at secondary school level ingeneral a fraction of the students (nearly 1/3) predict that the cube on the insulating surface willmelt first, at a higher level nobody doubts that it is the ice-cube on the conducting sheet which willmelt first, but the time difference is always underestimated (from 5 minutes to half an hour). Somestudents unfailingly try to avoid an answer by pointing out that the question is not well formulatedas some data are missing like the temperature of the cubes and of the environment, the dimensionsof the sheets. However the sheets are in front of them, and it is very easy to estimate (or measure)the two temperatures. (Again, it may come out with secondary school students that thetemperature of ice is undoubtedly 0°C.) Then the experiment starts and the students are asked toobserve carefully while the discussion on the developing process and on the variables needed todescribe it takes place. After a quarter of an hour, with the ice cube on the insulating sheet showingno signs of melting, attention can be suitably turned to the “behaviour” of the ice-cube on theconducting sheet. The discussion then brings in the possibility of reasoning in terms of conduction-convection-radiation, the need to focus on the thermal equilibrium of the cubes both with thesheets and with the environment, eventually the differences in the specific heats of the sheets inaddition to the conductivity coefficients, and to what will happen if the dimensions of the sheets arechanged.In the “lab-party”, due to the low level of students’ competence in Physics, sensory contact with thematerials opens an activity which has to be completed in the nine steps listed in the form: 1) Touch


the materials on the table. Write down three “qualities” of each one. 2) Indicate what you believeto be each one’s temperature. 3) Now imagine taking an ice-cube out of the freezer: how long doyou think it will take to melt completely? (Write down two of the reasons for making yourprediction.) 4) Now imagine placing one ice cube on a sheet of aluminium and one on a sheet ofpolystyrene. How long will it take for each one to melt completely? (Write down two of the reasonsfor making your prediction.) 5) Check your hypothesis, indicating which of the sheets you havechosen from among those available. Write down the resulting melting times. 6) Now choose twodifferent sheets and, based on the data resulting from the above experience, formulate a prediction onmelting times. (Summarise what you have based your prediction on.) 7) Check your hypothesis,writing down the melting times on the second pair of sheets. 8) List five terms (or brief definitions)that you believe necessary in formulating a “scientific” explanation of the event that you haveobserved. 9) Starting with the generating nucleus provided, expand the map so as to integrate theterms that you have listed in step 8 and, if possible, also the “qualities” you wrote down in step 1.Recordings of discussions and questionnaires completed by prospective primary school teachers havebeen analysed, which bring out expectations, points of view, cognitive obstacles, (mis)inter-pretationsand, as far as the “lab-party” is concerned, based on concept maps, profitability of lab-work.

3. More on “demo-show” experiments: the rolling containersContainers with different quantities of filling of the same material show a different behaviour ofmovement: when rolling down a curved guide they may approach the equilibrium position at thebottom with more or less damped oscillations.The experiment starts with the presentation of the material focusing the complete external identityof the containers. When the motion is shown, however, some containers oscillate normally with adamped amplitude while others reach rapidly (one or two oscillations) the equilibrium position. Itis then asked to provide an explanation of the different behaviours.In the following discussion of course “friction” is called in for the damping but soon it is clear to allthat the difference must be due to the different weight of the objects. Two positions then emerge:the different weight gives rise to a different force of friction between the surfaces of the containersand the slope (the majority) or to some effect related to internal friction (a minority which alsotries to infer on the quality of the material in the interior).In all the demo we have made (italian student teachers, chinese student teachers and researchersin physics education, mexican researchers in physics education) the conceptual scheme invoked forthe explanation is a “force scheme”. The didactical strategy may then direct the attention on oneside to the scientific methodology of getting informations on the internal structure of an object bypreforming other experiments and, on the other, to search for a conceptual scheme more useful forthe case. It is surprising that very few explicit the need of recurring to an energy scheme while allsuggest the need of having the experimental data on the weight.While this is recognized by the teacher as an important information it is asked to explicitpredictions on the influence of the weight on the damping rate. Such predictions are unanimous on:the damping rate is linearly related to the weight.Hand weighting the objects by some students disconferms the prediction: the damping rate ismaximum for some intermediate weight.The explanation in terms of the external friction force is then abandoned in favour of an internaleffect. Time is here needed for small group discussions which, usually, remain anchored at a forcescheme also if the suggestion of measuring the inside temperature may be advanced.This is a good step for the teacher to suggest the shift to the conceptual scheme of energyeventually focusing the difficulties of application of the force scheme to the case.Energy, potential and kinetic energy… internal energy.The introduction of this concept, apparentlyrelegated in the conceptual scheme “Thermodynamics”, for a mechanical phenomena seems tocome as a surprise for the majority who are then forced to reflect on the generality of the energyconcept while disconnecting the relation “friction then heat then temperature increase” toconceptualize the relation “dissipation rate in the external motion then internal energy”.


It is not easy to report qunatitatively the happenings in the discussion of a demo-show experimentand in fact it does not even make any sense. The important quantitative information may only berelated to the quality of the discussion itself and this is mainly related to the ability of the teacherto collect important ideas in the interventions of the participants and focus them on issues ofreflection on their knowledge.In our experience the discussions were always very lively while differing in their developmentaccording to the group of students. We may then conclude by mentioning some common directionsfor reflection:- energy of course: conservation and dissipation,- the difference between the mechanical and the thermodynamical point of view on the same

phenomenon,- heat and temperature,- the concept of time in mechanics and in thermodynamics,- the atomic model of matter.

From the work by Bandiera M. and Vicentini M. published in “Science and Technology Education: preparing futurecitizens”, Editor Nicos Valanides, Imprinta Ltd. Cipro, vol.1, pag.98-109”, Proceedings of the 1st IOSTE RegionalConference in Southern Europe, Paralimni Cyprus, 29 April - 2 May 2001.

GALILEI’S EXPERIMENTS ON INCLINED PLANE

Anna De Ambrosis, Physics Department, University of Pavia, Italy

The idea is to consider with students historical experiments and to provide them the opportunityof using also computer simulations of the original experiments. Despite the difference between areal and a simulated experiment, some important aspects, such as the necessity to plan whatmeasurements to carry out and the difficulty of finding significant relations among data, arecommon.Computer simulation encourage students in individual experimentation and give them thepossibility of familiarize with and compare various historical approaches with modern ones. Thisactivity can promote in the students a reflection on their own conceptual schemes favouring theirimprovement.

An example: historical experiments on the free fall of bodies.On this topic we prepared a package for students in introductory physics courses which includescomputer programs, teacher guide and student guide with worksheets. The guides contain ahistorical section with relevant original sources.The following experiments were considered:- Galilei’s experiments on inclined plane:

1. A qualitative study with bells2. Quantitative study with a water clock3. measure of velocity

- Atwood’s experiment(For the reconstruction of experiment 1 and 3 we followed Drake’s interpretation of Galilei’smanuscripts of 1607. For experiment 2 we referred to the original published source: Galilei’s“Discorsi e dimostrazioni intorno a due nuove scienze”. For the Atwood’s machine we referred tothe apparatus which is in the Physics Museum of our University).As an example the bells experiment is briefly described.Galilei did not have a mechanical clock to measure time intervals, but he could detect equalintervals of time as part of an acoustic rhythm. Thus, if a ball falling on an inclined plane touchedshiftable marks with bells, a series of sound was produced. The sounds could be produced at equal


intervals with appropriate positions ofthe marks. The analysis of the spatialdistances of the marks could give thesquare law.In the simulation we tried to reproducethe original situation in which Galileisang a refrain to find the position of theball at regular time intervals andadjusted the positions of the bells sothat the emitted acoustic signals wereregular. This experiment introduces theidea that displacements along theinclined plane are proportional to thesquare of corresponding time intervalsand it gives students the opportunity todifferentiate clearly the concepts ofposition, velocity and acceleration,which often, even in simple motion ofobjects, are not understood to be distinct.The simulation allows to perceive the fact that the motionis not uniform and that the velocity of the ball increases in successive time intervals. Worksheetsprovided to students are focused on the quantitative aspects of the motion. For example, bycalculating the displacement of the ball in successive time intervals, the students can recognize thatthe average velocity of the ball increases in a uniform way.Atwood’s experiment introduces a different approach to the sudy of motion. Behind thisexperiments lies the Newtonian theory. The comparison between Galilei’s and Atwood’s historicalapparatuses can be thus a key to understand different paradigms. Students can compare twodifferent historical approaches to understand the same physical concept and recognize why andhow particular methods were used to slow down the motion to be studied, allowing easiermeasurements of velocity.A reflection on historical experiments can help students to perceive the real difficulties in thedevelopment of the theory of motion, difficulties often hidden in textbooks and underestimated bystudents who see their difficulties as personal, rather than inherent in all such attempts toconceptualize the law describing physical phenomena.By observing students working with the simulated experiments and by analysing their filledworksheets it is possible to identify the obstacle that seem to prevent students’ understanding ofkinematics concepts.For example the bell experiments showed that it is difficult for student to evaluate a velocity if ameasure of the elapsed time has not been made with a clock. In their answers the students (fiftystudends in an introductory physics course) correctly referred to the the formal relation betweenspace and time writing v = _s/_t, but more than half of them could not apply the formula in thecontext of simulation. An analogous difficulty was evident in the evaluation of acceleration.We found that Galilei’s bell experiment is particularly effective in focussing on this problembecause it was conceived precisely in order to overcome the difficulties related to the measurementof short time intervals.Difficulties in understanding acceleration were shown in responses given in the worksheetsconcerning the Atwood’s machine. Half of the students could not reach correct conclusions onacceleration starting from their data. Even the students who correctly eveluated velocity could notrelate changes in the velocity and time intervals, thus revealing their difficulties in understandingthe meanings of the measured quantities.With some surprise students realized that their knowledge of kinematics was not sufficient to carryout the apparently simple experiments presented in the simulation. In many cases students realized


Fig.1. The inclined plane with bells reconstructed at the Museumof History of Science in Florence, Italy


that they had not really understood concepts they thought were clear in their minds.This awarenessled them to discuss their ideas with the teacher, to ask to use simulation again and to go back to thehistorical guide for a deeper analysis of the reported original sources.Of course, the students were given the opportunity of carrying out also laboratory activity on themotion on an inclined plane gathering experimental data and analysing them: they used an air-cushion rail, the one available in the laboratory.It appeared particularly useful to students comparing original experiments and modern ones: ithelped them, in particular, to recognize the technological constraints which influenced the designof the historical set-up.

From “Computer simulation and historical experiments” by Bevilacqua F., Bonera G., Borghi L., De Ambrosis A. andMassara M. published in Eur. J. Phys. 11, (1990), 15-24

WATER COOLING: HOW TO BUILD A PHYSICAL MODEL FOR AN EVERY DAYLIFE EXPERIMENT

A. Sconza, Phys. Dept. of Padua, University of Padua, Italy

By means of a temperature probe connected to a CBL interface and portable TI-89 computer weregister for one hour the temperature of the water contained in a small metal can. Furthermore wetake note of the weight of water at the beginning and at the end of cooling. Then we try toreproduce both the temperature time relation and the loss of mass by evaporation by successiverefinements of a physical model.These are the registered temperatures:

The other data of the experiment were:Mwater-initial = 380 g, M water-final = 365.9 g, Tambient = 22.0 °C = 295.1 K.Dimensions of the can: height 11 cm, base diameter = 7.4 cm, mass = 46.8 g

Total heat capacity :C = (Mwater cwater + mcan ccan + Cthermometer) = 385.4 cal/degree = 1613.3 J/ degree (neglecting thermometer capacity, ccan = 0.115 cal/(g oC)) From the temperature vs. time data we compute the energy loss in unit time (lost power) vs. timeor vs. the absolute temperature: -dE/dt = - C dT/dt.The lost power is rapidly decreasing with decreasing temperature, see next figure.The simplest model one can try assumes (conduction loss):

Lost power (watt) = - (dE/dt)cond = k (T-Tambient),but obviously it cannot reproduce the experimental data because in this model the power lossshould be linear with T: the fit (straight line of lowest inclination in the next figure) gives k = 0.64with a mean square error of 44.9.

Then we add the contribution due to radiation loss (Stefan-Boltzmann law):-(dE/dt)rad = ε σ S (T4-Tambient

4),where T and Tambient are in K degrees, S is the lateral surface of the can in m2, σ = 5.67 10-8 Wm-2 isthe Stefan-Boltzmann constant and ε is the emissivity (0< ε <1) of the can surface which is assumedequal to 1 (as for a black body) because the can was painted with black spray color.In our case S=0.11*π*0.074 = 2.557 10-2 m2 so that σS = 1.45 10-9 W.

It is important to fix, in some manner, the emissivity: if one tries to leave _ as a free parameter thefitting program chooses for it very big unphysical values.The quality of the fit (less inclined approximately straight line) increases a bit (mean square error= 38.7) but is always unsatisfactory.Finally we complete the model by adding the evaporation loss that is assumed proportional to thevapor tension of water, as given by Clapeyron formula:

pvap (T) = A exp( -m clatent /RT) ,


where m=18 g/mole is the molar mass of water, R = 8.31 J/(mole degree) = 1.99 cal/(mole degree)is the gas constant, T is the absolute temperature, clatent is the latent heat of evaporation (in cal/g)and A is a constant. An empirical relation gives furthermore: clatent = (796.3 – 0.695 T) cal/g.In conclusion we assume:

-(dE/dt)ev = µ exp( - (7203 – 6.3 T)/T),where µ is the parameter to be determined experimentally.Now finally the fit is very satisfactory (mean square error = 0.99) and the optimal values for thethree parameters are:

k = 0.15 Wε = 1µ = 3.2 107 W.

The relevance of the three energy loss mechanisms is illustrated by next graph: one discovers thatthe evaporation contribution is absolutely dominant at the beginning of the cooling down.

Then we try to see if the adopted model gives the right quantity of evaporated water (380 - 365.9= 14.1g). To compute the final mass of water one must integrate on time the mass loss given by theenergy loss due to evaporation divided by the latent heat of evaporation:

(dM/dt) = (dE/dt) ev / clatent = - (-dE/dt) ev / (4.186*(796.3-0.695*T)).The result of the integration is in very good agreement with the final experimental value of themass (365.9 g).



LEARNING PHYSICS VIA MODEL CONSTRUCTION

Rosa Maria Sperandeo-Mineo, Department of Physical and Astronomical Sciences, Universityof Palermo, Italy

Presentations and discussion in W2 have been mainly connected to two themes of the Seminar:1) Modelling the world: -Issues in developing imagined worlds and connecting them to the

phenomenal world.2) Mathematics: - Exploring the special case of developing physics through the descriptive

language of mathematicsThe starting point of W2 has been the Sperandeo-Mineo’s original proposal (R.M. SperandeoMineo:” Learning via Model Construction”) whose main objective was in shearing concerns, indefining th background of modelling in order to point out some relevant questions.The first part of discussion has been devoted to find common aims for the workshop outcomes.Participants agreed on two common objectives:1. to analyse Why and How a modelling approach may be aimed:-to make physics accessible to High School and College students-to make students aware of the various (and different) procedural aspects of scientific reasoning.2. to analyse Why and How Information and Communication Technology may support modelling

at different level of formalization.

1. Modelling and the process of physics knowledge construction All participants were aware that “models” are a central topic in discussion in contemporary scienceeducation; debates centred on the pros and cons of including a modelling perspective in sciencecurricula and on pragmatic strategies for designing classrooms that enable students to learn aboutscience as a modelling endeavour. This debate is, at the same time, very exciting and confusing: inour opinion, at the core of the confusion lie the different ways in which the term model is used. Inan effort to avoid contribution to the confusion, it has been helpful for us to clearly articulate whatwe mean by “model” and “scientific model”, i. e. to clearing the grounds, and in particular:1. modelling and models in the process of Knowledge Construction2. physics models and math models with reference to:

a)-the role of analogies and metaphors in their construction;b)-the different kinds of models used in physics.

Cognitive scientists have identified “mental models” as fundamental tools of human thought used tostructure our experience and thereby to make it meaningful. In fact, any experience, or system, we areinterested in, is commonly perceived as containing certain discrete elements related with each other,through some process we apply to them; in this way we develop the “model” for the experience.

(Giere 1990)….A model is an abstraction and simplification of a defined referent system or process,presumably having some noticeable fidelity to the referent system or process…. This fidelity is expectedwhether the model is intended to describe, predict, or explain elements of the referent system…

In order to have a more clear and useful definition of the term model, we need to border theexperiential field we are interested in. In scientific research, we generally think of models as beingsimplified representations of real processes or systems. We analysed some definitions of the terms“model” and “scientific model”:

(Apostel definition (see Bartels & Nauta 1969)…Any subject using a system A that is neither directly orindirectly interacting with a system B, to obtain information about the system B is using A as a model for B.(Bachelard 1979). “The model is not an imitation of phenomena, it represents only some properties ofreality”

The term model, then, implies a representation of reality, often in a simplified way that providesstructure and order. Cognitive scientists have identified metaphors and analogies as fundamentaltools of human thought used to build models of our experiences.

Discussion has been aimed to focus on some significant examples presented in the panel sessionsrelevant for the modelling approach.A paper (Smit 2001) has pointed out that the understanding of the “physics model” concept needsto clarify some topics:• The classification of the different types of models in physics.• The nature of models.• The role they play in physics and their functions.The discussion about the Harre’s taxonomy of models (Harre, 1970), mainly focused on the aspectsdifferentiating paramorph and homeomorph models, allowed the group to be aware of the need(proven by research) that the topic of models and modelling should receive far more attention inthe teaching of physics.The nature of analogical thinking and the relationships between methaphors and analogies havebeen discussed during all the workshop. Many examples have been reported; a well focusedexample has been reported by O.Levrini concerning the metaphor used in a “linguistic sense” or,in other words, as a “theory-constitutive metaphor” meant as Boyd did. She was referring to the useof the word “analogy” made by Weinberg in a wide-spread university textbooks about GeneralRelativity: “Gravitation and Cosmology”.Weinberg retains that the interpretation of the formalism of General Relativity has to bedeveloped on the analogy between gravitation (primary system) and geometry (the Gauss-Riemann theory of curved surfaces: secondary system). The two systems have already been viewedby Einstein as the terms of the analogy constructed on the identification of spacetime and thedifferentiable manifold. Such an analogy has been exploited to such an extent by the so-called“geometrodynamical interpretation” (Wheeler et al.) that the two systems have finally coincided andthe metaphor was dead. The more and more precise superposition of the two systems is an exampleof the process of transforming the metaphorical use of language into the literal one.In Weinberg’s opinion, the main reason to re-separate the two terms of the metaphor is that the literalidentification between geometry and spacetime for understanding gravitation introduces a differencebetween this interaction and the other fundamental ones. And this difference is unacceptable for ascientist who intends to unify natural interactions. Another important reason can be identified in thefact that the literal identification between spacetime and geometry (the Riemann surfaces) may leadto attach spacetime substantiality, in other words, that absolute character hold by Newton’s space thatEinstein, thanks to Relativity, wanted to remove from Physics.

2. Modelling and Physics EducationResearch on cognitive processes and epistemological analysis of the evolution of physics haveshown that a modelling approach can constitute a good frame of reference in order to designteaching-learning situations both faithful to the discipline and relevant for the learner. However,many pedagogical problems need clarifications and unambiguous choices. How a modellingapproach can be thought and/or learned?Results of an exploratory investigation regarding the personal experiences of a group of scientistson modelling in physics have been reported by Islas and Pesa (2001). Scientists analysed theirexperiences concerning: a)-their own learning modelling, b)-the ways they handle models withintheir research activities, and c)-the ways they handle models in the teaching of physics at university.Although the ways to construct knowledge in a scientist and in a student are not exactly the same,the results can give insights about the educational strategies we can activate in order to helpstudents to come closer to research in building knowledge of modelling. The obtained resultssuggest that traditional lessons on modelling would not be the best strategy to developunderstanding of this issues; active engagement and discussion about problems to solve seem topromote reflection on those topics that are sources of difficulties for students, that is: the linkbetween reality and its representations, the factual meaning of mathematical expressions and thebuilding of link between theoretical constructs and experimental activities.Islas pointed out some relevant results of her research that evidence the caution that we have tokeep in mind in using computer simulation: she found that sometimes students have difficulties to


distinguish model and reality.This sort of difficulties could be stimulated by the use of sophisticatedsoftware, because students tend to believe that they are watching “the reality” in the PC screen(maybe, they think that the images displayed in the screen are a kind of video of the targetrepresented there).To avoid this misunderstanding, the results of her work suggest that teachers should make explicitthe link between:- the portion of reality we are studying,- the model that physicists have elaborated to this target,- the cybernetic model (that constructed by computer experts on the base of the phsical model),- the software we are running.In order to point out some relevant characteristics of modelling strategies, some significantexamples presented in the panel sessions have been analysed from the point of view of modelling.They involved physics teaching at high school and university levels and analysed:i. How the modelling approach is used and how it is supposed to be used;ii. How scientific models interfere with pupils’ personal views of the world.Two examples have been reported : the first one concerning the teaching of Newtonian Mechanics(Campos, Jiménez, and Del Valle. 2001 )and the second one concerning the teaching ofElectrodynamics (Roa-Neri and Jiménez, 2001); both have pointed out the learning effectivenessand facilities offered by the modelling approach.Model-based reasoning are strongly based upon different kinds of hypotheses (Cambell 1920):empirical law hypotheses, that are summaries of perceived patterns in observations, and explanatorymodel hypotheses that introduce visualizable models at a theoretical level and often containcurrently unobservable entities. Molecules, field, waves are not simply condensed summaries ofempirical observations but rather are inventions that contribute new theoretical terms and imagesthat are part of the scientist’s view of the world and that are not “given” in the data. Examples havebeen reported in order to point out the two different kinds of models.D’Anna’s report evidenced as experimental observations and laboratory experiments can be usedin order to induce mathematical models that summarise correlations among relevant variable ofthe studied system. Moreover he showed as the modelling environment STELLA can be used inorder to construct the bridge between phenomenology and theory. The report induced a veryfruitful discussion concerning similarities and differences between Explanatory models andEmpirical laws, that are mathematical or verbal descriptions of patterns in observations, and howand why they are differentiated and/or have to be differentiated in our teaching.It has been pointed out that, according to Gilbert (1998), all explanatory models are formed by theprocess of analogy, that is the seeking of similarities and differences between a source (something whichis perceived to be somewhat like the phenomenon under study) and the phenomenon itself, which is thetarget. Such analogue models help scientists “make the unfamiliar familiar”.This suggests that analogicalreasoning may be an important no inductive source for generating such hypothetical models.Explanatory models are physical and mathematical models and their construction, at high schoollevel, involve the use of appropriate environments. To date, modelling languages can be dividedinto two kinds: so-called “aggregate” modelling engines (e.g. Stella, Model-it) and object-basedmodelling languages ( Starlogo, Agentsheets , )• Aggregate modelling languages use accumulations and flows and other graphical descriptors of

changing dynamics, eliminate the need to manipulate symbols, but they are strictly connectedto the mathematics of differential equations from the conceptual point of view. They do notrequire the formalism of differential equation, rather the formalism of finite differenceequations (more easy from a conceptual point of view). However, these languages are veryuseful when the model output needs to be expressed algebraically and analysed using standardmathematical methods.

• The second type of tools enables the user to model systems directly at the level of the individualelements of the system. The object-based approach has the advantage of being a natural entry


point for learners. In StarLogo, for example students think about actions and interactions ofindividual objects or creatures (rabbits each of which has associated probabilities ofreproducing of dying, or molecules each of which has associated a position and a velocity andother properties). StarLogo models describe how individual objects behave. Thinking in termsof individual objects seems far more intuitive, particularly for the mathematically uninitiated.

Sperandeo-Mineo presented some ways to use StarLogo in a teaching approach focused onmodelling.The discussion has been concluded by analysing the role of modelling within the curriculum and inparticular:• model construction versus model use

(student using already constructed models of phenomena versus students constructing theirown models).

• concreteness vs. formalism(critics to computer-based modelling about the concern that the activity of modelling on computer istoo much of a formal activity, removing pupils from the concrete world of real data).

• aggregate vs. object-based modelling languages(“aggregate” modelling engines (e.g. STELLA, Model-It, Modellus….) and object-based

modelling languages ( StarLogo, AgentSheets ,…. ).• the introduction of Dynamic System Modelling: -to connect micro-level and macro level and

deterministic effect,-to introduce the notion of levels of descriptions, (how representations,…..-to show how stochastic models can account for stability changes of rules on one level lead todifferent behaviours and patterns at another level).

In conclusion, the possibility to use tools allowing different representations and different levels offormalization is very exciting for teachers and researchers. Nevertheless, we need research in orderto point out the effective learning coming out from modelling procedures as well as the modes ofthinking stimulated by the different modelling procedures (diagrams, words, numbers, images,animations, algorithms……….).

3. List of contributions1) The learning of modeling: a scientists’ vision, S.M. Islas, M. A. Pesa (Argentina)2) Does physics formal knowledge really help students in dealing with real - world physics

problems?., N. Grimellini Tomasini, O. Levrini (Italy)3) Models, mental images and language in scientific thinking, J.J.A. Smit (South Africa) 4) Newton’s laws revisited, I. Campos, G. Del Valle (Mexico)5) Induced time-dependent polarization and the Vavilov-Cherenkov effect, Roa-Neri J. A.E.,

Villavicencio M., Jiménez J. L. (Mexico)6) Modelling physical processes: the example of a magnetic glider, M. D’Anna (Switzerland)7) Modelling physical processes: a fruitful combination of mathematics and physics, M. D’Anna

(Switzerland)8) Modelling physical reality: several ways of using computer simulations to help formalisation,

R.M. Sperandeo-Mineo (Italy).

ReferencesBachelard G., Il nuovo spirito scientifico, trad. It., Roma-Laterza, (1978).Bertels K. & Nauta D., Inleiding tot het modelbegrip, W.de Haan, Bussum, Netherland, (1969).Boyd R., Metaphor and Theory Change: what is “Metaphor” a Methaphor for? in Ortony AR. (Ed.). Methaphor

and Thought, Cambridge University Press, Cambridge, (1979).Campbell D., Physics: the elements. Cambridge University Press, (1920), republished as The foundations of science.

New York:Dover, (1957).Gilbert J.K., Boulter C., Models in explanations, Part1:Horses for courses? International Journal of Science

Education, 20, (1998), 83-97.Giere R. Explaining Science: A Cognitive Approach, Chicago: University of Chicago Press, (1990).Harré R., The principles of scientific thinking., University of Chicago Press, Chicago, (1970).



THE DEVELOPMENT OF FORMAL REASONING

Ian Lawrence, School of Education, The University of Birmingham, UKGuidoni Paolo, Department of Physical Sciences, University of Napoli, Italy

1. Understanding and reasoningCompensation was extensively discussed as one of the basic structures of reasoning. The need tobuild competence in this process upstream of formal mathematics. Several examples given ofcompensation reasoning at different levels - 2, 3 And 4 variable relationships. There are a numberof these structures of logical thinking, often made apparent though words or actions, before theyare made explicit by the children. For example: diversity only perceived by difference and ratio.Some debate over the extent to which the processes needed to be made explicit before the childcan be said to understand or said to show this kind of formal reasoning. Compensation was heldby some to be a highly formal process.M1(Tf-T1) = M2(Tf-T2) as an example of compensation – but it took one hundred years to arrive atthis formalisation from the starting position of Tf=(M1T1+M2T2)/(M1+M2). So seeing the patternmay not be all that easy if it takes adults that long – education is about compressed discovery!Hide and seek given as an example of children’s ability to compensate – how far to run in the timeavailable. An innate concept of distance covered in the time available means some understandingof velocity must be present. The need can then be to learn how to mobilise these abilities todevelop new skills developing the everyday into the formal: learning the specialised ways ofthinking that are formal thinking. This is not easy – ‘con futiga grande’ was Galilieo’s comment. Asan example this transition is not be helped by just defining velocity as displacement / time withoutreference to the existing abilities and experiences of the child – so showing the need forsympathetic teaching.Real proportionality is handled inside the head. So on the Piagetian scheme:• 2A: doubling and halving are possible• 2B: partitioning can be done• 3A: functional handling of V=IR• 3B: seeing 3 rearrangements of V+IR as a part of one relationship.And until we can do proportionality in the head we should not call it proportionality – so notformal reasoning at all!What is understanding?Understanding is nothing like a layered explanation – it is simply a recognition of commonality,much like recognising a face, so illustrating the central role of metaphor in understanding.An answer to the problem of what it is to understand must include aspects of mastery within onedomain and transfer to another.The examples of temperatures coming into equilibrium show on a graph and mobiles used byprimary school children to were both used as examples to show some pathways by whichacceleration into the formal could be explored.Formulae are a language for describing the world – just like seeing the world though spectacles –if you have well polished lenses, kept clean, then you see the world through the spectacles, not theformula themselves. It is the same with language – you want, most of the time, to see through thelanguage, not look at the language. A picture of languages used in science:Is there a big discontinuity between physics as a school subject and physics as a research subject –are they perhaps two quite different subjects sharing the same name?

2. A view of the use of metaphor and modelA help to move from the concrete to the abstract. Metaphor and model used quiteinterchangeably. Making a model is the same ass using a metaphor – we see as – do notunderestimate the heightened understanding that comes from involvement in generating the

metaphors. A leap of imagination is needed. Beware of asking children seemingly innocentquestions that cover up huge problems in modelling – eg about dissolving.Models come in two types only, reflecting a basic structure in language between nouns andadjectives with verbs as transformers.Correlation between variables and interactions between systems.Models as information compressors – a lot of konwledge about the worlkd storedina sall place.Some debate over the use of formulae.One view of the formulae as a declaration: a statement of the dimensionality of space in which youare working by declaring the number of variables that the relationship correlates. The constants/parameters give shape to the landscape. Spatial representations of problems are particularlyvaluable because of the ability to see the whole at one – you can see it as – so a gestalt likeapperception of the solution is possible. In this way it is possible to make meaning for learners.This is a metacognitive statement for teachers, not for transmission to students in any explicit form.

3. The place of metacognitionThe teacher’s phase space must exceed that of the students. In showing understanding to whatextent must we ask the students to make things explicit – if they know how, must they also be ableto talk about it in order to qualify for understanding? (when we are working hardest at learninghow to do something new, only sometimes is it helpful to also reflect on that something explicitly –and even then perhaps not for all learners). Suggested that we speak about cognitive growth notcognitive change (after all what changes?)

4. Teaching as sympathetic interaction: mediationThe need for sympathetic interaction – to make teachers operate in the zone of proximaldevelopment rather than a long way above it.Students are rather good at everyday reasoning, so we might need to mobilise this competence tohelp with the transition from the everyday to the formal and scientific. Effective techniques formobilising these competencies will be varied and sympathetic to the understandings andcompetencies of the children.

5. To work on:• Practical outcomes from this discussion• Where to read more• Concrete illustration of how to evaluate a model• The place of imagination, motivation in assisting transfer from concrete to formal operations.



TOYS FOR LEARNING PHYSICS

Christian Ucke, Physics Department E 20, Technical University Munich, Germany

The participants started with the question: What does formal thinking mean for learning physicswith toys? We did not discuss this for a long time but agreed that many toys can help to make afirst step to the far away goal of formal thinking. Toys are very important in the kindergarten,primary and secondary school and less important on university level. Nevertheless there are sometoys which are appropiate for developing formal thinking on a higher level. The talk fromProfessor B. Pascolo about ‘Mechanics applied to small great games’ in the Panel-Session CS04gave an example for that.Very important for toys is their affective and social aspect. Toys are known from outside of theclassroom, this means from the real world. They have to do with normal life. Physics experimentsin classrooms support formal thinking but children often think that they are valid only in theclassrooms. The transfer to the world outside the classroom is difficult.With many toys can be made quantitative measurements on different levels.This is a real challengefor teachers and children. Playing with toys means exploring the world. A careful leading forquantitative investigating toys together with the teacher can reveal more properties of the objects.The pedagogical difficulty is the balance between fun and playing on one side and eventuallyboring quantitative measurements and analysis on the other side.Teachers needs books, publications, workouts, stimulations what toys exist, where they can beobtained, how they can be used and on which level they can be used. This seems to be also alanguage problem. In English and German there is an extensive literature about these topics,although many books repeat very similar proposals and only very few have really new ideas (e.g.in German: Wittman, Josef: Trickkiste 1 + 2). There are books for teachers and for children. Buteven this literature is not always available (some good books are even out of stock) and issometimes expensive. Translations will be not made because it is not worth for presses. C. Uckementioned his website (http://www.e20.physik.tu-muenchen.de/~cucke), where he maintains afreely accessible database about literature dealing with physics toys. The database contains about1000 references; about 500 are in English.Even if good books for children are available, it is important to notice that children will needsupport from their parents or teachers. They often will not start theirselves to make experiments.The same is valid for proposals in the web which can be found more and more (e.g. in German:http://www.kopfball-online.de/ and also the website from C. Ucke contains many links to othersuggestions)Another problem is the availibility of toys. Many toys are only available for a certain time and onlyin certain shops in certain countries. The teachers must look carefully and regularly through thelocal shops to discover what is available. This is an additional straining.We discussed that this could be a chance and a task for the shops in science museums. They shouldoffer appropiate objects with good accompanying workouts. In this way they can fulfill animportant additional task. A problem seems for the museums how to organize this (manpowerproblem). Science teachers could cooperate with museums in this direction.K. Papp mentioned her way for examinations. She presents to students about ten toys. The studentmust select one toy and discuss and explain how she/he would use this toy for which purpose in aclass. This seems to be a very realistic approach how to examine students.We discussed the idea for a special seminar/workshop only devoted to toys. This should beprepared carefully but the need for such an event seems to be present.C. Ucke mentioned two links:Probably the best link in Germany for physics teachers is the “Teachers Page Physics”:http://wpex40.physik.uni-wuerzburg.de/~pkrahmer/home/index.htmlThis site is maintained from the physics teacher P. Krahmer.

1242. Special Aspects

http://www.grand-illusions.com/cd.htm (look for: Tim’s Wonderful Toys CD-ROM; many picturesand avi-files about toys)At the end we discussed to establish a database where you can find toys. Such a database canstimulate the exchange of information. The database should contain the names including tradenames (there is again a certain language problem), picture, short description (level, specialapplications, etc.), topic area (mechanics, optics, etc), reference/literature, source (producer), price,, contact person, entry date.The database must be maintained with a program which is generally available (e.g. Excel).C. Ucke declared to work out a first proposal with some examples (see table below).

ParticipantsSome persons attended only one session:M. Bertoncelj (Slovenia), A.G. Blagotinsek (Slovenia), A. Borgnolo (Italy), M. Cepic (Slovenia), D. Ferbar

(Slovenia), E. Gunacker (Austria), J. Holbech (Denmark), R. Martongelli (Italy), M. Matloob (Iran), A. Nagy(Hungary), S. Oblak (Slovenia), K. Papp (Hungary), P. Pascolo (Italy), G. Planinsic (Slovenia), N. Razpet(Slovenia), E. Stante (Italy), C. Woerner (Chile),

Interested:M. Michelini (Italy), G. Rinaudo (Italy)

e-mail [email protected]

name picture description keywordsreference/litera

turesource price

contact

person

entry

dateremarks

jumping toy, jumping

smiley face, Pop-Up

the toy jumps about 30cm

high; similar toys available

as jumping animal with

stronger springs, can be

used from kindergarten to

university level

mechanics,

spring,

acceleration

Carl H. Hayn,

Pedagogical

Interlude, Phys.

Teach. 28, 166

(1990), Dufresne,

R.J. et al.:

Springbok: The

Physics Jumping,

Phys. Teach. 39

(2001), 109-115

science

shops, toy

shops, Tobar

Ltd/UK

< 0.5 Euro C.

Ucke/Munich/

Germany

[email protected]

.de

set/ 01

whistler, (slovene:

živžga )

Any movements of the toy

along the geom. axis

produce whistling sounds.

No electrical parts. Can

be used as hands-on

accelration sensor.

mechanics,

acceleration,

sound

toy shops,

bookstores,

(UK)

1 Euro G.

Planinsic/Ljublj

ana/Slovenia/g

[email protected]

lj.si

set/ 21

popgun; (slovene:

pokalica)

wooden tube with piston

and cork stopper at the

end. When compressed

the stopper shoots out

and makes loud sound.

The pitch of the sound

depends on the length of

the air column.

mechanics,

thermodynamic

s, sound

Polish folk toy

bought in

street shop in

Zakpane,

Poland

1 Euro G.

Planinsic/Ljublj

ana/Slovenia/g

[email protected]

lj.si

set/ 21

bicycle pump that

utilizes creamer

charges

bicycle pump designed to

utilize compressed N2O in

creamer charges.Can be

used to demonstrate

adiabatic cooling and

change in whistle sound

due to lower c in N2O.

thermodynamic

s, sound

pump bought

in sports shop

in NM, USA.

Should be

addapted for

using

European

cream

charges

15 Euro G.

Planinsic/Ljublj

ana/Slovenia/g

[email protected]

lj.si

set/ 21 DO NOT

INHALE N20

GAS!! use child

rubber baloon

and whislte for

sound

demonstration


NEW TECHNOLOGY AND COMPUTER IN PHYSICS LEARNING

Laurence Rogers, School of Education, University of Leicester, UK

The thematic questions offered initially were:• How effective is IT for learning?• How does the teacher make IT more effective?To set the scene, Laurence Rogers presented a general survey of IT use based on common practicein England. IT uses were classified under the following headings:• Information systems (web, CD, multimedia, tutorial applications, databases)• Publishing tools (word processing, presentational software)• Visual aids (depending on graphics, images and audio technology)• Calculating, modelling and simulation tools (including spreadsheets)• Graphing tools (for presenting and analysing data from a variety of sources)• Measurement (data-logging, MBL, RTEI)• Communication (email, discussion groups etc.)For the purpose of identifying training needs and pedagogical issues, an analysis of software typeswas proposed, described by ‘software properties’, ‘potential learning benefits’, ‘operational skills’and ‘application skills’. The idea of ‘transformational skills’ was added to these descriptors inrecognition of the need for teachers to modify their practice to acquire application skills.In the initial discussion, workshop members declared their interests by suggesting a subdivision ofthe thematic questions into the following:

1. Why is the uptake of IT (computer-based measurement) so poor in schools after so manyyears of development and refinement of use by enthusiasts?Before this question could be explored it was necessary to query the evidence for the assertion oflow uptake and try to be clear about what criterion is used for making the judgement. The premisewas apparently not true in all countries, in particular the Netherlands. This special case ofsuccessful implementation was discussed in some detail and the principal conclusion was thatabout ten years ago the Netherlands adopted a systemic approach to education reform which tooksimultaneous action in four key areas:• curricular reform which not only integrated IT but also attempted to harmonise educational

philosophy across all curriculum subjects• reform of the examination system which incorporated IT targets in all subjects• development of a high quality resources for teaching IT across all subjects• providing compulsory training for all teachers in the use of ITThe absence of any one of the four components would have led to failure.The simultaneity of thesereforms was thought to be a key factor in their success. Moreover, it was significant that thecurriculum reform instituted a strong cross curricular context for IT rather than establishing aseparate IT curriculum ‘power base’ which might compete with individual subject interests.It was generally thought that it would be difficult to replicate these ideal conditions in manycountries, especially in the USA where local autonomy in the control of education was regardedas a fundamental human right! Consequently, methods of persuading teachers to adopt IT in their subject teaching was of greatconcern in most countries represented.

2. How can teachers be motivated to adopt IT?In the Netherlands and England, teachers are compelled to adopt IT by the national curriculum.In Italy IT is a compulsory part of the curriculum but not in a cross-curricular context. In Mexicoand other countries represented free choice prevailed and adoption is pioneered by teacherenthusiasts. One proposed approach to motivation was to demonstrate and convince teachers thatIT holds real benefits to their productivity and effectiveness as teachers. This prompted twofurther questions:

Are teachers aware of the benefits if IT?Do teachers value the benefits of IT declared by ‘experts’?

A suggested strategy for helping teachers cope with possible ‘technophobia’ is to rely more on theirpupils for expertise in operational skill. There are implications here for the teacher’s preferredteaching style and how modifying this would affect their self-esteem. This links to the nextquestion.A successful motivation strategy used in Denmark and England has been the inducement ofteachers by offering them free or subsidised laptop computers. Working at home, teachers put inmany more hours training effort than is affordable in normal school hours.

3. How do teachers find time to learn and prepare to use IT?The discussion questioned whether there was any difference between the time investment neededfor IT and any other innovation of method or tool; after all, IT is simply another educational tool.There is possibly a strategic planning issue here in that the provision of IT hardware in schoolsneeds to be accompanied by a realistic allowance of resources for teacher training, preparation andequipment maintenance. Some further suggestions linked back to the motivational issue, such thatif teachers could be convinced of the benefits of IT, they might then place a higher priority on theclaim that IT might have on the fixed time available for non-teaching activity.A variant of the time problem was raised: How can pupils be allowed sufficient time in thecurriculum for IT use? There is perhaps a paradox here in that computers sometimes/often enablemany classroom tasks to be performed quickly. It is necessary to analyse this in more detail infurther discussion. Perhaps it has to be recognised that the demands of any curriculum alwaysexceed the time available, such that choices have to be continually made about priorities; IT is justanother competing element.

4. What are the pitfalls in using IT that should be avoided?This question was proposed with some passion but discussion only just began to address the issue.It became linked to a questioning of the effectiveness if IT with statements like “the computershould not substitute a real experiment or the use of the blackboard or some other conventionaltool” and “the use of computers is burdened by the risk of pupils using them in an off-task mode”.The subsequent discussion gave thought to the principle of ‘appropriate use’ by which IT is notemployed for its own sake by it selected only when it is the most appropriate tool for the identifiedlearning objectives. The latter needs to consider not one but a variety of factors which include;context of use (experience and ability of pupils, curriculum topic, role of task as exploration orreinforcement etc.)

5. What is the role of the teacher when pupils use IT?There are potentially two risks associated with this question. First, a fear amongst teachers that thecomputer will displace the role of the teacher, and second that teachers might become sopreoccupied with technical problems with hardware and software that they forget, neglect or ignoretheir pedagogical role. Research suggests that the role of the teacher in making IT effective forlearning is crucial. But further questions are prompted:How does (should) the use of IT affect the style of teaching? (IT can transform the social climateof the classroom and requires re-evaluation of what ‘content’ is relevant.)Perhaps the setting of open-ended tasks and greater pupil-autonomy are better suited to IT thandidactic teaching approaches?Perhaps IT encourages a greater mixing if physics topics than traditional approaches?

6. How can teachers be educated to make IT more effective for learning?Can they be encouraged to adapt their well practised teaching skills? How might pupils’ computersat home be exploited? Further discussion needed.

7. How can the physics teaching community gather and share ideas for good practice?


Proposals so far: Use GIREP discussion page on internet.Compile list of time-saving uses of IT.Note references to the work of Alfred Bork (1978), Elena Sassi (2001), Priscilla Laws and others tobe identified.Evaluated physics websitesGuidance on how to design good activities with IT.Identify a few ‘brilliant’ examples of IT use. Present these at a workshop at Lund 2002.List research references.

8. A further question to address: What of the future?What is the next technology which will demand a response from the educational community? Willwe be led by and be reactive to the technology as it develops further, or should teachers be givinga lead in influencing future developments?

TEXTBOOKS AS AN IMAGE OF PHILOSOPHY OF TEACHING

Zofia Golab-Meyer, Institute of Physics Department, Jagellonian University, Poland

The turn over of Millenia is a good moment for reflection on changes in physics teaching, on a newtrends and fashions in it. Examining textbooks we can learn how a new trends, ideas are at least putin praxis. During the last few decades we learn a lot about way of thinking of children and adolescentstudents. New approaches (eg. constructivistic approach, whatever it means, inquiry methods) areproposed. New teaching and learning methods are known, new media in our hands. Under the verystrong social pressure the basic goals and priorities changed, from understanding a fundamental laws,to understanding environment, technical gadgets in everyday life. Relevance to everyday life is thekey word in the physics education.To see it one can look at the recommendations of Physics on Stage,Geneva 2000. As we know, in mass (but in particular cases!) scale traditional teaching failed. Newtrends and ideas became dominating. How is it reflected in textbooks? To have our discussion moreconcreat we discussed one elementary textbook (presentation of Leopold Mathelisch) and one highschool textbook (presentation of Krsnik).Among discussed topics were• Language of textbooks: everyday language is replacing scientific language. It is in hope that such a

language is understood and can transmit the correct message. Short slogans are replacingmathematical formulae, and have to be vehicle for grasping the essence of physical laws (e.g.“Waerme can man mischen”, from presented textbook). We discussed (just start of discussion)how much of misconceptions such a slogans can introduce. More profound investigations (taskfor the working group) are needed. Some expressions, formulations from Hewitt’s textbookConceptual Physics , can be subject of discussion in future (available for members for differentcountries).

• Targets of textbooks. Traditional textbooks are written both for students and for teachers. Forstudents they serve as manual for self study, they are written in narrative style, they explain laws,they conduct students in doing experiments, they help to draw conclusions, they offer problems forsolving. They contain summaries which help to remember. Some of those textbooks became verypopular, and are often used by students. Some of them are written in the way of direct dialogbetween author and reader. Such textbook have signs of individuality of its authors. Classicaltextbook play a role also as teaching guide for teachers. The aversion of reading commonly shownby students is a origin of so called “pocked textbook”, consisting only definitions and formulas. Onecan not learn from such a “textbook”. New generation of textbooks, especialy for young studentstake into account the different need of students and teachers. Textbooks for students are short,without much of narrative text. It is assumed that teacher will be very active in teaching process.Teacher is getting separate textbook, guide for teaching. Presented Austrian textbooks are exampleof such solution. It seems that in future will be like that.



• Problems. We discussed trends in going out from classical academic problems with welldefineddata, and one solution.A kind of narrative problems were presented by Martha Masa. Solving withstudents more difficult problems were presented by Italians.The number of questions arose duringour discussion. They are worthy of more profound discussion.

• Ethical responsability of authors of textbooks.• The ro le of history in teaching, especialy its role of making physics more human was also subject

of discussion.From our discussion we learned that new textbooks try to be friendly for students.That is new. In thepast transmission of knowledge was first priority. During discussion we realized that new trends inphysics teaching are present in textbooks.In examing textbook we have also to look:• how much students can learn from that was intended• what kind of potential misconception students can gain• how much students can learn how to learn• what kind of image of physics is created


3. Topical Aspects

3.1 Laboratory and Theory

USING EXPERIMENTAL LABORATORIES TO TEACH FORMAL PHYSICS

Slavko Kocijancic, Faculty of Education, University of Ljubljana, SloveniaColm O’Sullivan, Department of Physics, National University of Ireland Cork, Ireland

1. IntroductionDifficulties encountered by students in the use of mathematics in introductory physics courses hasbeen a constant theme in physics education debates in recent years, not least at GIREP meetings.In many cases, however, the problem turns out to be less to do with an absence of mathematicalskills per se than with a lack of familiarity with how we as physicists use certain simplemathematical techniques. For beginner students, in particular, this often involves difficulties inunderstanding how two specific analytical techniques, namely graphical and functional analysis, areused in physics.The problem of student difficulties in connecting graphs to physical reality is well known [1] as isthe value of laboratory experiments using real-time computerised data acquisition with graphicaldisplay to address these problems [2,3]. The ultrasound position/motion sensor (‘sonic ranger’) hasproved to be a particularly valuable tool in this context as described, for example, by Sassi [4] atthe GIREP/ICPE Conference in Barcelona. Figure 1 shows a simple and commonly used exampleof such an application in which the motion of a cart moving with constant acceleration is studiedby observing a real time plot of distance versustime.Most commercially available data acquisitionsoftware systems include a curve-fitting toolwhich enables data like that in Figure 1 to beanalysed so that the value of the acceleration canbe determined. This is achieved by asking thesoftware to analyse the data to find the best fit tothe function x = a + bt + ct2 using a, b and c asfitting parameters, from which the acceleration(= 2c) may be calculated.Experience of curve-fitting techniques has theadded advantage of helping students toovercome the other common difficultyencountered by beginners, namely that ofconnecting mathematical relationships (‘formulae’) to physical reality. For more complexrelationships than those arising in simple kinematic situations like that described above, however,the standard curve-fitting tools provided with most data acquisition systems prove insufficient.Curve-fitting features available on standard spreadsheet software, such as Excel, are also toolimited for many applications in introductory physics courses. Fortunately many flexible userfriendly curve-fitting packages are available which can be used to advantage in undergraduatephysics laboratories to enhance students’ understanding of the way mathematical functionalrelationships are used to describe the behaviour of physical systems. It is essential that the utilityadopted be uncomplicated and simple to use by students who may have little experience ofinformation technology at this level.It is also important that, prior to invoking such curve-fitting software to analyse experimental data,

Figure 1

students should have an opportunity to familiarise themselves with the package at least to theextent that they can convince themselves that the software does indeed execute what it purports todo. On the other hand, students do not need to know anything about the actual mathematicalcurve-fitting procedures running in the background, any more than they need to know how acalculator, a computer or a video display unit works in order to use these devices with confidence.

Example 1: simple harmonic motion

Figure 2(a) shows data from an experiment on an oscillating helical spring, again using theultrasound motion sensor to record and plot the extension of the spring as a function of time. Inthis case the data is then exported to a simple shareware curve-fitting package [5] which is askedto find the best fit (figure 2(b)) of the data to the function x = a sin(bt + c), with a, b and c as thefitting parameters. The value of b (ω = 2π/T) obtained may be used to calculate the period T of theoscillations.

Example 2: damped harmonic motionA somewhat more complicated function is involved if damping plays a significant role in theoscillations of the spring. Figure 3(a) shows data observed using the same helical spring as abovebut in this case a dash-pot has been added to provide damping. Figure 3(b) shows the data havingbeen fitted by a function of the form x = a sin(bt + c)exp(–dt) in which case the damping coefficient(d) may be determined as well as the period.

Example 3: magnetic field strength on the axis of a coilThe techniques described above may be used, of course, in any laboratory situation where onephysical quantity may be plotted as a function of another. In addition to the position sensor used

130 3. Topical Aspects 3.1 Laboratory and Theory

simple harmonic motion

time/sa

mp

litu

de

/cm

0.0 1.0 2.0 3.0 4.0 5.0-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

Figure 2(a) Figure 2(b)

Figure 3(a) Figure 3(b)

in the previous examples, low cost computerised sensors are now available for a wide range ofphysical quantities, the most commonly used being voltage, temperature, pressure, light intensityand magnetic field.

As a final example, Figure 4(a) shows a plot of the strength of the magnetic field as a function ofposition along the axis of a flat circular coil carrying a current. An ultrasound position sensor hasbeen used to determine the position of a magnetic field sensor, distance x being measured from thecentre of the coil. The mathematical relationship which describes the dependence of the magneticflux density on x, is


Figure 4(a) Figure 4(b)Figure 4(a) Figure 4(b)

( ) 23

22

2

2)(

ax

NIaxB

+=

Figure 5

where N is the number of turns in the coil, I is the current flowing and a is the radius of the coil.Once again the curve-fitting routine has been invoked to find the best fit to this function (Figure4(b)).An alternative approach in this case, because the values of all the relevant parameters are knownab initio, is to use a function-plotting routine in advance to predict the shape of the B versus x curveand to compare the results of this prediction with the actual measured data. Figure 5 shows the useof a simple freeware package [6] for this purpose.

2. ConclusionsSimple user friendly curve-fitting and function-plotting software routines used in conjunction withcomputerised data acquisition systems in the laboratory can be effective in helping beginner

students to overcome initial difficulties with understanding how graphical and algebraicrelationships are used in physics.

References[1] L.C. McDermott, M.L. Rosenquist and E. H. van Zee, Student difficulties in connecting graphs and physics:

Examples from kinematics, American Journal of Physics, 55, No 6, (1987), 503-513.[2] L. Newton, Graph talk: Some Observations and Reflections on Students’ Data-logging, School Science Review,

79, (1997), 49-54.[3] L. Rogers, New Data-logging Tools - New Investigations, School Science Review, 79, (1997), 61-68.[4] E. Sassi, Computer supported lab-work in physics education: advantages and problems, International Conference

on Physics Teacher Education beyond 2000, Barcelona, (2000) (CD-ROM).[5] The curve-fitting package CurveExpert is available from http://www.ebicom.net/~dhyams/cvxpt.htm[6] The freeware function-plotting package wgnuplot can be downloaded from ftp://ftp.ucc.ie/pub/gnuplot/

PHYSICS: FACING THE PRESENT TO FOSTER THE FUTUREThe Portuguese and the English Proposals for 16-19 years old students

Teresa Lobato, Margarida Saraiva-Neves, Escola Secundária de Fonseca Benevides, Lisboa,Portugal

1. IntroductionThe importance of learning Physics is recognized for scientists and science teachers in general. Inspite of its importance on developing techniques and devices to make more comfortable our dailylife, students are more and more unmotivated to follow courses on Physics, and those who darestudying it get, generally, poor results.In the beginning of a new millennium, it is imperative that teaching and learning Physics becomemore exciting tasks so students, engaging on its study more enthusiastically, become better-informed citizens and/or scientists able to contribute to the welfare of society.Several countries recognized the importance of restructuring the programs of Physics on anattempt to render its learning more appealing to students.This agrees with the position of authors as Robitaille et al (1993) that consider curriculum as animportant variable to explain differences among national school systems and to account fordifferences among students’ outcomes.From a constructivistic point of view, students should play important roles in the building of theirknowledge. Computers and computers’ software may be used in science education as means ofpromoting Meaningful Learning. However, pupils must have teacher’s guidance to accomplishcomplex tasks.

2. Mode of InquiryIt was done a documental analysis of the Portuguese official programme of Physics-A, 10th gradeand of United Kingdom curriculum – Advancing Physics.

3. Outcomes3.1 Some characteristics of the Portuguese curricular proposalThe curriculum of Physics recently projected in Portugal, regarding secondary level education,states, as one of main objectives, the contribution for a democratic citizenship. Students’ motivationfor further studies on science and related carriers is also an important concern. To accomplish this,conceptual dimension of Physics is faced not as an end, but as a powerful mean to accomplishmentioned aims. Science education means learning concepts, laws and theories, as well as thinkingabout the nature of knowledge and its importance in modern daily life.The methodology suggested appeals to several means of communication, a strong component oflaboratorial work and team discussions. Though proposed experiments deal mainly with currentmaterials, the use of new technologies is highly recommended. Mathematics is regarded as a meanto a better way of understanding Physics, instead of an obliged boring subject.


All curriculum organization is focused on the understanding of Science and its relation withTechnology and Society. Problem solving is the chosen way of performing STS curriculum.“Conservation and Degradation of Energy” is the unifying subject of Physics-A 10th gradePortuguese programme, allowing the study of concepts included in Thermodynamics, Mechanicsand Electricity. Contents are organized in four units, each one according to a theme: “From energysources to the conscious user”;“From the sun to the heating”;“Energy towards the movement” and“ From electrical power plants to the consumer”. Intended learning level is specified by objectivesdefined for all units.Main characteristics found are the following:• Science is knowledge in permanent construction• Negotiation of meanings through team work is highly encouraged• Impact of physical knowledge on society is recognized• Experimental activities are considered main promoters of Meaningful Learning• All curriculum has a STS orientation

3.2 Some characteristics of the English curricular proposalSocial issues and the promotion of team discussions are considered important in new curriculumAdvancing Physics. Students’ interests for recent technology seem to be the starting point oflearning. The criterion to choose learning subjects does not exclude traditional topics but presentsthem with a different approach.The new course is designed to reflect up-to-date aspects of Physics in action, including relevantsocial issues. The need to adapt to a changing world where electronic devices, digital images andquicker communications are forcing new life styles, is a main concern.It is expected that students understand Mathematics as an important body of knowledge for abetter comprehension of physical laws. This matches Ogborn (2000) position that considersMathematics must not be seen as a “necessary evil to complain about”, but as fundamental toenlarge the pleasure and power that Physics can offer.Students are encouraged to build and execute their own projects, which will be part of theirassessment, which includes the ability to communicate, and the comprehension of science andtechnology in a wider context.Main characteristics found are the following:• Team discussions are encouraged• Up-to-date physics and social issues are relevant• Practical work is central and has new perspectives• Maths is intended to be regarded as a support to develop Physics• Computers are commonly used• Students are encouraged to develop their own projects

4. Conclusions and ImplicationsBoth new Portuguese and English curricula have similar concerns: attract more students to thestudy of Physics; relate school studies with daily life problems; minimize the fear involving the needof knowing Mathematics; approach the subjects through recent technology and use the study ofscience to build more responsible citizens and better human beings.Essential differences are revealed with a closer look on the way the all process starts. Englishapproach largely appeals to latest technologies such as computer data and CD-ROM support ofinformation, side by side with traditional books.In Portugal, although computer techniques are becoming more and more common, the projectedcurriculum has a more classical approach.According to Teodoro (2001), the use of exploratory software, which considers computer as a“thinking tool”, agrees with the constructivist perspective of the teaching-learning process.However being regarded as a wishful tendency in Portuguese official curriculum, the use ofcomputers are still very far from being a common practice in the classroom environment.


Something must be done about this, as it is not possible to go on ignoring technology when “theyoung of today are Homo-Zappiens: they Zap from channel to channel.They live a life with remotecontrols, cellphones and Internet” (SjØberg, 2001).Another striking difference is that 40% of English project contemplates themes which arededicated to recent issue; Portugal is still very far from modern times, when classes of Physics areconsidered.English projected Physics program allows students to have a considerable degree of freedom tochoose subjects according to their own interests. In the Portuguese Physics curriculum thatpossibility is not foreseen, but it must be stated that there is a curriculum area, named “ProjectArea”, where students may choose and carry on their own projects.Finally, a very similar concern in both projected curricula: the continuous formation of teachers.But, unfortunately for Portugal, the two countries reveal very different ways to deal with thisproblem. In UK, 300 teachers were involved in the development of the innovating course, and anetwork of local teachers’ groups were created to attend regular meetings to discuss problems,seeking advice, recording progress and, sometimes, complain loudly. In Portugal, so far, teachers areessentially spectators of the changes happening in curricula that will be implemented on 2002/2003school year. Up to this date, there is no public knowledge of any attempt to perform innovatingcourses to help teachers to deal with and feel these changes as theirs.Though not as daring as the English, the Portuguese project is an interesting one and itsimplementation could represent a very important step on improving Physics teaching and learning.As the success of the proposed changes strongly depends on the will and performance of teachers,no significant improvement will be achieved without their collaboration. This should not be aproblem as we can count on very good professionals who are deeply committed to face thechallenge of educating today’s youth, fostering a better world.In Portugal, studies on Science Education are very recent. However pacing slowly, they are movingwith determination. A lot of researchers are carrying out investigations in several fields anddeveloping educational materials, such as computer software, being a good example Modellus –Interactive Modelling with Mathematics. With this software students may design and test models,introduce alterations, predict outcomes and quickly visualize their own ideas. This will, eventually,help them in the process of transition from concrete to abstract thought.If you want pupils to perform complex skills, then teacher must orient the work proposals with thatobjective (Mulder & Ellermeijer, 2001).Teachers should be aware of the importance of curriculum innovations and have access to specialinformation/formation. Then, new technologies could become part of teachers’ strategies, aimingthe development of formal thinking in Physics.

AcknowledgementsFor the financial support we thank the Portuguese Foundation for Science and Technology.

ReferencesHobson, A., Physics – Concepts and Connections. New Jersey: Prentice Hall, (1995).Ministério da Educação, Revisão Curricular do Ensino Secundário. Lisboa: DES, (2000).Mulder C. & Ellermeijer, T., In Proceedins of the 3rd International Conference of ESERA on Science Education

Research in the Knowledge Based Society, Thessaloniki, (2001).Ogborn J., Changing the Curriculum in Physics, Unpublished paper presented at the GIREP conference “Physics

beyond 2000”,Projecto de Programas de Física e Química – A (2000). Lisboa: Ministério da Educação.( http://www.des.min-edu.pt/ver_curricular/programas/física.pdf )Robitaille D. F., Scmidt W. H.; Raizen S., Mcknight C., Britton E. & Nicol C., Curriculum frameworks for Mathematics

and Science, TIMSS Monograph nº 1, Vacouver, Canadá: Pacific Educational Press, (1993).Silva T. T., Teorias do Currículo, Porto Porto Editora, (2000).SjØberg S. Why don’t they love us anymore? – Science And Technology Education: A European high priority

political concern, In Proceedins of the 3rd International Conference of ESERA on Science Education Research inthe Knowledge Based Society, Thessaloniki, (2001).

Teodoro V. D., Modellus 2.01, Faculdade de Ciências e Teccnologia, Universidade Nova de Lisboa, (2000).Teodoro V. D., Experiências Matemáticas: um exemplo com oscilações. Unpublished paper. Faculdade de Ciências e

Teccnologia, Universidade Nova Lisboa, (2001).


TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR

Samo Lasič, Gorazd Planinšič, , Faculty of Mathematics and Physics University of Ljubljana,SlovenijaGiacomo Torzo, Department of Physics, University of Padova, Italy

1. Introduction: applicability of the didactical apparatusThe main purpose of talking about nonlinear phenomena in school is to gain the students abilityfor distinguishing the nonlinear system from the linear one by qualitative observations of thesystem dynamics. The complex and qualitatively different behavior of nonlinear systems should bepointed out as an remarkable consequence of non-linearity that happens to occur very often innatural systems. Formal thinking plays an essential roll when basic features of a nonlinear systemand its dynamics are compared to the linear system. A qualitative approach is particularly fruitfulfor developing the basic idea of the phenomena and gaining a deeper understanding of both linearand nonlinear systems.The simplest example of nonlinear oscillator is the physical pendulum. The torsion pendulum [1, 2,3] instead is considered as a didactical example of a nonlinear system that performs all the basicfeatures of a dynamical system. According to the equation of motion the torsion pendulum ischaracterized as a Duffing oscillator. The Duffing equation is often numerically solved to analyzeits dynamics [4, 5]. There are also known electric circuits that behave according to the Duffingequation [6]. Such systems are very precise and easy to analyze but they are abstract and hard touse for developing formal thinking.The basic idea has been initiated and first experiments performed by prof. dr. Giacomo Torzo atUniversity of Padua. The theoretical analysis as well as design and construction of improvedapparatus has been done at Physics Department, University of Ljubljana as BSc thesis [7].The apparatus can be used as a didactical tool for demonstrating:• the amplitude dependence of the period of oscillation,• the hysteresis phenomena in the response of sinusoidaly driven pendulum and• bifurcation of stable orbits and the transition to chaotic motion.Our construction is based on theoretical predictions and numerical simulations. The goal was tobuild a mechanical oscillator of large dimensions that would help the observer to recognize andunderstand the phenomena. The main problem was to find the design that will optimize therequirements mentioned above. For this reason we have chosen the construction that leaves all thefree parameters to be varied to some extend.

2. Description of the apparatus


Building parts:

1. Oscillatory rod2. Mass3. Damping disk4. Damping magnet5. Steel tape6. Driver with adjustable eccentric arm7. Windscreen wiper motor8. Buzzer9. Movable bench10. Spring for fine adjustment of zero angle11. The pendulum head12. Inertia rod with variable moment of inertia

Fig. 1. Vertical torsion pendulum

The oscillatory part of the pendulum consists of a weighted rod (Fig. 1 – part 1, part 2 ) fixed in thependulum head (Fig. 1 – part 11 ) together with inertia rod with two symmetrically placed weights(Fig. 1 – part 12). The weights can be fixed on different positions enabling the center of mass andthe moment of inertia to be changed respectively. The pendulum rod can be fixed either in thedirection upward or downward. In this way two types of nonlinear restoring force can be achieved.The driving mechanism consists of a windscreen wiper motor (Fig. 1 – part 7) and the rotating discwith eccentric driving arm (Fig. 1 – part 6) which transforms the rotation in approximatelysinusoidal oscillation. The amplitude of driving oscillation can be varied in the range from 0° to 45°by changing the eccentric fixing of the arm.The excitation is transferred to the oscillatory part of the pendulum through torsion distortion ofthe steel tape (Fig. 1 – part 5). The driving mechanism and the fixing of the tape to the driving armare fixed on a movable bench (Fig. 1 – part 9). This allows using tapes of different length l, enablingone to change the tape torsion coefficient which is proportional to l-1.The viscose damping can be applied by bringing the magnet close to the 4 mm thick aluminum disc(Fig. 1 – part 3, part 4).The fine adjustment of the zero equilibrium angle has been done by using two symmetrically placedlinear springs (Fig. 1 – part 10). attached to the strings that are winded around the pendulum head.Each spring is fixed on a screw in the support to allow independent tension adjustment. In additionthe springs help to reduce the torque on the motor.The buzzer (Fig. 1 – part 8) gives a sound signal once per driving cycle. This is an excellent remedyto help the observer following the phase difference between pendulum and the driving oscillation.The buzzer is particularly useful for observing the period doubling transition to chaos.

3. Mathematical model of the torsion pendulumIn order to design a pendulum that will satisfy the demonstration requirements one shouldcarefully choose the parameters. So let’s take a look at the torsion pendulum model.If the oscillation of driven torsion pendulum is confined to moderate angles (smaller then say 30°)the motion may be accurately modeled by the Duffing equation [8]

where c is the damping, coefficients α and β determine the linear and nonlinear part of therestoring force and F cos(ωt) is the forcing term.The sign of the coefficient β depends on the direction of the restoring force and determines thecharacter of the pendulum behavior. In case where the pendulum rod is fixed in downward positionβ<0 (regular torsion pendulum), while for the rod in the upward position β>0 (inverted torsionpendulum).The remarkable difference between the two types of torsion pendulum is seen from the form ofpotential energy that determines equilibrium and stable equilibrium angles (Fig. 2). In the case ofregular torsion pendulum (Fig. 2.a) the potential energy is of the form

and in the case of inverted torsion pendulum (Fig. 2.b)

where k is the torsion coefficient, m is the mass at center of mass distance R and θ is the anglebetween the pendulum rod and a vertical line.


)cos(3 tFxxxcx =+++ &&& , (1)

cos2

1)( 2 mgRkU = (2)

cos2

1)( 2 mgRkU += , (3)

Fig. 2. Potential energy form in the case of regular torsion pendulum (a) and in the case of inverted torsion pendulum(b).

In the case of regular torsion pendulum the only equilibrium is a stable equilibrium at zero angle.In the case of inverted torsion pendulum instead we must consider two different situations. If theratio k/mgR>1 there is only one stable equilibrium at zero angle similarly to the regular case. Butfor k/mgR<1 the zero equilibrium is a labile one. Beside this there are two symmetrically placedstable equilibrium angles

4. Period dependence on amplitude and the response to sinusoidal excitation When the non-linearity is small the higher harmonics can be neglected and harmonic type ofoscillation can be assumed. With this assumption we can calculate the period dependence onamplitude (Fig. 3) and the response to the sinusoidal excitation (Fig. 5).To the first order in β the period dependence on amplitude is given by

where ω is the circular frequency.


(a) (b)

±=± mgR

k16 . (4)

22

4

3A+= , (5)

2222

322

4

3)( AcAAF ++= (6)

Fig. 3. A schematic ω(A) diagram for freeDuffing oscillator. The dashed line representsthe harmonic oscillator.

Fig. 4. Measured data for free oscillation of theinverted pendulum.

The period variation is particularly evident in the case of inverted pendulum where the periodincreases when the oscillation is damped (Fig. 4).To calculate the response of the driven pendulum the Duffing iterative method is often used. Theresult can be written in the implicit form:


with the phase shift given by:

243tan

A

c

+=

2. (7)

The response diagram shows the hysteresis phenomena. There is a region in the response diagramwhere two physical states of the system are possible. The resulting state depends on the initialconditions or on the value of the past driving frequency. When the driving frequency slowlyincrease the amplitude and phase jump occurs from point 1 to point 2 on the diagram but when thefrequency decrease the jump occur from point 3 to point 4 (Fig. 5). To see the effect one must waitfor transient oscillations to die out. The assumptions we took for the calculus of the responsediagram seem realistic as the results (Fig. 6) confirm.

Fig. 5. The hysteresis in the response diagram forβ<0. The jump occurs from point 1 to 2 when thedriving frequency increases and from point 3 to 4when the frequency decreases. The dashed linerepresents free oscillations. With the dotted line wemarked the non-physical solution.

Fig. 6. Measured data confirm the hysteresis in theresponse diagram. Here is an example of invertedtorsion pendulum (β>0).

5. The transition to chaos for inverted torsion pendulum The complexity of the response increases with non-linearity. This is especially evident in the caseof double potential energy well. The labile equilibrium at zero angle brings into the system theweak causality thus onsets the system for the chaotic motion. For some combinations of systemparameters α, β, c, F, ω there is no periodicity and no evident order in the response oscillation.Choosing the parameters α and β which are determined by the pendulum dimensions one canchange the damping c, the driving amplitude F or the driving frequency ω to enter the chaoticregion. However, the transition to the chaotic region is gradual. Bifurcations of stable orbits followthe Feigenbaum scenario of period doubling finally reaching the chaotic motion.With the didactical apparatus it is possible to vary any of the system parameters in order to achieveclear transition to chaos [4]. In our case the frequency can be easy adjusted by varying the voltageon the driving motor. We have chosen to enter the chaotic region by slowly decreasing the drivingfrequency. At relatively high frequencies the response repeats after one driving cycle (Fig. 7) whatis usually called a one-period motion. If the frequency is gradually decreased one-period motiontrajectory becomes unstable. The motion in phase diagram is now attracted to two differentintersecting stable orbits. After one driving cycle the motion reaches the point of intersection andchanges the orbit. In this case the response repeats after two driving cycles thus called a two-periodmotion (Fig. 8). The similar bifurcation phenomena with period doubling from two-period to four-period motion (Fig. 9) occurs when the frequency is further decreased. The driving frequencyintervals from one doubling to another are geometrically decreasing and finally lead to the chaoticresponse (Fig. 10).

6. ConclusionsExperiments with the apparatus evidently confirm the theoretical predictions. The prototype canbe used for developing an efficient didactical tool. The large dimensions result in slow and evidenttransients, easy observing the phase shift and the instability of periodic orbits in the transition tochaotic motion, thus making possible to gain a quality insight into the physical phenomena. Withlarge dimensions the phenomena is fascinating, especially in the case of chaotic motion. However,too slow demonstrations are not efficient in classical lessons. For this reason the dimensions shouldbe appropriately reduced.

References[1] B. Duchesne, C. W. Fischer, C. G. Gray, K. R. Jeffrey, “Chaos in the motion of an inverted pendulum: An

undergraduate laboratory experiment”, Am. J. Phys., 59, (1991), 987-992.[2] C. L. Olson, M. G. Olsson, “Dynamical symmetry breaking and chaos in Duffing’s equation”, Am. J. Phys., 59,

(1991), 907-911.[3] R. D. Peters,“Chaotic pendulum based on torsion and gravity in opposition”, Am. J. Phys., 63, (1995), 1128-1136.[4] H. J. Korsch, Chaos-a program collection for the PC, Springer-Verlag, (1994).[5] Internet pages for non-linear systems with simulations:

www.mcasco.com/pattr1.html.www.apmaths.uwo.ca/~bfraser/version1/index.html.www.chaos.engr.utk.edu.html,http://monet.physik.unibas.ch/~elmer/pendulum/bif.htm.

[6] B. K. Jones, G. Trefan, “The Duffing oscillator: A precise electronic analog chaos demonstrator for theundergraduate laboratory”, Am. J. Phys., 69, (2001), 464-469.

[7] S. Lasic, The thesis work: “Didactical treatment of torsion pendulum with the transition to chaos”, University ofLjubljana, Faculty for Mathematics and Physics, (2001).

[8] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, (1950).


Fig. 7. Measured data: one-period motion (drivingfrequency ν=0,83 Hz).

Fig. 8. Measured data: two-period motion (drivingfrequency ν=0,63 Hz).

Fig. 9. Measured data: four-period motion(driving frequency ν=0,56 Hz).

Fig. 10. Measured data: chaotic motion(driving frequency ν=0,47 Hz).

ELECTROSTATIC MOTOR

Nevio Miklavčič, Srednja Pomorska Sola Portoroz, Slovenija

1. Construction We used the instructions that we found on the internet page www.eskimo.com/~billb/emotor/emotor.html .We built a similar electromotor using 3 plastic bottles of 1,5 liter, and we glued them withaluminium foil. The two bottles that are used like stators are entirely glued with sheets of foil,leaving a 5 cm foil-free space at the bottom (Fig.1). They play the role of two capacitors that areoppositely charged.

In the central bottle that plays the role of the rotor,we made a hole in the bottom and put through it asharpened rod, so that the bottle could spin freely.We glued two foil strips around the bottle so thatthey do not touch each other. Each of the foils ischarged alternatively with the charge of the statorthat is nearer.The two stators are connected to a source of highvoltage direct current (7000 V). The charges thatare accumulated on the stators are brought by twocommutators or brushes (pieces of aluminium foil)on the rotor.This central bottle is charged positivelyin that region that is nearer to the positively chargedstator and negatively on the other side. A couple of

opposite electric forces makes the rotor spin around.It begins to spin with a constant acceleration until the friction forces that are bearing resistance areequal to the electric ones.

2. Measurements with the electromotorWe measured the frequencies (f) at which the rotor was spinning and how they depended on thetension (U) applied.

The spinning of the bottle was varying from day to day and some days the electromotor wasworking with a frequency higher than 3 Hz, but anyhow we needed a voltage higher than 3kV orsometimes 4kV for its spinning.Now I will try now to explain the simple mechanism that makes the bottle spin.Measuring the rate at which the bottle was stopping when there was no electrical force, we haveestimated the angular acceleration (α) and the torque (M) due to the friction forces.From the formula ϕ=α*t2/2 for the accelerated spinning, measuring the number of turns (N) thatthe bottle made while it was stopping and the time (t) needed to stop, we calculated the angle (ϕ)and then the angular acceleration:α=2*(N*2π)/t2

and the result was α = -0,38 1/s2

Then we estimated the torque. We measured the mass of the bottle (m=50g) and assuming that


Fig. 1.

first day second day U(kV) f(Hz) f(Hz) 4 0,7 1,0 5 1,3 1,7 6 1,7 2,5 7 2,0 2,9

almost all of it is disposed at the distance of the radius of the bottle (R= 4,3cm), we calculated themoment of inertia (J) with the formula: J=m*R2, and obtained the result J=9,2*10-5kg*m2.The torque (M) was: M= J*α = 3,5*10-5Nm.Let us now take in consideration a simple model of 4 chargeswhich are disposed as in Fig.2. The two charges in the middle areconnected so that they can spin around the central axis.The torque exerted by the two external charges (e) on the systemof the two inner charges (e) is equal:M=2R*F,where F=√2*Feand Fe is the electrostatic force between two charges:Fe= e*e/(4πε0*r2)Using this equation we can find out that if we put for R=4,3cmand r= √R, the torque of 3,5*10-5N would be exerted by chargesof e=1,1*10-8 As.The rotor is a capacitor and if we use the formula C=ε0*S/l, although the two aluminium foils arenot parallel, we find out that its capacity is equal to C= 4,3pF (In this case we used for l the valueof 4,3 cm as the average distance between the two aluminium sheets that have a surface ofS=210cm2.)If such a capacitor should accumulate a charge of 1,1*10-8As, the voltage U=e/C would benecessary: U=11nAs/4,3pF; U = 2,6kVThe result is very near to the lowest voltage at which the rotor begins to spin. Experiments showedthat the rotor moves a little at the voltage of 2kV, but can not rotate. It begins to turn around atvoltages higher than 3kV or sometimes 4kV.We can estimate the power developed by the electromotor when it is spinning at the frequency of2,9 Hz or with the angular velocity ω:P=M*ω = 0,6 mW.Another way to estimate the power is using the formula P=U*I.The current I=e/t is calculated estimating the charge that is transmitted every turn of the rotor:I=2e*f= 2ε0*(S/l)*U*f=160nAThe electrical power is then:P=U*I = 7kV*160nA = 1,2mW.These results are very approximate because of the simplified formulas used and because the realcurrent that goes through the motor was not measured. Every spark means a loss of energy and areal good electrostatic motor should work without sparks.

3. Electrostatic motors in the past and in the futureThe first electrostatic motor (and also the first electric motor) was constructed by B. Franklin in1748. It was a spark-type electrostatic motor and consisted of a wooden rotor carrying thirty spikeswith brass thimbles on their ends and two oppositely charged Leyden bottles as high voltagecapacitors. The first motors required about 20kV to function and consumed a current of about 1µAat full speed and their power was thus of the order of 20mW. The diameter was approximately 1 m.The second type of electrostatic motors that are more efficient are the corona motors. In a coronamotor, the rotor carries no electrodes and acquires its charge not through a spark but through amore convenient and efficient corona discharge. The first one was constructed by the Germanphysicist, J.C. Poggendorf in 1870. His motor required voltages over 2000 V. In his time theinvention of electromagnetic motors quite completely stopped the developing of electrostaticmotors. In 1958 an experimental version of a corona motor was constructed by the Russianengineers (Yu. Karpov, V. Krasnoperov and Yu. Okunev). Operating from a 6kV power supply, themotor turned at 6000 rpm and developed the power of 5W. Somewhat different motors weredeveloped in the next years.


Fig. 2

Although electrostatic motors are much weaker than electromagnetic motors, they have some goodproperties: they are extremely light in weight and use inexpensive materials. Electrostatic motorsare a good choice for applications in space and in control systems of various automation devices,especially in those where no magnetic materials or equipment may be present. Recently inventedmicroscopic motors built of silicon and constructed on the surface of integrated circuit chips areelectrostatic motors. In fact also muscles are a class of motor called a linear motor and newelectrostatic motors are developed and applied also in Nanotechnology.

References[1] www.eskimo.com/~billb/emotor/emotor.html[2] O. Jefimenko, D.K.Walker, Electrostatic Motors, The Physics Teacher, March (1971), 121

HOLOGRAPHY: A PROJECT-TYPE APPROACH FOR CONTEXTUALIZEDTEACHING OF OPTICS

Pedro Pombo, João Pinto, Department of Physics, University of Aveiro, Portugal

1. IntroductionExperimental work has been increasingly pointed, by science educational researchers, as a keystrategy in the teaching of physics [1]. New curricular proposals put emphasis on the use ofexperimental work at the classroom in physics courses [2]. However, experimental work should notbe used exclusively in a merely academic and demonstrative way or as a “closed” activity wherestudents just follow the “steps” in a protocol [3,4]. It should be used on an open problematicsituation in a researching perspective based on problem solving and conceptual change strategies[1,5-7]. The whole process should be contextualized, relating science, technology and society.The fact that students get a product after performing experimental work or find some currentapplication of their experimental activity has been shown to be pedagogically positive.Experimental work should help the student to learn science, to learn about science and to “do”science [8].With the technological development and the large application of laser light in nowadays’ society,holography has come to show a great potential in several areas, namely in science education. Itsapplication to education started in the late 80’s in the USA [9] and spread to other countries.Researchers point holography as a teaching tool with a great success and highly motivating forexperimental teaching of optics [10,11].In Portugal, we started the research on educational holography at high school level in 1997 througha pilot study with science teachers and students [12]. The aim of this project was to promoteexperimental teaching of optics and to motivate students to the study of physics. Given thecharacteristics of the holographic technique, it was shown to be a teaching strategy based onexperimental work in an investigative perspective quite in tune with the current suggestions ofscience educational researchers. After the first results, we spread the study to a network of schools,working with teachers of several areas and with both science and art students [13,14].In this paper, we present a work that aims the contextualized teaching of optics through theapplication of holography for the study of mechanical, thermal and vibrational deformations. Thistask implicates experimental work, problem solving and relates several physics concepts. We alsopresent the educational results concerning other groups of students who started to work onexperimental holography.

2. Educational strategyA project type activity usually involves different topics in a same area and shows somemultidisciplinarity.When some activity of this kind is performed, this should be contextualized and,if possible, generate a final product with some interest to the student. The applicability of this


activity or of its final product at a technological or social level promotes student motivation [4,15].The purpose of this study is to contribute to a more effective learning of optics and to a betterunderstanding of its relationship with other areas of physics, through a strategy of contextualizedteaching, based on experimental work and problem solving.The experimental technique used is interferometric holography [16] and it is intended that studentsapply the technique studied in order to give importance to science and its interaction withtechnology and society. As the students had made several experiments in optical holographybefore, namely reflection holography, they chose to use a Denisyuk configuration [17] and havemade reflection interferometric holograms, using double exposure holography for mechanical andthermal deformations and time average holography for vibrational deformations.Having a well defined purpose, students worked experimentally on the starting hypothesis, havingto solve problems that appeared during the experiments, reflecting and analysing the obtainedresults so that they could progress until getting to the proposed objective. In this work, studentsverified the importance of holography and its utility for technological applications with potentialimpact on quality of life in our society.First of all, students must dominate reflection holography and optimise hologram processing sothat they can apply this technique in the referred context. In this specific case students had to makeprevisions and to remember and/or discuss several concepts of physics. Thus, besides dealing withoptics, they are also dealing with many other topics such as shifts, compressions, tensions, elasticity,temperature, oscillations, standing waves and vibration modes. In what concerns optics, theapproached topics are spatial frequency, interference fringes, fringe analysis and also those alreadystudied in experimental holography: reflection, refraction, absorption, interference, diffraction,vision, image formation, laser light, waves and coherence.This program runs after class time during two months and involved 16 years old science students.The experimental work was performed at the High School’s Holography Laboratory. The workingschedule was as follows: (1) Theoretical introduction to interferometric holography, (2) Testperforming, (3) Fringe analysis with test performing and discussion, (4) Experimental work and (5)Final report. The objects used were metal bars, soda cans and loudspeakers.Initially, the choice of the kind of material and object to use in each kind of analysis was discussed.During the vibrational deformation analysis, it was necessary to choose both the appropriatediameter of the loudspeaker and the ideal frequencies to work with.Afterwards, it was necessary to solve some technical problems, namely to develop stable fixedsupports for some of the objects, to develop a technique to manipulate objects at high temperaturesin the holographic system, to practise the interaction with an object keeping it static in theholographic system and to paint some objects so that they would become more light diffusing.In the shift analysis, an exposure of the object in its initial state and another exposure of the objectsubject to a shift were made. For each interferogram, only the shift caused in the object, controlledby micrometric screws, was varied. In the compression analysis, an exposure of the object in itsinitial state and another exposure of the object subject to compression were made. Compressionwas made putting a calibrated weight on top of the object. For each interferogram, only thecompression caused in the object was varied. In the tension analysis, an exposure of the object inits initial state and another exposure of the object subject to the tension of a rubber band were made.For each interferogram, only the tension caused in the object was varied. In the thermal transferenceanalysis, a long exposure of the object subject to high temperature was made. For each interferogram,only the initial temperature of the object was varied (200ºC and 280ºC). In the vibration analysis, a longexposure of a loudspeaker subject to a signal with a certain frequency was made. For eachinterferogram, only the frequency of the loudspeaker vibration, controlled by a frequency generator,was varied. Results were then analysed comparing the fringe patterns of the respective interaction(shifts, compressions, tensions, initial temperature and vibration frequencies).Based on the initial tests, students were encouraged to predict which kind of pattern should beobtained in each analysis.


In addition, some other groups of students were starting to work in experimental holographyfollowing a working plan and a teaching strategy similar to those discussed in previous papers[13,14].These groups of students performed all their experimental work at the Physic’s Departmentof the University of Aveiro during a Summer Course.In order to facilitate the analysis of results, the students of the deformation analysis project madea final report and the students who started to work on experimental holography have answered twoquestionnaires, one before and another after workshops and continuous assessment of all students’performance during experimental sessions was done.

3. Experimental workThe equipment used during experimental work was the following: He-Ne laser (10 mW, 632.8 nm),optical table, shutter, spatial filter, iris, mirror, photometer, supports, frequency generator,loudspeaker, calibrated masses and oven.The holographic support used during all the experimental work were Slavich holographic plates (7µm, 633 nm, 100 µJ/cm2, 3000 lines/mm) and the chemicals solutions were developer SM-6 andbleacher PBU-Amidol.The experimental configuration used by all students was a reflection holography set-up [17], asshown in the following figure:


Table 1: Developer SM-6 chemical formulae. Table 2: PBU-Amidol bleach chemical formulae.

Ascorbic acid 18.0 g Potassium persulphate 10.0 g Sodium hydroxide 12.0 g Citric acid 50.0 g Phenidone 6.0 g Cupric bromide 1.0 g Sodium phosphate dibasic 28.4 g Potassium bromide 20.0 g Water 1000.0 ml Amidol 1.0 g

Water 1000.0 ml

Slavich plates PFG-01 were processed using Geola’s process [18], i.e., developed in SM-6 andbleached in PBU-Amidol to obtain phase holograms.The processing scheme for Slavich plates was the following:

1- Developer: SM-6 3 min. 3- Bleach: PBU-Amidol until clear2- Rinse: deionised water 3 min. 4- Wash: running water 15 min.

Each group of students prepared their chemical solutions and proceeded with all holographicchemical processing. Solutions’ chemical formulae [18] are shown in the tables 1 and 2:

Fig. 1: Diagram of theexperimental setting forinterferometric holography.

4. ResultsThe experimental results obtained for the deformation analysis were 18 reflection interferograms,of which 15 were positive and 3 were negative. The exposure times range was from 3s to 25s. The

students were able to make 15 white-light reflection interferometric holograms with high quality.Images are clear with good visible fringes.The educational results obtained show that students havedone a correct qualitative analysis of the fringes in the interferograms. They also made a correctrelationship between patterns in objects subject to the same physical interaction, but with differentvalues. A valid association between the kind of pattern and the kind of physical interaction wasmade and a correct understanding of the fringe pattern formation in the holographic process wasachieved.The experimental results obtained by the groups of students who started the work on experimentalholography were 30 white-light reflection holograms, of which 24 were positive and 6 werenegative.The exposure times range was from 2s to 10s.All the positive results are master hologramswith high quality holographic images. Two of them were diffraction gratings and one was a doubleexposure hologram. Educational results are shown on graphs 1 and 2. In what concerns holographyconcepts, students’ knowledge showed an increase from 42% to 92% and in optics concepts,students’ knowledge showed an increase from 45% to 78%.


42%

92%

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After Experimental Holography

45%

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Before Experimental Hologra

After Experimental holograph

Graph 1: Students’ progress on holo-graphy concepts.

Graph 2: Students’ progress on opticsconcepts.

5. ConclusionsIn general, students got very enthusiastic and curious about physics. Some scientific skills andpersonal attitudes have been promoted such as team work, problem solving, scientific inquiry andself-confidence. An interdisciplinary effect has been achieved linking different areas of knowledgelike physics, chemistry and visual arts.Relating the deformation analysis project, a relationship between different physics areas has beenachieved and the interaction between science, technology and society and its importance in our lifestyle has been explored. Students showed a great control on reflection holography and hologramprocessing and they have learned with a successful level this holographic application:interferometric holography. Students’ knowledge showed an increase on optic’s concepts namelyspatial frequency, interference fringes formation, fringe analysis and relationship between fringepattern and physical interaction.An important achievement was students’ scientific attitude duringexperimental work.When students perform their experiences a variety of variables could influencethe result, so they kept all constants and only varied one at time in a systematic way. Data analysisand problem solving was fundamental on their work, which has promoted their scientific inquiry.During this project students needed to understand some general physics concepts such as shifts,compressions, tensions, elasticity, temperature, oscillations, standing waves and vibration modes.Relating the groups that have started working on experimental holography, a significant increaseon students optic’s concepts has been achieved namely reflection, refraction, absorption,interference, diffraction, vision, image formation, laser light, waves and coherence. Students’knowledge showed a significant increase on holography concepts namely on theory of holography,differences between holography and photography and holographic applications. Some students

have joined the Physics Club of their School and have started a project-work on holography.In conclusion, experimental holography at high school teaching may be considered an efficient toolfor contextualized teaching of optics.

AcknowledgmentsThe authors gratefully acknowledge the financial support of “Programa Ciência Viva” from“Ministério da Ciência e Tecnologia”.

References[1] R. Lazarowitz, P. Tamir, Research on using laboratory instruction in science, In D. L. Gabel (ed.) Handbook of

Research on Science Teaching and Learning (New York: Macmillan), (1994), 94-128.[2] Ministério da Educação, Departamento da Educação Básica, Programas do Ensino Básico e Secundário,

Ciências Físico-Químicas, (1995).[3] A. Cachapuz, I. Malaquias, I. Martins, M. F. Thomaz, N. Vasconcelos, O Trabalho Experimental nas Aulas de

Física e Química: uma Perspectiva Nacional, Gazeta da Física, 12, (2), (1989), 65-69.[4] R. White, The link between the laboratory and learning, International Journal of Science Education, 18, (7),

(1996), 761-773.[5] D. Hodson, Practical work in school science: exploring some directions for change, International Journal of

Science Education, 18, (7), (1996), 755-759.[6] Izquierdo, Mercè, Sanmartí, Neus y Espinet, Mariona, Fundamentación y diseño de las prácticas escolares de

ciencias experimentales, Enseñanza de las Ciencias, 17, (1), (1999), 45-59.[7] G. Pérez, P. Valdés Castro, La orientacíon de las prácticas de laboratorio como investigación: un ejemplo

ilustrativo, Enseñanza de las Ciencias, 14, (2), (1996), 155-163.[8] D. Hodson, Practical work in school science: exploring some directions for change, International Journal of

Science Education, 18, (7), (1996), 755-759.[9] U. J. Hansen, J.A. Swez, Holography in the high school laboratory, SPIE Proc. Education in Optics, 2525, (1995),

173-181.[10] D. Olson, Real and Virtual Images Using a Classroom Hologram, The Physics Teacher, 30, (1992), 202-208.[11] P. John, Advanced Holography in High School, Holography 2000/Proceedings of SPIE, 4149, (2000), 296-302.[12] P. Pombo, R. M. Simões, J. L. Pinto, Ensino Experimental de Holografia, Proc. Física 98: 11ª Conf. Nacional de

Física, 8º Encontro Ibérico para o Ensino da Física, ed. SPF, (1998), 151.[13] P. Pombo, R. M. Oliveira, J. L. Pinto, Experimental Holography in High School Teaching, Holography

2000/Proceedings of SPIE, 4149, (2000), 232-238.[14] P. Pombo, J. L. Pinto, Experimental Holography as a Teaching Tool, Physics Teacher Education Beyond 2000,

International Conference on Physics Education, (2000), 591-594.[15] J. W. Zwart, J. L. Vande Voort, T. Yogi, Playground physics, Science Teacher, 61, (5), (1994), 29-31.[16] R. L. Powell, K. A. Stetson, Interferometric vibration analysis by wavefront reconstruction, Journal of the

Optical Society of America, 55, (1965), 1593-1598.[17] Y. Denisyuk, Photographic reconstruction of the optical properties of an object in its own scattered radiation

field, Sov. Phys. Doklady, 7, (1962), 543-545.[18] Y. Sazonov, P. Kumonko, D. Ratcliffe, M. Grichine and G. Skokov, Holographic Materials Produced By The

“Micron” Plant At Slavich, Geola Internal Report, (1997), 1-9.

EXPERIMENT AND THEORISATION: AN APPLICATION OF THE HYDROSTATICEQUATION AND ARCHIMEDES THEOREM

Santos Lucília, Talaia Mário, Departamento de Física, Universidade de Aveiro, Aveiro, Portugal

1. Introduction

Today’s teaching is still under the influence of the old habit of “packing” knowledge in differentand non-interacting fields, and, therefore, it becomes too split in different disciplines. This basicmalfunction seems to be one of the reasons to explain the separation between formal teaching andpractical everyday issues, or between the administered teaching and the student’s interests andexpectations.As suggested by Hodson (1996) the reconceptualization of experimental work must be based onthree essential points, namely, to help students to learn science, to learn about science, and to learn


to do science. This new perspective of facing experimental work settles a contribution to researchactivity and to a better teaching/learning achievement.Barberá e Valdés (1996) have shown that most of practical work (laboratory and/or field work)performed in schools can be considered only as illustrative because it leads to experiments of the“recipe” kind, and because it causes some sort of apathy or, at least, very low motivation on thestudents.In this experimental activity the students “establish”, by themselves, from registered observations,concepts and scientific theories that were ministered before, in formal teaching, in differentdisciplines.In this study students understand basic concepts of physics of fluids (mass, weight, pressure andimpulsion, see, for example, Massey, B.S. (1983) that affect their everyday life, and also may dealwith the interaction of theory and experience.

2. TheoryIn this work it is our interest that the experimental activity may raise some unexpected questionson the student. Namely, it is our aim to analyse the influence of the reading process in a Umanometer, when different density liquids are used (on the manometer and on the tank).This research activity makes use of the Fundamental Law of Hydrostatics and of the ArchimedesTheorem.The Fundamental Law of Hydrostatics, or Stevin’s Law, states that the pressure difference betweenany two considered points inside a liquid in static equilibrium, under the influence of gravity, isnumerically equal to the weight of a column of the liquid, that has a unit area in the base and is ashigh as the vertical distance between the two considered points.Generally speaking, the pressure p on a certain point in the liquid is given by

where p0 is the pressure exerted by the atmospheric air on the free surface of the liquid, ρl the liquiddensity, g the gravity acceleration and h the depth of the point inside the liquid.Expression (1) indicates that the pressure inside a liquid increases with increasing depth and is thesame in every point located at the same level inside the liquid.Archimedes Principle, states that every body that is dived into a liquid in equilibrium receives, fromthe liquid, a vertical upwards impulsion that is numerically equal to the weight of a volume of liquidequal to the volume of the immersed body.The mathematical relation that translates Archimedes Principle, for floating bodies, is given by theexpression

Here P represents the weight of the body and Sh the volume of the immersed part, S being thestraight section of the container, and h the level difference between the two surfaces, read on thetank that contains the liquid.On the other hand


ghpp l+= 0 (1)

gShP l= (2)

MgP = (3)

where, in our case, M is calculated from

+=i

iMMM '0 (4)

In expression (4), i

iM ' represents the summing up of the loads placed on the upper top of the

recipient with mass 0M (see figure 1).

Through mathematical treatment it is possible to write

From expression (2) the pressure of the air in the recipient can be evaluated.The same value shouldbe measured by the differential U manometer. Again through mathematical manipulations eq.(5)can be rewritten as


MS

hl

1= (5)

''

hhl

l= (6)

where 'l is the manometric liquid density and 'h the level difference registered by the differential

U manometer.

3. Experimental set-upIn Figure 1 the experimental set-up is schematically represented. It is composed of three maincomponents: a water manometer (a tube of 10mm internal diameter), a recipient (10cm size and50cm height) and a tank (22cm size and 70cm height). On the top of the recipient there is a tubesystem that allows the student to use an automatic data acquisition system (ADAS) and/or a watermanometer.On building this experimental set-up care was taken on using low cost, market accessible materials,so that it could be an affordable device.As can be seen on Figure 1, connection C allows simultaneous data registration by means of anADAS chart and a water manometer, when valves A and B are open. The researcher can chooseone of the two alternatives. Exemplifying, he can use only the ADAS chart, when valve A is openand B is closed or only the water manometer, when valve A is closed and B is open. Financialavailability may be the criteria to decide the alternative to be used.

4. Results and discussionIn the experience different density liquids were used on the tank.Figure 2 shows the influence that can be observed on the difference of level of the liquid, h, when thetank contains oil (ρl = 923 kgm-3), ordinary water (ρl = 1000 kgm-3) or salty water (ρl = 1146 kgm-3).

The graph on Figure 2 shows that, for every liquid,there is a good agreement between the experimentalrecorded data, and the adjust straight line. It is alsoobserved that, for a given value of h, impulsionincreases on liquid with the higher density,as expected.The adjustment straight lines for the experimentaldata are also indicated in the Figure 2 and they aregiven for

Oil: h = 0.1093M (7)Water: h = 0.1002M (8)Salted water: h = 0.0865M (9)

Being M expressed in (kg) and h in (m).On the graphs of Figures 3 to 5 we present theexperimental data of the level difference, read onthe water manometer for every liquid on the tank,equation (5), experimental data read for the levelFigure 1: Outline of the experimental device

difference of the liquid on the tank, and theexpected values for the level difference if themanometric liquid was the same as the liquidon the tank.One must emphasise the relative positioning ofthe experimental data read on the watermanometer to the theoretical straight line.Thisobservation indicates the influence of thedensity of the liquid on the tank on the valuesread on the water manometer.Also on the figures, as expected, theexperimental data read on the tank present anexcellent agreement both with equation (5),and when the use of a manometer with thesame liquid as the tank is proposed.

5. ConclusionThe experimental set-up allows acomprehension of basic notions such as mass,weight, pressure and impulsion.The study shows that there is a differencebetween the readings on the level difference ofthe liquid on the tank and the level differencefor the water in the manometer when theliquid in the tank is different from the one onthe water manometer. This fact contributes toa better interpretation of the data, due to thedensity difference of the liquids.The analysis of the results shows the goodagreement between the observed data and theexpected ones. We think that, trough the use ofthis experimental device, in a practical scienceclass, an opportunity is given to the teacher toinclude, in is practice, activities that give hisstudents “hypothesis” of explicitly speak abouttheir own concepts. In this way, conditions willbe created so that student’s deal withalternative conceptions they may have andchange them. In this sense teaching will haveto be adaptable, and this implies thatindividual differences on the evolution oflearning are taken into consideration and thata standard way of teaching is avoided.

ReferencesBarberá O. and Valdés P., El Trabajo Prático en la

Ensenanza de las Ciencias: una revision, Ensenanzade las Ciencias, 14, (3), (1996), 365-375.

Hodson D., Practical work in school science: exploringsome directions for change, International Journal ofScience Education, Vol 18, (7), (1996), 755-759.

Massey B.S., Mechanics of Fluids (5 th. Edition), VanNostrand Reinhold (U K) Co. Ltd., Molly MillarsLane, Wokingham, Berkshire, England, (1983).


0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

M (kg)

h (

m)

Experimental data: oilExperimental data: waterExperimental data: salted waterEquation (7): oilEquation (8): waterEquation (9): salted water

0.0

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0.0

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0 1 2 3 4 5 6

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h (

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Experimental data: difference in water manometerEquation (5): waterExperimental data: difference liquid (water) in the tank

Figure 2: Experimental data. Theoretical equationsfor every liquid on the tank

Figure 3: Experimental data. Liquid on the tank: oil

Figure 5: Experimental data. Liquid on tank: saltywater

0.0

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0 1 2 3 4 5 6

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h (

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Experimental data: difference in water manometerEquation (5): salted waterExperimental data: difference liquid (salted water) in the tankPredicted data: difference in salted water

Figure 4: Experimental data. Liquid on tank:ordinary water


PROBLEMS IN THE PHYSICS OLYMPIADS

Giuliana Cavaggioni, A.I.F. Committee responsible for the Physics Olympiads in Italy

1. Some guidelines in organising the participation in the IphOs in ItalyItaly has participated in the Physics Olympiads since ‘80s, sixteen years ago. Initially the membersof the competing teams were selected with a national exam among the students indicated by theirschools as excellent in physics. They were excellent indeed, as far as their extensive knowledge inall areas of school physics was concerned. Nevertheless the results of the first Italian teams werevery disappointing. Knowledge alone appeared to be not enough to find the solutions tochallenging problems.Therefore the A.I.F. Group, entrusted by the Ministry of Education with thistask, decided to establish new guidelines for a selection process which was intended to helpstudents to measure their progress in applying their knowledge to problem solving.A selection procedure started therefore which consists of four phases, held during one school year,the first phase taking place in December and the last one early in June. The students are supposedto practice problem solving mainly as personal work. As a matter of fact problem solving skills arescarcely taken into account in the regular physics courses in the schools. The participation in thePhysics Olympiads however was an opportunity for many schools to offer the most interestedstudents a special programme oriented to physics problems. In this task schools often receive helpfrom the Physics Departments of the local Universities.In this way some improvements were made. But it was also clear that the process of learning aboutproblem solving needs time: for instance the contestants who had already taken part in the firstphases of the Olympiads in the previous years were more likely than others to succeed later inreaching the third or the fourth phase of the competition.With an overall participation of 500 schools and 30.000 students the level of excellence in the firstphase of the Italian Physics Olympiads is the same as that of good students one could find in anyclass. Of all the contestants about 2000 are usually passed to phase 2 and then 90 to the third phase.Finally, the winners, ten in number, participate in the last competition in which the Italian team forthe IPhOs is selected.Other initiatives were developed to help the students both in theoretical and in experimentalproblem solving. In the period between the third and the fourth phase the top 15 students attenda one week seminar on problem solving in physics organised with the collaboration of theInternational School for Advanced Studies and the Physics Department of the Trieste University.In this contest the students work as a co-operative group and it is possible to make someobservations on how these students tackle problem solving.We noticed for instance that they oftenput much more confidence and care into sophisticated mathematical calculation than into theunderstanding of the physics in the text of the problem. The programmes of the Italian PhysicsOlympiads Summer School are also performed on the basis of co-operative groups. This is a fiveday initiative of full immersion in physics problems and hands on activities. Finally a discussiongroup on physics problems started this year on internet between the contestants of the third phaseof the Italian Physics Olympiads. This group is led by the Italian Physics Olympiads organisers andsome participants in the previous Italian IphOs teams are acting as tutors in what is intended as an“almost” peer to peer way.

2. Problems in the Physics OlympiadsThe problems that the contestants meet in the Physics Olympiads are unusual for manyparticipants. Instead of the exercises normally presented in the school textbooks these problemsare not directly related to the formulas just treated and often they can have different answers,

3.2 Problem Solving

some of them not needing elaborate calculations. Students use a heuristic approach to find the mostconvenient and elegant solution.While the first contest in Italian Physics Olympiads is based on a multiple choice paper the others requirethe solution of problems.The problems given in these phases have a different level of complexity but theyare in any case designed so that the planning of some strategy is required to solve them.It is not easy to measure the difficulty of the competition tasks: the most common approaches in thePhysics Olympiads are based on the scores of the participants and so a coefficient of difficulty canbe defined only after the competition.An objective approach was introduced in early InternationalPhysics Olympiads defining a so called requirement level: to define this numerical coefficient,parameters such as the number of areas of physics in the problem, the formulas needed to solve itand the kind of mathematical calculations involved are all taken into account. However thisrequirement level was connected to a particular solution of the problem and it doesn’t fit problemswhich can be solved in different ways. Moreover defining the requirement level other importantaspects are neglected, regarding the text of the problem, the way in which the data are given or thepresence of explicit assumptions and boundary conditions.In closing this review let me add that preparing good problems is indeed one of the mostchallenging tasks in the Physics Olympiads: problems have not only to fit the needs of thecompetition well but they must also be interesting enough for the contestants to feel it worthwhileand be motivated to solve them. At the addresses written below problems given in the national andin the international contests can be found.

Referenceshttp://www.jyu.fi/tdk/kastdk/olympiads (Official IPhOs Internet Address)http://members.nbci.com/links2ipho/regional.html (links to national and regional Ph. Ol. Contests)http://www.cadnet.marche.it/olifis (Italian Physics Olympiads Official Internet site)

PROBLEM SOLVING ACTIVITIES IN TRAINING THE ITALIAN CONTESTANTSOF THE PHYSICS OLYMPIADS IN A SUMMER SCHOOL

Dennis Luigi Censi, Italian Physics Olympiads National Group of the Associazione perl’’Insegnamento della Fisica

1. The italian physics olympiadsThe Italian Physics Olympiads have been organised since 1987 by a group of members of theAssociation for Physics Teaching on behalf of the Ministry of Education. They include variousenterprises ranging from the competitions – organised in four consecutive stages – for selecting thestudents forming the Italian Team at the International Physics Olympiads, to the theoretic andexperimental competitions reserved to students being at a beginner level in the study of physics, tolocal and national enterprises aiming at deepening the learning of physics by the students of theHigh School. Many students and teachers take part in these activities, as the Italian PhysicsOlympiads are rooted in a network of schools; in each school there is a teacher who is theresponsible for the co-ordination of the activities related to the Olympiads organised in that school.In developing the activities of the Olympiads, one of the most urging needs is to offer to the youngpeople taking part in the competitions learning supports aiming at deepening the knowledge of thesubject acquired at school. The enrolment in the Physics Olympiads is let to students’ individualfree choice, and gathers the applications of those who, even if at different levels, are bent for theirown choice to the study of physical sciences for which they have a particular interest and curiosity.

2. The physics summer schoolThe physics summer school is for high school students who have been admitted into the secondphase of the competitions for selecting the team for the Olympiads.


Acknowledging the educational importance of not wasting the potential abilities often displayedby the interest that the students show in a subject, the summer school is organized with a playfulcharacter whose aim is to motivate understanding with particular attention to the development ofcreativity through activities of planning and of problem solving.Since 1998 the summer school has been held at the end of august, for one week, in Sassoferrato, alittle town lying among the hills of central Italy. Teachers and learners are lodged in a boarding-house and students are involved in the management of the house.The teaching activities take placein the classrooms of the local scientific lyceum.

3. The participants in the physics summer schoolAmong the students who in past years took part in the Italian Physics Competitions, the needs tofind a reference point and a support for the individual activities in deepening the study of physicsthey perform outside their school courses, were often noticed. Moreover, when competitions havea residential character, as it occurs in the third level competition, the Physics National Competition,students have always shown to appreciate much the possibilities of communication and to performactivities jointly with their colleagues having the same interests. For this reason, in forming thegroup of participants in the Summer School, a criterion assuring a good level of likeness in abilitiesand interests has been followed.Therefore the students who have been admitted into the second level of the competitions forselecting the team for the Olympiads and have not yet finished the High School, are allowed toapply for attending the Physics Summer School. About 20 participants are selected among thosewho have particularly distinguished themselves in the physics competitions. The participants areaged between 16 and 18, and are attending the third and fourth year of high school. The geographicprovenience and the school level of the applicants reflect the distribution got in the participationin the Italian Physics Olympiads and there are students from all the parts of Italy.According the results of a questionnaire given at the end of the Summer School the interests of theparticipants lie mainly in the scientific and mathematical field, but other cultural and learning fieldsare not excluded, even if in these the interests, as it is to be expected, are very different.At the end of the activities all the students are given certificates of attending the School.

4. Working contentsThe physical fields dealt with during the Summer School are those provided normally for in theschool programmes and the whole attention is centred on the deepening and the working methodsaiming at the active involvement of the students.An important – in our opinion – aspect of the Summer School which we want to point up duly isthe criterion according with which the contents have been selected and the working way has beenorganised in every section.In the summer school, more emphasis is placed on the student’s activities rather than the teacher’sinstructions. The students develop a given theme, problem solution or laboratory pratical activity,working by themselves or in groups depending on the particular circumstances and theirpreferences.At least two teachers are always present to suggest and, if necessary, to co-ordinate thework. When it is needed, the teacher intervenes in order to give information or explanations.Formal activities are developed in 11 sections, which last three hours each, alternating betweentheoretical and experimental activities, among them:• technological activities are developed, on site, jointly with firms• other activities are hands-on, performed outdoors

Each section provides for moments in which the students can explain to the others the results oftheir own work and the observations they made during the activity. In the activity of problemsolution the teachers’ task has been to briefly introduce the theoretical references, to propose aseries of problems taken from the Italian and international competitions of the Physics Olympiads,

152 3. Topical Aspects 3.2 Problem Solving

and to co-ordinate the activity, favouring, when it was the case, the interactions among the students.Experimental activities are based on semi-structured proposals, both to draw up hypotheses and tochoose experimental equipment. They can be realised with inexpensive or sophisticated materials.It has been noticed that for the activities on the ground in the technological field a longerintroduction was needed aiming at pointing the more relevant characteristics of the systems and ofthe installations. Usually, in fact, the students ignore almost completely technological devices andsolutions. Anyway, in this case too, the activities in which the students can “see and understand thetechnology in action” with the support of technicians is pointed up.As for the hands on activities, the students have free choice in formulating problems and practicallysolving them with the materials available on location. Here are some of the problems proposed: Ifwe put an egg on the top of a water rocket, what happens to the egg when the rocket starts fallingdown? Which maximum height does the rocket arrive at and which maximum height does the eggarrive at? If we want to recuperate the egg unbroken, how can we make and mount an effectiveparachute?

5. The participants’ opinionsAt the end of every edition of the summer school the participants declare their anonymousopinions about some aspects of the summer school. The answers could vary on five levels, from anupmost usefulness perceived equal to 5 to a judgement of essential uselessness equal to 1.From the analysis of the participants’ opinions appear that they take part at the summer school:• to practise physics outside of school • to prepare themselves for the physics olympiadThe participants’ average opinions about the summer school activities are (5=max; 1=min):• problem solving = 5• experimental activity = 5• technological activity = 4/5• astronomical observation = 4/5• outdoor physics = 4• lessons = 4and about school organization (5=max; 1=min):• collaboration among students = 5• materials to study and work = 4/5• teachers’ assistance = 4/5• meal = 4• transportation = 4• lodgings = 2/3In their opinions, the participants declared some positive aspects about the school. The moresignificant among them is:It’s important that there is always a close relationship between teacher and student to work in adifferent method from the detached and less stimulating way used at the schoolAnd in the participants’ opinion the points to change or add are• more theoretical comments and/or depth• consider some improvements on the school’s logistics

6. ConclusionsThe Summer School of the Italian Physics Olympiads, come to its fourth edition in 2001, is the firstsummer school in the physical field which has organised in Italy for students of the secondaryschool. Nevertheless activities like these are little customary during the summer holidays theSchool has met the interest of the participants and comes out as a consistent means with the aimsof the Italian Physics Olympiads, to spread the interest in the study of physics, to accustom to thestudy through problem solving, to favour the exchange of experiences between students on themes


of common interest and to offer to young people opportunities of deepening the subjectknowledge acquired in school courses. In the plans of the organising committee of the ItalianPhysics Olympiads there is the intention to continue to develop this will. The experienceachieved in the four years’ activity of the Summer School suggests to redefine the space devotedto technology, to lengthen the time available for the experimental activity, and to reduce to theleast the formal intervention by the teacher during the activity of solution of theoreticalproblems.

More information about the physics summer school of the Italian Olympiads is on:http://www.cadnet.marche.it/olifis.

HOW TO START A PROBLEM

Francesco Minosso, Board of Olympiads, AIF

1. Teaching with problemsSome students make many mistakes when I ask them to solve a problem. If the problem is moredifficult than usual, also clever pupils make mistakes they usually don’t.In order to help them I try some techniques used with less skilled students in the very first approachto a problem: reading and comprehension of the text. I’m showing you first the schedule of the fullactivity. Second, as an example of the text reading and comprehension, I’ll show you a problem Iused during the training of the Italian team for the International Physics Olympiads of last year.The problem was given in 1989 in Warsaw at the International Physics Olympiads. It speaks abouttwo unmisceable boiling liquids.

2. The schedule of the activitySTEP ONE: approaching the problem• Individual work: students have to read and decompose the text into sequences.• Working group and discussion: teacher asks questions, presses answers, opinions, incomplete

explanations and interpretations.At the end of the Step One, students can define:- the physical system- interactions- transformations and quantities useful to describe transformations

• The teacher makes a synthesis on the blackboard

STEP TWO: sketching and modelling• Modelling may be the definition of a mathematical model or the sketch of a diagram, or a

transformation table (initial conditions, parameters variation, final conditions) • Defining the subject, for example the principles someone thinks to use getting the goal• The teacher makes a synthesis on the blackboard drawing and writing down propositions and

equations This is the most important step to solve the problem. Discussions often start at this point. Theteacher works hard, asking questions and finding answers together with the students.

STEP THREE: it’s almost always a problem of calculations• Setting up calculations• Solving calculations• Finding numerical results controlling their consistency as to the data and the problem

(compulsory)It looks like a simple technical fact, but success is not sure! It’s an individual work, but the teacheris ready to help if necessary.


STEP FOUR: reassembling and documentingIt’s an individual work. Students dislike this because they have already got the solution. They thinkit’s not important: and it isn’t, for the present problem. While they reassemble and document thesolution of the problem, they ask many questions and really learn problem solving for the nexttime.

3. Text decomposition into macro and micro-sequences (Fig. 1 and 2)Now I’ll show you the example of what I did for the training of the Italian Team at the PhysicsOlympiads.Fig.1 and 2 show the text of a problem about two no misceable boiling liquids issued at the IPhOin Warsaw in 1989. It’s a very interesting problem about an unusual situation for the Italian pupils.Square and curly brackets show the text decomposition into principal macro-sequences. They areoften the same as paragraphs. Slashes divide the text into micro-sequences.As you can see, the decomposition is not univocal and has to be discussed with students. Thisdecomposition was made with their help and doing so they could answer STEP ONE.

4. Specific words (Fig. 3 and 4)In the next two figures you can see the specific terms of the Physics. Looking at these, students canrealise what they know about the physics of the problem and what they are asked to learn to solvethe problem. Also they can remember where and when they studied these subjects and answerSTEP TWO.

5. Connective elements (Fig. 5 and 6)Connective elements are words as “like”, “and”, “if”, “that”, “beginning with”, “you can see that”,“than”, “then”, and so on that introduce assumptions or explanations. They mark the crucial pointsof the text and if there are many, surely there is a difficult passage in a problem, as you can see inthis example. Looking at these words helps in modelling, stating equations and planningcalculation: STEP TWO and THREE.

6. Synthesis (Fig. 7 and 8)The synthesis, on the side of the text, aims at reassembling the sense of the problem guiding theschedule of the solution: STEP THREE and FOUR. Just not to get lost in the pure linguistic analysisof the text, and just to remember that reading the problem is not to solve it, but only to approach it.

7. ConclusionSometimes the solution strategy of a problem coincides with the very same solution, as it happensabout the problems at the end of chapters of a school-book: these problems have not beendiscussed in this account.We have seen that students, used to work at the problems of the end of chapters, after a carelessreading of the text, proceed to write and solve the equations directly.This method is rather effectivein case of simple problems and sometimes works in the solution of more complex problems too, butthe degree of reliability of this system seems to be correlated with the degree of the masteryachieved by students.It seems that the development of the problem solving skills is not linear but it proceedsdiscontinuously.The analysis of the mistakes made by the cleverest students, in a complex situation,is a confirmation of that. During the training, a separation of the problem solution, into phases, iseffective to improve an approach to physical difficult problems.The strategy of training can be applied in all those teaching situations critical in passing from alevel of mastery to a higher one and, so, more than a particular kind of problem, the strategy isconnected with the learning situation peculiar to students.This way teaching problems teaching is interesting since students do not learn only in imitation, butthey are guided to reflect on the resolutive processes they put into effect becoming, in the end,more skilful. The observation of their behaviour during these activities confirms what we said



Fig. 1


above. The interactions assumed by students in three times of their work are significant forunderstanding the kind of learning developed by this work and making clear in which moments theprocesses of formalization take place.Applying the strategy of training above-mentioned, it has been observed that students, after astarting indifference suggested by their habits of work, begin to ask some questions.At first, during the reading of the text, their questions are directed to the comprehension of themeaning of sentences and words about their content, but, afterwards, we can observe students beginto speak more about the physical content of a problem, to consult their manuals and to refer toteachers as to a consultant.A second time, that in the traditional practice seems automatic and, on the contrary, is crucial, isthe one in which the solution strategy is determined and one proceeds to the writing of equations.It is a more delicate and abstract phase of formalization. It has been observed that in this phase,too, students put questions more accurate and technical than before. Often it happens that otherstudents assume interaction with their teachers who always maintain the role of consultant. We seesome students, at this point, look up in their manuals or note-books to search partial solutions tosimilar problems in order to reutilize them in the new context.The third time in which one can observe further changes, as regards the learning in imitation, is thephase of documentation of the solution. Students ask questions of a different kind as to theprevious ones. Mostly they are questions intended to value the substantial character of reasonings,the coherence of results with analysis, principles and phenomenology. In this case, too, students donot necessarily assume the interaction and it is in this phase that, in substance, strategies of problemsolving are set and students become more and more skilful.By appropriate times of discussion and synthesis, teachers who manage this work transform theinteraction with their students into an effective work of cooperative learning.

Fig. 2

Specific words (Fig. 3 and 4)In the next two figures you can see the specific terms of the Physics. Looking at these, students can realisewhat they know about the physics of the problem and what they are asked to learn to solve the problem.Also they can remember where and when they studied these subjects and answer STEP TWO.


Fig. 3


Fig. 4

Connective elements (Fig. 5 and 6)Connective elements are words as “like”, “and”, “if”, “that”, “beginning with”, “you can see that”,“than”, “then”, and so on that introduce assumptions or explanations. They mark the crucial pointsof the text and if there are many, surely there is a difficult passage in a problem, as you can see inthis example. Looking at these words helps in modelling, stating equations and planningcalculation: STEP TWO and THREE.


Fig. 5


Fig. 6

Synthesis (Fig N.7 and 8)The synthesis, on the side of the text, aims at reassembling the sense of the problem guiding theschedule of the solution: STEP THREE and FOUR. Just not to get lost in the pure linguisticanalysis of the text, and just to remember that reading the problem is not to solve it, but only toapproach it.


Fig. 7


Fig. 8

DOES PHYSICS FORMAL KNOWLEDGE REALLY HELP STUDENTS IN DEALINGWITH REAL-WORLD PHYSICS PROBLEMS?

Nella Grimellini Tomasini, Olivia Levrini, Physics Department, University of Bologna, Italy

1. The research frameworkFor some years now, several studies have been carried out on problem solving and, in particular,some of them aimed at examining ways of overcoming the ‘mechanical’ behaviour that still seemsto be the dominating attitude in tackling problems. In their attempts to explain why so manystudents fail in their problem solving, Gil and his collaborators underline the importance of lookingspecifically into the assumptions implicit in traditional problem solving activities (Gil et al., 1988).These authors feel that, in order to improve problem solving skills, the whole ‘methodology of thesuperficiality’ must be called into question as main responsible for so many failures. Thismethodology leaves no space for questioning, or for thinking about alternative solutions to fast,safe, peer-popular answers, and includes a hasty analysis of different situations undertaken withoutseeking overall coherence. “It is consistent with coming to conclusions based on just qualitativeobservation without control” (Gil & Carrascosa, 1985). The researchers shift the focus of theirattention from traditional Physics exercises to open-ended problems, “those situations that presentproblems with no ready-made solutions.” Dealing with these situations means inventing hypothesesand working out strategies for interpreting them. The open-ended problems referred to in theseworks are problems that can be adapted from traditional texts by removing the simplifyinginformation and conditions that exercises normally contain. It is then up to the solver to redefinethe problem critically so as to be able to solve it formally.The Spanish researchers maintain that such problems can be helpful in bringing about significant“conceptual and methodological change” regarding the learning of Physics as seen from theconstructivist point of view (Gil & Carrascosa, 1985).In the present study sterilised real-world qualitative problems will be considered. Unlike the open-ended problems, these ones do not require solvers to invent strategies, but unlike the usualexercises they do not allow a mechanical recognition of the right formula on the basis of the givenquantitative data. In a sense, an intermediate category of problems will be taken into accountcoherently with the following specific aims of the study:• Investigating what sort of correspondence Physics students recognise between a simple real

phenomenon and the algebraic formalism and what meaning is attached to such acorrespondence;

• Investigating the implicit or explicit criteria that lead students to choose one specific strategy fordealing with an actual phenomenon and, in case, the origin of the difficulties found by them;

• Assessing the potentialities of sterilised real-world problems as teaching tools for introductoryactivities aimed both at encouraging students to reflect on the role played by Maths contentknowledge in the choice of a problem solving strategy in Physics and at making students moreand more familiar with the process of formalisation peculiar of Physics knowledge construction.

2. The studyThe population considered in the study is composed by 20 voluntaries students attending thesecond year of the Physics graduation program of The University of Bologna (A.Y. 1995-96).The data source are individual semi-structured interviews anchored to a sterilised real-worldproblem conceived by Lawson & McDermott for an experiment aimed at “assessing studentunderstanding of the so called work-energy and impulse-momentum theorems by students’performance on tasks requiring the application of those relationships to the analysis of an actualmotion” (Lawson & McDermott, 1987). The students are asked to compare the changes inmomentum and in kinetic energy of two frictionless dry-ice pucks with different masses (the ratiowas 1:10) as they move rectilinearly under the influence of the same constant force for the samedistance. No information about the masses is given to he students: they can only see different


dimensions and different colours (the smaller puck is made of aluminum and the bigger of brass).According to our aims, the interview protocol has been designed so as to give the problem the roleof starting point for developing a more general discussion with each student about the role and themeaning of the formalisation processes in Physics. The protocol is composed by three differentparts:• in the first part, very similar to the first part of the American protocol, a researcher asks each

student to look at the motions of the two pucks (implemented by another researcher) and tocompare both their momentum and their kinetic energy. This part aimed at collecting data aboutanswers and arguments provided by the students before any interviewer intervention.

• the second part aims at encouraging students to make their reasoning strategies explicit. Typicalquestions of this part of the interview are: How did you arrive at your answer? What did youobserve? What quantities did you considered relevant? In what relationship did you put them?

• the third part aims both at encouraging students to reflect about the role and the meaning of theprocesses of mathematical formalisation in Physics and at collecting some information abouttheir image of Physics and Maths. Meaningful questions of this part are: what is for you amathematical formula? In the following three formulas does the sign “equal” have the samemeaning or not: T=mv2/2; F=ma; mgh=mv2/2? In your opinion, what role is played bymathematics and observation in Physics knowledge construction?

The collected data (audio-recording and transcripts of the interviews) have been analysed atdifferent levels, aiming respectively at:• pointing out the answers and the arguments provided by the students in the first part of the

interview and comparing them with the American results (descriptive analysis);• understanding the peculiarities of the reasoning strategies laying behind the answers and the

arguments provided by students in terms of the role played by observation, the role played byformulas, the meaning attached to momentum and kinetic energy. This analysis has been carriedout keeping into account mainly the first two parts of the interview (first level of the interpretativeanalysis);

• understanding what correlations can be found between the reasoning strategies used by thestudents and the image of Physics and of Maths held by them. This last step of the analysis hasbeen carried out looking for interrelations between the third part of the interview and the firsttwo parts (second level of the interpretative analysis).

3. ResultsDescriptive analysisFigure 1 and 2 sum up the results obtained through the descriptive analysis, i.e. the answers and thearguments provided by students before interviewer’s intervention (first part of the interview). It isworth noticing that, unlike the American students, not all the Italian students arrived at the rightanswers by applying directly the two theorems:“Pb>Pa inasmuch as the brass puck received a largerimpulse than the aluminium puck, being the force applied longer” and “Ta=Tb, inasmuch as thesame work has been done over the two pucks, acting the same force along the same distance”. Forsome of them the actual situation did not remind them of the two theorems, nevertheless on thebasis of a appropriate reflection on the structure of mechanics they were able to correctly developthe second law.Other results confirm what the American researcher consider the most relevant evidence: severalstudents look at momentum and kinetic energy as mere combinations of mass and velocity.Coherently, they answer on the basis of a “compensation argument” of this kind: “The twomomenta are the same because the heavier puck runs more slowly” or “Since the heavier puck runsmore slowly and speed counts more, Ta is larger that To”.


Very close to this argument is what we call “Lack of quantitative data”: “I saw the heavier puckrunning more slowly; so I could say, as far as momentum is concerned, that they are equal. But Icannot conclude unless I measure. Analogously, since speed counts more than mass within kineticenergy formula, I could say that Ta is larger that To but, again, I cannot conclude without data”.Another relevant category concerns the argument “same force applied”:“The momenta (or kineticenergies) are equal because the pucks received the same push, being the applied force the same”.The category “Confused Discussion/Other” collects arguments such as: “Since I know thatmomentum is conserved in isolated systems, I can say that… I do not know” or “Since work is equalto the change of kinetic energy, I should look at which puck does a larger work, maybe which oneheats more the table”.


Momentum task

Pb>Pa

Largerimpulsereceived

Pb>Pa

Other correctstrategies

Pa=Pb

Same forceapplied

Pa=Pb

Compensationargument

?Lack of

quantitativedata

Pa>Pb Pa<Pb

No frictionVa>Vb ma<mb

?Confuseddiscussion

/ OtherS1 XS2 XS3 XS4 X (F=ma)S5 XS6 XS7 XS8 XS9 XS10 XS11 X (Ta=Tb)S12 XS13 XS14 XS15S16 X (Ta=Tb) XS17 X XS18 X XS19 X XS20 X X

7 3 4 3 1 1 110 4 7 3

Kinetic energy task

Ta=Tb

Sameworkdone

Ta=Tb

Othercorrect

strategies

Ta=Tb

Sameforce

applied

Ta=Tb Ta>Tb

Compensationargument

?Lack of

quantitativedata

Ta>Tb Ta<Tb

No frictionVa>Vb ma<mb

Ta<Tb

Energyreceivedlonger

?Confuseddiscussion

/ OtherS1 XS2 XS3 XS4 XS5 XS6 XS7 XS8 XS9 XS10 XS11 XS12 XS13 XS14 XS15S16 XS17 XS18 X XS19 XS20 X X

6 1 1 3 2 1 17 2 4 4 4

8

Figure 1: Results of the descriptive analysis concerning the momentum task

Figure 2: Results of the descriptive analysis concerning the kinetic energy task

The argument called “No friction” has been taken apart from “Confused discussion” in order tostress some students’ difficulties in dealing with an actual sterilised motion. The lack of friction,indeed, led them to answers totally counter-perceptive: “Because of the lack of friction the twospeeds are the same (!) and Ta<To because ma>mo,.” or “Because of the lack of friction the mass isnot relevant, so Pa>Po and Ta>To because va>vo,”. The collected data are not enough for pointingout the roots of these arguments but they reveal a certain confusion among lack of friction, lack ofair and, maybe, lack of gravity (Torosantucci & Vicentini, 1991).

First level of the interpretative analysisWhat is there behind each answer? What is the role played by observation and formal thinkingwithin each reasoning strategy? • Behind the correct answers a strongly structured formal knowledge can be observed: Maths is

seen first of all a way of organising Physics concepts and of managing the relationships amongthem. The mathematical structure of mechanics – organised in definitions, axioms, theorems andcorollaries – allows these students to use formal thinking as “guide” for inquiring real-word.Theirformal attitude toward Physics and Physics learning leads also them to see observation as theory-laden: learning Physics means also to re-educate observation in the light of content knowledge:“I. Looking at the motion, what did you observed?S9: The time […] for having an idea about their mass. […] force and mass,[..] and of course thecovered space. [..] and as a consequence the acceleration of each body.I: What about velocity?S9: No, velocity is secondary with respect to the others […]I. Why?S9: [because I have] studied classical Physics. A non-Physics student maybe would have looked atvelocity and volume, like Aristotle… […] but in Newton’s law there is acceleration…

• Behind the argument “same force applied” a phenomenological attitude toward Physics can berecognised: these students are moved by the need of giving a physical form (i.e. linked to the ideaof force) to the intuitive idea of push. In this case, observation is not guided by contentknowledge but focused on the cause of the motion and formulas are used as tools for giving astructure to an intuitive and qualitative reasoning. For these students the interview representeda chance for improving their understanding because the discussion allowed them to reflect on theconnection between a vague idea of push and the physical concepts of momentum and kineticenergy. They showed to master the necessary content knowledge for solving the problem but tofind it difficult to fit it in with their own cognitive demands.

• Behind the strategies based on a “compensation argument” or on the refusal of answeringbecause of the “lack of quantitative data” there is no structured content knowledge. Kineticenergy and momentum are seen as quantities defined in terms of mass and velocity and theformula is mainly a tool for quantifying. They had never thought of what observing means: forthem observing is strictly linked to the idea of “measurement” and/or to the collection ofquantitative data.

• More and more weakly structured is the content knowledge behind the other reasoning strategies.The lack of an individual elaboration of Physics leads students to enter blind alleys and fall downinto traps created by the evocative role played by some key-words or slogans learnt by heart.

Second level of the interpretative analysisThe first level of analysis has already allowed us to recognise that different reasoning strategies canbe connected with different images of Physics and Maths.In particular a comparison between the formal cognitive style behind the “correct strategies” andthe phenomenological one behind the “same force applied strategy” enables us to identify twodifferent ways of looking at Physics: on one hand, Physics is seen as a formal system of conceptsand relations among concepts elaborated by humans in coherence with a rationalistic way of


looking at natural world, on the other hand, as a process of knowledge construction characterisedby a continuous dialogue between humans and nature aimed at formalising a intuitive, qualitativeapproach to natural phenomena.The critical attitude showed by these two groups of students toward Physics cannot be foundamong the students whose answers have been situated within the categories “compensationargument” and “lack of quantitative data”.The latter, indeed, showed to hold a naive image of Physics seen as a collection of mathematicalformulas according to which:• theory and experiment are naively related: experiments serve for measuring, theory for

organising quantitative data;• quantitative data are predominant over the qualitative reasoning;• formula is only a tool for quantifying:

“For me a formula is a manner of putting together some quantities, that must be not only wellidentifiable, but also quantifiable, because it is in such a way that a formula is successful.” (S10)

4. Concluding remarksThe wide-spread use of close-ended quantitative exercises can be somehow justified within someepistemological positions based on a strong overlapping between Physics and Maths and it cannotbe denied that such a practice can lead some brilliant students to outstanding performances.Nevertheless, the study shows that the effectiveness of formal teaching of Physics is notindependent from the actualisation of extremely particular conditions:• the formal teaching approach being resonant with a formal attitude of students toward Physics

and Physics learning; an attitude that leads them naturally to recognise Maths also as logicalstructure and as rational net of relationships among concepts;

• students being able to recognise by themselves the implicit processes of formalisation, gainingawareness of the cognitive and epistemological meaning of “observing”, “schematising” and“mathematically modelling”.

The study, carried out with 20 “good” Physics students, shows how rarely are these conditionssatisfied. When they are not, the traditional exclusive use of exercises may lead to undesirededucational results such as:• encouraging students to look at Physics as an established collection of unquestionable formulas

that in fact prevent them from transforming content knowledge into a cognitive tool forinterpreting real-world phenomena;

• hindering the development of an individual approach to Physics learning and of an individualway of looking at Physics.

The study suggests the necessity of taking into account different problem solving typologies sincethe first years of schooling. Real-world sterilised problems – and real-world open-ended problems(Levrini, 1996) – can contribute to the creation of a learning environment characterised by amultiple-approach to Physics which can:• encourage each student to find his/her own path for cultural, cognitive, emotional growth;• emphasise a strict overlapping among problem solving, epistemological reflections and lab-work,

i.e. a strict connection between the development of intellectual abilities and inquiring ones.

ReferencesGil Pérez D. & Carrascosa J. Science learning as a conceptual and methodological change. European Journal of

Science Education, Vol. 7, No.3, (1985), 231-236.Gil Pérez D., Martinez-Torregrosa Y, Senet F. El fracaso en la resolución de problemas de Física: una investigatión

orientada por nuevos supuestos. Enseñanza de las Ciencias, Vol. 6, No.2, (1988), 131-144.Lawson R.A. & McDermott L.. Student understanding of the work-energy and impulse-momentum theorems.

American Journal of Physics, 55(9), (1987), 811-817.Levrini O., Un problema “reale” per capire la fisica, La Fisica nella Scuola, XXIX, 2, (1996), 59-63.Torosantucci G. & Vicentini M., Il fenomeno della caduta, in Grimellini Tomasini N. & Segrè G. (Eds.), Conoscenze

scientifiche: le rappresentazioni mentali degli studenti, La Nuova Italia, Scandicci (Firenze), (1991), 109-137.


INFLUENCE OF NARRATIVE STATEMENTS OF PHYSICS PROBLEMS ON THEIRCOMPREHENSION

Elena Llonch, Marta Massa, Patricia Sánchez, Elisa Petrone, Facultad de Cs. Exactas, Ingenieríay Agrimensura, Universidad Nacional de Rosario, Rosario, Argentina

1. IntroductionNarrative text has a closer correspondence to everyday experience than does expository text. Itinvolves dynamic events that imply characters, goals and intentions; expositive ones include staticcontents such as concepts, descriptions and arguments. Many knowledge-based inferences aregenerated during the comprehension of narrative text, requiring the activation of knowledgestructures and their integration to conform a meaning representation of the text. This fact is typicalof those cases in which the subject ought to solve a real problem, generally ambiguous, and writtenin a narrative style.Problem solving can be seen as involving two processes: comprehension through which the subjectorganizes a mental model of the situation, and searching of possibilities, evidences and goals inorder to generate a strategy for its solution (VanLehn, 1998; Johnson-Laird, 1983). We assume thatthose processes are determined by the type of statements and by the way a subject constructs aninternal representation. They will orient the selection of either an algorithmic solution or a specificheuristics as an alternative procedure to solve it (Nickerson et al., 1998).Problem solving involves processes that Baron (1991) identified as characteristics of reasoning,such as assumptions, inferences, predictions, arguments and the selection of examples and counter-examples to validate or refute statements. Galotti (1989) has stated that subjects perform differenttypes of reasoning according to the type of problem they are asked to solve.Formal reasoning is generally associated to well-defined situations, where all the relevant data aregiven. It is based on logical inferences where the initial premises imply implicitly a conclusion.Informal reasoning, generally associated to “open” situations, is not restricted by logical operationsas it may include inferential processes developed, sustained and evaluated by a system of beliefs orby common sense.Problem solving lies on a situational modelling (Perkins et al, 1991) whose adequacy depends onthe recognition of the demands of the task (Johnson-Laird et al., 1988). Often the situationalmodelling is clearly provided to the student who ought only to understand it and apply principlesand laws.When problems are written in a narrative style, the situational modelling is a crucial activitythrough which the specific problem or sub-problem is defined. The subject has to recognize therelevant data and to decide possible solving strategies.Within a research project dealing with reasoning on Physics problems, we analyse the problemspace 1that students construct when they read, interpret and solve problems written in a narrativestyle. We analyse how students solve them in order to identify indicators of their comprehension,possible bias in the interpretation of premises and the features that orient the situationalmodelling.

2. TheoryThe Mental Model Theory (Johnson-Laird, ibid.) assumes that discourse models are constructed onthe basis of inferences from general and specific knowledge. Particularly, the comprehension of aproblem is seen as the construction of an initial mental model by means of the syntactic andsemantic processing of language and the activation of the subject’s prior knowledge and beliefs.This process requires that the subject recognizes the text as a coherent, consistent and plausible


1 Problem space includes three components: the initial problem state, some operators that can change a problem stateinto another and some efficient test for whether a problem state constitutes solutions.

discourse, within an appropriate temporal, spatial, causal and intentional framework. Theassociation to previous specific knowledge leads to generate new models. In a determined stage ofthis process, the subject evaluates the plausibility or the credibility of the model according to thesupporting evidence and conclusions. If the subject recognizes invalid conclusions, then new modelsmay be constructed or a previous one may be re-structured. Therefore, reasoning failures may beattributed to an insufficient search for relevant models, to omissions of counter-examples or to thelack of evaluation of the facts implied by all the models constructed in the process. The processfinishes when refutation is absent, and the subject defines the model.The procedures applied by a subject when he searches for information may reproduce strategiespreviously used in similar or different contexts, or may be created by analogy or association. Twobasic criteria for the evaluation of situational models are the bias that determine a unique sense tointerpret arguments, and the completeness while considering all possible arguments.Availability and representativeness are heuristic principles present in problem solving. Availabilityappears when the subject considers certain information and disregards others. Representativenessoccurs when the subject focuses on irrelevant information and disregards the trustfulness andpredictive power of data.When these principles are applied to relevant information the task will besuccessful, but if they are applied to superficial information mistakes will be made (Salmon, 1991).

3. MethodSubjects and Research Design: The participants were 20 college students of the first Physics courseat the National University of Rosario, Argentina, whose ages were between 18 and 22. The activitywas performed at the end of the semester, when mechanics contents had been studied. Prior to thestudy, they had worked with different kinds of problem statements with increasing degrees ofdifficulties.The participants had to read and solve individually different problems, written in a narrative style,that referred to realistic everyday situations. Type A (extracted from Nickerson et al., 1998) hadseveral details and included only verbal data; Type B included verbal and numeric data given as intraditional scientific problems; Type C included verbal and numeric data and its format resembledan encounter situation. A set of 84 problems (defined as the individuals) with complete solutionswas finally analysed

Type A: One morning, exactly at dawn (7 a.m), a monk began his way to the top of a hill.A narrow and spiralroad, half a meter wide, went to a temple at the top of the hill. The monk walked and stopped several timesalong the way to rest. He reached the temple at dusk (6 p.m). After spending several days in the temple hestarted his way back along the same road, leaving at dawn (7 a.m), walking with varying speed and makinga lot of stops along his way. His downward speed was, naturally, greater than his mean upward speed.Demonstrate that there is a certain point along the road where the monk is going to be, in both travels, at thesame time of the day.

Type B: An archer applies a force to put the arrow in the launch position. After keeping it in that position fora few seconds, she launches the arrow with a velocity v0. The arrow reaches its target: an apple which standson a pillar of height h at a distance d from the launch position. As a result of the impact, the apple breaks intotwo similar pieces, with velocities v1 and v2 of the same modulus. These two pieces reach the ground atdistances d1 and d2 on each side of the pillar.a) analyse the process, from the moment the archer puts the arrow in the arc until the arrow reaches theground,b) how can you obtain the velocity of the arrow at the moment it touches the ground?,c) determine the position where the arrow touches the ground,d) suppose that the archer launches the arrow from the same initial position, but in such a way that the arrowalso rotates around its longitudinal axis. Will the arrow hit the apple if the latter is located at the same positionas in the previous situation?

Type B: Jane, whose mass is 50 kg, is being threatened by a lion when she is by a river full of carnivorous fish.Tarzan, whose mass is 80 kg, is looking at the scene from the opposite bank and decides to help her. He runs


with a speed v0, grasps a rigid log of 60 kg mass, sticks it in the bottom of the river, and, as the wind blowswith a force F, he swings and falls on the other bank. Finally the scared lion runs away.-Analyse the situation -Find the velocity of Tarzan when he reaches JaneIn order to get back home Tarzan and Jane hang from a liana of negligible mass and length L over the river.The wind has stopped. Which is the minimum initial speed they must have if they want to reach exactly theother bank?

Type C: A road crosses the railway at an angle of 60º. A car, whose mass is 1000 kg, moves along the road at80 km/h. When it is at 150 m from the cross, a train leaves a nearby station which is located 85 m away fromthe cross, with an acceleration of 2 m/s. The driver, whose mass is 65 kg, thinks he can pass the railway beforethe train reaches the cross. Do you think he is right? Explain your answer.

Solutions were compared to the ones rendered to the following narrative problem which required onlymathematical skills.

Type B: Mary tells her friends Bob, Peter and John that she is a “psychic” and to prove it she puts 24 similarchips on a table. Then she covers her eyes and asks one of the boys to take one chip, another to take two andthe last one to take three chips.Without having seen who has taken each amount of chips she promises to guessit. But she says that in order to do so, she needs Bob to take as many chips as he has taken before, Peter totake twice the number of chips he has and John to take four times the number of chips that he has taken. Oncethey have done so, she asks her friends to put away their chips and then she uncovers her eyes. Suppose eachboy has done exactly what Mary asked. Will Mary be able to guess how many chips took each boy in the firstplace? ¿How can she do that?

Data collection and analysis: Fourteen variables were used to analyse the problem solving activities:data identification, data type, data format, data location, modelling level, solving proposal,graphical support, type of graphical representation, inclusion of data in graphics, adequacy ofgraphs, solving stages, stages organization, temporality and solution format.The modalities for eachvariable were defined during the analysis, according to the different features detected. As a result,we obtained a data matrix of 84 files (individuals)× 14 columns (variables), that enclosed a set of66 modalities. Therefore, we selected a multivariate statystical analysis, applying multiplecorrespondence analysis and mixed cluster processing (Lebart at al., 1985). The SPAD software(C.I.S.I.A., 1988) was used. The data matrix is represented as a cloud of points (individuals) in the 14-dimensional space of the variables. The distance between them is the statistical χ2. The softwaretransforms the data matrix into a variance and covariance matrix. Then, the software solves aneigenvalue problem to obtain the principal directions, called factorial axes, of the topologicalconfiguration. The eigenvalues measure the dispersions along each principal direction. Therefore, thefirst factorial axis is related to the direction of maximal dispersion of the data, and its percentage ofinertia measures the contribution of this axis to the interpretation of the initial data matrix.The secondprincipal axis, orthogonal to the first one, is oriented in the next greatest dispersion direction.Similarities among individuals were identified through the classification process, as shown in Fig. 1,where the coordinates are the projections of the points over the factorial axis.

4. Results and conclusionsThe classification analysis of the protocols allowed the identification of three classes (see Fig. 1),whose features are as follows:Class 1 (64.28 %): includes protocols corresponding to the solution of Type-B narrative situations.Some relevant data are not recognized, but the students construct an adequate situational model tosketch solutions. Solving processes are done using similar verbal, graphic and symbolic representations,with multi-step solutions, though incomplete and without chronological organization. Graphicrepresentations are, generally, schematic, with partial correspondence to the situations depicted in thestatements. Although the solving processes are sometimes biased by the use of representativenessheuristics, some stages were successfully solved, using Mechanics principles and laws.


Class 2 (17.86 %): includes protocols corresponding to the solution of purely narrative situations,like Type A.Verbal data are recognized, either in the description of the situation or in the questionsattached to the statements. The solving proposals keep a chronological organization. Solvingprocesses are done in a narrative way, mostly without graphic representations and showingdifficulties to find the relationship between the narrative text and the heuristic of “encounter”, asa conceptual physical structure to arrive to a solution. It should be stressed that only two studentsexplicitly used this heuristic and one student said: “I can’t solve it”.Class 3 (17.86 %): includes protocols related to Type-C problems. Explicit numeric data, eitherrelevant or accessory, are recognized. Similarities between the data activate previous knowledgethat is retrieved as an heuristic of accessibility. This fact accounts for the right and quickperformance in the recognition of the demands of the task. A further abstraction, within the sameheuristics, leads in a few cases to the substitution of the two-dimensional problem by a one-dimensional model, with a relevant cognitive economy.The number of constituents of the classes represents almost strictly the number of protocols thatbelong to each type of problem. The features of each class show that the data were well identifiedregardless the format. Nevertheless, when the text is purely narrative, students demand furthernumeric or at least symbolic information in order to initiate solving processes. We may concludethat the initial model is constructed considering the explicit information, although implicit relevantdata is not always detected (as occurs with events that take place in short time intervals or whenconsideration of all possible situations is required). Relevant implicit inferences are omitted. Thisfact prevents the student from developing successive models, with a progressive transformation ofrichly described events and processes to simplified models related to a set of sub-problems, toarrive to a solution. In this process, the student “translates” the narrative text into a scientific one.Particularly, if the student is restricted to a chronological sequence, then the representativenessheuristics should not operate to provide an immediate solution (e.g., to solve problem Aconsidering it as an “encounter” problem of two ideal monks moving simultaneously downwardand upward).When narrative problems include numerical or graphic data, provided as in traditional scienceproblems, it is easier for the student to develop an appropriate situational model. This fact allowsthem to apply of heuristics and specific principles to arrive to the solution of at least some stages,as was recognized in class 3.


Fig.1: Affinityclassification of the

protocols based on thecomprehension and

solving tasks. (Class 1 / 3includes 54 protocols,while the classes 2 / 3

and 3 / 3 are integratedby 15 protocols)

Therefore, pure narrative problems offer additional difficulties. They involve people performingactions in pursuit of goals, the existence of obstacles that interfere and emotional reactions that thereader has to understand. Many inferences, generated during the comprehension of narrative textsand added to the specific demands of the task, require a higher cognitive effort.We may also conclude that purely narrative statements activate informal reasoning patterns thatlead students to use systems of beliefs that bias the solution with the demand of unnecessary dataand with the addition of difficulties to the situation.

ReferencesBaron J., In Voss, Perkins and Segal (eds.), Informal reasoning and education, chap. 8, New Jersey.LEA, Hillsdale,

(1991).Lebart L., Morineau, A., Fenelon, J., Tratamiento Estadístico de Datos, Barcelona, Marcombo, (1985).C.I.S.I.A, SPAD N Integré, París, (1998).Salmon M., Informal reasoning and informal logic in Voss, Perkins and Segal (eds.), Informal reasoning and

education, chap. 8, New Jersey: LEA, Hillsdale, (1991).Johnson - Laird, P. N., Mental Models, Cambridge (Mass.), Harvard University Press, (1983).Johnson – Laird, P. N., Anderson, T., Common sense inference, Cambridge: MRC Applied Psychology Unit, (1988).Galotti K. M., Approaches to studying formal and everyday reasoning, Psychological Bulletin, 105, (1989), 331-351.Van Lehn K., Problem Solving and Cognitive Skill Acquisition, in Michael I. Posner, ed., Foundations of Cognitive

Science, Cambridge (Mass.), The MIT Press, (1998).Nickerson R.; Perkins, D., Smith, E., The teaching of the thinking, N.J., LEA, (1998).Nickerson R.S., Reflections on reasoning. N.J., LEA, Hillsdale, (1986).Perkins D. N., Faraday, M., Bushey, B., Everyday reasoning and the roots of intelligence, in Voss, Perkins and Segal

(eds.), Informal reasoning and education, chap. 5, New Jersey, LEA, (1991).

NotesAs a reference for the reader, we attach briefly orientations to solve the proposed problems.

Type A: One morning, exactly at dawn, a monk…The solution is obtained applying a graphic heuristic: drawing a graph of the monk’s ascent and descent.The graph can take any shape because nothing is said about the hourly progress. As the departure andarrival take place at the same hour but on different days, the problem is equivalent to another one wheretwo people traverse the same mountain path at the same time (7 a.m) on the same morning, but one in anupward direction and the other downward. The solution is easily obtained if we move and superimpose thegraph of the monk’s ascent and descent.


7 a.m

(some

days

later)

x

t7 a.m

(1st day)

6 p.m

(1st day)

top of the hill

Monk’s problem transformed problem

x

t7 a.m tencounter

top of the hill

Type B: An archer applies a force…To solve the problem, the process needs to be divided into five stages and to discuss the assumptions thatsupport the applied model:(1) the archer puts the arrow in the launch position: Work done = Ep arch(2) the arrow leaves with a velocity v0 : Ep arch = 1/2 m v0

2

(3) the arrow describes a parabolic trajectory towards the target: d = vo te ; h = 1/2 g te2 the arrow’s velocity

v just before reaching the target has two components: vx = vo ; vy = g te(4) the arrow impacts the target and breaks it into two similar pieces, with velocities v1 and v2

perpendicular to v0 with conservation of linear moment: marrow v = m1 v1 + m2 v2 + marrow v’

or marrow (v0 , g te, 0) = m1 (0,0,v1) + m2 (0,0,−v1) + marrow (v’x, v’y, v’z)(5) the arrow and the two pieces move along new parabolic trajectories with initial velocities v1, v2 and v’

until they reach the ground, while the center of mass describes another parabolic trajectory with aninitial velocity vCM = marrow (v0 , g te, 0)/(m1+ m2 + marrow). Therefore, the position where each one willreach the ground can be obtained.

If the arrow also rotates around its longitudinal axis, it has also rotational kinetic energy:Ep arch = 1/2 m v0

2 + 1/2 I ω2. Therefore, it will not reach the target.

Type B: Jane, whose mass is 50 kg, is being threatened by a lion…To solve the problem, the process needs to be divided into three stages and to discuss the assumptions that

support the applied model:(1) Tarzan runs with a speed v0, grasps a rigid log (a plastic collision with angular moment conservation):

mTarzan v0 h = I(Tarzan + log) ω0 (where ω0: angular velocity of the log; and h: river’s depth) (2) Tarzan swings to the other bank while the wind blows (the moment of the wind’s force produces the

variation of the angular moment of the system (Tarzan + log ): r × F = dL/dtTherefore, the velocity of Tarzan (v = ωr) when he reaches Jane can be obtained assuming, for example, that

F is constant.(3) Tarzan and Jane grasp a liana and cross the river: ∆Ep = −∆Ek

Type B: Mary tells her friends Bob, Peter and John that she is a “psychic”…This is a combinatorial task. The solution requires to recognize six possibilities as follows:


Chips = 24 Possibilities 1 2 3 4 5 6 friends B P J B J P P B J P J B J B P J P B 1st extraction 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2nd extraction 1 4 12 1 8 6 2 2 12 2 8 3 4 2 6 4 4 3 number of extracted chips 23 21 22 19 18 17number of chips on the table 1 3 2 5 6 7

Mary can guess when she sees the number of chips that are lying on the table

Type C: A road crosses the railway at an angle of 60º…The car passes the cross 6.75 seconds later (t = distance to the cross/speed of the car). During this time thetrain has moved a distance x = 1/2 at2 = 45.57m. Therefore, the driver is right.

DIFFICULTIES ON INFERENCIAL PROCESS. A STUDY ON THERMODINAMICPROBLEMS

Marta Massa, Marta Yanitelli, Susana Cabanellas, Facultad de Cs. Exactas, Ingeniería yAgrimensura, Universidad Nacional de Rosario, Rosario, Argentina

1. IntroductionDifferences on problem-solving performance between experts and novices have been studiedduring more than two decades. Attention has been focussed on mental processes to organiseknowledge (Chi, et al., 1988; Chi, et al., 1982) on specific areas such as: categories that the subjectsuse to organise knowledge; relationships between these categories and problem-solving abilities;identification of similarities between problems; degree of mental processing required to producethe categorisation tasks (Smith, 1992). The latter has received less attention and the degree ofmental processing has been analysed by measuring the time required to complete the tasks. Resultshave shown that there is not a direct linkage between expert behaviour and successful solving

procedures, as experience sometimes may be associated to automatic processing rather than toproblem-solving ability (Schoenfeld and Herrmann, 1982).In physics courses at University, students perform problem-solving activities that demand similarconceptual knowledge with quite different success. Frequently, teachers ask themselves if theobserved differences are due to features that Chi, Feltovich and Glaser (1981) attributed to theproblem’s “surface structure” (literal physics terms, objects, described physical configurations), thestructure of the statements, the language and style, or to the cognitive process that are demandedfor the situational modelling.Among the procedural abilities required during problem solving, the production of inferences hasnot been extensively studied. Its relevance lies on the fact that it is an important activity ofinformation processing: internal representation of data, integration to previous knowledge,transformation in order to construct new meanings and relationships (Riviere, 1986).Many researches have shown that a procedural knowledge of algorithmic is a necessary conditionbut not a sufficient one for the comprehension of physics concepts (Solaz et al., 1995; Chi et al.,1981; Kempa, 1991). So, an effective and successful problem-solving activity would lie on the wayof representing connectives between concepts involved in mental models.Specifically, a remarkable different performance was observed when students solved problemsrelated to the model of an ideal gas if: (a) it is applied in the analysis of a process or (b) it is requiredto determine its validity in a real situation. Although both activities imply the same concepts, theformer is related to an algorithmic procedure, while the latter involves inferences and conceptsprocessing. In fact, it was significant that 37% of the students that were successful when solving tasklike (a), failed on solving (b).The purpose was to analyse patterns of reasoning when students make inferences to solve problemsexpressed in conditional statements. The research sought data to answer the following questions:1. Which is the reasoning structure employed by a student to solve the reference for a conditional

statement?2. How does a conceptual model, such as the ideal gas one, relate to mental models representing

a real or possible situation involving a specific gas?

2. MethodSubjects. The subjects of the study were 64 students that attend an engineering basic physics coursedealing with Thermodynamics. Classes were developed according to the conceptual structurepresented in Statistics Physics and Thermodynamic (Jancovici, 1976). They were asked to solve aset of classical problems dealing with the framework of the kinetic theory of gases and the first lawof thermodynamics, as a partial evaluation of the learning process. The problems were similar tothose presented in the class and the textbooks that the students used as a complement referencefor their studies. We specifically analysed one of these problems stated in a conditional form,extracted from Fishbane (1994):

1 mol of helium gas is contained in a cubic recipient of 50 cm each side. In this condition its internalenergy is 3600 J. If it were possible to apply the model of ideal gas to the air in normal conditions, wouldit be possible to do the same with the described helium system?

Data collection. We used as protocols only 26 pencil-and-paper tasks (41 %) belonging to thesubjects that intended, successful or unsuccessfully, to solve the problem. During the first stage, weanalysed the protocols considering: i) selected data and their initial transformation; ii) principlesand laws applied to calculate the relevant variables; iii) conclusion statement; iv) justification.In a second stage, we attempted to identify similar features in order to establish possible categoriesof resolution style. Finally, we made an interpretative analysis of possible inferential processesinvolved in the detected categories.

3. ResultsThe protocols were examined in order to find similarities in (a) the identification of data, (b)


variables that were calculated to arrive to a conclusion and (c) the justification stated to make theconclusion consistent. As a preliminary result, 14 students concluded that “the model of ideal gases isapplicable”, showing the three mentioned stages while reasoning; 7 students considered that “themodel was not applicable”, after the three stages, and 5 did not arrive to any conclusion, with scarcecalculus of variables.Table I shows the resultant categories for the students that arrived to a conclusion.

4. DiscussionThe analysed protocols suggest the existence of different stages to arrive to a conclusion. Only fewsubjects, belonging to p-V-T (Yes) category, proceeded as Fishbane (1994) did. Consequently, theirreasoning may be assumed as an “expert” one. As a first approximation, the stages involved inarriving to a conclusion may be interpreted as following:


CategoryCategory Analysis

Model

validation

Justification characterisation Frequency

p-V-T

Yes

He – Air comparison using the functional relationship between volume, pressure and temperature (V-T or p-T). In some cases the comparison is made only between pressure values. The criterion is based in the proximity ofthe stated numerical values.

5

No

He – Air comparison using the p-T variables but looking for coincidence with the normal conditions values. One case takes into account the different molecular composition. One case presents no justification.

4

Density YesIt is stated that the density is low without establishing a comparison parameter. In one case the comparison is made between molecular density of He and Air.

4

Internal energy

YesComparison of Internal energies only. The functional relationship between p-V-T is ignored. The criterion is based in the proximity of the stated values.

2

NoComparison of Internal energies looking for coincidence between numeric values. The functional relationship between p-V-T is ignored. Misconceptions are detected.

2

-p-T Yes

In one case the comparison is made between the three variables. The other one resorts to the qualitative definition of ideal gas stating “if air is an ideal gas at normal T and p, then it is a low density gas”.

2

U-T NoHe – Air comparison using the U-T variables but looking for coincidence with the normal conditions values. Misconceptions are detected.

1

-U YesHe – Air comparison based on y U .Confusions over the molecular composition of the air are detected.

1

Table I: Description of the identified categories.

1º) Representation of the problem statement initial models. A representation of the content isconstructed by integrating the text components and refining specific information of previousknowledge related to the molecular structure of gases. It may be supposed that “bridging”inferences are made. They do not strictly derive from the linguistic properties of the text butcontribute to solve the reference, i. e.: mol→ Avogadro´s number; helium → monatomic.

2º) Interpretation of the question stated as a conditional clause. This premise comprises threeelements: air at normal conditions, ideal gas and a conditional statement. The disposableinformation does not allow to establish in which direction the process is made, but the subject issupposed to organise progressive models and to make new “bridging” inferences: air → mixture;normal conditions → pressure = 1 atm; temperature = 0º C = 273 K; ideal gas → low density, diluted; U= Ek av. and other related concepts. A more complex task would be, perhaps, the analysis of the syntaxto discover the conditional clause and to transform the premise to a logic format “if p then q”.

3º) Substitution of transformed premises. Substitution of (a) an entity by the class1 it belongs to and(b) the normal condition by values proximal to it. In this way the premise extends its meaning asshown in the figure presented below.


1 mol of helium gas is contained in a cubic recipient of 50 cm each side. In this condition its internal energy is 3600 J

model of the premise

Previous knowledge about gases

He

1 mol of helium gas is contained in a cubic recipient of

50 cm each side. In this condition its internal energy is

3600 J...

Previous

knowledge

modelof pair

modelof q

Idealgas

If air is in normalconditions then it is an ideal gas

g p

If it were possible to apply

the model of ideal gas to the air in normal conditions,

¿would it be possible to do

the same with the described helium

system?

Previous

knowledge

Idealgas

modelof p

air

If air is in normalconditions then itis an ideal gas

if a gas state is near to normalconditions then it is an ideal gas.

model of p*

model of q

1 The class pis an abstract entity that represents a group of elements p* having a common property. In this case, the twoentities (helium and air) are interpreted in the general sense of a gas.

4º) New model construction which implies “renewing” the interpretation of the first premise tointegrate it to the model of the second one and a “re-ordering”. In addition, a complex selectionprocess of the necessary and sufficient conditions related to the “proximal to normal conditions ofthe gas” is performed to define a bridge to compare the class entities. Then, the helium state isevaluated in order to configure the following modus ponens figure, from which the requiredconclusion arises:

If a gas state is near to normal conditions then it is an ideal gas.Helium state is near to normal conditions.Helium may be considered an ideal gas.

5. Conclusion and implicationsA significant proportion of the students (see Table I) consider Density as a relevant variable. Thisfact shows an availability bias when, probably, the qualitative definition of an ideal gas isrecuperated: It is said that an ideal gas is the state to which all gases tend when their density is verylow (Jancovici, 1976). The subjects would be moving through the mentioned stages to elaborate an“helium model” making use of their previous knowledge of ideal gas as a diluted one.Likewise, it may be interpreted that subjects included in the Internal energy category select thisvariable because of the influence of its presence as a datum in the first premise of the problem.Then, this fact shows a bias of representativeness that takes them to set up the followingconfiguration:

If a gas has U equal or proximal to that of air in normal conditions then it is an ideal gas.Helium has U=3600 J.Helium may be considered an ideal gas.

Students belonging to ρ-p-T category would seem to reason as those included in the p-V-Tcategory, but considering density as relevant variable. This may be attributed to the influence of thetextbook definition. Finally, the two latter categories, with a scarce number of individuals, share thecharacteristic features of Density and Internal energy ones.In summary, the different types of reasoning identified seem to be related to the processes requiredby the third and fourth stages to transform a premise in spite of a proximity criterion and todetermine the necessary and sufficient conditions to validate the model application. The study hasallowed the identification of some organisational patterns for the information and elaboration of aconditional statement, which deserve further analysis.Nevertheless, the evidence suggests that students are more apt to look beyond the logical form ofa proposition and consider alternate hypotheses in contexts that they are able to restructureconcretely. Teaching should be geared toward assisting students to achieve this concreterestructuring. Likewise, considering the interplay among a variety of variables in a given context iscrucial for the generation of viable hypotheses and reasoning about the situation. Students shouldbe taught to carefully consider the relevant variables in a given situation and the necessary andsufficient conditions for the applicability of a given conceptual model.

ReferencesChi M. T., Feltovich P. J., Glaser R., Categorisation and representation of physics problems by experts and novices,

Cognitive Science, 5, (1981), 121-151.Chi M. T., Glaser R., Farr M.J., The nature of expertise, Hillsdale, NJ: Erlbaum, (1988).Chi M. T., Glaser R. Rees E., Expertise in problem solving, In Sternberg, R. J. (Ed.), Advances in the psychology of

human intelligence,.1, Hillsdale, NJ, Erlbaum, (1982).Fishbane P., Gasiorowicz S., Thornton S., Física para Ciencias e Ingenierí, 1, Prentice-Hall, (1994).Jancovici B., Physique Statistique y Thermodynamique, Premier Cycle, Mc Graw Hill, (1976).Kempa R. F., Students’ learning difficulties in Science. Causes and possible remedies, Enseñanza de las Ciencias, 9,

(1991), 119 – 128.Riviere A., Razonamiento y representación, Madrid: Siglo XXI, (1986).


If p then q p*q

Schoenfeld A. H. and Herrmann D. J., Problem perception and knowledge structure in expert and novicemathematical problem solvers, Journal of Experimental Psychology: Learning, Memory and Cognition, 8, (1982),484 - 494.

Smith M.U., Expertise and the Organisation of Knowledge: Unexpected Differences among Genetic Counsellors,Faculty, and students on problem Categorisations Tasks, Journal of Research in Science Teaching, 29(2), (1992), 179-205.

Solaz J., Sanjosé V., Vidal-Abarca E., Influencia del conocimiento previo y de la estructura conceptual de losestudiantes de BUP en la en la resolución de problemas, Revista de Enseñanza de la Física, 8, Nº 2, (1995), 22-28.

MATHEMATICS OF DIMENSIONAL ANALYSIS AND PROBLEM SOLVING INPHYSICS

Decio Pescetti, Dipartimento di Fisica, Università di Genova, Italy

1. IntroductionAs is well known, the qualitative methods, based on the application of the principles of dimensionalhomogeneity, continuity and symmetry, offer the opportunity for a truly fertile analysis of thephysical systems prior to their complete mathematical or experimental study [1-3]. In problemsolving, the qualitative methods enable us to deduce useful information about the dependence of aphysical quantity (the unknown) on other relevant quantities (the data) [4-8].The complete description of a real physical system would require a great number of parameters.Wemust select and take into account the important quantities and ignore those which have relativelysmall effects. A given physical quantity is expressed by a number followed by the correspondingunit of measurement. The radius of the Hearth is 6.4 · 106m, but it is also 2.1 · 10-10; from this pointof view it has no meaning to speak of big or small numbers. The order of magnitude of the relativeeffect, on the value of a specified unknown X, of a neglected quantity can be expressed by properdimensionless products (pure numbers) of the system’s characteristics. For instance, consider adamped simple harmonic oscillator. A body of mass m is acted on by an elastic restoring force F=-kx, where x is the deplacement from the equilibrium position and k is the force constant. Whatis the effect, on the period τ, of a viscous damping force, where r is a constant.


dtrdxF /viscous =

What is the effect, on the period , of a viscous damping force, dtrdxF /viscous = , where r is a

constant. As already remarked, to speak of small or big value of r has no meaning; a given damping constant of value, for instance, skg /10 3 , has also the value amu/s2310024.6 . As is well

known, such an effect is expressed by the dimensionless product 2/1)/(mkrP = , whose physical meaning is: the ratio between the maximum value of the damping force and the maximum value of the elastic restoring force. We show that the origin of the information, on the problem’s solution, obtained by the qualitativemethods, is made more transparent by making a clear distinction between the mathematicaldimensional analysis results and the phenomenological assumptions and laws of nature relatingsuch results to the physical world. Obviously, wrong phenomenological assumptions plus correctmathematical dimensional analysis will lead to erroneous results, as discussed in the comment toexample 4 of section 4.Let us remark that the examples of dimensional analysis given in introductory physics textbooksare likely to be misleading. The reader may be left with the impression that dimensional analysisis a routine procedure. The practice of dimensional analysis requires a great deal of insight andexperience. For instance, such insight enters in a crucial way into the initial selection of variablesto be included in the analysis. Often considerable penetration is required to recognize when aparticular dimensional constant, such as the acceleration due to gravity, may be required. Failureto include a relevant variable or dimensional constant will lead to an incorrect result.

Unfortunately, there is nothing in the nature of the mathematics of dimensional analysis to tell thepractitioner that a crucial variable has been omitted.In section 2 we discuss the mathematical bases of dimensional analysis. In section 3 we present aset of exercises in linear algebra, which are proposed as an help to understand the rational of thepaper: the necessity of a clear distinction between purely mathematical results and physicalassumptions. In fact, the answers of such exercises are involved in the qualitative solution of thephysical problems discussed in section 4. Section 5 is devoted to concluding remarks.

2. Dimensional analysis and continuity principleThe dimensions of physical quantities are represented by vectors in an abstract finite-dimensionallinear vector space, referred to as the dimension space. In Mechanics the dimension space has abasis of three elements: length L, mass M, time T. The extension to electromagnetism requires abasis of four elements (L, M, T, electrical current I). Finally, the extension to thermal phenomenademands a five element basis: (L, M,T, I, absolute temperature Θ). For instance, in the linear vectorspace of mechanics, the quantities length, mass, time, density, velocity, acceleration, force, viscosityand elastic force constant are represented by the vectors (1,0,0), (0,1,0), (0,0,1), (-3,1,0), (1,0,-1),(1,0,-2), (1,1,-2), (1,0,-1), and (0,1,-2) respectively.The principle of dimensional homogeneity (PDH) states that in any legitimate physical equation thedimensions of all terms which are added or subtracted must be the same.


The problem description indicates that a physical quantity X (the unknown) is a function of other quantities nAAA ...,,, 21 (the data):

)...,,,( 21 nAAAfX = . (2.1)

The PDH allows us to study a function of fewer arguments: )...,,,( 21 mx PPPP = , (2.2)

Substantially, the Buckingham theorem is the following theorem of linear algebra: if the rank of the matrix qjniaij ...,,2,1and...,,2,1,)( == , associated with the n vectors )...,,,( 21 iqii xxx is r<q, there are

exactly r vectors which are linearly independent while each of the remaining m=n-r vectors can be expressed as a linear combination of these r vectors. In other words, it can be formed a complete set of m independent linear combinations between the n vectors )...,,,( 21 iqii xxx , such as

0)(...)()( 2211 =+++ nn xhxhxh For every set nhhh ,...,, 21 , there corresponds an unique

dimensionless product between the data nAAA ...,,, 21 : nhn

hh AAAP = 2121

A complete set of independent Px is formed by m+1 elements. The principle of continuity states that small causes produce small effects. In teaching physics, this principle is tacitly taken for granted. Usually, its statement is not explicitly given. Perhaps because there are exceptions [1,9], especially for complex systems (butterfly effect). In fact, arbitrarily small causes might produce finite effects, but this cannot be the rule. The idealizations of physics such as frictionless planes, inextensible string, massless pulleys etc. find their justification and validity in the continuity principle. Such idealizations imply that we may have experimental conditions in which the friction of the plane, the extensivity of the string, the mass of the pulley etc., have negligible effects, with respect to those of other causes and, of course, with respect to the finding of a well defined unknown. Let us consider, for the sake of simplicity, the particular case m=1 in eq. (2.2),

)(PPx = .

where Px is a dimensionless product between X (elevated at the first power) and some of the data,and P1, P2, ..., Pm are a complete set of independent dimensionless products between the datathemselves. One has (Buckingham’s theorem): m=n-r, where r is the rank of the matrix formed bythe dimensional exponents of the data.


Let us suppose, without loss of generality, P<<1. We say that xP has an essential type dependence

on P if

=0

)(lim0

PP

.

On the contrary, the dependence is non-essential if cP

P=)(lim

0 ,

where c is a dimensionless positive constant.

Example1. (Harmonic oscillator) A body of mass m is acted on by an elastic force kxF = , where x is the deplacement from the equilibrium position and k is the force constant.. The body is relaxed from the rest at an initial position x0 . The body executes an oscillatory motion. Find the period . Solution: The relevant parameters are m, k, x0 ; so

),,( 0xkmf= .

Exercise 1. Find the coordinates of the vector (0,0,1) relative to the basis (0,1,0), (0,1,-2), (1,0,1). Answer: (1/2, -1/2, 0). Exercise2. Find the coefficients hi of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0), (0,1,-1) such as: 0)1,1,0()0,0,1()2,1,0()0,1,0( 4321 =+++ hhhh .

Answer: hhhhhhh 2,0,, 4321 ==== .

Exercise 3. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,-1). Answer; i) (0,0,1)=1/2(0,1,0)-1/2(0,1,-2); ii) (0,0,1)=(0,1,0)-(0,1,-1); iii) (0,0,1)=-(0,1,-2)+(0,1,-1). Let us remark that answer iii) is not independent of answers i) and ii). Exercise 4. Find the coefficients hi of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0), (0,1,0) such as: 0)0,1,0()0,0,1()2,1,0()0,1,0( 4321 =+++ hhhh .

Answer: hhhhhh ==== 4321 ,0,0, .

Exercise 5. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,0). Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2). Exercise 6. Find the coefficients hi of a linear combination of the vectors (1,0,0), (1,0,0), (0,1,0), (1,0,-2) such as: 0)2,0,1()0,1,0()0,0,1()0,1,0( 4321 =+++ hhhh .

Answer: 0,0,, 4321 ==== hhhhhh .

Exercise 7. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (1,0,0), (1,0,0), (0,1,0), (1,0,-2). Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2).

Examples

The criteria for the evaluation of the orders of magnitude [7,8] are: a) The dimensionless constants c are of the order of unity. b) A dimensionless product P very different from unity is irrelevant to the solution of the problem. The criterion b) is satisfied under the following conditions: i) continuity of function ; ii) a non-essential dependence of on P; iii) a correct choice of the

dimensionless product xP , that is in xP should appear the dominant data. By dominant data, with

respect to the prediction of the unknown X, we mean a subset of s (dimension of the linear space) relevant parameters for which the phenomenology under study is preserved, even if all the other parameters are absent (in the sense that they are not influential).

3. Linear algebra exercises


By PDH and answer of algebra exercise 1, one finds,

kmc /= , where c is a dimensionless constant. The period is independent of motion amplitude x0 . By criterion a) for the evaluation of the orders of magnitude, the constant c is of the order of the unity. As is well known, the exact value of c is 2 .Example2. (Damped harmonic oscillator) Find the effect, on period , of a viscous damping force dtrdxF /= . Solution: The relevant parameters are m, k, x0, r. So,

),,,( 0 rxkmf= .

By PDH and answer of algebra exercise 2, one has hmkrP )]/([ 2= . A proper choice of the constant

h is h=1/2. In fact 2/12/1= kmrP is the ratio between the maximum value of the damping force and the maximum value of the elastic restoring force. [cause: friction) ]0[0 P .By PDH and answers i), ii) and iii) of algebra exercise 3, one has:

kriiirmiikmi /),/),/) .Representation i) is correct. In fact, the problem of finding the period makes sense also in absence of the friction force. Representations ii) and iii) are senseless from the physical point of view. Let us remark that a complet set of independent dimensionless products Px is formed by two elements. For instance: )//(),//( krrm . Obviously, there are infinite other possibilities; for instance:

)]/(),//([)]//(,//[ 5/15/35/4 krmkrorrmkm . The correct choice of Px is a matter of physics; it does not follow from the mathematics of dimensional analysis. In conclusion

)(Pk

m= .

The dependence of PP on)( is non-essential . As is well known, the complete exact mathematical solution of the problem leads to:

2/12 ])2/(1[

2)(

pP = .

Example3. (Harmonic oscillator with mspring ) A body of mass m is placed on an horizontal air track. The body is attached to an horizontal spring of force constant k and massa ms. The body is released from the rest at a position in which the spring is stretched by x0 . The body executes an oscillatory motion. Find the period .Solution: The relevant parameters are m, k, x0, ms. So,

),,,( 0 smxkmf= .

By PDH and answer of exercise 4, one finds that there is the following dimensionless product between the data: h

smm )/( , where h is a constant. A proper choice of h is h=-1. Then: P=m/ms .

0)(0]mass)spring:[(cause P .

By PDH and answer of algebra exercise 5, one finds: kmi /) and ii) kmi s /) .

The dominant parameters are m, k, x0 . Reprensentation i) is correct. In conclusion, one has

)(/ Pkm= , where is a dimensionless undetermined function of P. There is a non-essential dependence of

)(P on P: constantpositive)(lim0

=PP

.

A complete exact mathematical analysis yields to: 2/1)3/1(2)( PP += . Example 4. (Flexible chain) A flexible chain of total length l and mass m rests on a frictionless

4. ConclusionsThe application of the qualitative methods in problem solving is put into execution according to thescheme: [(1) Phenomenological behaviour of the system] ⇒ [(2) Identification of the unknown Xand of the relevent parameters A1, A2, ..., An] ⇒ [(3) linear algebra (Buckingham’s theorem)] ⇒[(4) Complete set of independent dimensionless products P1, ..., Pm; complete set of Px

independent dimensionless products] ⇒ [(5) Proper choice of P1, ..., Pm; correct choice of Px] ⇒[(6) Physical exploitation of the qualitative solution].The mathematics of dimensional analysis is involved in steps (3) and (4) only. The preliminary step(1) requires sound information about the system’s phenomenological behaviour. Step (2) demandsthe knowledge of the physical relations that must be invoked to work out a complete quantitativestudy of the system. In step (5) the mathematical dimensional analysis results are coupled with thephysics of the problem, and finally in step (6) an explicit interpretation of the qualitative solutionis given.


table with length x0 overhanging the edge. At time t=0 the system is released. Find time after which the entire chain leaves the table. Solution: The relevant parameters are: gmlx gravityand,,0 . So,

),,,( 0 gmlxf= .

By PDH and answer of algebra exercise 6, one finds: hlxP )/( 0= . A proper choice of h is h=1.

Then lxP /0= .

By PDH and answer of algebra exercise 7, one finds: gx /0 or gl / .

In conclusion, the correct representation is )(/ Pgl= .

The sliding time does not depend on the mass m of the chain. It appears quite natural, from the physical point of view, to predict the following special cases: a)

])([)0( PP , b) ]0)([)1( PP . Therefore there is an essential dependence of)(P on P.

The solution can also be written ))/( 1

4/120 Pglx= ,

where )()( 4/11 PPP = . The essential dependence on P is preserved. The parameters x0 , m and

g are a set of dominant parameters, but condition ii) for the validity of criterion b) for the evaluation of the orders of magnitude is not satsfied. The function )(P is an “universal” function. The function )(P cannot be obtained by the qualitative methods. The function (P) can be found: i) by the analytical solution of the equation of motion F=ma;ii) by numerical solution of the equation of motion F=ma;iii) experimentally. One finds: 0)1(;467.0)9.0(;32.1)5.0(;99.2)1.0(;)0( ===== .Remark: [(wrong phenomenological assumptions)+(correct mathematical dimensionl ansalysis)]

[erroneous (senseless) information on the problem solution].Wrong assumption: the motion of the chain is periodic; wrong question: find period . By PDH etc. : lxPPgl /;)(/ 0== .

Erroneous conjecture: cPP

= constantpositive)(lim0

. In conclusion: senseless result: period

glc /= .

Let us remark that, when studying a physical system, the information obtained by the qualitativeanalysis might be increased by breaking the problem into subproblems, even if additionalquantities must be introduced [5].The notion of similarity is closely related to dimensional analysis and model building. One of thereasons that dimensional analysis is a useful method in physics is that many variables are combinedinto one or few dimensionless numbers (products). It is significant that these dimensionlessproducts serve often as a quantitative measure of similarity.It is surprising that no introductory university algebra textbook mentions the dimension space ofthe physical quantities, as an example of linear vector space.We hope that this paper will contributein filling the gap between mathematics and physics in the teaching/learning of this important topic.The sistematic resort to qualitative methods along the lines presented in this paper should be thesource of a fertile reflection on the mathematical modeling of physical systems and on the model-reality relationship.

References[1] G. Birkkoff, Hydrodynamics (Princeton: Princeton Un. Press), (1960).[2] A. B. Migdal, and V. Krainov, Approximation Methods in Quantum Mechanics (New York, Benjamin), (1969).[3] L. I. Sedov, Similarity and Dimensional methods in Mechanics, (Moscow: MIR), (1982).[4] M. Hulin, Eur. J Phys., 1, (1980), 55.[5] J. M. Supplee, Am. J. Phys., 53, (1985), 549.[6] E. A. Deslodge, Am. J. Phys., 62, (1994), 216.[7] D. Pescetti, Qualitative Methods in Problem Solving, in Bernardini C et al (eds), Thinking Physics for Teaching,

(New York, Plenum Press), (1995), 387-399.[8] D. Pescetti, Small Numbers and Mathematical Modeling of Physical Systems, in Pinto R and Suriqach S (eds),

Physics Teacher Education Beyond 2000, (Paris: Elsevier Editions), (2001).[9] J. Gleigk, Chaos (Viking Penguin), (1987).

DEVELOPING THINKING IN PHYSICS THROUGH PROBLEM SOLVING

Sabina, Sawicka - Wilgusiak, Faculty of Pedagogy, Warsaw University, Poland

Problem solving dates from J. Dewey, from the moment he has published his book entitled „Howwe think?”. In this book J. Dewey has defined the entire act of thinking through the followingstages logically connected with themselves and namely:- perception of the difficulty;- identification of the difficulty and its definition;- possible solution;- drawing conclusions from the proposed solution;- further observations and experiments leading to accepting or to refusing the conclusions [1].A man begins to think when he meets a difficulty.The idea of a „problem” is defined by the generaldidactics as:- a structure of incomplete data; [2].- tasks that require overcoming some difficulty of theoretical and practical character with the aid

of the subject research activity;- problem, that is task, which pupils cannot solve by means of their own knowledgeProductive thinking, that enriches this knowledge, is required [3].Problem tasks from the field of physics can be split up into three groups:1) tasks for forecasting the physical phenomena;2) tasks for designing the experiments;3) tasks for explaining the phenomena.


To point 1)The following experiment may be used as an example of the problem task for forecasting thephysical phenomena [4].

Experiment 1The exits of the glass tube are closed with metal plates, e.g. metal elements from two electrophones(Fig. 1).Plates located in this way form some kind of a flat capacitor. We put inside the tube a table-tennisball with its surface metallized. We connect the metal plates with the poles of an electrostaticmachine and we put it into operation.What kind of effect do we expect to find? Will the ball remainin its state of rest or will it begin to run alongside the tube?


E = Sd

U

8

2

Solution:The ball will move between the covers of the capacitor An oscillating movement of the ball willtake place until the capacitor is discharged. With the distance between the capacitor platesremaining unchanged, the number of runs made in a given time depends on the voltage applied tothe plates, so it characterizes the field energy.The electrostatic field energy in the space between the flat capacitor plates amounts to:

A Glass tubeB Table-tennis ball with conducting surfaceC, D Metal plates.E, F Insulating holdersG, F School stands

Fig. 1.

where:U = voltage between the capacitor plates;d = distance between the capacitor plates;S = area of the capacitor plates;ε = dielectric constant of the substance between the plates.This energy can be converted into mechanical energy that can be observed in the experimentpresented.A theoretical example for forecasting phenomena might be the following one: “What wouldhappen on Earth if the Sun would stop to shine?”

To point 2)The following experiment may be used as an example of a problem task for designing research

Experiment 2The rotor of the shape shown in Fig. 2 has been put onto an horizontal axle. In which way, usingelectrostatic phenomena, can such a rotor be put in rotary motion?

The rotor can be made of eight table tennis balls painted with a conducting paint, six rods made of organicglass should be used with a length of 13 cm and a cross section of 4x3 mm. Glass pipes can be also used.

1863. Topical Aspects 3.2 Problem Solving

A Insulating standB Table tennis balls covered with conducting paintC Rod of organic glass

Fig. 2.

A, B Table tennis balls covered with conductingpaint

C Horizontal axle with conical bearingsE, D Insulating rodsG, F Insulating posts

Fig. 3.

Fig. 4.

The balls are pushed onto the ends of the rods . The rods are attached to the horizontal axle withconical bearings (Fig. 3) between two parallel posts. In case of glass pipes they can be fixed withthe aid of cork and two gramophone needles as the horizontal axle. The rotor has to be balancedin order to obtain a uniform rotary motion; it can be put into rotation by placing it between twometallic bodies supported on insulating stands. Those bodies are electrified by connecting themwith an electrostatic machine (Fig. 4).

A RotorB ConductorC Insulating stand

To point 3)Many problem tasks can be given to explain physical phenomena:- Why is the sky blue?- How long should day and night last on Earth so that a body would weigh nothing at the equator?- Why do the ships not sink?- What part of a wooden cube of specific weight D=0.6 G/cm3 will be under the water surface,

when it floats?- Two resistors of value 20Ω and 50Ω have been connected in series and in parallel (Fig. 5). In

which circuit will the largest amount of heat be issued and why?


20

50

20 50

220 V

220 V

Fig. 5.

Summing up it can be stated that putting to pupils problem tasks for forecasting physicalexperiments, designing researches and explaining such phenomena will educate both logical andintuitive thinking. It has an influence on the motivation to learn.The pupils start to like physics andto learn it with more interest and the knowledge thus gained is lasting and operative.

References[1] J. Dewey, How we think. A restatement of the relation of reflective thinking to the educative process. New York,

(1910).[2] W. Okon, Wprowadzenie do dydaktyki ogólnej. Warszawa, PWN, (1987), 227.[3] K. Kruszewski, Sztuka nauczania. Czynnosci nauczyciela. Warszawa, PWN, (1998), 112.[4] J. Eisner, B. Jackowski, P. Labuz, D. Tokar, Wybrane cwiczenia z fizyki. Opole (1969), 31.


3.3 Modelling

MATHEMATICAL FORMAL MODELS FOR THE LEARNING OF PHYSICS: THEROLE OF AN HISTORICAL EXAMPLE

Giorgio T. Bagni, Department of Mathematics, University of Roma, La Sapienza, Italy

The History of Sciences is an important tool for Didactics: in fact in this paper we study theintroduction of the concepts of work and of kinetic energy according to an historical example. Ofcourse, first of all it is worth noting that frequently historical development of a concept is notsuitable in order to plan curricula, although sometimes we can point out analogies between stagesof the historical development and corresponding educational stages: educational work can bebased upon the results achieved in the full historical development, and we particularly underlinethat the History of Sciences makes it possible to point out mathematical formal models that canbe used in Didactics of Physics by analogy.The 17th century is characterised by a great cultural vivacity. The question about which the debateregarding the “vis viva” took place was the following: what is the physical magnitude that causesthe motion? According to René Descartes (1596-1650) (Michieli, 1949), such magnitude would bethe “quantitas motus”, i.e. the product of the mass of the considered body and its speed. On thecontrary, Gottfried Wilhelm Leibniz (1646-1716) published the paper entitled Brevis demonstratioerroris memorabilis Cartesii, et aliorum circa legem naturalem, secundum quam volunt a Deoeamdem semper quantitatem motus conservari; qua et in re mechanica abutuntur1. He described asimple experiment and concluded that “quantitas motus” cannot be considered the cause of themotion: so it is necessary to define a new “vis motrix”.Such “vis motrix” was introduced by Leibniz himself in the work Specimen dynamicum proadmirandis naturae legibus circa corporum vires et mutuas actiones detergendis et ad suas causasrevocandis (1695); but Leibnitian “vis motrix”, or “vis viva”, was not clearly defined: it is notproportional to the speed of the considered body (as “quantitas motus” is), but it is proportionalto the square of such speed2.Of course, Leibnitian ideas too cannot be considered totally correct, from a modern point of view:Leibniz considered implicitly his “vis motrix” as a real force: the modern concept of work was stillignored.In the development of the question about the “vis motrix” we must consider Gian Maria Ciassi(1654-1679), who wrote Tractatus physicomathematicus, published in Venice in 1677 (Ciassi, 1677;Nicolai, 1754; Pellizzari, 1830; Rambaldi, 1863; Michieli, 1949; Bagni, 1991, 1992 and 1993); in thiswork we can find some interesting notes.Ciassi was born in Treviso, Italy, on march 20, 1654, and studied in Padua (Favaro, 1917); he died,

1 Let us quote Leibniz himself: “... suppono, primo corpus cadens ex certa altitudine acquirere vim eousque rursusassurgendi, si directio eius ita ferat, nec quicquam externorum impediat... Suppono item secundo, tanta vi opus esse adelevandum corpus A unius librae usque ad altitudinem CD quatuor ulnarum, quanta opus est ad elevandum corpus Bquatuor librarum, usque ad altitudinem EF unius ulnae... Hinc sequitur corpus A delapsum ex altitudine CD praecisetantum acquisivisse virium, quantum corpus B lapsum ex altitudine EF. Nam corpus A postquam lapsu ex C pervenit adD, ibi habet vim reassurgendi usque ad C, per suppos. 1, hoc est vim elevandi corpus unius librae (corpus scilicetproprium) ad altitudinem quatuor ulnarum. Et similiter corpus B postquam lapsu ex E pervenit ad F, ibi habet vimreassurgendi usque ad E, per suppos. 1, hoc est vim elevandi corpus quatuor librarum (corpus scilicet proprium) adaltitudinem unius ulnae. Ergo per suppos. 2, vis corporis A existentis in D, et vis corporis B existentis in E, sunt aequales”.And Leibniz concludes: “itaque magnum est discrimen inter vim motricem, et quantitatem motus, ita ut unum peralterum aestimari non possit” (Leibniz, 1768).2 Ciassi, 1677, p. 57. By “work” we translate Ciassi’s “vis”.

only 25 years old, in Venice. His physical work was based upon the use of mathematical tools, e.g.geometric proofs. In Tractatus physicomathematicus Ciassi’s aim is the justification of somestatements exposed in Meditationes de natura plantarum, Ciassi’s previous work. The Authorcompares the situation of a lever to the study of the equilibrium of a fluid in communicating vessels,and surely this is the most interesting part of Ciassi’s Tractatus.Let us report some remarks, in Latin original text: “Immo haec ipsa altitudinis linearum a motiscorporibus descriptarum reciprocatio cum gravitate ipsorum prior causa est, aequalis momenti,quod Galileus non advertit. Etenim corpus cum alio in hac reciprocatione constitutum unamtantum unciam gravitans, ut elevetur ad quatuor pollices, eandem vim requirit, ac corpus gravitansquatuor uncias, ut elevetur ad unum pollicem tantum. Puta ut corpus G unam tantum unciamgravitans attollatur per lineam EA, cuius altitudo sit quattuor pollicum; requiritur eadem vis, ac utcorpus F quatuor uncias gravitans attollatur per lineam DB, cuius altitudo sit tantum unius pollicis.Quia scilicet cum in altitudine lineae EA sint quatuor partes, quarum unaquaeque est aequalisaltitudini DB totius; licet ad elevandam corpus G ad singulas harum quatuor partium requirereturalias tantum quarta virium pars, quae requiritur in elevatione corporis F ad equalem altitudinemtotius DB; in omnibus tamen simul quatuor partibus EA requiritur quadrupla vis; quia quater eaquarta virium pars replicatur” (Ciassi, 1677, pp. 57-59).So Ciassi states that a bodyG, in the point E, to be liftedup to the point A requests“vis” as a body F in D liftedup to the point B if and onlyif the weights G and F areinversely proportional to GCand FC, so to the virtuallycovered segments AE andBD.In order to appreciate the realimportance of Ciassi’s con-clusions,let us underline that thestatement that P and P’ areinversely proportional to h andh’ is equivalent to the statementof a similar proportionality tothe squares of the speeds v, v’referred to respective motionsof considered points (in fact h isproportional to the square ofthe speed,being 2gh).So Ciassi’sstatement implies theproportionality of the “vis” (ofcourse,today,we should refer tokinetic energy) acquired by adropping body to the square ofits speed and it can be considered in the theoretic frame of the problem of the “vis viva”, according toLeibnitian point of view.It is important to underline that the publication of Ciassi’s Tractatus physicomathematicus took place in 1677,so nine years before the publication in “Acta Eruditorum Lipsiae” (1686) of the celebratedLeibnitian work about the “vis motrix”: it could be interesting to investigate if Leibniz knew, in1686, Ciassi’s research and results. Let us compare, for instance, some words by the consideredAuthors:


A

B

G H C I F

D L

K E

Figure 1 (Ciassi, 1677, p. 54)The Author geometrically proves that: AC : CD = AE : BDso, being: GK = HE = AE/2 and FL = ID = BD/2,it follows:AC : CD = GK : FL

Ciassi in 1677 wrote:

“A body weighing one ounce, considered withanother body in such lever, and lifted up to fourinches requests the same work requested by a bodyweighing four ounces lifted up to one inch” 2.

Leibniz in 1686 wrote:

“I suppose that the same work is necessary either inorder to lift up a body weighing one pound to theheight of four yards, or in order to lift up a bodyweighing four pounds to the height of one yard” 3.

190 3. Topical Aspects 3.3 Modelling

Of course, in this case, the similarity can be referred just to a secondary statement; moreover,previously mentioned Ciassi’s main result, too, is not accompanied with clear references to theopposition between Descartes’ ideas and Leibnitian solution of the problem of the “vis motrix”.So we don’t want to give Gian Maria Ciassi full credit for the direct solution of the consideredproblem. However, Ciassi’s studies can be considered surely important and historically interesting.According to Y. Chevallard’s terminology, as we noticed previously, we can state that the Historyof Sciences is an important tool for the transposition didactique. The well known triangle ofChevallard visualises a really frequent situation: the academic knowledge (the so-called savoirsavant) is sometimes far from the process of teaching-learning (Chevallard, 1985). So we must“draw up” the savoir savant to the classroom practice, to the process of teaching-learning by thetransposition didactique: it can be achieved by the use of some historical examples, too. Once againwe must remember that the historical development is not always suitable in order to plan curricula,although sometimes it is possible to point out an analogy between the stages of the historicaldevelopment and educational stages.As regards the savoir savant, the historical development of a concept can be considered as thesequence of (at least) two stages: an early, intuitive stage and a mature stage; in the early stage thefocus is mainly operational; the structural point of view is not a primary one. From the educationalpoint of view, a similar situation can be pointed out (Sfard, 1991): in the early stage pupils approachconcepts by intuition, without a full comprehension of the matter; then the learning becomesbetter and betters, until it is mature. Of course, processes of teaching-learning take place nowadays,after the full development of the savoir savant. So the transposition didactique, whose goal isinitially a correct development of intuitive aspects, can be based upon the results achieved in themature stage, too, of the development of the savoir savant.Moreover the process of teaching-learning and the transposition didactique must consider thatpupils’ reactions are sometimes similar to corresponding reactions noticed in the History; thiscorrespondence and, of course, the knowledge of historical examples themselves are importanttools for teachers: epistemological skill is needed, and this is a matter related to teacher training.From the educational point of view, it is worth noting that the quoted historical example deals withthe analogy between different situations, as geometric features of a lever and energy or work. Ofcourse, as regards analogical reasoning, we must underline that the really different propensity forself-correction should be considered, e.g. when we compare research scientists and young students:frequently scientists employ analogical reasoning in formulation of a conjecture, whose soundnessmust be verified; on the other hand, generally students do not perform this meta-discursivemonitoring (for instance, some mathematical examples are discussed in: Bagni, 2000)4.

The author thanks Prof. Marisa Michelini of the University of Udine, who kindly and continuously offeredevery possible help.

3 Leibniz, 1768, III, pp. 180-181. By “work” we translate Leibnitian “vis”.4 A clear educational problem consists in the uncertainty about the effects upon the learning of teachers’ choices. Thisuncertainty concerns particularly the cognitive transfer (Feldman & Toulmin, 1976; D’Amore & Frabboni, 1996) thatmust be stimulated by the teacher; and effects upon the learning must be carefully verified (D’Amore, 1999): so the useof historical examples can be useful to introduce some important topics: however their effectiveness must be carefullycontrolled in order to obtain a correct, full learning.

ReferencesBagni G. T., Gian Maria Ciassi fisico trevigiano, Teorema, Treviso, (1991).Bagni G. T., (1992), Gian Maria Ciassi (1654-1679) fisico trevigiano, Atti e Memorie dell’Ateneo di Treviso, (1991-

1992), 141-158.Bagni G. T., La matematica nella Marca: Vincenzo, Giordano e Francesco Riccati, Edizioni Teorema, Treviso, (1993).Bagni G.T., “Simple” rules and general rules in some High School students’ mistakes, Journal fur Mathematik

Didaktik, 21, 2, (2000), 124-138.Chevallard Y., La transposition didactique, du savoir savant au savoir enseigné, La Penseé Sauvage, Grenoble, (1985).Ciassi G.M., Meditationes de natura plantarum et Tractatus physicomathematicus De æquilibrio praesertim fluidorum,

ac de levitate ignis, Benedetto Miloco, Venezia, (1677).D’Amore B. & Frabboni F., Didattica generale e didattiche disciplinari, Angeli, Milano, (1996).D’Amore B., Elementi di Didattica della Matematica, Pitagora, Bologna, (1999).Favaro A., I successori di Galileo nello studio di Padova fino alla caduta della Repubblica, N. Archivio Veneto, 65, 1-

3, (1917).Feldman C.F. & Toulmin S., Logic and the theory of mind, Cole, J. K. (Eds.), Nebraska Symposium on Motivation

1975, Univ. of Nebraska Pr., Lincoln, London, (1976).Leibniz G.W., Opera Omnia, Fratres de Tournes, Genevae, (1768).Michieli A.A., Le sventure di uno scienziato trevigiano (G. M. Ciassi), Atti dell’Istituto Veneto di Scienze, Lettere ed

Arti, CVII, II, (1949).Nicolai G. B., Lettera da Trevigi, a data 9 novembre 1754, sulla scoperta da lui fatta del libretto del Ciassi, Memorie

per servire all’Istoria Letteraria, IV, V, (1754).Pellizzari I.A., Discorso in lode di G.M. Ciassi trivigiano, Giulio Trento, Treviso, (1830).Rambaldi G. B., Iscrizioni patrie, Longo, Treviso, (1863).Sfard A., On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of

the same coins, Educational Studies in Mathematics, 22, (1991), 1-36.

MODELLING PHYSICAL PROCESSES:THE EXAMPLE OF A MAGNETIC GLIDER

Michele D’Anna, Liceo cantonale Locarno, Switzerland

1. IntroductionThe teaching of physics urgently requires renewal: that is the conclusion expressed and shared bya growing number of those in the know. There are even those who think that the issue has become aquestion of the survival of physics at school. However, it is obvious we are not talking about simplyrenewing the contents or the didactic methods; instead we should aim at the reassessment of theformative function of science in general and that of physics in particular, making use of the newinstruments at our disposal in a wider educational project.Already in 1973 Arnold Arons lucidly stated:

“Wider understanding of science will be achieved only by giving students a chance to synthesizeexperience and thought into knowledge and understanding. Such a chance is not available in the delugeof unintelligible names and jargon precipitated at unmanageable pace and volume in so large aproportion of our college courses, and it is not available in the absence of humanistic, historical, orphilosophical perspectives within these courses. Neither will salvation be found in topical courses oncurrently “popular” matters such as the energy crisis, environmental problems or societal impact - solong as these problems are plunged into without any genuine prior understanding of the underlyingscientific ideas.” (Toward wider public understanding of science, Am.J.Phys., 41, June 1973)

The use of modern technologies has therefore to be considered in the wider context ofmodernisation of the science teaching. With respect to physics, it is hoped that it will stimulate newconceptual approaches and that is offers opportunities to promote an interdisciplinary approach,especially with mathematics. Many colleagues during this seminar will analyze the generalformative objectives of the teaching with modern instruments.The present contribution is about a concrete example of combined use of on-line data acquisitionand modeling with a general purpose software. The student is offered the possibility of step by stepunderstanding both the structure of the physical laws and the fundamental role of mathematics.This is an essential instrument to express precise relations between ideas and not an instrument to


perform calculations.This approach enables the study of experimental phenomena well beyond thetraditional simplified (linear stationary physics) problems of the secondary school: a welcomepossibility of dealing with more realistic situations. Not a way of increasing the already full list ofarguments, but a new viewpoint on some of them, together with the mastering of new generalmethodologies. Thus we hope to enhance the quality the learning process of the student.Let’s consider the well known experiment of the fall of a magnetic dipole into a metal tube: it is afascinating demonstration, but experimental measurements are not simple. The usual treatmentlooks at the change in the force sustaining the tube with and without the moving magnet: becauseit equals the weight of the magnet, you can reasonably argue that a constant limit velocity is rapidlyattained. However the direct experimental study of the motion poses many practical problems.The proposed experiment deals with an almost identical physical situation, but because of theduration of transient position and time are easily and precisely measured. Moreover we can changea few significant parameters and investigate the relations between physical quantities. Theexperimental setting is very simple: add two magnets to the sides of a glider free to move down anair track of gentle slope.

2. Description of experiment: qualitativeWhen you let the glider go, a simple look is persuasive of constant velocity. The behavior differsfrom standard accelerated motion and makes it easy to guess the presence of a friction force, whichcompensate the force parallel to the incline. How can we proceed toward a quantitative study? To make explicit the didactical context and the student’s prior knowledge in mathematics andphysics, let’s suppose no calculus but some acquaintance with numerical methods for the first andsome mechanics and no electromagnetism for the second.For our experiment this implies that we have no physical hypothesis about the force of friction:because of this we recommend each student to move the glider by hand up and down the inclinewith different speeds and get the feeling of the different forces necessary to do this.The first step is now to find the dependence of magnetic force from velocity. Time intervals aremeasured with two photogates and after some trials it appears (for fixed mass and angle of incline)that toward the end of track the velocity is fairly independent of time and position (Fig. 1a e 1b).


Fig. 1 a: The photogates are placedin the middle of the track; theglider accelerates, as shown bycomparing transit times.

Fig. 1 b: Now the photogates areplaced near the end of the track;the glider no longer accelerates, asshown by comparing the almostidentical transit times.

We can now investigate the dependence of the limit velocity from mass and angle, which are thetwo parameters fixing the value of the parallel force, and we find a power-like behaviour of themagnetic friction force as function of velocity (Fig 2). A linear relation is easily established and thevalue of the proportionality constant calculated.


sin( ) = 0.0504 x = 2,00 cm

transit time glider mass limit velocity

(msec) (kg) (m/sec)

78.3 0.257 0.255

72 0.277 0.278

67.5 0.297 0.296

65.1 0.307 0.307

61.2 0.327 0.327

57.4 0.347 0.348

55.8 0.357 0.358

52.7 0.377 0.380

49.9 0.397 0.401

48.9 0.407 0.409

45.7 0.437 0.438

43.6 0.457 0.459

41.7 0.477 0.480

39.8 0.507 0.503

limit velocityas function of mass

y = 1.0015x + 0.0002

R2= 0.9993

0.100

1.000

0.1 1

glider mass (kg)

lim

it v

elo

cit

y(m

/s)

The same procedure can be used to investigate the dependence of the limit velocity from the angleof the track, with similar conclusions.

3. On-line data acquisitionOnce this question has been settled, it’s obvious to ask about the transient, that is how the limitvelocity is reached. The on-line data acquisition plays an essential role for this part of theexperiment and is achieved using a device known as a motion sensor, which essentially is anemitter / receiver of ultrasonic waves and measures distances. Of course automatic calculationsgive also velocity and acceleration, or any other function of the data which may be useful for latercomparison.

Fig. 2: Limit velocity as function of mass.

4. Modeling the experiment: quantitativeLet’s now analyze the experimental results the way physicist do it: a first possibility is to solve theNewton’s equations of motion by numerical methods and obtain acceleration, velocity and positionas functions of time. Because of not constant forces entering the problem, this is the only way forthe student to compare experimental data and theoretical predictions. Recall our “referenceframework“ the student has some acquaintance with both iterative algorithms and numericalmethods made with pocket calculators or software like Excel. To model this problem we usedStella.This software uses a graphic window to build the model and another window to write down theprogram lines. The student literally “draws” the model choosing the physical quantities and therelations between them (see Fig. 4a).


Graph Display

Time (s)

0 0.5 1.0 1.5 2.0 2.5 3.0

00.5

1.0

1.5

2.0

Run #

1

Positio

n (

m)

00.2

00.4

00.6

0

Run #

1

Velo

city (m

/s)

v(t) = v(t - dt) + (dv\dt) * dt INIT v = vo INFLOWS: dv\dt = a y(t) = y(t - dt) + (dy\dt) * dt INIT y = yo INFLOWS: dy\dt = v a = (Fparall+Fmagn)/M alpha = 0.06rad Fmagn = -k*v Fparall = M*g*SIN(alpha) g = 9.81m/sec^2 k = 0.357*9.81*0.06/(0.496) N.sec/m M = 0.357kg vo = 0m/sec yo = 0.531m

measured_position = GRAPH(TIME) (0.00, 0.531), (0.0405, 0.532), (0.0809, 0.533), … measured_velocity = GRAPH(TIME) (0.00, 0.001), (0.0405, 0.019), ..

avo

Fmagn

g

v

dv\dt

yo

y

M

dy\dt

alpha

k

Fparall

~

measured position

~

measured velocity

Fig. 3: On-line data acquisitionwith Motion Sensor II CI-6742and Interface 500 (CI-6760).Graphs were obtained withScience Workshop Software, allproduced by Pasco.

Fig. 4 a: The integration of motion (position and velocity as function of time) is obtained with STELLA 5.1 software,produced by High Performance Systems Inc.

The outputs of Stella are graphics and numerical tables, which of course make very straightforwardthe comparison between experimental data and the predictions of the model (see Fig. 4b e 4c):


Fig. 4 b:The comparison betweenexperimental data (graph 2) andtheoretical model (graph 1)shows a good agreement forposition as a function of time

0.00 0.86 1.72 2.58 3.440.00

1.00

2.00

1: y 2: measured position

1

1

1

1

2

2

2

2

Fig. 4 c: Comparison betweenexperimental data (graph 2) andtheoretical model (graph 1) forvelocity as a function of time.

0.00 0.86 1.72 2.58 3.440.00

0.30

0.60

1: v 2: measured velocity

1

1

1

1

2

2

22

An important didactical point is the active choice of relations made by the student: the role ofStella is only to perform the calculations of the instructions received. The proposed comparisonwith experimental data is significantly well beyond the best fitting procedure and can be used as asimulation tool. The physical relevance of changing the value of a parameter is rapidly visualized:in our example we can investigate relations between magnetic force and velocity other than linearand their effect on the motion.But the main feature, the one that best shows the potential of this method, is the possibility tomodel different conceptual approaches to the description of physical phenomena. As a firstexample we propose a model based on the balance of momentum (Newton’s quantitas motus),which of course will give the same graphical-numerical results for kinematical quantities.

As a second example we propose the analysis of the process in terms of energy flows: the modelrepresents the transfer of energy in the unit time (that is power) performed by the parallel forceand by the magnetic force. We obtain the amount of energy exchanges, and from the knowledge ofvelocity, we obtain an independent evaluation of the kinetic energy increase (see Fig. 6a).


p(t) = p(t - dt) + (Fparall + Fmagn) * dt INIT p = vo INFLOWS:Fparall = M*g*SIN(alpha) Fmagn = -k*vy(t) = y(t - dt) + (dy\dt) * dt INIT y = yo INFLOWS:dy\dt = v alpha = 0.06rad g = 9.81m/sec^2 k = 0.42 N.sec/m M = 0.357kg v = p/M vo = 0m/sec yo = 0.531m

vo

g

p

Fparall

yo

y

Mdy\dt

alphak

Fmagn

v

Fig. 5: Position and velocity as function of time: the balance of momentum (exchanges).

v(t) = v(t - dt) + (dv\dt) * dt INIT v = vo INFLOWS: dv\dt = a Wgrav(t) = Wgrav(t - dt) + (Pgrav) * dt INIT Wgrav = 0J

INFLOWS: Pgrav = Fparall*v Wmagn(t) = Wmagn(t - dt) + (Pmagn) * dt INIT Wmagn = 0J

INFLOWS: Pmagn = Fmagn*v a = (Fparall+Fmagn)/M alpha = 0.06rad Ecin = M*v*v/2 ENERGY_BALANCE = energy_exchanges-Ecin energy_exchanges = Wgrav+Wmagn Fmagn = -k*v Fparall = M*g*SIN(alpha) g = 9.81m/sec^2 k = 0.357*9.81*0.06/(0.496) N.sec/m M = 0.357kg vo = 0m/sec

avo

Fmagn

g

v

dv\dt

energy exchangesENERGY BALANCE

M

alpha

k

Fparall

MWmagn Wgrav

PmagnPgravEcin

Fig. 6 a: Modeling energy flows.

Looking at the graphic (v. Fig. 6b) we conclude that:a) in the long run, energy exchanges balance: the power entering and leaving the system glider are

equal and this relates with the constancy of velocity;b) in the short run, it’s evident that more energy enters the system than leaves it, the difference

remains in the system as kinetic energy.


Fig. 6 b: Power flows.0.00 0.86 1.72 2.58 3.44

Time

1:

1:

1:

2:

2:

2:

-0.16

0.00

0.16

1: Pgrav 2: Pmagn

1

1

11

2

2

22

Fig. 6 c: The balance of storedenergy and exchanged energy.

0.00 0.86 1.72 2.58 3.44-0.30

0.00

0.30

1: BALANCE 2: Ecin 3: Wgrav 4: Wmagn

1 1 1 122

2 2

3

3

3

3

4

4

4

4

Looking at the graphic (which represents the complete balance of energy, traditionally referred toas the conservation of energy) students may wonder where this “result” come from, because it wasnot explicitly put into the model.

5. Class implementation and conclusionThis example has been implemented in a class of 16 students during the last two years of uppersecondary school.The course was named Physics and Applied Mathematics and from the beginningparticular attention was paid to experimental work and modeling in the laboratory: both theteacher of physics and mathematics were present most of the time.The teaching resources dedicated to the training of students with Stella and Science Workshop arelimited. After two or three lessons dedicated to simple examples of algorithms and models theybecome quite autonomous, much before they can properly handle higher mathematicalinstruments: eventually the exercise of modeling can help to understand the deep meaning of thelink between physics and mathematics.

Compared with the traditional lesson “teacher versus students”, this approach requires an activeparticipation which triggered more interest in the subject and, consequently, a better school results.Of course, exercises in modeling are commonly proposed in written tests: this year’s finalexamination also contained a problem about the magnetic glider experiment.To sum up, we hope to open new possibilities of contents and methods both for teaching andlearning. The combined use of on-line data acquisition and modeling may help to reduce the gapbetween the real world processes and traditional teaching.We believe this approach opens a concrete perspective of change of didactical methods.

MODELLING PHENOMENA IN VARIOUS EXPERIENTAL FIELDS: THEFRAMEWORK OF NEGATIVE AND POSITIVE FEEDBACK SYSTEMS

C. Fazio, A. Giangalanti, G. Tarantino, I.P.“Enrico Medi”, Palermo, ItalyR. M. Sperandeo-Mineo, Dep. of Phys. and Astr. Sc., University of Palermo, ItalyGRIAF, Research Group on Teaching/Learning Physics, University of Palermo, Italy

1. Introduction It is widely accepted that the general aim of physics is to interpret and predict the physical world.Physicists frequently rely on model-based reasoning to simplify and classify complex phenomena,to predict trends and to explain mechanisms and processes. Many research studies (Gilbert et al.,1998) (Andaloro et al., 1991) have pointed out that model building can be a formative pedagogicalactivity, since it allows pupils to better understand many content areas, enabling them to seesimilarities and differences among apparently different phenomena. Consequently, a teachingapproach focusing on modelling procedures can contribute to construct a unitary view of physicsas well as to unify the scientific approach to many problems.Definitions of a “model” and of what constitutes the modelling process abound in the literature(Berry et al. (eds), 1986). We intend as modelling of an experience, or object, or system, the processby which:• some of their relevant discrete elements are perceived to bear relationships among each other;• these relationships are expressed through equations.In this context, modelling represents a deeper or fuller understanding of a problem and is differentfrom solving equations and/or fitting experimental data. As a consequence, we intend as a “model”of an experience (or a system): a representation of that experience capable to be validated and tobe used for further study of that or analogue experiences.In this paper, we present a teaching approach where different physical phenomena are analysedusing the same point of view that involves an analysis of the interaction between a system and theenvironment. The analysed systems can be studied using a framework that involves the possibilityto define systems as feedback systems, i.e. systems where a retroaction tends to reduce or improvethe output.Feedback systems can be understood using a causal loop approach (Prior D., 1986) that describesthe relationship between two parameters chosen in order to describe the system evolution (see Fig.1a). In it, each arrow represents the influence that the tail parameter exerts over the head one. Eacharrow carries a positive or a negative sign:• a positive sign means that any change in the tail parameter of the influence link results in a

change in the head one which is in the same direction as the change in the tail parameter; thusa decrease (or increase) in the tail parameter results in a decrease (or increase) in the quantityassociated with the head parameter.

• a negative sign means that any change in the tail parameter of the influence link results in a


change in the head one which is in the opposite direction to the change in the tail parameter;hence a decrease (or increase) in the tail parameter of a negatively signed causal link results ina increase (decrease) in the quantity associated with the head one.

The combination of the two causal links of the loop model allows us to classify the nature of thefeedback at work in the model as negative or goal seeking and positive or growth promoting.

2. The case of negative feedback systemsAs an example, let us consider the cooling of a cup of hot coffee left on the table.A verbal descriptionof this experience points that the hot coffee interacting with the environment will cool to reach theenvironment temperature. Using experimental results, the temperature decrease can bemathematically described. However, a fitting procedure is not directly connected with theunderstanding of the system behaviour: to explore the major relationships, perceived by the observersas responsible for the behavioural characteristics of the system under analysis, is an essential step. Inour case this involves the need to perceive the cooling process of a body at temperature Tb in anenvironment at constant temperature Ta<Tb as a negative feedback system: in fact a decrease in thetemperature difference (Tb-Ta) (the parameter 1 of our loop) results a decrease in the cooling rate (theparameter 2 of our loop) and an increase in the cooling rate results a decrease in the temperaturedifference between the body and the environment. In other words, the model affirms that the rate ofcooling does not increases as the cooling proceeds; instead the process continually seeks to achieve astate of equilibrium. The causal loop for the cooling system is reported in Fig. 1b.Other different kinds of phenomena dealing with systems evolving to equilibrium can be analysed:body falling in a viscous fluid, emptying of a water reservoir through a small section tube, capacitydischarging process and radioactive decay can be some examples. These phenomena can all beinterpreted in the frame of negative feedback systems, i.e. systems where a retroaction that tendsto reduce the output in the system is present. It can be shown that systems are approaching theequilibrium condition in such a way that the rate of variation of the variable y, characterising thesingle phenomenon, is related to y itself according to the equation:

where k is a real positive constant and the system time evolution is described by the solution ofequation (1).This approach makes us able to treat many different phenomena with the same mathematicaldescription and to use a single model that describes feedback interactions.

3. Didactic approach to modelling physical systemsThe starting point of class work is the observation of phenomena. Then, the first relevant point forthe model formulation phase is the choice of the relevant features of the observed system to takeinto account. This first stage of the process of abstraction involves the construction of a mentalimage of the system and the first formulation of a descriptive model by using verbal or iconiclanguages.As second step, experiments are performed in order to show that a characteristic variable of thesystem exhibits a clear decreasing exponential-type time dependence. Fig. 2a is an example of thecooling process. Exporting data to a spreadsheet and using a fitting procedure, the decreasingexponential function, y = e-kt, is found to be actually the best choice.A graph of the rate of variationof temperature as a function of the temperature can also be plotted (Fig. 2b); this graph clearlyshows that ∆T/∆t is directly proportional to the temperature difference, T, with a negative slope(that, obviously, means that we are dealing with a cooling process). As a consequence, themathematical model describing the cooling process can be expressed in the form


)()(

tkydt

tdy= (1)

kTt

T= (2)

In order to construct a more intuitive physical model that can account for the decreasingexponential-type time dependence of a variable y, we analyse, using a spreadsheet, the emptying ofa cylinder of section S, initially full of water up to the height h0. We monitor the height of the liquidin the hypothesis that the cylinder is emptied by connecting it to another one, of smaller section s(see Fig. 3 ). Then, the smaller cylinder is cleared out and the procedure is repeated until the bigcylinder is empty.By thinking carefully over the conditions of emptying, pupils may understand that the quantity ofwater that is every time subtracted is proportional to the quantity of the liquid contained in the firstcylinder. This example can introduce pupils to the negative feedback system concept, without usingthe usual calculus techniques and convince them that the function y = e-kt is correlated tophenomena where a relevant variable y is proportional to its rate of variation. Moreover, usingfinite difference equations, students can try to analytically write this last result as

where y represents the height of the liquid in the cylinder. This equation is, obviously, the finitedifference equivalent of eq. (1) and is the general form of eq. (2).Using this approach, we can formulate an equation for the phenomena already studied by findingout appropriate physical variables replacing the generic one, y, of equation (3).In Table 1, different phenomena dealing with systems evolving toward the equilibrium areconfronted in terms of physical variables, equations and characteristic times.


kyt

y= (3)

Physical

phenomenon

Physical

variable

Equilibrium

condition Equation Solution

Characteristic

time (1/k)

Cooling of a hot

body of mass m and

specific heat c in an

environment at

constant temperature

through an exchange

surface S

Difference of temperature between the body and the environment

T

Thermal equilibrium

T = 0

kTt

T= kteTT = 0

hS

cm

(obtained by Newton’s law

of cooling)

Discharge of a

capacitor C through

a resistor R

Difference of potential across the capacitor

V

Electrostatic equilibrium

V = 0

kVt

V= kteVV = 0

RC(obtained by

circuitequation)

Body of mass m

falling in a fluid of

coefficient of

viscosity

Net force acting on the

body F

Dynamical equilibrium

F = 0

kFt

F= kteFF = 0

m

(obtained by motion

equation) Emptying of a R-

radius cylindrical

reservoir, containing

a liquid with density

and viscosity ,

through a narrow

pipe of lenght L and

radius r

Difference of height

between the end of the

pipe and fluid surface

h

Idrostatic equilibrium

h = 0

kht

h= ktehh = 0

gr

LR4

28

(obtained by Poiseuille law)

Tab. 1

4. Model extension: the case of positive feedback systemsPositive feedback systems can be classified in two different categories, involving their relevantproperties:1) systems where one or more properties are monotonically growing with time2) systems in equilibrium (where properties are periodically dependent from time)In order to represent the firs kind of positive feedback interaction, a non physical example can giveuseful insights: a bank account. Moreover, the same framework is applicable to the phenomenon ofthe avalanche production of neutrons in a fission process. Using the causal loop schema, a possibleverbal description of this phenomenon is “A rate of neutron production which always tends in adirection so as to meet the neutron number, which always adjusts to meet the rate of neutronproduction”.The causal loops for the first kind of positive feedback systems is represented in Fig 4a: an increaseof the relevant parameter y results in an increase of the speed of variation of y and an increase ofthe speed of variation of y results, as well, in an increase of y. Type a) causal loops are suited toschematise the verbal description of phenomena in which there is an exponential growth of therelevant parameter (i.e. the number of neutrons in the fission process);In the second kind of positive feedback systems (see Fig. 4b) an increase of the relevant parametery results in a decrease of the speed of variation of y and an increase of the speed of variation of yresults in a decrease of y. Type b) causal loops can be used to schematise the verbal description ofphenomena regarding energy conservation, where an increase of the displacement from theequilibrium position of a body results in a decrease of the body speed and an increase of the body


Fig. 1: a) Causal loopdescribing the relationshipbetween two parameters in anegative feedback system.b) Causal loop describing thecooling process

Fig. 2: a) Cooling of warmwater in melting ice. Theexperiment is performed bycooling 10 cc of water at 30 °Cin a bath of melting ice.Measures are taken with acommercial MBL system and asemiconductor temperatureprobe; sampling speed is 0,5samples/sec.b) Rate of cooling of 10 cc ofwater at 30 °C in a bath ofmelting ice with respect thetemperature differencebetween the water and the icebath.

speed results in a decrease of the body displacement from the equilibrium position. It is easy tothink about processes that can be schematised by type b) causal loop, as the vertical up and downmotion of a ball in a gravity field, the pendulum swing or the mass-spring oscillations, whendamping is not taken into account.It is interesting to note that real oscillating processes involve interactions between the system andthe environment connected with negative as well as positive feedback. This is clearly due to thecontemporary action of conservative and non conservative forces; so, type b) positive feedbackloop does apply only to the description of true conservative systems.

5. ConclusionsThe didactic approach, here described, has been tested in courses for pre-service teacherpreparation, in order to make prospective teachers aware of an unitary approach to the study ofinteractions between a system and the environment. Several parts of the modelling approach havealso been tested in some Italian High School in the context of the I.MO.PHY. (Introduction toModelling in PHYsics Education) Project (Sperandeo-Mineo, 2000). Teachers, who haveexperimented the approach in their classrooms, have been trained through a Net-Course (includingon-line discussion) concerning the use of new technologies as well as the various modellingprocedures. The effectiveness of their classroom activities have been analysed using qualitative aswell quantitative methods (Sperandeo-Mineo, 2000). Teachers’ comments and criticism allowed usto deeply modify and improve the approach as well as the Student Guides of the proposedactivities.The evaluation procedure has been deepened through the direct observation of pupils’ activities


Fig. 3: Modelling of phenomena wherethe variation of a variable y isproportional to y itself: a cylinder ofsection S, initially full of water toheight h0 is emptied by connecting it toanother one, of smaller sections, thenclearing out the smaller cylinder. Thisprocedure is repeated until the firstcylinder is empty.

Fig. 4: Causal loops describing therelationship between two parametersin two different types of positivefeedback systems.

(performed by ourselves in some classrooms of Palermo High Schools) and the analysis of pupils’log-books and class-works. Pupils’ understanding of physical concepts has been evaluated usingtests and problem solving. A preliminary analysis of the results allows us to infer that the reasoningprocedures used in order to obtain a physical model of real systems have been understood by themajority of pupils. Moreover, the focusing on the features of interactions, between the system underscrutiny and the environment, through the construction of the feedback model has made pupilsaware of the unitary way of physics to look to the reality. Last but not the least, mathematicalproperties of exponential and logarithmic functions can effectively be introduced or thoroughlyinvestigated by studying physical phenomena that can be described in terms of a generic variablewhose rate of variation is proportional to the same variable.Many relevant points of the modelling procedure have been understood also by younger students;in particular, using analogies among different phenomena, some of them became moreunderstandable than in the traditional approach; models can be easily implemented and the role ofdifferent parameters pointed out through the use of a simple spreadsheet.

AcknowledgmentThis work is supported by M.I.U.R., S. e C. i F. project (cofin 99).

ReferencesAndaloro G., Donzelli V. and Sperandeo-Mineo R. M., Modelling In Physics Teaching: the role of computers

simulation., International Journal of Science Education, 13, (1991), 243-254.Berry J. S., Burghes, D. N., Huntley, I. D., G. James D. J. and A. O. Moscardini (Eds) Mathematical Modelling.

Metodology, Models and Micros. (New York: John Wiley & Sons), (1986).J. K. Gilbert, C. Boulter and M. Rutherford, Models In Explanations: Part 1, Horses for courses?. International

Journal of Science Education, 20, (1998), 83-97.D. E. Prior, A New Approach To Model Formulation. In Berry, J.S., Burghes, D.N., Huntley, I.D., James, D.J.G. and

Moscardini,A.O. (eds), Mathematical Modelling. Metodology, Models and Micros. (New York: John Wiley & Sons),(1986).

R. M Sperandeo-Mineo, I.Mo.Phy. (Introduction To Modelling In Physics- Education): a Netcourse SupportingTeachers in Implementing Tools and Teaching Strategies, Proceeding of the GIREP Conference: “Physics TheacherEducation Beyond 2000”: Selected papers, (London: Elsevier Editions), (2000).

THE LEARNING OF MODELING: A SCIENTISTS’ VISION

Stella Maris Islas, Departamento de Formación Docente, Facultad de Ciencias Exactas-Universidad Nacional del Centro de la Provincia de Buenos Aires, - Campus Universitario, –Tandil, ArgentinaMarta A. Pesa, Departamento de Física, Facultad de Ciencias Exactas y Tecnología, - UniversidadNacional de Tucumán, Tucumán, Argentina

1. Why to study modeling in a population of expertsIf we are to attempt a model-based teaching and learning approach (Clement, 2000), it is necessaryto clarify learning questions from different points of view. We consider that the way in whichexperts in models work on the topic with their students, as well as their experiences in the learningof modeling are data that can guide one of the possible ways of facing this problem (Nersessian,1995).Since scientists elaborate and use models in a very conscientious way, knowing the ways ofmodeling they use can enrich the wealth of knowledge on this topic.The cognitive demands of modeling on students (Harrison-Treagust, 2000) are not the same as onan expert; for that reason we should be careful when extrapolating results from this population.Thepurpose of this report is to offer information given by these experts in modeling, without suggestingthat their experience should be taken as an example a learning process to be followed.


2. What we mean for scientific modelHaving in mind the multiple meanings of the word “model” (Bunge, 1985, p. 455; Samaja, 1993, p.246) we begin this presentation by making explicit the vision from which our work has beenapproached. More detailed epistemological explanations can be found in Islas-Pesa (2000)A scientific model is a type of theoretical construct that -together with the other components of thebody of a theory – guides the observation of reality, the posing of a problem, and othercharacteristic strategies of scientific research. We can mention Bunge (1985), Laudan (1986),Toulmin (1964), Bachelard (1991), among the thinkers that accept this notion. Other similar viewsconcerning this topic may be found in Halloun (1996), Van Driel - Ver Loop (1999), Nersessian(1995), Snyder (2000), among others.For the analysis of scientific models construction, we start from the notion of mental model: “themental model is an internal representation of information that is similar to the informationrepresented” (Moreira, 1998). The analogous characteristic of this correspondence explains why itis possible to have several models for the same object, system or event, and it depends on themodelers’ experiences on that part of reality, as well as on their beliefs and intentions.As every internal representation, mental models are idiosyncratic, so they cannot be directlyexplored. Their peculiarities can be inferred studying the expressions generated from them, whichcan be verbal, symbolic and/or material (Halloun, 1996; Nersessian, 1992).Experimental designs, algebraic expressions, graphics, etc. are some visible manifestations of themental model that a scientist elaborates in order to solve research questions. When producing suchmanifestations theoretical and experimental revisions become at stake. This, together with theconsensus conditions within the scientific community show the scientist the most appropriateexpression format to share his/her model with their colleagues. In this way, he/she builds a model(“expressed model”) which is useful for his work and it may become a new “scientific consensualmodel” if his colleagues agree with him (J. Gilbert et al, 1998).This pathway should not be interpreted as linear. As every process of knowledge construction, itsuffers modifications at different stages. There is feedback from the researcher himself (naturalevolution of any mental model; Moreira, 1999) and also from the scientific community (colleagues’opinions and referees, reports obtained in publications, congresses, etc.). The scientific modelsevolve along time and there are several different acceptable models for a certain fact orphenomenon at the same time (Grosslight et al, 1991; Snyder, 2000).Following J. Gilbert terminology, we consider ‘a teaching model’ as: “a specially constructedexpressed model used by teachers to aid the understanding of a given consensus model” (J. Gilbertet al, 1998). When designing a teaching model we should consider students’ previous knowledge, aswell as their interests and cognitive abilities.When students show mental models consistent with the scientific model to which they refer, thereis a certain guarantee that the use of teaching model has achieved its purpose. The teaching modelacts as mediator between the scientific model and the mental models that students generate. Thesemental models can be explored analyzing the answers given by those students to the questionsposed in class.In this report we will present results related to the way in which some scientists express themselveswhen referring to the teaching models they use in class, and their experiences with modeling. In thenext section we briefly describe the characteristics of the research that frames the work hereexposed.

3. Brief review of the research carried outThe sources of data for our study were the manifestations registered during semi-structuredindividual interviews, carried out at the Hard Sciences Faculty at Universidad Nacional del Centro,located in Tandil (Argentina). We carried out a substantive sampling (Samaja, 1993) in thementioned Faculty selecting: eight working scientists, six advanced students from the Physicscourses, and six teachers who graduated there and are now schoolteachers. In this report, the


attention will be exclusively placed on the scientists’ subjects.The interviews were based on protocols with open questions that required the interviewees to talkabout their concept of model (“variable 1”) and about their personal experiences in modeling(“variable 2”). In this way we obtained data on both variables, then we processed this informationfrom literal transcriptions of the interviews using a system of categories (Dey, 1993).

4. The Exploratory character of the researchWe call this piece of research exploratory (Samaja, 1993) because there is no systematizedknowledge from a theory on the learning of models yet (Gobert - Buckley, 2000). In the specializedliterature there are interesting discussions about the way in which some eminent scientists such asMaxwell have made models (see Nersessian, 1992), but reports on modeling in current scientificpopulations (Grosslight et al, 1991; Islas-Pesa, 2000) are rare. It is still more difficult to findinformation on the process of construction of the concept of scientific model in the currentscientific community.Our purpose is to present data gathered in a certain population, which could be integrated in thebody of knowledge concerned with the learning of scientific models. Although in the last years thistopic is being recognized (Clement, 2000) it still lacks a theoretical systematization (Gobert-Buckley, 2000).We also hope the results of this exploratory investigation are able to generate new hypotheses andnew research on this matter (in this objective we coincide with Aiello-Sperandeo, 2000).

5. ResultsFrom the data gathered from the group of scientists for the second variable, we will detail thosethat are most strongly related to the process of construction of the notion of the scientific model.Those are their own learning experiences, and the didactic strategies that they use to help theirstudents in such a process.

6. Personal experiences related to scientific modelsWith the purpose of analyzing them, we can divide these experiences in three groups:1) The experiences of these scientists when learning modeling.

• They don’t remember any classroom situations in which the topic of modeling was specificallyexplained when they were university students. Rather, their initiation in the use of models isclosely related to the environment of research tasks.Although they don’t say anything about finding it difficult to use models in their current work,they remember having had problems when they were students, especially in the first years atuniversity. Not all of them agree on the effectiveness of explanations of models in the classroom,especially when students have scarce conceptual preparation and rudimentary mathematicaltools. This result must be interpreted keeping in mind that they are referring to traditionallessons; those they attended, in which the teacher’s role was to offer explanations, but they didnot use any other strategies.All of them coincide in that modeling becomes comprehensible for them when they facedresearch problems. They start practicing research during the last year of their degree course ofstudies. This was done under the supervision of an expert. Progress in the understanding ofmodeling occurs when they carry out research, till they reached a point when they were able todesign their own models.• They think highly of the expert’s contribution, mainly when they have just become membersof research teams. The gathered information allows us to suppose that none of them conceivesthe learning of modeling without the guide of a competent scientist.

2) The handling of models within their research activities1.


1 As this report focuses on learning, we refer briefly to these results to offer the reader a framework to understand thecentral topics.

• The main function of models in these scientists’ activity is to guide all the stages in the solutionof research problems. During such stages each model is corrected by experimental control. Themodels are also useful to explain and describe what is observed.• The scientist establishes the limits of validity of his model according to the variable selectionpracticed and the verification of the systemic frame of the model within an accepted theory.These limits are quantified through experimental control, by measuring and Calculating errors.• Some of them spontaneously direct the dialogue to the control of research results within thescientific community. They give relevance to the opinions of those who evaluate theirpublications and to contributions that they receive from colleagues (work teams, meetings,congresses, conferences, etc.)

3) The handling of models in the teaching of Physics at university.• The teaching models that they use in university lessons could be described as a simplifiedversion of the models they use in their research work. To find a model which is simple enoughfor the cognitive abilities of their students they select those variables that allow a simplermathematical treatment than those used in professional research. One of the interviewees pointsout another difference: the teaching models are less flexible and creative than those used inresearch: he uses only consensual models.• To facilitate the understanding of modeling, they use references to accessible models (e.g.:ideal gas model) to introduce some topics and they take examples from the history of science.In both cases, the basic idea is to sequence learning from the simplest thing to the most complexone.• The handling of models is more intensive and fruitful when done with advanced students thanwith younger ones because the former have better conceptual and mathematics tools. Somenovice students, who lack those tools, do not interpret the physical meaning of mathematicallanguage.This is the biggest concern for these scientists as regard the handling of models in class.They try to overcome this problem asking students to: 1) explain the way in which they linkmathematical entities to the physical phenomena represented, 2) analyze the physical meaningof the problem results they obtain, and 3) discuss the error intervals implied in the mathematicaldevelopment and/or in the experimental task.• Students usually change the direction of the model-reality relationship, considering that realityshould obey the rules settled by the model. From this vision the model regulates nature. Theresearchers point out that tasks done using computer simulation can reinforce this idea. If theyare not cautious enough in this respect students may think that simulation is the “should-be” ofreality (we agree on this with Harrison-Treagust, 2000).

7. Final commentsRevising the way in which these scientists evoke their learning of modeling, the role of researchpractice appears to be the most important factor. Having in mind that the ways to knowledgeconstruction in a scientist and in a student are not exactly the same, we should not expect studentsto do the same kind of research scientists do. But anyway, we should help students to come closerto research to build knowledge of modeling.The results we obtained suggest that traditional lessons on modeling would not be the best strategyto develop understanding of this issue. None of these scientists remember having had that sort oflessons, and neither of them admit doing so as a university professor. Discussions on each of theproblems they solve promote reflection on those topics that are sources of difficulties for theirstudents. These are: the link between reality and its representations, the factual meaning ofmathematical expressions and the building of links between the theoretical construct and theexperimental activity.

ReferencesAiello-Nicosia M. L., Sperandeo-Mineo R. M., Educational reconstruction of physics content to be taught of pre-

service teacher training: a case study. International Journal of Science Education, 22, (10), (2000) 1085-1097.


Bachelard, G., La formación del espíritu científico. Siglo XXI, Editores México, (1991).Bunge M., La investigación científica, Ed. Ariel – España, (1985).Clement J., Model based learning as a key research area for science education International Journal of Science

Education, 22, (9), (2000), 1041-1053.Dey I., Qualitative Data Analysis, Ed. Rotledge, USA, (1993).Gilbert J., Boulter C., Rutherford M., Models in explanations, Part 1: Horses for courses?. International Journal of

Science Education, 20 (1), (1998), 83-97.Gobert J., Buckley B., Introduction to model-based teaching and learning in science education., International Journal

of Science Education, 22, (9), (2000), 891-894.Grosslight L., Unger C., Jay E., Understanding Models and their Use in Science: Conceptions of Middle and High

School Students and Experts, Journal of Research in Science Teaching, 28, (9), (1991), 799-822.Halloun I., Schematic Modeling for Meaninful Learning of Physics, Journal of Research in Science Teaching, 33, (9),

(1996), 1019-1041.Harrison A., Treagust D., A typology of school science models. International Journal of Science Education, 22, (9),

(2000), 1011-1026.Islas S., Pesa M., El manejo de modelos científicos en las clases universitarias, Memorias del V Simposio de

Investigadores en Educación en Física, Santa Fe, Argentina, (2000).Laudan L., El progreso y sus problemas., Ed. Encuentro, España, (1986).Nersessian N., How do Scientists think? Capturing the Dynamics of Conceptual Change in Science, In: Giere, R. (ed)

Cognitive models of Science, Vol. XV, (Minnesota Studies in the Philosophy of Science), University of MinnesotaPress, Minneapolis, (1992).

Nersessian N., Should Physicists Preach What They Practice?, Science & Education, 4, (1995), 203-226.Moreira M., Modelos Mentais. Investigaciones en Enseñanza de las Ciencias, 1, (3), (1998), 193-232.Samaja J., Epistemología y metodología, Eudeba, Buenos Aires, (1993).Snyder J.,An investigation of the knowledge structures of experts, intermediates and novices in physics, International

Journal of Science Education, 22, (9), (2000), 979-992.Toulmin S., La Filosofía de la Ciencia, Ed. Compañía General Fabril Editora, Buenos Aires, (1964).Van Driel I., - Verloop N., Teachers’ knowledge of models and modelling in science, International Journal of Science

Education, 21, (11), (1999), 1141-1153.

MODELLING WITHOUT NUMBERS

Ian Lawrence, School of Education, University of Birmingham, UK

Within information and communication technology (ICT) use in the UK there has been anevolution that has seen many choose to take up ICT, seeing it as a tool to allow more time to bedevoted to higher order thinking during science lessons. Whilst this is not universal, it has been adistinctive trend over the last few years. As ICT becomes less special, and more pervasive, so usewithin science classes approaches the pattern for other pieces of laboratory apparatus, beingselected on merit when it is the appropriate tool for the job. One exemplification of this trend,special to science laboratories, is in the use of ICT to display and capture experimental data. In theearly days it was enough to get the graph on the screen. Now this is just the beginning; the screenbecomes a playground for shared ideas, trying to encourage learners to shape the data to exploreshared conceptions of the world. This plasticity extends to capturing several sets of data underslightly different conditions and looking for patterns in the data, always seeking to actively makesense of the world rather than just inspecting what is there. Although, since Galileo, and perhapseven before, the power of numbers has been explicit within the sciences; these activities can nowtake place without the explicit mediation of numbers. We allow the computer to mask these fromus, dealing only with the strategic decisions, looking for patterns in the numbers. ElsewhereLaurence Rogers has written of the magic of graphs, seeking to emphasise the patterns that arisefrom measurement, without necessarily looking at the numbers produced by the measurementsthemselves. The computer can relieve the pressure on the tactical decisions, so allowingconcentration on the higher order skills of seeking and then exploring the patterns.Now I think the time is ripe for another step. Without concentrating on the numbers that underlie


so much of science we can allow access to more of the creative aspects of science, emphasisingsynthetic skills. I mean that we should be able to use the computer as a medium to think with inreasoning about the world. A combination of a supportive didactic approach with the necessarysoftware and hardware resources can allow children to explore, modify, and even construct theirown functioning models of the world and then to compare these models with the world. Thiscomparison is vital, yet constrained and constraining. At the frontiers of science much comparisonis done by comparing the numerical predictions of a theory with the best measurements that wehave. For students in the early years of secondary education, say 11-14 years old, this is not apossibility. These students are not yet in a position to make authentic measurements, even in apedagogically contrived context, nor are their skills with numbers sufficiently well developed thatthey can develop fully quantitative models. Both of these point to a solution that mixesappreciation of the correct qualitative behaviour of a model with the quantitative features comingout about right. This domain of thought might be labelled the semi-quantitative, having parallelswith back of the envelope calculations and with everyday reasoning about the rough and readymagnitudes of things. This latter is very well developed, evolutionary and cultural pressures havingdeveloped this model of thought to such an extent that this everyday reasoning is extremelyeffective in allowing us to survive and even prosper. In fact we might see the computer as anprosthetic, allowing the mobilisation and extension of these leaned behaviours in the service ofgetting started on the processes of reasoning in the sciences.In post 16 education there has already been a significant amount of work done in modelling, botha school and university. In the context of school based innovations in physics significant steps mightinclude elements such as the use of Modellus in Advancing Physics [Lawrence, I and Whitehouse,M (eds) (2000), Lawrence, I and Whitehouse, M (eds) (2001)] and the use of the DynamicModelling System in the revised Nuffield Physics[Harris,J (ed) (1995)]. Both of these allow modesof thinking in classrooms that are qualitatively different from what went before, by providing thisintellectual prosthetic. Yet in these cases effective use of the aid requires competence witharithmetic and algebra. A lack of competence and confidence in these areas reduces the impact ofthe modelling process, impeding the thinking and often reducing the modelling experience to anexperience of observing a simulation, perhaps preceded by some heavily guided typing. Thisexperience of the simulation may be valuable in itself, in that a functional understanding of thesystem might be reached, but the mechanisms will remain out of reach as the learners will not bemanipulating or effectively interacting with the rules of the system, only looking at inputs andoutputs.So to modelling and the prospects that it offers for the engagement of younger students with themechanisms that might underlie the world in which they live. What prospects are there forenvironments in which they might create functioning worlds, expressing their own ideas abouthow these worlds interact? Inspection of the computer packages on work that has been done sofar [ Starlogo, Agentsheets, Stagecast Creator, WorldMaker, LinkIt and Stella], looking carefullyat the structures of physical theories [Barbour, J (1999)], and taking advice from those who havethought about learning in the sciences [Ogborn, J. (2000)] makes it likely that all modelling canbe usefully described in terms of one of two ontological categories. All modelling environmentsin which the worlds can be constructed at least within the target domain, fall into one of twocategories.Some environments have as their building blocks objects and their associated properties, Otherspresent variables and their relationships as the fundamental building blocks. Drawing attention toeither of these pairs of fundamental building blocks may help students to see the structures of thesciences, laying bare the elements put together in order to construct understandings of the world.Seeing these fundamentals in action can allow intuitions to develop about the nature of the wholewithout too much abstract theorising.In addition it is possible to see how sensitive routes from everyday thinking displayed by childrenof the target age group can be mobilised and developed to build up competences in theorising in


the sciences. This is not to say that such developments will be easy, but that such a route is worthyof exploration.There is some evidence that children do in fact think of rough and ready quantities and therelationships between them, as a part of their everyday experience. [Ogborn, J (undated)].Analysisof everyday discourse of adults supports the notion that this mode of reasoning continues into laterlife. From this line of thought comes the scientific development that becomes reasoning aboutprinciples. This is a specialised sub-domain of the whole region in which people reason aboutordered quantities, with its own carefully constructed rules and premises, carefully mapped onto,and perhaps helping to create, the worlds that they describe.A second mode of thinking grows from the importance of narrative in learning, met in storiescreating explanatory worlds and heard in many different cultures. Engaging with meaningfullearning in the sciences is much like sitting listening to one of these fireside stories, engaging theimagination to see how the actors within the story develop and interact to account for theexplanandum of the story. In modelling the parallel process is refining objects (getting to knowcharacters better and better) as you add more and more properties to the basic actor, so bothdeveloping and constraining possible actions. In the sciences the specialised sub-domain of thismode of reasoning is the appeal to causal mechanisms, a chain of sufficient causes resulting inactions that mirror events in the real world. The scientific and everyday modes of thinking pointtowards the possibility of constructive learning when the thinking process is based on objects andproperties.So modelling in the sciences seems to require two complementary approaches; one based onobjects and rules by which they interact, perhaps captured by their properties, and another, built ona fundamentally different basis, that allows reasoning with quantities that need not be fullyquantified, linked by relationships that affect the values of those quantities.As a sample of these two approaches consider models that might be made with two separatecomputing packages, both under development and both realised in Java.First WorldMaker, modelling the world of buses and anti-buses. producing jets where these collide.Buses move left to right, antibuses right to left. Collisions result in excited states which can thenresult in jets, that move off at right angles to the incident particles. Altering the luminosity of thebeams and the collision cross-sections allows the frequency of jet production to be modelled.Objects are subjected to rules in order to characterise their interactions. The model outcomes canthen be compared with real world outcomes in order to see how well the rules embedded in themodel might mirror the regularities in our world.

Next VnR, modelling the actions of a pair of antagonistic muscles:


The biceps and triceps are in opposition, respecting the leverage each provides. The net pull, whichcan be positive or negative, drives the rate of change of position of the forearm.In a more complex example, VnR models simple harmonic motion. A series of pictures shows howthe model evolves over time:


Please note that this is an early version of the software, so that the feedback loop does not lie asone might like it to, in order to make the diagram as readable as possible.In both of these we see the essential features that make the computer such a valuable tool inimplementing these environments.• A shared public space for thinking. What one learner writes, another can read back, and then

comment on, and modify.The process of constructing the model is both interactive and iterative.• An environment which can present partially or fully completed models, so allowing the

modelling environment to be connected to the domain in which the models are to be built. Apart of the domain can be [Rader, C et Al (undated)] expressed. Content free tools may benecessary for flexibility, but teacher controlled scaffolding is essential to help connect studentsto the domain in which their skills and aptitudes are to be engaged in the service of learning. Itis also true that both of these programs developed on the basis of a good deal of classroomexperience, so allowing space for the atmosphere of the classroom to invade the design of thesoftware [Hinostroza, J.E. & Mellar, H (2001)]. In both cases the approach to the modellingprocess can be mediated by the teacher by providing appropriate routes into the modellingactivity and by some preparatory work within the modelling environment itself.

• Ease of reading of the model - The chains of thought that went into constructing the model beeasily read back so that fruitful interactions, including discussion, can take place between thelearners working around and on the screen. Furthermore the learning required to make progressin reading back these models is not so large - the modelling environments are accessible withoutmaking large investments of time in learning to read the language necessary to be an effectiveagent in the environment.

Most of the progress in making modelling tools for the target age range has been focused onmodelling tools developed using objects and properties, so in am attempt to compare the two kindsI have spent some time developing an alternative based on variables and relationships. Here is ashort description of some thoughts and some indications of progress so far.Variables are represented by bars, showing values from as large as can be imagined (positive only,or both positive and negative) to zero.• These variables can be altered, by dragging the top of the bar, or linked via a selection of

relationships, to other variables:• The strength and combinations of these relationships can then be altered, to make different

models.


• Build and Run toolbars control the process, allowing interaction with the model whilst theinferred difference equations are calculated, allowing the model to show what has beenprogrammed.

In effect this is a development based on the work of Sampaio [Sampaio (1996)].The next step will be to compare the differences in the learners thinking when using the twosystems, seeing if the differences in the ontology produce different strands of reasoning in thelearning process. In the same way that expert thinkers in their fields can switch modes so as tomatch the task at hand, so learners might also need this flexible facility.

ReferencesOgborn J., (2000) Round table. Girep Phyteb (2002). Comments from the table.Hinostroza J.E. & Mellar H., (2001), Pedagogy embedded in educational software design:report of a case study

Computers &Education, 37, (2001), 27 –40.Hitchins D.K., (2000), System Thinking.

www.hitchings .co.uk./systhink.html [11/06/2001]Rader C. et al (undated) Rader C., Cherry G., Brand C., Repenning A. and Lewis C.: Designing Mixed textual and

Iconic Programming Languages for Novice Usershttp://www.cs.colorado.edu/~crader/VL98/VL98.html [ 20/03/2001].Ogborn J., (undated) Cognitive Modelling and Qualitative Development. Personal communication 06/1998.Lawrence I. and Whitehouse M. (eds) (2000) Advancing Physics AS CD. Bristol. Institute of Physics Publishing. ISBN

0-7503-0735-8Lawrence I. and Whitehouse, M. (eds) (2001) Advancing Physics A2 CD. Bristol. Institute of Physics Publishing.

ISBN 0-7503-0737-4Harris J., (ed) Revised Nuffield Advanced Physics. Harlow. Longman, (1985).Barbour J., The End of Time, London, Weidenfield and Nicholson, (1999).Sampaio FF, LinkIt: Design, development and testing of a semi-quantitative computer modelling tool, Unpublished

PhD Thesis. Department of Science and Technology, Institute of Education, University of London, (1996).

LEARNING DATA ANALYSIS

Dario Moreno, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, Mexico, D.F.Gabriela del Valle, Division de Ciencias Basicas e Ingenieria, Universidad AutonomaMetropolitana, Azcapotzalco, Mexico, D.F.

1. IntroductionIn physics we use probabilistic tools to interpret experimental data, but as is well known,probabilistic reasoning requires the higest level of formal thinking.This presents a difficult problemto university physics teachers, since most of their beginning students have not reached this stage ofcognitive development.To promote the passage of students to this higher level of understanding, we have devised asequence of lab and computational activities to offer the students the opportunity to acquire somestochastic experience and we want to share with you this teaching strategy that we have been usingfor several years.We make our students start by playing real games of chance (dice, cards, roulette) and then, in thespirit of the pioneering work of Janez Ferbar [1] and George Marx [2], they simulate part of“the game that nature plays in the labs”.Our first aim is to show students that in physics any measurement produces not a single number,but a distribution. It is easy to see that this variability may be hidden by using the wronginstrument; for example by using a calendar to measure the time needed to run 100 meters; but wemay also disguise variability when we use digital instrumentation.Usually the distributions students find belong to a family with a hump in the middle and ourfirst task is to learn how to describe them by using just a few parameters. As an indicator oflocation we introduce the usual mean value, X. Although we could start by using variances as a


measure of dispersion, for didactical reasons we prefer to use the range, the difference between thelargest and the smallest observation [3].Using the computer “to generate observations”, students see that out of chance and chaos, someregularities emerge. These regularities refer mainly to the stability of the relative frequencies withwhich certain results occur. These relative frequencies (or proportions) lead in a natural way to theintroduction of the concept of “probability”.Chance is a peculiar mixture of two apparently contradictory ingredients: stability and variability;and extrapolating from our own experience, we believe that we must allow students the timenecessary to get familiar with randomness. Before the advent of computers, this was impossibleto do in a short time, but now computers are so fast that we have enough time to play withrandom-walks, Markov processes, chaotic dynamics and other games that contribute to our funand stochastic education.Once the students recognize that the outcome of an experiment is a distribution, computers allowthem to generate their own data.For this, we use the function RND that is available in most computer languages. The uniformdistribution in (0,1) that RND provides is the “mother distribution” for many others andconstitutes the starting point to introduce averages and dispersion parameters. We do not want touse new technology just to teach the same old formulas! Our aim is not numbers, but insight.To be realists, we must recognize some hard facts: 1) in the elementary labs, one rarely takesmore than 6-8 measurements of the same magnitude, 2) nobody knows the exact shape of thedistribution of student’s measurements (we only know for sure that these distributions are notstrictly gaussian); 3) there are quick and easy ways, using the range and other order statistics, to getconfidence intervals for most statistical parameters.Last but not least, students can get a feeling of the confidence they can have on their results, byusing the “betting interpretation” of probabilities; i.e. we say that we are reasonable sure of someoutcome if we can bet on it 20 against 1 without going broke.We want not to just describe variability, but we want to make predictions in spite of it. Then thefinal task emerges: to find an interval where we can be pretty sure the experimental results willland if some one repeats our experiment using the same methods and equipment. The end result isa procedure that gives confidence intervals to be used to predict what will happen if our experimentis repeated many times somewhere else.Former GIREP Proceedings are very useful even for people that did not attend the events, and weare trying to reach that level of usefulness, even if that means starting at a too elementary level andrepeating things that most of you may know. Sorry about this.

2. Using the function RNDThe function RND produces list of numbers in the open interval (0,1). In QuickBasic, the order

For j=l to 12PRINT RND

Next j

produces a dozen of numbers of the type 0.44588, 0.84512, etc.If we start with the order RANDOMIZE TIMER, every time we run the program we will get adifferent dozen.The most important property of the numbers produced by RND is that although they are producedin no special sequence, they end up by being uniformly distributed in the interval (0,1).The students discover all the properties of RND by writing their own programs. As a sub product,they come in contact with the limitations of the arithmetic that computers use. This is animportant cultural byproduct, helping them to put in perspective the phrase “the computer says”,so common among illiterate people.


3. Generation of pseudo-resultsStarting from RND, we can simulate a student making measurements of a physical magnitude.The main problem is to decide wich distribution to use. In fact nobody knows actually whichdistribution student’s measurements follow. Many teachers think that it is a perfect gaussian, butthat is not true. If this were true, then when you measure the diameter of a pencil, then therewould be a probability different from zero that your pencil has a negative diameter. The mainreason to use the gaussian has being “historial simplicity”, but nowadays computers are allowing usto be more realistic.

4. Generating dataSome people have studied which distribution fits the measurements made by students and theyhave found that it is more like a beta distribution, with a and b larger than zero. Even withcomputers this is a complication, so the suggestion has been made that the distribution to useshould be a superposition of two gaussians with different variances. Conforted by the existence ofthe central limit theorem, we have used the distribution arising from the sum of twelve RND (sothat the variance is unit) and shifting it in such a way that its mean is zero.If you are a believer in normality, you may start from a flat distribution in (0,1) and use atransformation like Box-Muller or an algorithm like Marsaglia’s, but most students will notunderstand you.

5. Analizing dataEven in Mexico, you can buy software that either fits straight-lines to your data or does otherstatistical duties, like calculating averages and variances. The trouble is that most of this softwareproduce worthless results. For example, experimental points are assumed to have uncertainties onlyalong one axis and in the case of a well known commercial package, ... some variances turn up beingnegative! What is worse, some students may report this nonsense, showing that their understandingis nil.This is one of the reasons we have decided to use the range as a measurement of dispersion and the“twenty against 1 confidence interval” as a final assessment of reliability.As we have said, we do not want that the students just use a magical recipe —derived bysomeone else— to asses their data. In a certain sense, every student writes his own program to finda 95% confidence interval.They have experimentally found that the raw range grows as the samplesize grows but since they are not trying to be absolutely sure but only reasonably sure, clipping therange is fair enough.This is a sample of the program used by students to get the boundaries of their intervals ofconfidence. We have suggested a program that everybody can understand. Besides, it is fast enoughfor our teaching purposes.

Randomize timer

GOSUB CONSTANTES GOSUB LABELS FOR LL = 1 TO 10

FOR J = 1 TO NTGOSUB GENERATE IF Al < 0 AND Bl > 0 THEN Q = Q + 1

NEXTjIF Q > 9500 THEN HIGH = HIGH + 1IF Q < 9500 THEN LOW = LOW + 1LOCATE , 20: PRINT Q; “ HITS”Q = 0

NEXT LLPRINT HIGH;” HIGS “; LOW;” LOWS” END


CONSTANTES:CLS : CONST DOCE = 12Cn = 0.51NS = 5NT = 10000

RETURN GENERATE:

R = 0FOR BB = 1 TO DOCE

R = R + RNDNEXT BB

X= R-6 RETURN LABELS:

PRINT “NUMBER OF HITS IN INTERVAL (X - Cn*RANGE,X + Cn* RANGE) “PRINT “TOTAL OF TRIALS NT = “; NT;” SERIES OF NS =”; NS;” MEASUREMENTS EACH”PRINT “ TESTING C(“;NS;”) =; Cn: PRINT

RETURN

The final product of using this program, is a table giving us the value of Cn to be used to find theborders of the confidence interval that allows us to bet 20 against 1:

(X - Cn Range, X + Cn Range)

This is part of the table (the most important part):n 2 3 5 10 20Cn 6.4 1.3 .51 .23 . 12

6. Final commentsPlease notice that we are not preaching a new testament to replace the old one. Usuallyexperimental uncertainties are so badly handled in the introductory labs, that their main purposeseems to please the lab instructor. We are not proposing a new recipe —to please a newinstructor— but we are trying to offer some experience with stochastic processes. What we want isto provide students with the opportunity to “play with chance” as a first step to begin “thinkingprobabilistically”.After using this approach many times with our students, we feel reasonably sure that most of them,at the end of the semester, handle numbers that they do understand... in 19 cases out of 20.

References[1] G. Marx, Games Nature Play, Physics Education[2] B.J.T. Morgan, Elements of simulation,Chapman and Hill, (1984), Chapter 4.[3] F. Mosteller, R.E.K. Bourke, Sturdy Statistics, Addison-Wesley (1973), Chapter 14.

MODELS - THE BASICS OF PHYSICAL THINKING. CONCLUSIONS FORMULTIMEDIA

Romanovskis Tomass, Institut für Experimentalphysik, Universität Hamburg, Germany, on leavefrom Faculty of Physics and Mathematics, Latvia University, Riga, Latvia

1. Models in physicsA look in the high school and university textbooks of physics shows that models are very importantin teaching physics. Mass point, elastic and inelastic collision, rigid body, blackbody, ideal gas, idealengine, incompressible liquid, laminar flow, crystal lattice, the raisin cookie model of the atom, the


planetary model of the atom, Bohr model of the atom, Solar system, uniform motion, free fallmodel, harmonic motion, wave, electric charge, electric field, photon are a few to mention.Regretfully, models and the role they play in understanding the physical world are not muchexplained in the textbooks. Models are rather neglected even in didactics of physics.The Russian physicist N. Umov was the first to make serious comments concerning the role ofmodels in physical thinking and knowledge. In 1909 he wrote that our perception of the world, fromthe very simplest everyday experience to its most sophisticated contents, is a collection of modelspresenting a more or less successful reflection of reality [1].

2. Galileo’s four stages of physical reflection Galileo was the first to propose a model of advancement of rational thinking. He formulated fourstages of physical reflection: acquisition of facts, modelling, drawing conclusions, and verifying thelatter by experiment [2]. These stages can be recognised in writings of many physicists. The Russianscientist Razumovsky has put them in a simple cyclic model of rational thinking [3] .

A striking manifestation of theevolution of physical thinking theauthor found in Descartes’ paperabout the rainbow [4]. Descartesannounces that he has found a methodof obtaining new knowledgeconcerning the rainbow. Essentiallythe method is modelling a rainbow bya single drop of water presented as aglass sphere filled with water.Descartes made a numericalsimulation (likely the first one in thehistory of physics) calculating theangle between light rays going in

and coming out of the drop after refraction, internal reflection and second refraction at the surfacefor tens of rays. Assuming a refraction index of n=4/3 the conclusion was made that the anglebetween the rays going in and coming out cannot exceed 41.5 degrees in the main rainbow and isat least 51.5 degrees in the additional rainbow. Experiments with a natural rainbow and a ball ofwater confirmed the validity of Descartes’ rainbow model.Newton has also appreciated Galileo’s model of evolution of rational thinking (I.Newton. Opticks.Dover Publications, Inc, 1952, Preface, p. XXIV):“For the best and safest method of philosophising seems to be, first diligently to investigate theproperties of things (1. Facts) and establish them by experiment, and then to seek hypotheses (2.Model) to explain the properties of things (3. Conclusions) and not to attempt to predeterminethem, except in so far as they can be aid to experiments (4. Experiment)”

3. Two levels of physical reflection. Feynman’s four stages of physical reflectionA rapid advancement of theoretical physics started after Galileo. The Huygens-Fresnel principlewas discovered, Fermat’s principle of shortest time, conservation of momentum and energy,d’Alembert’s, Legendre’s, and Hamilton principles. Feynman’s reflections on the subject discussedon the example of geometrical optics [5] may help to understand this period in the history ofphysics.“Although geometrical optics is just an approximation, it is of very great importance technicallyand of great interest historically. We shall present this subject more historically than some of theothers in order to give some idea of the development of a physical theory or physical idea.”Feynman noticed that everyday experience in a natural way favours the ray model. Then heconsiders reflection and refraction experiments and turned to the advancement of physicalthinking providing evidence for Galileo’s 4-stage model of rational thinking.


3. CONCLUSIONS 2. MODELS

1. FACTS 4. EXPERIMENT

Fig. 1: Galileo’s four stages of physical reflection

1. First we observe an effect, then we measure it and list it in a table;2. Then we try to find the rule by which one thing can be connected with another. The rule,

found by Willebrord Snell, a Dutch mathematician, is as follows sin θI = n sin θr. For waterthe number n is approximately 1.33.

3. Equation (26.2) is called Snell’s law; it permits us to predict how the light is going to bendwhen it goes from air into water.

4. Table (26.2) shows the angles in air and in water according to Snell’s law. Note theremarkable agreement with Ptolemy’s list.

However, Feynman considered a further advancement of physical reflection at a higher levelunpredictable at the time of Galileo.Actually we encounter a new cycle of advancement of physicalreflection where facts are not separate observations and numbers any more but physical modelsand laws, as confirmed by Feynman’s words:

1’. Now in the further development of science, we want more than just a formula. First wehave observation, then we have numbers that we measure, then we have a law, whichsummarises all the numbers. But the real glory of sciences is that we can find a way ofthinking, such that the law is evident.

2’. The first way of thinking that made the law about the behaviour of light evident wasdiscovered by Fermat in about 1650 and it is called the principle of least time or Fermat’sprinciple.

3’. Now let us demonstrate that the principle of least time will give Snell’s law ofrefraction… However, the importance of a powerful principle is that it predicts newthings. It is easy to show that there are a number of new things predicted by Fermat’sprinciple. ... Another prediction is that if we measure the speed of light in water, it will belower than in air. It is a brilliant prediction.

4’. Later measurements of the speed of light in water experimentally confirmed thatconclusion.


3’. NEW PREDICTIONS

4’. EXPERIMENT

2’. PRINCIPLE / THEORY

1’. MODELS / LAWS

Fig. 2: Feynman’s four stages of physical reflection on the highest level

4. Models and computer simulationsThe main computer experience physicists have acquired in numerical modelling and it has its effecton the software offered for physics education. The major number of software packages is relatedto simulation [6]: “Software teaching packages are increasingly being seen as a way for students toobtain more knowledge quicker and in a more exciting way. Computers eliminate the tedium ofsolving equations in favour of a more interactive and user-friendly approach. With the computerdoing the hard, time-consuming and “boring” work, the learner is more likely to have a wish toinvestigate the physics of a system.”Setting the model in the front of physics education the authors of simulation software do notrealise that the infrastructure – teachers, textbooks, programs, listings of required knowledge andskills – is not prepared to use simulation software in teaching.

5. Interactive exploration in multimedia and four stages of physical reflectionSince evolutionary stages of physical thinking are clearly manifested in the works of Galileo,Newton, Descartes, Feynman and other prominent physicists, Razumovsky has suggested to followthem in the practice of teaching and has developed methodics for teaching a series of topics inaccordance with the model of physical thinking [3]. However, planning lessons based on theprinciple exclusively would restrict creativity of the teacher and his guidance in the teachingprocess, which is different from following the principles in revelation of physical models, especiallyin multimedia.The 4 stages of physical thinking are associated with certain mental activities: observing andmeasuring, comparing, concluding, experimenting and testing. The modern computer techniquesoffer interactive multimedia tools to each of them: observing videoclips/animation and measuringon the screen, calculation and presentation in graphs, controling screen experiments (video,simulation).Simulation software enables to change a parameter (elasticity of the spring, oscillation amplitude,etc.) making the student a passive observer. The lack of a exploration feedback is one of majordisadvantages of simulation. It is very important that interaction does not end with the change ofthe parameter and is supplemented with mental activities of 4-stage model of rational thinking:observing and measuring, comparing, concluding, experimenting and testing.For example, the Cabri-Géometrè II software allows preparing of simulated reflection andrefraction phenomena as a CabriJava applet for exploration on computer screen (see Fig. 3).Demonstration mode: The teacher may seize the incident beam with the mouse (marked with acircle) to change the angle of incidence. The student has to observe the changing angles ofincidence and refraction and the ratio of sinus functions to test Snells’s law (observing, comparing,concluding, and testing activities).Student mode (without automated calculation of ratio of sines): student may seize the incidentbeam with the mouse, write the angles down and calculate the ratio of sines to test Snell’s law(observing and measuring, data processing, concluding, and testing activities).The novel means of multimedia allow to include in physics education a number of interestingphenomena too complicated for a traditional physics course. First, these are simulations and video

sequences of real phenomena providingmeasurement data directly on the computerscreen. Interactive exploration allowscultivating all stages of physical reflection:observing, measuring, and data processing forconclusions and tests. For example, in the“Multimedia Motion” [7] movies one canexplore the motion of sportsmen and objects,collisions of cars, launching of a missile, afalling chimney and a rotating hammer. Themeasurements are obtained to an accuracy byfar exceeding the accuracy available in thehigh school or college laboratory.The “Coach5” [8] software for video clipsoffers efficient measuring and mathematicaltools (data processing for conclusions andtests) for the exploration of motion in videosequences of real phenomenas.The teaching scenario of R.Carlson in his“World-in-Motion” software [9] is ofparticular didactic interest since it iscompletely consistent with 4-stage model of


Fig. 3: Simulated reflection and refraction phenomena forinteractive exploration prepared with Cabri-Geométrè IIas CabriJava applet.

rational thinking: Discussion, The Model, Marking the Video, Graphical Analysis.Most of the authors of multimedia products do not offer any didactic solutions to use the productfor teaching. At the same time the authors of popular classical textbooks realise the trend towardmultimedia. Cutnell and Johnson (Physics 5/e), Halliday, Resnick and Walker (Fundamentals ofPhysics 6/e) offer annotated web bibliographies [10] to their textbooks. The user can select achapter for links to web sites featuring supplemental tutorials, illustrative examples, applicationsand working JAVA Applets. This is an appreciable approach since the authors provide instructionswhere and how to match the multimedia to the textbook.In the traditional teaching environment (blackboard, textbook, paper) the role of models isundervalued because of the lack of efficient modelling tools while modelling in multimedia isexaggerated (simulation without feedback) and mainly used at the first level of physical reflection(Galileo’s model). More attention should be paid to interactive exploration of simulated and realexperiments promoting all stages of physical reflection within both Galileo’s and Feynman’sinterpretations.

AcknowledgementsThe present study has been made partly within the project “Physik multimedial”. This project issupported by BMBF (ZukunftsInvestitionsProgramm). The author thanks M.Tilgner/Hamburg forconsultations on the culture of Ancient Egypt.

References [1] U. Grinfelds, T.Romanovskis, E.Silters. Models in teaching math and physics. Riga, Zvaigzne, (1983) (in Latvian

language).[2] M. Gliozzi. Storia della fisica. Storia delle scienze, volume secondo, Torino, (1965).[3] V.G. Razumovsky. Developing students creativity. Prosveshchenie, Moscow, (1975) (in Russian language).[4] Oeuvres de Descartes. Publ. par C.Adam, P.Tannery. t.1-12, Paris, 1897-1913.[5] R. Feynman, R. Leighton, M.Sands. The Feynmans lectures on physics. Addison Wesley.[6] A. Gillies, B.Sinclair. Simulations for students. Physics World, July, (1997), 47-51.[7] Cambridge Science Media: http://www.csmedia.demon.co.uk[8] Software Coach 5: http://www.cma.beta.uva.nl[9] R. Carlson. World-in-Motion Physics Software. http://members.aol.com/raacc/wim.html[10] Cutnell and Johnson, Halliday, Resnick and Walker. http://purcell.phy.nau.edu/Wiley/

MODELS, MENTAL IMAGES AND LANGUAGE IN SCIENTIFIC THINKING

Jan J.A. Smit, Potchefstroomse Universiteit vir CHO, Potchefstroom, South Africa

1. IntroductionModels play a fundamental role in the development of physics. A literature review reveals thecomplexity of models in physics. A logical first question when dealing with models is: What is amodel? or How does one define a physics model? Literature reveals that any attempt to give acomprehensive definition of a model ends up with a definition that is so inclusive and general thatit says virtually nothing. An example of such an inclusive definition is the one by Apostel (Bertels& Nauta 1969).Any subject using a system A that is neither directly or indirectly interacting with a system B, to obtaininformation about the system B is using A as a model for B.This definition is so inclusive that even a telephone directory can be regarded as a model for thetelephone system of a region. The author’s experience is that the best way to developunderstanding of the model concept in physics is to investigate• The classifications of the different types of models in physics.


• The nature of these models• The role they play in physics and their functions.We will in this presentation first attend to the three topics stated above.

2. Classifications of the different types of physics modelsDifferent classifications of models are reported in literature for example Harré (1970, 1991),Santema (1978) and Leatherdale (1974). It is appropriate to start a discussion of physics modelswith the classification of all models by Santema and Klaus (Santema, 1978).They classified all models in two categories (Figure 1): models of existence or being and subjectivemodels. Models of being are according Plato (Santema, 1978) models the Godlike demiurge usedto create the world. Subjective models on the other hand are human creations. Santemadistinguishes between two types of subjective models: knowledge models and make models. Themodels of physics belong to the type of knowledge models. Make models are the models used byengineers for example to construct artifacts. According to Santema (1978) knowledge models aremodels that help the scientist to know the world. They can be seen as interfaces between man andreality. Knowledge models in physics can again be classified into different types (Harré, 1970).Figure 2 displays the famous taxonomy of Harré.


Models

Models ofExistence/Being/

Plato

Subjective models(human creations)

Make models(engineer) Knowledge models

(scientist)

Figure 1: Classification of models by Santema and Klaus (Santema, 1978)

Figure 2: Harré’s taxonomy of models.

Knowledgemodels

Homeomorphs Paramorphs

Micro- andmegamorphs

Teleomorphs Metriomorphs

Idealisations Abstractions

In this taxonomy models are classified according to their relation to the source of the model. If thesource is the object under modelling the model belongs to the class of homeomorphs. If the sourceis not the object under modelling the model falls in the class of paramorphs. Homeomorphs canagain be subdivided into different classes (Figure 2). Examples of homeomorphs are models of thesun, moon, earth, twin stars and Halley’s comet. Examples of paramorphs are the planetary modelof the atom, the sunwind and the model of the earth’s magnetic field (analogous to that of a barmagnet). Homeomorphs are divided into three classes: micro- and megamorphs, teleomorphs andmetriomorphs. Micro- and megamorphs have to do with scaling. Micromorphs are models wheredownscaling of the entities are done for example a model of the Universe. Megamorphs has to dowith upscaling of very small entities. A teleomorph as the name implies is to a certain extend animprovement of the object under modelling. Harré (1970) subdivides teleomorphs intoidealisations and abstractions. Idealisations can be described as follows. Suppose a scientist wantsto model an object or system with properties P1, P2, P3,….. Pn relevant to his investigation.An idealisation is then a model with properties P1, P2, P3,… Pn.Each of the n properties of the model is on some or other scale of values more perfect than thecorresponding property of the object under modelling.An example of an idealisation is an ideal gas. In a real gas the molecules have volume (P1), thecollisions are not perfectly elastic (P2) and the intermolecular forces are not zero (P3). In an idealgas the molecules are assumed to have no volume (P1), the collisions to be perfectly elastic (P2)and the intermolecular forces to be zero (P3).Abstractions are teleomorphs where the model has fewer properties than the source object. If thesource object has n properties, P1, P2, P3,…Pn, then the abstract model has less than n properties.The model of a body in linear mechanics serve as an example of an abstraction. A real body forexample has mass, dimensions, colour and a temperature. The only property of concern in lineardynamics is its mass.At the heart of any metriomorph is a mathematical calculation. One often read of a city with ametriomorphical family of say 2,63 children.An example from physics is the fission of the uranium-235 nucleus yielding 2,47 neutrons per reaction.Paramorphs are analogue models. The analogy relates the object or process under modelling tosomething the physicist has more knowledge of, or understands better. According to Mary Hesse(1966) every analogy has a positive, negative and neutral part. The positive part gives properties ofthe model and object under modelling that relates and the negative one the differences.The neutralpart constitutes the properties of the model and object that we do not have sufficient knowledgeof to classify them as positive or negative. They produce fruitful ideas for research.It needs to be remarked that there are composite models that bear characteristics of more one typeof model in Harré’s taxonomy.Harré (1991) formulated another very simple classification of models. He classified physics modelsinto three types:Type 1 : The model represents a real existing entity. Examples of models belonging to this Type aremodels of atoms, gasses and the Earth. A question in this context is: When is an entity real?Physicists believe an entity to be real when its existence can be proofed experimentally.Type 2 : Models of this type are of hypothetical entities. The entity may or may not exist. There areclues of its existence, but not experimental proof. An example is the Higgs boson. A few years agoit was the top quark. If the Higgs boson is detected its status will change to Type 1. If its non-existence is proofed it will shift into the history of physics.Type 3 : Models that do not represent any real or hypothetical entity belongs to the third type.Examples are models of the ether, caloric and phlogiston from history. Some type three models arefunctional and serve instrumental purposes. An example is the conventional current model. A


function of this model is to assist in the quantitative description of energy transfer in electriccircuits.

3. The nature of models in physics• It firstly needs to be noted that models are creations of the human mind and that most authors

(on models) believe that a model cannot be derived from data in a logical way. Leubner (1989)says a model is an invention of the human mind. Ramsey (1964) states: The model arises in amoment of insight.

• An example to verify this view is the relation between the work of Tycho Brahe and Kepler’smodel. Brahe made many observations of the planets’ motion, but could not see any structurein it. It was Kepler who, in a moment of insight saw the model.

• A Model summarises and gives structure to scientific knowledge on a certain topic (Van Oers,1988). It deals with a limited aspect of reality (Park, 1988) and is a simplification of the objectunder modelling (Kollaard, 1991).

• A model brings together knowledge – sometimes of vastly different aspects of reality. Atomicmodels for example link phenomena of heat, chemical bonding, light emission, electricity anddensity.

• A model can be a representation of a real entity, but is not the real thing. It is a token, a symbolof the real thing ( Van Oers, 1988).

• Models do not occur in nature, but the objects of modelling do.• Models in physics are in general not replicas, copies or real representations of the modelled

entities. Models do not look like the real thing. It is not a picture of the real thing.• Models are temporary by nature. As new knowledge emerges the model is either adapted or

rejected. An example is the series of atomic models.• If more than one model is possible for the same entity, the simplest model suitable for the task

under modelling is preferred (Park, 1988).• Physics models are either abstract mathematical models with no spatial image associated or a

model that can be visualised. Mental pictures can be formed of the latter group of models. Themodels lend for mathematical examination.

• Physics models are community property. They belong to the community of physicists and aretherefore known as canonic models. All scientists carry more or less the same mental picture ofa canonic model

• Models must fit into the structure of science. Any model must co-exist with other models (Smit& Finegold, 1995).

• Models form part of theories. In any mature theory, Leubner (1989) and Harré (1970)distinguish between three elements: a set of real phenomena, described by experimentalobservations; a model with the associated mathematical relationships and a set ofcorrespondence rules to interpret results derived from the model by mathematical investigationin terms of real phenomena.

4. Functions of models• The fundamental function of any physics model is to give scientists knowledge of reality

(Santema, 1978). We know reality through models. The model constitutes an artificial reality thatcan be investigated on mental, visual or material level. (Author’s translation, Van Oers, 1988).

• An important function of any model is the explanation of phenomena. Each model has afunctional domain. Take as example the particle and wave models of light. The functionaldomain of the particle model includes for example the photo-electric and Compton effects. Thephenomena of interference and diffraction lie within the functional domain of the wave model.An investigation by Smit and Finegold (1995) revealed that the mixing of models gives rise toepistemological problems by learners.

• Models play a key role in the prediction of entities/phenomena. The predictions of the planet


Neptune, the neutron, neutrino and many other elementary particles were based on existingmodels. It is thus clear that models play an important and essential role in the development ofphysics.

• At the frontiers of physics scientists sometimes have different models of the same entity. Anexample was the two different models of the atom held by the contempories Dalton andBoscowich.

In the next paragraphs models are related to scientific thinking.

5. Scientific thinkingAccording to Harré (1970) there are only two carriers of scientific thinking: mental models(images) and words. The visual physical models form an important subset of a physicist’s mentalmodels. Nouns and verbs in general serve to tag models of objects and processes the models areinvolved in. For example, the tag atom (noun) associates with a mental picture that can be drawnon paper. The verb combine tags a process in the statement: two atoms combine.Conceptualisation in physics involves that learners form scientific acceptable mental images ofphysical entities and processes associated with the corresponding verbal tags.The author found in his research that students often have alternative conceptions with regard tothe nature and functions of physics models and that these alternative conceptions give rise toconceptual problems.The following examples contextualise what is stated above.– Students hold the perception that the model is a picture or replica of the object or entity under

modelling (Smit & Finegold, 1995). This leads to conceptual difficulties with regard to theparticle and wave models for radiation and sub-atomic particles.

– Students view all models as belonging to Type 1 (Harré, 1991). This leads to problems with theunderstanding of the conventional current in DC electricity (Smit & Nel, 1991).

6. ConclusionA conclusive remark is that little attention is given in syllabi and textbooks to the topic of models.Reference to what models are, is often made in introductory paragraphs, but no in-depthdiscussions of the different types of models, the nature, functions or role of models in physics isgiven. Research has proven that for proper understanding of physics this topic should receive farmore attention in the teaching of physics. The same applies to the relationships between words andmental images in scientific thinking. In teachers’ training courses little attention is given to thisbasic principles of scientific thinking. It is the author’s experience that even at school leveldiscussion of the principles of human thinking bears fruit.

ReferencesBertels K. & Nauta D., Inleiding tot het modelbegrip. (W. de Haan, Bussum, Nederland), (1969), 167.Harré R., The principles of scientific thinking. University of Chicago Press, Chicago, (1970).Harré R., Private communication. Oxford, (1991).Hesse M.B., Models and Analogies in Science, University of Notre Dame, Press, Notre Dame, Indiana, (1966).Kollaard U.H., Didactisch vertalen. Vrije Universiteit, Amsterdam, (1991).Leatherdale W.H., The role of analogy, model and metaphor in science. North-Holland Publishing Company,

Amsterdam, (1974).Leubner C., The Structure of Scientific Theories. Innsbruch University, Innsbruck, Austria, (1989).Park D., The How and the Why. Princeton University Press, Princeton, New Jersey, (1988), 74.Ramsey I.T., 1964 Models and Mystery, 13, quoted in W.H. Leatherdale, (1974), The role of Analogy, Model and

Metaphor in Science (North Holland) , 62.Santema J.H, Modellen in de Wetenschap en de Toepassing ervan. Delftse Universitaire Pers, Delft, Nederland, (1978)

186-193.Smit J.J.A., & Finegold M., Models in physics: perceptions held by final-year prospective physical science teachers

studying at South African universities, International Journal of Science Education, 17(5), (1995), 621-634.Smit J.J.A., & Nel S.J., Perceptions of models of electric current held by physical science teachers in South Africa.

South African Journal of Science, 93 (1997), 202-206.Van Oers B., Modellen en het Ontwikkeling van het (Natuur-) Wetenskaplijk denken van leerlinge, Tijdskrift voor

Didaktiek der Beta-Wetenschappen, 6(2), (1988), 115-143.



3.4 Hands-on/Toys

FROM PLAYING WITH TOYS TO MEASUREMENTS

Maja Bertoncelj, Ana Gostinčar Blagotinšek, Faculty of Education, University of Ljubljana,Slovenia

1. Talking teletubbiesThe picture of Talking Teletubby is shown in Figure 1. We used Talking Teletubbies to classifymaterials into conductors and insulators. Objects, made of different materials, are put between twocontacts, shown in Figure 2. If the object is a conductor, the Talking Teletubby tells us his name. Ifthe object is an insulator, the Talking Teletubby remains quiet.Talking Teletubbies are suitable onlyfor qualitative observations but they are very sensitive – they talk even when more people holdingtheir hands touch the contacts. Results of our experiments are presented in Table 1.

object material conducting non

conducting

coin metal

rubber gum

pencil lead graphite

toy umbrella dry dry wood

toy umbrella wet wet wood

coffee cup ceramic

bracelet gold

wire constantan

LEGO brick plastic

child skin

Table 1: Classifying materials into conductors and insulators with Talking Teletubby.

Figure 1: Talking Teletubby.

Figure 2: Contact Points.

Figure 1: Talking Teletubby.

2. BounceabilityWe investigated two of the variables that affect how high a ball bounces: the material the ball ismade of and the surface, which the ball is bounced off. We dropped the balls from 1 m andobserved the rebound heights. Results are presented in Figure 3. What determines how high theballs bounce on different surfaces? During the bounce, both the shape of the ball and the shape ofthe surface are deformed. The height of the bounce is determined by how much energy ofcompression is returned as the shape of both the ball and the surface restore to normal. Each balltype and surface type interact differently, producing a unique result. Even so, some surfacesproduce fairly consistent results with all types of balls. For example, all the balls bounce on theexercise mat much the same, as can be seen from Figure 4. The mat deforms more than the balls,acting much like a trampoline. In contrast, if the surface stays deformed as the Styrofoam may, than

Table 1: Classifying materials into conductors and insulators with TalkingTeletubby.

the energy that went into causing the deformation does not return to the ball. Happy/UnhappyBalls are Butadien- Butylcaoutchouc balls which feel very much the same when squeezed, butbehave very differently when bounced. One ball bounces quite high (like a super ball) while theother bounce just a little (like a ball of clay). The balls in this activity are suitable both for semi-quantitative and quantitative measurements.We can just compare the rebound heights for differentballs on the same surface by dropping them at the same time and list the balls from highest bouncerto lowest bouncer or we can measure the rebound heights and present results in a graph.


0

20

40

60

80

100

metal ball squash ball unhappy ball happy ball tennis ball

Type of Ball

Rebound H

eig

ht

[cm

] stone

metal

exercise mat

wood

Styrofoam

Figure 3: Rebound Heights for Different Balls.

0

20

40

60

80

100

stone m etal exercise

m at

wood Styrofoam

Type of Surface

Re

bo

un

d H

eih

gt

[cm

]

m etal ball

squash ball

unhappy ball

happy ball

tennis ball

Figure 4: Rebound Heights on Different Surfaces.

3. Plastic spring toyWe examined the properties of a Plastic Spring Toy (shown in Figure 5) by hanging weights onspring and measuring the elongation of the spring. The weights, which we used, were Table-ClothWeights - Fishes (which are rather more fun than traditional metal masses). When a graph ofdistance versus weight was plotted, a straight line was obtained, showing that the spring did obeyHook’s Law. A spring constant of 1,8 N/m can be determined from a graph, shown on Figure 6.

4. Ramps and carsIn this activity, we investigated relation between height of the ramp and the distance the Toy Car(shown in Figure 7) travelled on horizontal ground after descent. The car rolls farther with asteeper ramp because it is moving faster when it reaches the bottom of it. The faster the car ismoving, when it leaves the ramp, the more distance it can cover before friction slows and stops it.Initial potential energy of the car (Wp = mgh) is transformed into kinetic energy at the bottom ofthe ramp (Wk = H mv2). Amount of kinetic energy is slightly smaller than initial potential energydue to resistant forces. Assuming that resistant forces are constant along the horizontal track thecar travels, distance it reaches is in the first approximation prop ortional to square root of initialheight of the car on the ramp. Experimental results are presented on Figure 8.

2263. Topical Aspects 3.4 Hands-on/Toys

0

10

20

30

40

0 5 10 15 20 25

Added Mass [g]

Length

of th

e S

pring [cm

]

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14

Height of the Ramp [cm]

Dis

tance T

ravelle

d [c

m]

Figure 5: Plastic Spring Toy and Table-ClothWeights - Fishes.

Figure 7: Toy Car. Figure 8: Effect of Height of the Ramp on Distance Travelled.

Figure 6: Effecf of Added Mass on Lenght of the spring

5. Pull-back carsBy pulling back the Pull-Back Car (shown in Figure 9) we store energy in the spring. When werelease the car, the stored energy is transformed into kinetic energy of the car.We investigated howthe pull-back distance affect the distance travelled by the pull-back car before friction slows andstops it. Various models differ significantly in reliability. Several models were tested to obtainconsistent results. Results for our model are shown on Figure 10.

6. Walking turtleIn this activity we used Wind-up Walking Turtle Toy shown in Figure 11. We collected distance dataevery 3 seconds (presented in Figure 12). It can be immediately seen that the distance covered inconstant time intervals decreases with time. Obtained data enable us to calculate average velocityand average acceleration for each time interval. Velocity is the rate of change of position, andacceleration is the rate of change of velocity. When a graph of velocity versus time was plotted(Figure 13), a straight line was obtained showing that the velocity of the Turtle was decreasing atconstant rate. From a graph of acceleration versus time (Figure 14) it can be seen that the Turtlewas walking with a negative acceleration (or a deceleration) of approximately –0,45 cm/s2.


0

20

40

60

80

100

0 2 4 6 8 10 12 14

P u ll-B ack D is ta nce [c m ]

Dis

tan

ce

Tra

ve

lled

[c

m]

Figure 9: Pull-Back Car. Figure 10: Effect of Pull-Back Distance on Distance Travelled.

0

20

40

60

80

100

0 3 6 9 12 15 18 21

Time [s]

Po

sitio

n [

cm

]

Figure 11: Wind-up Walking Turtle Toy. Figure 12: Position-Time Graph for a Walking Turtle

0

2

4

6

8

10

0 3 6 9 12 15 18 21

Time [s]

Ve

locity [

cm

/s]

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0 3 6 9 12 15 18 21

Time [s]

Accele

ration [cm

]

Figure 13: Velocity-Time Graph for a Walking Turtle Figure 14: Accelerationo-Time Graph for a WalkingTurtle

7. ConclusionsPhysics is an exciting science and it can be made fun through the use of ordinary children’s toys.Toys can be utilised at all grade levels from kindergarten through college by varying thesophistication of the analysis. Toys bring part of play – time joy in the classroom and are valuableas motivation and experimental equipment. Depending on situation, simple qualitativeobservations, semi-quantitative or quantitative measurements can be made using simple toys. Andbecause children are familiar with toys, we can also use them to build the links between everydayknowledge and scientific knowledge.

ReferencesTaylor B.A.P., Porh J., Portman D.J., Teaching Physics with TOYS: activities for grades K-9, Terrific Science Press,

Miami University Middletown, (1995).Turner R.C., Taylor B.A.P., Physics and toys – Physics fun for everyone, in Proceedings of an ICPE - GIREP

International Conference Hands on – Experiments in Physics Education,– Didaktik der Physik, University ofDuisburg, pp.138-147, (1998).

Radford D., Science from toys: Stages 1 and 2 and Background, A Unit for Teachers, Macdonald Educational, (1982).

HOW SCIENCE CENTERS AND MUSEUMS CAN SERVE THE FORMAL LEARNINGIN THE SCHOOLS

Pietro Cerreta, ScienzaViva, Associazione per la divulgazione scientifica e tecnologica, Calitri(Av); Gruppo di Storia della Fisica dell’Università di Napoli; Sezione AIF di Calitri (Av), Italy

1. Multiplicity of formal thinkingPhysics is not a science like the other ones: it is a mathematicized or formalized knowledge whichdescribes, interprets and foretells the phenomena of nature.We know that there are many types of theories in physics. But often we forget that they do not usethe same mathematics. Likewise, we often ignore that they don’t have the same conceptualorganization [1]. The classical mechanics uses the mathematics of infinitesimal analysis and it isstructured on principles, but the classical thermodynamics makes use of a much simplermathematics and it is organized on only one central problem: the impossibility of perpetualmotion.The geometrical optics is organized on principles, but it does not use mathematical analysis:it only needs Euclidean geometry.Notwithstanding this variety of forms in which the physical theories present themselves, when wetalk about the formal thinking of physics, we often make a drastic reduction of them, we refer onlyto the model of classical mechanics: that is, to the infinitesimal analysis ( “in act” infinity ) and tothe hypothetical-deductive structure. It is not a mystery that physics has been dominated by theNewtonian paradigm.A deeper historical knowledge could aid us to consider the multiplicity - developed during the time- of the physics-mathematics relationship and to recognize the plurality of “formal thinking” inphysics.Therefore it is useful to know that Galilei used a mathematics with only rational numbers and hefaced the arguments in a discursive, not axiomatic way: it is sufficient to think of the Dialogo. Hisfollower Cavalieri had already introduced into physics irrational numbers, with the Theory ofindivisibles, that is also the first example of infinitesimal analysis [2]. This theory was not shared byGalilei probably because, placing among the rational numbers also the irrational ones, it impliedmental operations that clashed against his constructive and experimental vision of the concepts.After a few decades, Newton founded infinitesimal analysis and he put it at the basis of hisPrincipia with an organization of the physical concepts structured in a hypothetical-deductive way,precisely on “principles”. The formal thinking of this new physics was decidedly different from

228 3. Topical Aspects 3.4 Hands-on/Toys

Galilei’s one. So Galilei was not Newton’s forerunner, as the positivistic and linear vision of historywants, the same vision which inspires many text-books [3]. Afterwards, in spite of the great successof the choices made by Newton, not everyone accepted their validity. During the period of theFrench Revolution, for example, Lazare Carnot propounded a mechanics with completelyantagonist choices. Sadi Carnot, with identical choices, founded the classical thermodynamicswhich, as we have referred to before, doesn’t make an essential use of the mathematical analysisand it doesn’t let itself be dominated by the differential equations. I could mention other, also morerecent, examples, but here I would like to refer to Faraday, whose work in electromagnetism wasrealized with such an elementary mathematics that it almost appears without a formal thinking [4].At the end of this historical reflection, someone could object that - for example - in the schoolcourses we find no trace of the formal thinking of the mechanics of Lazare Carnot, although it isalternative to the Newtonian one. And he could conclude that in physics there has been a sort ofDarwinian selection of the formal thinking that has given us the strongest.This is what Kuhn, a veryfamous science historiographer and philosopher, asserts. According to him, the formal structure ofthe mechanics is the historical product of the cognitive evolution [5], passed through successivelinguistic stratifications, also contrasting, and that we have to take it as it is, because it is the bestpossible result given by the history of science.

2. Does the formal thinking of the mechanics represent all the physics?But, let’s ask ourselves, does the formal thinking of mechanics represent really all the physics? Yes,apparently. In fact it externally appears with a compactness and with a simplicity of enunciationwhich doesn’t leave doubts. The only criticisms are the relativistic ones, which, however, don’tdiscuss its validity inside the limit of the low speed. And yet, if we stop to consider the firmness ofits internal logic, we discover that its concepts and its laws leave much to be desired [6].The inertialprinciple is not demonstrable. The concept of force is circular with the second principle, which isexactly the principle which should give the predicting basis to the whole theory. In other terms: thisimportant concept results to be a metaphysical one. The concept of mass, moreover, can be definedcorrectly by using only the experimental basis contained in the third principle. If it is not in theinternal logic, therefore, where is the force of this theory?The surprising fact is that all these problems, have been known for some time [7]. Mach and Hertz’scriticisms of the nineteenth century are well-known, to mention only the most famous [8]. Still moresurprising is that physicists implicitly entrust to this mechanics the task of representing the formalthinking of the whole discipline. In fact we know that the students, from the secondary school tothe University, pass through its conceptual structure when being initiated to physics. This stridentcontradiction is almost always put at the margin of pedagogic discussions. Specialists in educationprefer to observe the mote of the misconceptions in the students’ eyes, who - poor fellows! - bringwith them the misunderstanding of the common language, rather than observe the beam in thephysicists’ eyes! It is astonishing to hear what Kuhn says on this problem. He maintains that thelanguage which expresses the concepts of the mechanics and the laws of nature they would like toexplain are related by a “indivisible mixture” [9] and that is a natural fact. Instead, according to me,this fact is not natural. It is here that we find the origin of the drama of thousands of students whoare forced to pass through the narrow door of an unpleasant physical thinking. This drama, whichinduces a large part of the young people to give up the scientific studies, compels us to stop andreflect [10].

3. Formal and informalBut, in order to do this, we need to move out of the narrow circle of the physics specialists, who -as we have seen - haven’t produced, in the course of the time, appreciable improvements to thequality of their thinking. Modern sensitivity asks us to widen our horizon and to consider thisproblem not only as scientific but also as pedagogical and, so, as civil [11]. Let’s see, therefore, whatformal thinking means in this larger cultural ambit. First of all let’s start by asking ourselves what


the adjective “formal” [12] means. It is a synonym of tidy, logical, sequential, but also ofmathematical, expressed by symbols, and sometimes it is identified with “scholastic”. Its opposite,“informal”, instead denotes the episodic, the casual, the extemporary, the partial, the intuitive andthe out-of-school. The adjective “formal” also assumes derogatory meanings like the ones of rigid,abstract and abstruse, and so the “formal thinking” is also the one difficult to understand, that is theeducationally problematic one. Consequently, “informal” stands for easy, simple to grasp,educationally elementary, and sometimes even banal.In the press and in the debates of cultural politics the binomial formal-informal is just what suppliesthe most frequent categories for the discussion about the learning of the sciences in general and ofthe physics in particular [13]. People even assert that the children who normally frequentplanetariums and interactive museums are also the ones who better understand the importance ofmathematics for the study of physics and of the other sciences in general, and so of the formalaspects of the scholastic subjects. People assert, moreover, that a common person gets half of hisscientific culture through informal procedures. Even the new expression “informal science” hasbeen coined to denote a science which is offered by out-of-school institutions, for example by thescience centers and the scientific museums, structures which compete in a strong way with theschool itself on the level of the educational efficiency [14]. This fact lead us to think that, actually,there exists an informal thinking which we must recognize. This kind of thinking could be imaginedas a thinking which, being contrasting to the dominant one, has passed through history, out of theparadigms of the scientific community and which is still living in the wider human community. Athinking which evolved at the margin of the niche and of the slang of the specialists.

4. Does an informal thinking really exist?This idea is much more than an hypothesis of work. Its consistence emerges very distinctly if weconsider a pedagogic and scientific problem the United States are now facing.The National ScienceFoundation (NSF) has learnt that the American students’ attainments in mathematics at the end ofthe high school studies result unsatisfactory. The data come from an international research on themathematical skills of young people, divided according to their age, published by the NationalCenter for Education Statistics [15]. Well then, it results that American young people, though theystart with good results in mathematics at the primary school, in the course of the studies - startingfrom the teens - progressively they remain behind in the classification of excellence, in comparisonwith the ones of their same age in other countries. The NSF has charged with this problem all theeducational agencies of the United States. Even Dimensions [16], the bimonthly of the ASTC(Association of Science-Technology Centers), which is primarily interested in informal teaching ofsciences, has opened the debate on the argument, dedicating one issue to the mathematics whichmust be discovered in the science centers. The same attitude has been expressed by TheExploratorium Magazine [17], the quarterly of the famous science museum of San Francisco,entitling its last issue “Math Explorer” and underlining that this publication has been supportedby a fund from the NSF.Mathematics is that part of the science which is, by nature, typically formal. Then it is reallyremarkable that the task of contributing to the solution of the educational problem of themathematics has been assigned to the community of those professionals of the so-called “informalscience”. Therefore people are looking for a mathematical thinking out of the tightly scholasticconceptions. Is this an implicit recognition that the interactivity and the “hands-on” are believablevehicles of education for mathematics? Traditional teachers say that the interactive scientificexhibits are only external hints for the essentially intellectual scientific education, does thisprejudice fall? The answer is still open.In the meantime, however, the prompt reactions of Dimensions and of the ExploratoriumMagazine to the problem show how the “experimental” resources of the science centers are alreadycapable of presenting interesting solutions. The exhibits, having been built to stress naturalproperties, educate to analyze singular problems of a physical nature in which converge, all


together, the logical aspects, the geometrical ones and the ones of calculation, aspects which theprogressive distribution in the school time doesn’t render equally effective.The physical variables of the phenomena produced by the exhibits and the hands-on experiencesare so numerous that they can constitute a real mine of mathematical ideas. What the twomagazines propound is to proceed to the re-examination of these phenomena to make evident“where” the mathematics in the physics is. In this way we can trace the mathematical thinkinginside physics, which is nothing but the formal - really effective - thinking of physics. This way iscompletely different from that indicated by the text-books, which are the teaching instruments stillnow preferred by the scientific community, and which bear the physics formal thinking almostexclusively by the algebra of the formulas.

5. The brain leads the hand, but the use of the hand shapes the brainThis “looking for” the mathematics in the interactive equipments also opens another path. The onewhich induces us to ask ourselves if this mathematics, different from the one learnt through pen andpaper and from the abstract one, is also different in substance (it derives from differentfoundations). If it were so, a mathematical “quality” would be on the point of being introduced; itcould be preferred because of the impact it has on those who learn.The general problem is to establish if every physical process corresponds to a mathematicalconstructive algorithm and vice versa.If this correspondence were demonstrated, the non-constructive algorithms - the ones of themathematical analysis which are based on the infinity in act - would no longer be necessary. Wewould have a turning point after three hundred years of classical mechanics.A starting point for this work is certainly the scientific datum that between the brain and the handthere is a bilateral dependence: the brain leads the hand, but the use of the hand shapes the brain.This means that the formal thinking of those who enter physics only through the descriptions andthe formulas of the text-books is a type of thinking we must consider unbalanced and thereforeincomplete. To complete it he would have to review the procedure using the “manual” dexterity(the effective realizability of the algorithm). Therefore the interactivity contained in the exhibitshelps to build the physical thinking, also in a formal sense.It is what my experience as a Fellow at Lemelson Center for the Study of Invention and Innovationof Smithsonian Institution of Washington has recently taught me.But why through the exhibits and not directly through the physics laboratories? The professionalequipment of laboratories is not educationally equivalent to the exhibits. The exhibits favour thenecessary exploration of nature without creating, in those who makes the first steps in science,strong anxieties about the functioning of events which are very far from common life.

6. How science centers can serve the schoolsThe variety of the exhibits offered by the science centers and by the museums has a didacticfecundity more dense than an “ad hoc” experiment, supposing that this has really been proposedin the school lessons. Only one experiment, like only one exhibit, although it concentrates theattention of the spectator on a particular phenomenon, never exhausts the mathematicalcomprehension of the physical laws there are in it. Frank Oppenheimer, the founder of TheExploratorium, wanted collections of exhibits to represent a certain physical law, for example theharmonic motion or the waves. He was convinced that we build our scientific knowledge only whenwe catch what is shared by families of apparently different phenomena. Thirty years of successprove he was right.Therefore the “cornucopia” of different phenomena is another service to formalthinking that is offered us by science centers and by science museums.The school of the future can’tgive up this lesson [19].Moreover, a collection of exhibits can supply what hasn’t been understood sufficiently yet, that isthe comparison of the theories. This exigency starts becoming more and more urgent once thephase of the astonishment and of the surprise given by the exhibits has been passed. Let’s think


about the concept of ray of light which is at the basis of the geometrical theory of light, and let’sthink about the interpretation of the images obtained by mirrors or by lenses through geometricalconstructions. To understand this theory well what normally is done it is not sufficient: because itonly gives one answer to the various questions. In fact we normally emphasize that the concept ofray explains the phenomena presented by the exhibits well and that the Euclidean geometrysupplies mathematics to calculate distances and enlargements of the images. It would be moreinstructive, instead, to show how an alternative hypothesis to the one of the ray of light wouldfunction. On this subject, The Exploratorium, for years proposed the Image Walk [20], that is a walkamong the exhibits having as a guiding principle another basic concept, the “spot of light”. Thiswalk forces the physicists to admit that a new fundamental idea, the one of an elementary cone oflight (just to understand each other, the cone of the pinhole) can improve the quality of theexplanation of the phenomena very well. With this new concept we can explain all the phenomenaof the traditional optics, starting from the everyday experience of the circular form of the shadowsproduced by the leaves of a tree, difficult to be explained with the concept of ray of light. As thiswalk has been made famous by an artist, an outsider of physics, its language appears to the visitorslike a curiosity. But not for The Exploratorium physicists [21], who have already confessed theembarrassing situation of having to prefer the ideas of the “walk” to the ones of the text-books onwhich they studied. If we analyse the differences between the two basic concepts, we note twodifferent mathematical conceptions of physics: one in which it is imagined that a beam of light canbe assimilated to a straight line (a cone thinned several times through splits); the other, to a conicalspot. The first is a very strong abstraction, the second is a less strong abstraction, moreunderstandable because it represents the experience of the pinhole well. In conclusion, thisexample suggests how to utilize the exhibits to go into the depth of the theories. This would suggestmoreover what they are made of and say what their models consist of. And also this can be re-proposed at school, obviously at a new school.ScienzaViva, the no-profit Association I am here representing, is committed to demonstrating thatthese efforts are really possible, realizing the “Interactive Science” Project, a program recentlyfunded by the Italian Ministry of Scientific and Technological Research.

References[1] A. Drago, “Storia delle teorie fisiche secondo le loro due scelte fondamentali: la matematica e l’organizzazione

della teoria” in S. D’Agostino, S. Petruccioli (eds.): Atti del V Congr. Naz. St. Fisica, Accademia dei XL, Roma(1984), 365-373.

[2] A. Drago,“La nascita del principio d’inerzia in Cavalieri e Torricelli secondo la matematica elementare di Weyl”in P. Tucci (Ed), Atti del XVII Congresso Nazionale di Storia della Fisica e dell’Astronomia, Como, (1997).

[3] P. Cerreta, “Il confronto tra le storiografie di Kuhn e di Koyré”, in AA.VV., Alexandre Koyré. L’avventuraintellettuale, a cura di Carlo Vinti, Edizioni Scientifiche Italiane, Napoli, (1994), 653.

[4] T.K. Simpson: Faraday’s Mathematics.On Getting Along without Euclid, Faraday Conference: Lecture, St John’sCollege, Annapolis, March 30, (2001).

[5] T.S. Kuhn: Dogma contro Critica, Raffaello Cortina Editore, (2000), 116.[6] P. Cerreta, A. Drago: “L’ideologia nella didattica della fisica: i principi della dinamica, suggerimenti per il loro

insegnamento”, in A. Drago (ed.), Fisica, Didattica, Società, CLU, Napoli, (1975), 82-98; S. Sgrignoli,: “Insegnarela fisica senza partire dalla dinamica” in Epsilon, Paravia, Anno I, n°3, (1988), 34-38.

[7] L. Eisenbud: ‘On the Classical laws of Motion’, Am. J. Phys. 26, 144, (1958), R. Weinstock: ‘Law’s of ClassicalMotion: What’s F? What’s m? What’s a?’, Am. J. Phys. 698, (1961).

[8] E. Mach: La Meccanica nel suo sviluppo storico-critico, Boringhieri, 1977; H. Hertz: Principles of Mechanichs,The Macmillan Company, New York, (1899) .

[9] T.S. Kuhn, Dogma contro Critica, op.cit, 116.[10] S. Tobias: “Math anxiety and physcis: Some toughts on learning ‘difficult’ subjects”, Physics Today, June (1985).[11] G. Delacote, “Putting Science in the Hands of the Public” in Science, Vol 280, 26 June (1998), 2055-56.[12] P. Cerreta: Apprendimento formale e apprendimento informale delle scienze, Quipo Web X@X, Edizioni

Project, (1999), 27-33.[13] Inverness Research Associates: An Invisible Infrastructure. Institutions of Informal Science Education,

Association of Science Technology Centers, Washington, (1996).[14] R. J. Semper: “Science Museums as environments for learning”, Physics Today, November (1990), 50-56.[15] NCES 1999 - The Third International Mathematics and Science Study (TIMSS), Overview and Key Findings


Across Grade Levels, NCES 2001- Highlights from the Third International Mathematics and Science Study-Reapeat (TMSS-R), Office of Educational Research and Improvement, US Dep. Of Education.

[16] Dimensions, Bimonthly News Journal of the Association of Science-Technology Centers, March/April (2001).[17] The Exploratorium Magazine, The Exploratorium, San Francisco, Vol 25 n°1, (Spring 2001).[18] F.R. Wilson, The Hand. How its use shapes the brain, language, and human culture, Pantheon Books, Random

House Inc., New York, (1998).[19] P. Dohetry, D. Rathjen, Exploratorium Teacher Institute, Gli Esperimenti dell’Exploratorium, a cura di P.

Cerreta, Zanichelli, (1997).[20] Bob Miller’s Image Walk, Exploratorium Quarterly, The Exploratorium, San Francisco, Winter, Vol 11, Issue 4,

(1987).[21] Ibidem, p. 27.

TOYS IN MOTION

Ana Gostinčar Blagotinšek, Faculty of Education, Ljubljana, Slovenia

Playing with various toys, pushing and dragging them around, children can experience a fact, thatan external force is necessary to start motion and to change the direction or velocity of motion. Inevery-day situations, when friction and air resistance are not negligible, force is also necessary tokeep an object moving. Doing it themselves, children can actually feel the force. Both bodies, theone that exerts the force and the other, being acted upon, are visible and obvious. So are theconsequences of the exerted force.This is an excellent starting position to investigate moving of toys along inclined paths and todevelop the concept of gravity. Doing it this way, it is simply searching for the force that causedmotion.Observing the motion of toys and motion of their parts is also very interesting and informative.Simple linear motion and very complicated composed motions can be recognized while observinga simple toy-duck and her eyes, body and legs in motion.Toys as valuable tool in teaching physics were recognized also by late professor dr. Janez Ferbar.He used the possibility to help children developing some very difficult concepts through every-dayexperience with toys. Why not use toys to learn while playing?

1. Toys, which can be dragged or pushed aroundA collection of various toys, which can be dragged around, as in Fig. 1, is examined first.They wouldnot move on horizontal surface unless somebody drags them around.Children can recall their previous experiences or try the activity.They can feel the force and see theconsequent changes in speed or direction of the toy’s movement. The body and activity, whichcaused the motion, are easily recognized.


Fig. 1: Toys, which can be dragged around. Fig. 2: Toys, which can be pushed around.

Because air resistance and friction are not negligible, toys slow down and stop if the dragging forcevanishes. Dragging force is necessary to start and sustain motion, and also to change speed ordirection of movement in everyday conditions, can be deduced from this and followingexperiments.Toys, which can be pushed around, provide similar experience. Two examples are presented on Fig.2. Pushing force and the smoothness of the surface influence performance of both toys.The duck on the left of the Fig. 2 is interesting also because of its legs. They help us show, how smallchanges in initial conditions can strongly influence outcome of the experiment. Dependant of theinitial position of the feet, it “walks” with both legs simultaneously or moves left and right leg in

turns, as shown on Fig. 3.Children like to play with such toys, becausethey can experiment and change them, andthis makes the toy interesting for longertime.With such, “animal toys”, we can also makeinterdisciplinary connection with biology.Discussion about which is the way that realanimals are walking encourages describingtheir observations and stimulatesaccurateness. With this duck it is possible todistinguish between the motion, which issimilar to the motion of live animal (left andright leg in turns) and the motion that onlytoy-duck performs (both legs movedsimultaneously).

2. (Free) Falling toys and toys, moving down inclined pathsAll sorts of balls, and also an interesting woodpecker, which is shown on Fig.4, are handy tools to introduce gravity. Observing toys, falling towards theground, and recollecting previous experiences with pushing and dragging thetoys, children are ready to accept gravity as a force, pulling all objects on Earthvertically downwards. Using previous experiences they know that an externalforce is needed to initiate the motion of the toy. As there is no obvious body,which could be identified as a source of the necessary force, planet Earth canbe identified as a body, exerting the force on falling objects. It is wise also totake notice of the fact, that no contact between the two interacting bodiesis necessary.

Which force pulls theduck, shown on Fig. 5,down the slope? This question can beused to evaluatepupils understanding. In my experience, evensome students at faculty level need somereminding of their experience with other, freefalling toys, before they give the correctanswer.

3. Different types of motion Observing toys in motion, one can notice, thatdifferent parts of the same toy move very


Fig. 4: Descendingwoodpecker.

Fig. 5: Duck, walking down the slope.

Fig. 3: Toy-duck can move its legs simultaneously or inturns.

differently. Goose’s eye of the on Fig. 6a moves linearly, while dots on the wheels move cycloidal.Attaching some “glow in the dark” stickers and observing the goose in the dark is fun and surprisesmost children of very different ages. Movements can be analysed and explained while observing thetoy move. Very few students would draw the path of the marks on the wheels correctly even afterobservation.Duck on Fig. 6b is even moreinteresting to observe and playwith. While moving, its headmoves left - right, wings go upand down and it even makessound like cackling. All this iscaused by motion of the wheels.It is interesting to investigatethe connections betweendifferent parts and the transferof motion. After carefulobservation and appropriateexperiments children can findout that movement of the headand wings is caused by rotationof the front pair of wheels,while cackling is caused bymotion of rear pair of wheels.

4. Connection with every-day lifeIf the toy-duck on Fig. 6b is not working properly, some improvements can be suggested and carriedout. Connections with real life problems, such as increasing low friction between wheels and theground, are easily found.First idea children come up with is to add a load on the duck to make it heavier. They can try it andfind out that it helps to improve performance of the toy. But does it help on a slippery road? Itwould in some cases, but it is not very practical.Another suggestion is also easily found and carried out: Change the properties of the surface theduck moves on.The children suggest sandpaper, rubber and different textiles and all are better thanthe smooth surface of the desk. Parallel with winter road conditions is to sand the road. But eventhis is not always done in time.It is usually only at the end that children remember that something can also be done to the wheelsto improve the grip on the road. After elastic bands are wrapped around the wheels of the duck, itmoves the head and the wings and cackles happily even on a smooth surface of the desk.

5. Problems ahead The danger of such approach to concept of force is to stress only the dragging or pushing force,acting on the toy. Feeling it and causing it with their bodies, children are aware of it. But resistantforces remain unnoticed unless attention is not focused on them. Children could easily developconviction of uniform motion under constant external force. One way to avoid this is to focus alsoon the interaction between the toy and the surface it moves upon, as it was described in previousparagraph.

6. ConclusionsForce as abstract concept becomes concrete experience for the children, who have the opportunityto play with suitable toys. This helps them to develop the concept of force on concrete and laterabstract level of thinking. Introducing gravity and later electric and magnetic forces can benefitfrom this approach.Other interesting features of motion can be observed while playing with toys. But it is important to


a) b) Fig. 6: Goose (a) and duck (b).

choose appropriate toys and activities to support learning, get rid of some misconceptions and notgenerate new ones.Another important aspect of toys in the classroom is that the use of them increases motivation forlearning and helps to connect learning with every-day life outside the classroom. Some ideaschildren get during lessons they can test in their free-time with their own toys and so develop anduse scientific approach to problems outside the classroom, too.And, at last, but not least, playing is fun. So, playing time is never wasted time.

Comment:Activities, described in “Toys in motion” were developed in cooperation with late professor JanezFerbar, who realized how valuable toys are in process of education and showed us many usefulexamples.

PHYSICS IS MIGHTY AS IT IS EASY

Katalin Papp, Anett Nagy, Miklós Molnár, János Bohus, University of Szeged, Department ofExperimental Physics, Hungary

1. Do Hungarian students really like the physics least?The aim of our survey was to investigate the system of high-school students’ science relatedmotivations and attitudes, especially those related to physics (Papp 2000). We examined thechanges of these systems between the beginning and the end of secondary education. The results ofthe study have also confirmed the unfavourable position of physics.During the secondary school the attitude and its change towards physics characteristically differentfor male and female students.To show the tendency we gave the quotient of attitude for two grades(last column in Table 1).The greatest number of statistically significant gender differences were found in physics, with boysliking physical science more than girls did. Students were asked about the frequency with whichtheir teachers demonstrate an experiment or with which they themselves do an experiment or


TABLE 1. Changes of subjects preference index (PI) of boys and girls

BOYS GIRLS

SUBJECT 9. grade

12.grade

rate(12./9.gr.)

SUBJECT 9. grade

12.grade

rate(12./9.gr.)

computer classes 4,04 3,80 0,94 biology 4,17 3,73 0,89 history 3,94 3,75 0,95 foreign language 4,11 4,07 0,99 biology 3,82 3,35 0,88 Hung. Literature 3,92 3,84 0,98 foreign language 3,82 3,72 0,97 history 3,91 3,73 0,95geography 3,82 3,51 0,92 computer classes 3,51 3,09 0,88mathematics 3,66 3,70 1,01 geography 3,49 3,41 0,98 Hung. Literature 3,59 3,18 0,89 mathematics 3,44 3,28 0,95 physics 3,50 3,29 0,94 Hung. grammar 3,39 3,20 0,94chemistry 3,31 2,95 0,89 chemistry 3,25 2,79 0,86Hung. Grammar 3,06 2,66 0,87 physics 2,96 2,68 0,91

Remark: the arrows show the significant change: p<0,01; p<0,05

practical investigation in class. We have found, thereis a difference between the attitude toward physics.There is a better attitude where students report highfrequencies of teacher demonstrations and studentsexperiments too (Fig. 1).

2. Rain-sensor I.In TV advertisements we can see windscreen-wipersin modern cars which automatically work if somewater beats against the screen. We can construct asimple device modelling the rain-sensors built inthese modern cars. Using the fact that waterconducts electric current in a small degree, evensome drops of water falling to a metal conductor canbe detected with a sensitive device. The detector asindicated in Fig.2 is a connection with a transistorbuilt on a printed circuit.


112581699064 8227819513355N =

very oftenrarenever

27

26

25

24

23

22

21

grade

9.

12.

Physics

attitude

Figure 1. Physics attitude as a function offrequencies of teacher’s experiments

Figure 2. Structure of rain-sensor I.

3. Rain-sensor II.Another rain-sensor can be constructed using the law of the refraction of light. A laser-pointer, asemi-cylinder shaped prism, a photodiode and an optical rail which holds the previous objects areneeded (Jodl,Eckert 1998). After fixing the semi-cylinder shaped prism in a way that the laser-beam is totally reflected on the surface of the prism, this high-intensity laser-beam enters thephotodiode. Its effect can be detected with a voltage meter. The ray of light can be easily seen ifafter the laser pointer it is horizontally spread out (for example with a glass stick placed verticallyto the ray of light).If the upper surface of the semi-cylinder becomes wet the ray of light change its direction oftransmission as it does not suffer total reflection any more. As a consequence, the photodiode isnot lighted at all. The measured voltage remarkably decreases indicating that some water is on thesemi-cylinder.

4. Electric candleTo realise electric candle the following electronic network is needed. To the collector-emittercircuit of a transistor place an electric battery (4,5 V), a flashlight bulb, a switch, and a photodiode(F) between the base and A point as it is shown in Figure 3.Closing the circuit with the switch the bulb is not illuminating. Light the photodiode (for examplewith the light of a match) and the bulb is illuminating. If the intensity of the lighting is low or if thediode is not lighted at all, the bulb finishes lighting. If we make sure that after having lighted the

+ - Amplifier Ammeter

- +

Electrodes

candle with a match, the bulb itself can light the photodiode (F), the circuit becomes self-supporting: the bulb lights until its light reaches the photodiode (F).

5. Graphite messagesUsing a simple DC amplifier and a message etched by a graphite pencil a set-up can be constructedto demonstrate how resistance depends on both the length and cross-section of the conductor. If theinscription is “physics”, say, we can even demonstrate how physics conducts.The inscription, which will be part of a circuit, is traced out on an A4 sheet of paper with a graphitepencil without any discontinuity. It requires some time to do as we need quite a thick layer of graphiteto significantly reduce the resistance. In addition to the inscription we can draw two lines, a narrow anda wider one to show that the resistance depends not just on the length but on the diameter of theconductors as well. The resistance between the two ends of the graphite circuit is about 2 MΩ.First connect the conductor with a 5 V, then measure the electric current flowing through theamplifier with a bulb. The brightness level of the light bulb provides a measure of the change inresistance of the graphite layer.Doing experiments in physics lessons need not cost much. Simple, everyday items and materials canbe used for demonstrating physical phenomena. Hungarian physics teachers like Ányos Jedlik,József Öveges, Miklós Vermes, Árpád Csekõ, Károly Jeges in the past showed this, along with otherHungarian scientists who followed in their footsteps. They were clever at devising experimentspractically from nothing. Do we still need these types of experiments in this day and age? Are theseexperiments able to divert the attention of pupils who are daily bombarded with information fromthe media? Experience gained over the years and the results of several detailed surveys seem tobear out our view that a physics education containing experiments made from simple, cheap,everyday objects and a little zeal can indeed motivate young minds. The next two experiments areoffered as examples.

6. Heki and his houseThis is a house made of paper with a simple idea behind it (Papp, Nagy, Bohus 2000). Whenconnected up, a circuit with a battery and an electromagnet attracts a laminated spring. Close to thelatter sits a dog called Heki. The circuit is closed by a little triangle-shaped contact attached to oneof the paper walls of the house. This metal strip connects two small metal-plates on side, theseplates being joined together by a capacitor to avoid sparking (Figure 4).When we shout something near the house the metal strip swings out for a moment as the soundwaves cause the circuit to break.Then the electromagnet ceases to be a magnet and does not attractthe laminated spring. As a consequence the spring moves away from the electromagnet and pushesthe dog out from the house. To make it work as before just push the spring onto the electromagnet


F

C B

E

- + K A

Figure 3. The circuit of electriccandle

(the metal strip on the wall closes the circuit) and shout the name of the dog or “Cats”, say. This willmake Heki, who is ever alert, come out from his house.In another version of the house a small contact attached to the “ceiling” (rather than a metal strip onthe wall) closes the circuit. Sound waves make the contact move slightly, causing the circuit to break,and once again the spring makes the faithful Heki peek out from his house (Siddons 1988, p. 22).

6. Unburnable paper-moneyIf we set fire to dry paper money we can be sure it will burn. At least everybody expects it tohappen, like those with money to do so. But as we shall see this is not always the case. Specialconditions can change things somewhat.For this surprising experiment you need some alcohol (approximately 40% - rum, brandy,schnapps). Fill a glass with some warmed brandy, strike a match and sprinkle a little salt over it soas to give the flames a colour (orange) because it contains sodium. Afterwards soak a banknote inthe alcohol with the help of a pair of tweezers or tongs, then leave it in the glass for a couple ofseconds till the whole paper money is wet from the alcohol. After dousing the flames and takingthe money out of the glass smooth it out and hold it above a clean tray. Then try setting it alightwith a match or lighter. For a few seconds the money will be on fire. Then it will go out and you getback your wet but otherwise unharmed banknote, so you will not be poorer for the experience.The reason for this unexpected result is simple enough if you remember that drinkable alcoholcontains water as well, which wets the paper. The ethanol burns merrily away but the wet banknotedoes not. Once the ethanol is used up, the fire goes out. (Note: the whole note should be soaked inthe brandy glass of alcohol, otherwise I cannot be responsible for the outcome!)To make the experiment a little livelier you can demonstrate your true magical power to protectpaper money from burning. Just add some magic powder („salt” to you and me) for effectiveresults. The odd magic word like „Shazam” or „Abracadbra” can work wonders not only with thepaper money but young pupils as well. Budding magicians out there might think of adding this oneto their repertoire of magic tricks.Modern technology everyday subjects and toys are all suitable for making students curious andinterested in physics. Perhaps after these experiments even students will accept that physics ismighty as it is easy.

References1. H. Jodl, B. Eckert Low-cost, high-tech experiments for educational physics Physics Education 33, (1998), 226-235.2. C. Siddons, Experiments in Physics (Basil Blackwell Ltd.), (1988).3. K. Papp, A., Nagy Bohus J, Két elfelejtett kísérlet (OKSZI Módszertani lapok, Fizika) .4, (2000).4. K. Papp 2000 Do Hungarian students really like the Physics least? Proceedings of International Conference

Physics Teacher Education beyond 2000, Barcelona Aug. 27-Sept.1, (2000).


Figure 4. Inside and outside of the house

a = 20 cm b = 13,5 cm c = 7 cm m = 15 cm h= 9,5 cm

TEACHING MECHANICS AND BIO-MECHANICS

Paolo B. Pascolo* Applied Mechanics professor, University of Udine and director ofBioengineering Department of CISM, Italy

I apologize for my short and probably ingenuos intervention. Some ideas about use of toys areshowed in the previous poster. Every toy can contain topics.When Prof. M. Michelini told me about this important conference I start with a short story.Once upon a time in Rome, at of one of my friend’s house, a special friend’s son.To understand the kind of child I like to remember that this boy, at only nine years old, wrote toMargherita Hack and me, about materials to build a missile.-At eleven years old he asked me, what’s mechanics ?Looking around I saw a toy. I took it and I started to expain kinematics.A few days later, when I was in Udine, I received a present: a collection of toys (Fig. 1) and a shortletter.“Caro Paolo, per i tuoi studenti. (for your students), Ciao, Lorenzo.”


Fig. 1

Fig. 2

With monkeys it is possible to introduce several topics (see poster “Teaching Mechanics withtoys”).With chicken (Fig. 2) you can descrive a drop hammer, a mallet (i.e. the impact forces),

with chupa- chupa (Fig. 3) two kinds of rotisms, it is able to introduce the ordinary and the epicyclictrain and the consequent Willis formula.

with a boat-man (Fig.4) ………, a four bar-linkage.


Fig. 3.

Then the pendulum (Fig. 5), a shearing machine (Fig. 6), a two pistons engine (Fig.7), ……

Fig. 4.

Fig. 5.

Fig. 6.

Very interesting is this kind of horse (Fig. 8).


Fig. 7.

Fig. 8.

Look at the positions of the legs, rather perfect.To study its kinematic I did X-rays (Fig. 9), because it was glued all together.Its kinematics, in relative motion, can be easy described with a multi-body code (Fig. 10).

Fig. 9.. Fig. 10.

The motion, correct, is the result of a sistem very different to the biological one, so it’s possible, forexample, to design different kinds of solution for artificial fingers.The first example is the result of actuators (Fig. 11).


Fig. 11.

Fig. 12.

The second represents the biological tendinons scheme (Fig. 12).

It is possible to go on for much longer, but I’m convinced to have transmitted my opinion in thisfield. We should learn to introduce a teaching method that sets as one of its aims the purpose toeducate students able to recognize the different topics of mechanics in every day life objects, evenin toys.

SIMPLE EXPERIMENTS HELP IN GAINING A BETTER UNDERSTANDING OFPHYSICS CONCEPTS

Gorazd Planinšič, Physics Department, Faculty for Mathematics and Physics, University ofLjubljana, Ljubljana, SloveniaMiha Kos, Ustanova Hiša eksperimentov, Slovenian Hands-on science centre, Ljubljana, Slovenia

Hands-on experimentsIt is generally known that the most effective experiments in teaching physics are often the simplestexperiments. In the past, teachers used only demonstration experiments that were allowed to beoperated only by them. Later, the laboratory practical work for students was introduced intoschools, but it never offered an opportunity for explorative work guided by individual curiosity.Doing simple experiments became popular after the Second World War, but at that time theseexperiments could not find the way from homes to the strict and threatening atmosphere of theschools. In the sixties, the term ‘hands-on experiments’ (HEX) became more frequently used inconnection with the arrival of hands-on science centres. Though the experiments in hands-onscience centres are typically far from being cheap, they are often based on simple and clear ideasthat came from simple and low-cost experiments. When we speak of low cost, pocket, string-and-sticky-tape or toy-based experiments it is usually their hands-on character that makes them souseful in schools.Today it is generally accepted that learning by the personal (hands-on) experiencehas several benefits, such as• development of observational, manipulative and other transferable skills• testing personal capabilities and limitations• creating motivation for learning science and appreciation of scientific achievements• broadening general knowledgeHowever, the bare accomplishment of HEX will rarely lead to the following results:• understanding the principles of nature• learning logical reasoning • identifying analogies, similarities, symmetries...The listed cognitions can only be achieved if the experiments are “wrapped” in something else -something that provokes, demands and guides rational actions in parallel with experimentalexperience. This “wrapping” can be promoted by specific design, but it is eventually created by theverbal or written communication. For this reason all science centres pay a lot of attention to writingthe instructions for every experiment. Composing such instructions is not an easy task; too long atext will keep visitors away from reading it, but too short and brief information might not make thepoint. The more effective alternative is verbal communication, such as discussion that can beinitiated by the explainers at the exhibits or during the interactive science shows.How about using simple experiments in school? Obviously in schools one cannot afford to set upthe kind of experiments that are seen in science centres, nor can we use the conventionaldemonstration experiments for hands-on activities. The school experiments can be used as hands-on if they are cheap (toys for example) and/or are easy to set up by the teachers or students. Thereare several resources where the ideas for such experiments can be found (see for example the listof selected resources in [1]).But in the same way as discussed above, the bare accomplishment of the experiments in schoolactivities will not explain the phenomena. Many experts in the past few years have expressed thisopinion. It is embedded in the approach of Lillian McDermott and the Physics Education Groupfrom Seattle, Washington. The PEG’s approach is based on a lively discussions in small groupscentered on simple experiments and initiated by carefully posed inquiry questions [2,3]. Thereflections of several English authors on how the experiments can (or cannot) help understandingscience has been edited by J. Wellington [4]. Perhaps the following quote from his chapter showsthe main point: “Discovery learning [by doing experiments] may work for teaching knowledgethat...but cannot be expected to teach knowledge why ...”. The desired knowledge can be



Discussion and reflections Pictorial presentation of experiments

"If I blow in this orange corrugated tube it sings - hear this--I will do it again, but this time very gently--the tube does not sing now! Can you think of a role what I have to do to make the tube sing?"

"You have to blow in the tube strong enough and it will sing" "Now I will do the same with this wider corrugated tube. It also sings when I blow in it. It looks like this is the feature is common to all corrugated tubes." "I am going to do another experiment now: I will rotate the tube like this-- and it starts to sing again!--What about the wider tube --this one emits even more clear and loud tone. So I'll continue to play with this one.""Let's see what we have observed until now: the corrugated tube sings when I blow through it and it also sings when I rotate it. Maybe there is a connection between the rotation and air blow through the tube. What do you think?"

successfully taught if the experimentation is enriched by stimulating conversation and discussionby posing inquiry questions, working with the ideas, concepts and principles, and facing theconfrontation of different opinions and arguments. The references above are quoted more as beingworks that inspired us and not as approaches that we would directly follow. However useful andvaluable these programs may be to other countries, the specifics of individual school systems,cultural, ethical and other characteristics of different countries require the design of new programsto be done from the beginning.Our own experiences and those of others show that in addition to the enrichment discussed above,the HEXs are a far more efficient teaching tool because (or if) they can be associated with freshideas, names, appearances, items, inventions or problematic matters that one encounters every dayfrom the news, TV, internet, commercials etc. In addition, it is a great advantage if one can presentthe experiment as a completed story. All these ingredients will make the experiment more relevantto the world everybody knows, and therefore more attractive and worth spending more time towork on. This approach has been used in the recent Advancing Physics AS course [5] and is alsothe main secret of the success of Bloomfield’s How things work [6]. However, the design of theHEXs in school has to be guided by the physical concept we want to show, not by the mere desirefor creating an attraction.Some points from the reflections above are illustrated in the following two examples.

The singing corrugated tubesMany of you have already discovered that corrugated tubes (like those used for electricalinstallation) emit a loud and clear tone when one blows air in or out of the tube. The question ofhow the sound is produced is intriguing but the correct explanation goes far beyond the secondaryschool level [7,8]. However, the corrugated tubes offer a great tool to demonstrate scientificreasoning to younger children (age 10 to 14). The teacher’s script goes like this (the typicalchildren’s answers are in italics):


(In general children will agree. Older children can be encouraged to predict the connection by themselves.) "Suppose that we are right - what do you think will happen if I cork up the end of the tube and rotate it again?""I think the tube should not sing now." "Let's try.--You were right! So we have good reason to believe that the rotation causes the air blow through the tube. One thing we don't know is in which direction does the air blow: from this end where I am holding the tube up to the free end, or maybe in the opposite direction. What do you think?" (Some children will say that the air is blowing from the free end to the end where we hold the tube. They are misled by the experience that wind blows into the driving car when we open the window.) "Can you suggest the experiment which will show us what is the direction of air blowing through the rotated tube?" (Several ideas may be suggested. The teacher tries to guide ideas towards the simple experiment.)"If I fix tight this plastic bag at the end of the tube, what will happen with the bag when I rotate the tube?" "If the air blows towards the bag, the bag will fill up and if it blows in the opposite direction it will squash."

"Done. --The bag has squashed and the tube stopped singing. So we have proved now that the air was blowing from where my arm holds the tube to the free end. And this is the same direction in which the small piece of paper will travel if I put it into the tube and rotate the tube. See this--Now we know a lot more about this corrugated tubes." "But--now, one day we come across with this green corrugated tube...which does not sing whatever I do with it. Why the green tube does not sing? What is the important difference between the two tubes that makes one singing? Take the two tubes and try to find out by yourself."

Soap bubbles: the colour of one and many Isaac Newton (1642-1727) observed how a prism bends a beam of sunlight, and the emergent beamcomes out of the prism as a fan of spectral colours. He found that he could re-combine the coloursback into white light by passing a spectrum through a second similar prism and a lens. He also leta single coloured light from the first prism to pass through a hole in a screen to a second prism.Thisdid not produce more colours. With these experiments he proved that the prism does not add anycolour to the light but it splits the beam of white light into a spectrum. The whole story is seldomtold in schools, while the experiments on re-combination of colours are even more rare. The factthat white light is composed of colours is not proved by a single prism experiment, and themisunderstanding of this phenomenon before Newton is good evidence for this. The followingsimple experiment can help the teacher to show the splitting and re-combination of white light infront of a large audience.Beautiful spectral colours can be observed as white light reflects from the soap bubbles. Cut themask from the cardboard as shown in the figure below.Here one can see the mask for showing the reflected colours of thesoap bubbles on the overhead projector. The arrow shows theorientation (top view) of the mask during projection.Cover the semicircular opening with a sheet of white paper. Put themask on top of the overhead projector as indicated in the figure.Make sure that the mask is at the level of the pupils’ eyes. Put thelarge petry dish on top of the mask and pour some soap bubblemixture into it. Blow a soap bubble by using a wide drinking straw. In the darkened room the wholeclass should be able to observe the beautiful spectral colours reflecting from the back side of thebubble. The turbulences caused by the heat from the overhead projector will make the event evenmore magical.At this point, the teacher tells the story of how the colours are formed and asks what experiment


(The inner surface of the green tube is smooth while the rest of the tubes have wrinkles inside as well as outside. Typically children will say that the difference in tube diameter or length might be the reason. Teacher can bring more tubes to show that this is not the right answer. Most of the children will not try to touch also the inner surface of the tube. But even the youngest will never think that the colour of the tube could have anything to do with the generation of the sound.) "So, it seems that if there are no wrinkles on the inner surface the tube will not sing! Now, the last question - what kind of a tube do we need to prove that the colour of the tube in this case was not the reason why the tube did not sing?" "We need the green coloured corrugated tube with the wrinkles inside" "Here is one--and it sings. Congratulation!"

h i h fl d l

will convince everybody that white is made of spectral colours that are seen in this very experiment.Finally we all agree that if we were able to mix the spectral colours that we see, and we could obtainwhite, we would have good proof. But how can we mix the colours from the bubble? We can try bymaking a lot of bubbles that reflect the spectral colours in all directions, thus mixing the light. Andhere comes the experiment.Take another straw with a thin (1mm) layer of sponge wrapped aroundthe end of the straw. Dip this end into the bubble mixture and blow into the straw. The foam full ofsmall bubbles will start to grow from the end of the straw, and soon the foam will take the shapeand the size of a big bubble from the previous experiment. The foamy dome will appear perfectlywhite from wherever in the class you observe it.

References[1] G. Planinšič, Pocket experiments - a step towards more relevant and more attractive physics in school, GIREP

2000, Barcelona, conference proceedings on CD, August (2000).[2] L. C. McDermott, Physics by Inquiry vol. 1 and 2, John Willey & Sons, NY, (1996).[3] L.C. McDermott et al, Tutorials in Introductory Physics, Prentice Hall, NJ, (1998).[4] J. Wellington (editor), Practical work in school science - Which way now?, Routledge, London, (1998).[5] J. Ogborn and M. Whitehouse (editors), Advancing Physics AS, IOP, Bristol, (2000).[6] L. A. Bloomfield, How things work, John Willey & Sons, NY, (2001).[7] Y. Nakamura, N. Fukumachi, Sound generation in corrugated tubes, Fluid Dynam. Res. 7, (1991), 255-261.[8] L. H. Cadwell, Singing corrugated pipes revisited, Am. J. Phys. 62, (1994), 224-227.e-mail [email protected]

LEARNING BY PLAYING OR PLAYING BY LEARNING

Nada Razpet, The National Education Institute, Ljubljana, Slovenia

1. Classifying and ordering chipsA set of chips is distributed on a table by chance. Two values colour and shape can be assigned toeach chip.Children can classify chips by shape and distribute them into groups. The group with a specificvalue of shape is placed into a separate spatial region. This is a “many to one” mapping.


squares circles triangles

Because the classification is done by a single variable only, the distribution of groups can berepresented on a line from the left to the right.Children can divide groups of the same shape into subsets of chips with the same colour anddistribute them again.The table is filled with groups of the chips with the same colour and the sameshape. In this case, when the classification by two variables was performed, the resultingdistribution can be presented on a two-dimensional plane.

Rows define x-axis and columns define y-axis. Now, we can assign letters to each value and writethem into a table:S for square, C for circle, T for triangleG for green, B for blue, Y for yellow and R for red.The first letter (representing shape) is the same within each column and the second letter(representing colour) is the same within each row of the table. This method can be used tointroduce basic concepts of a table and a graph (every point in the plane is described by twoparameters). It is recommended to write headings in the table and the graph presentation.The chips of the same colour and shape can be put into towers where their heights are related tothe number of chips. The children can compare towers and tell weather there are less circle yellowchips than red square chips. The histograms (bar graph) could be introduced.A set of chips can be divided into groups or subsets by the rejection method. If a classificationby shape and colour is to be done the chips that match chosen criteria are put into one groupwhile the others are rejected. We distribute chips by answering a question “Is this one the sameas...?” (the equivalence relation). On the other hand when we would like to order a group ofobjects, it is not enough to check the equivalence, but every two objects should be compared bya certain value. For example, chips of different intensity of the same colour can be ordered bythe colour intensity. Chips of different colours could be ordered by colours where a certaincolour sequence is agreed (or for the oldest children - comparing the wavelengths of colours. Achip with greater value of wavelength can be placed on the right side of a chip with a lowerwavelength value.)

2. Recognition of missing chipsWhile children close their eyes, a group of chips is removed from the table. The next question couldbe asked: ” Which chips are missing? Why is it possible to recognise the missing group of chips?”The rest of chips are scrambled. “Is it possible to recognize missing chips now?”


YS YC YT

BS BC BT

RS RC RT

GS GC GT

3. Linear attribute gamesThe sequence with three different chips is put on the table.The attributes are shape and colour.Thechildren could find the differences between first and second chips and then with second and thirdchips. Then they can try to continue the chain with appropriate chip.One attribute difference domino gameThe chips have different shapes (but same colour).


Two attribute difference domino game (shape and colour)

Two-dimensional games (one attribute difference)

From the left to the right in the same row, the chips differ in shape (one attribute difference), if welook the rows they differ in colour (one attribute difference). The chips cover a plane (two -dimensional games). If the first three chips in each row form a base of a series, then children knowwhich chip follows (the forth chip is the same as the first one in the row).If the difference between the chips in horizontal and vertical way is the same (for example shape),than the plane is like the picture below:

We could help children (with asking them questions) to continue the patterns on the right end oron the left end (it is harder, because we are used to write from the left to the right, but it isimportant in future education, for example in solving mathematical equations). The questionscould be: Which chip is before triangle? (circle) Which is before rectangle? (triangle) Which isbefore circle? (rectangle). Which is after circle? (triangle)

The set of the objects is simple in the above case, because there are only two values that can beassigned. The example from the real live could be more complicated as there could be more valuesand it is usually not obvious how to select the variable by which the classification is done.Additional variables might confuse children. In each subset (each group) the objects match in achosen variable although they are not equal.

Example 1I asked my students to:• find variables by which I have classified the

post stamps (see figure with stamps below) • find the variables to divide these stamps into 3

subsets.Each subset of post stamps could be divided intwo groups by their format (portrait or landscape)or by the motive on the stamp. If a stamp iscompared to other stamps of the same group theyshare the same motive, but they are not equal.

Example 2A few sheets of paper for making paper planeswere given to students. They were asked to makethree planes that differ in one attribute. Most ofthe students found this task to be very difficult.

Example 3Students from four schools take a part on the regional physics championship. The leader of thejourney prepares a list of participants. The participants on the championship are divided into fourgroups A, B, C and D. Prepare the tables, make the headings and fill the table. Choose the nameand the group in which participants compete.

4. MappingWhen two sets are given, to an element from the first set (the set of the original), an element fromthe second set (image) can be found by following the rule.Typically, mapping games are card gamesand domino. From the introducing mappings, the concept of a function can be developed.

Domino gamesEach domino has two fields with different number of points (from zero to six). A continuous linecan be build by adding a domino with matching numbers of points in a field. It is not necessary toknow how to count to play the game.Dr. Ferbar liked to play this game with special dominoes. It this case, the players have to be able tocount.


“Cvetni bimino“ (Flower “bimino”)Each domino has two fields like the classical domino. But thefields on dominoes differ in:• sort of flowers• number of flowers• blooming state (have flower bud or not)• background colour• background pattern (there are stars or not)The game can be played using the different rules:• the adjacent fields must be equal in one attribute • the adjacent field must be different in one attribute The rule could be set at the beginning of the game. One playeris chosen to be a leader. He or she • explains the rules with words • shows the rule without words by placing two (or more)

dominoes.The leader tells to other player if they draw the right dominoin every step of the game. If the player doesn’t draw the rightdomino he or she has to remove it from the line. The winner isthe player, who first finds out the rule or who first gets rid of all his or her dominoes.

Card gamesA matching pair to a card can be found in many different games.• a similar pattern (fish and roof, the tail of peacock and Christmas tree)• a sort of a tree, a fruit or a leaf, bark (in this case three properties should be matched)• a profession of a person in a card and what he or she is doing


“Cvetni bimino“ (Flower “bimino”

Cards with lines and curves

Triangle card games:The rules are for example:• the lines (curves) meet at the adjacent

sides of triangles (the colour is notimportant)

• the lines (curves) meet at the adjacentsides of triangles in same colours.

Rectangle cards games


a) b)

For example: a) no rule b) the number of lines at the adjacent sides of the square should differ byone.Below, there is a mapping between the number of points on a dice and a piece of the rabbit jigsawpuzzle. There are 4 rabbits in different colour. All pieces are put on the table. The player throwsthe dice and chooses the piece with suitable points. The winner is the player who puts togethermore of the puzzle.

?

5. Cards for learning physicsCards with photos of physical phenomena and their simple explanations can be put as motives of cards.

6. The importance of gamesThe games described above are an introduction for recognizing variables and their mutualrelations. In some games, it is expected to find a relation between objects, while in others playersshould set the relation or the rule.It is advantageous to choose those motives that children are most attached to.Themes can differ according to the age and interests of children and according to the knowledgethat we have achieved.

Playing cards without written rules is important to encourage children to find their own rules. It isnot a trivial task. A chosen rule might seem well defined at the beginning of the game, but mightlead to a confusing situation or even a fight when the game continues. Such conflicts show whyscience laws have to be verified by experiments.

ReferencesJanez Ferbar, From diversity and variability to variables, unpublished paper, (1998).Janez Ferbar, private communication.Janez Ferbar, Domen Ferbar, Ana Gostinčar Blagotinšek: Razvrščanje (Classifying), Pedagoška fakulteta, Ljubljana,

(2000).Breda Jontes, Barva k barvi (Colour to colour), Državna založba Slovenije, Ljubljana.Grega Stušek, Tristrani domino (Three side dominoes), DID delavnica d.o.o.Janez Ferbar, Cvetni bimino (Flower bimino), Dr.Mapet, Ljubljana, (1996).Fojž A. Zorman, Črni Peter, Ljubljana, (1990).Boža Jelen, Črni dimnikar (The black chimney sweeper), Dopisna delavska univerza, Ljubljana.Nada Razpet, Z računalnikom brez računalnika (With a computer without a computer), Seminarsko gradivo, Zavod

republike Slovenije za šolstvo, Ljubljana, (2001).

DEVELOPING FORMAL THINKING THROUGH TOYS AND EVERYDAY OBJECTS FORTHE FORMATION OF FUTURE PRIMARY SCHOOL TEACHERS

Allasia Daniela, Montel Valentina, Rinaudo Giuseppina, Department of Experimental Physics ofthe University of Torino, Italy

1. IntroductionIn the usual presentation of physics in Secondary Schools, the formalization process consists oftenin starting from the formal mathematical law which describes the process and then understandinghow to apply it to the problem being examined and eventually to other problems. It is essentially a“contents based didactics”, which is very efficient from the point of view of the results to beobtained, but requires that the students master already the physical concepts involved in thecontent being examined and have a solid mathematical background.Both conditions are rarely fulfilled in students of the university courses for the formation ofprimary school teachers (Scienze della Formazione Primaria, SFP); besides, for these students, it isessential to approach the physical concepts and their formalization in a way ready to be transferredto the future pupils. To take into account both aspects, we based our approach on the use of toysand every-day-life objects, the idea being to associate in a tight way a concrete object with anabstract concept by giving to the concept a formal expression: the fact of seeing physically aconcrete object helps to give a form to the abstract concept, but, at the same time, the formalizationhelps to understand the implicit physics hidden in the object.What we call “formalization” is indeedvery rarely a mathematical relation, more often it consists just in the use of the proper words or ina graphical representation. The idea of using toys, games, every-day-life objects to approachscientific concepts is not new: there is already an extensive literature on this argument and thepositive [1] as well as the negative aspects [2] have been widely discussed. Our opinion is that, witha careful selection of the objects used, the benefits prevail over the drawbacks. Indeed not all theobjects as well as not all the concepts are suited for this approach. In the choice, one has to takeinto account at least a few important features, in particular:• the toy or the object must be of common use, hopefully the pupil should make or find it, so that

each student can have his own object in his own hands, in order to manipulate it, take it home,etc.: the appropriation is an important aspect of any durable learning;


• if the toy or the object is made by the pupil, it must be of cheap and safe material; its use mustbe easy and the results of the “experiment” must be certain and reproducible;

• the implicit physics concept must be simple, fundamental and not completely new to the student;it is important that the new knowledge be directly related to the previous knowledge, in orderto allow a spiral growth of the concepts.

In the following sections we will discuss in detail how we used some simple toys to introduce theideas of force and energy.

2. Simple toys to start formalizing the concept of forceThe formalization of the concept of force is certainly one of the most discussed items in the physicseducational literature[3]. The usual way to formalize it is to refer to Newton’s second law ofdynamics: this is just the type of abstract and formal approach which causes difficulty ofcomprehension and does not help to clarify the concept of force. Indeed, in most textbooks, thesecond law provides more a way to measure the force than to clarify what is a force, that is to clarifythat the force is the expression of the interaction which causes the variation of velocity.In our approach we start directly with a concrete and familiar object which helps to clarify first ofall the idea that the force implies an interaction. Many toys have these features, for example, thesling. In the following we describe a typical sequence of activities to formalize progressively themain aspects of the concept.


With this simple activity:• the pupil applies directly the force, thus he has a direct and physical intuition of its meaning,• it is also rather evident that there is an interaction between the finger and the rubber and

between the hand and the sticks,• the interaction is always between two objects, that is the finger needs the rubber to apply the

force and the hand needs the sticks,• each force can thus be given a name, which indicates the two objects that interact,• one can apply larger or smaller forces, and the effect will depend on the intensity of the force.It may appear to be a trivial formalization, but it is fundamental, because, by giving a name to theforce, the student is trained to consider the force as a property of the interaction and not as aproperty of his hand or of his finger.Once he has understood this fundamental aspect of the force in this particularly clear situation, it

The sling Ask the pupils to build their own slings to launch small balls: they can simply tie together two pencils at one end to form a V and fix at the opposite ends a rubber band. The balls can be made with paper. With his sling in his hand, each pupil should :

• describe the forces that he can recognize, • give to each force a «name», which recalls the two

interacting objects, that is the source of the force and the subject to which the force is applied (in the figure Ffr isthe force of the finger on the rubber, Fhs is the force of the hand on the sticks),

• apply larger or smaller forces and describe the effect on the rubber and on the distance reached by the paper ball in the launch,

• draw a picture of the sling and write the name of each force near the point where it is applied, as in the figure.

Ffr

Fhs

Figure 1

will be easier to him to discover the two “actors” of the force also in other cases, where they are notevident (for example, in the case of the force of reaction), and thus understand that the force doesnot depend only on the “force source” but also on the “force subject”.The next step is to understand that the force has a direction and that one needs to specify it todescribe the force fully.With a sling, the idea of direction is very evident, because the pupils observe very soon that theresult of the launch depends strongly on the direction.


With his sling in his hand, each child should :• try to launch the ball in given directions, until he

understands how to orient the rubber in the application of the force,

• draw a picture of the sling and represent the direction of the force with an arrow with its tail near the point where the force is applied, as in the figure.

Ffr

Fhs

Figure 2

This kind of formalization is simple, but is the basis of what we call the “mathematics of thearrows”. The rules of this mathematics are:• arrows must be drawn in the correct direction,• the tail of the arrow must be as close as possible to the application point of the force,• the length of the arrow must be proportional to the intensity of the force.There will be a fourth rule, to be added later using other examples, on the composition of forces.

3. Formalizing energyThe concept of energy is certainly central in physics. There is a continuous debate on whether, foryoung students, energy is a more intuitive concept than force, since generally the application of amuscular force is intended to obtain a given “useful result”, which is just the characteristic of anenergy transfer.It is well known that force and energy are not well separated in the intuition of the students, who,in the description of a simple fact as the launch of a ball with a sling, would oscillate between aforce-based and an energy-based model: one can therefore ask whether it would be moreconvenient to start from the concept of energy rather than from that of force.Our experience, with students of SFP, is that, though energy might be more intuitive, it is moredifficult to formalize, in the sense described above, that is it is more difficult to relate the formaldescription of the concept of energy to specific aspects of the object or of the action: as aconsequence, energy remains a vague concept, difficult to attach to something concrete.In our approach, we start with concrete facts and objects that can be associated which energy, forexample the launch of a ball, since the idea of energy is naturally associated with that ofmovement. Indeed, in order to set something in motion, one need to transfer energy from some“source”: how does this happen? Many toys are useful to investigate the transfer mechanism and to formalize it. We will discuss, asan example, the toy-gun and the sling.The toy-gun is shown in Figure 3. It is evident that, in order to launch the ball, it is necessary tomove different parts while applying the force: to load the gun, the piston must be moved tocompress the spring; when the trigger is released, the end of the piston, moved by the compressedspring, puts the ball in motion. Both the forces and the displacements are needed.


The same analysis on the force and on the displacement can be done with the sling (Figure 4). Thesling is more flexible than the gun, because it is possible to apply different stretches and appreciatetheir effect on the launch and it is also possible to prepare slings with rubbers of different stiffnessand compare the launches: it is immediately clear that, to optimize the distance of the launch, oneneeds, besides the force, also a reasonable stretch.

The toy-gun Ask the pupils to bring toy-guns such as the one shown in the top figure, in which the spring is clearly visible and it is also clear how the spring is put in tension and how the ball is launched. Examining the toy-gun, the child should:• observe the springs and understand how they

operate,• describe the force Fft - "force-finger-trigger"- used

to load the gun, shown in the bottom picture (thebottom pictures were taken against the light to show better the spring displacements),

• imagine the force Fsp that the spring applies to the piston and the force Fpb that the piston applies to the ball when the trigger is released,

• give to each force a «name», which recalls the two interacting objects, that is the source of the force and the subject to which the force is applied,

• describe all the movements and the displacements of the force application points when the gun is loaded and then when the trigger is released,

• draw a picture of the gun, in which are clearly indicated, as in the figure: - the name of the forces, - the arrows which visualize the force directions, - the movements of the different parts and the

displacements of the force application points.

Figure 3

spring end points

toy-gun

piston

Fft

spring end points

force

displacement

loaded

piston

spring end points

piston

unloaded

Fsp

Fpb

Ffr

beforeafter

displacement of theforce application point

With the sling in his hand, the pupil should:• observe and describe, besides the force, also

the length of the stretch of the rubber,• try to stretch more or less and describe the

effect that this has on the launch,• draw a picture of the sling, in which are

clearly indicated, as in the figure:- the name of the force applied to the rubber

and the arrow which visualizes its direction,- the movements of the different parts and the

displacement of the force application point.Figure 4

At this level, the formalization of the energy concept consists essentially in the words used in thedescription:– the word “transfer”: by compressing the spring in the gun or by stretching the rubber of the sling,

something is transferred, which serves to set the ball in motion. Let us call it “energy”;– the word “store”: the energy is not transferred directly to the ball, but it is first transferred to the

spring of the gun or to the rubber of the sling, and remains stored there temporarily, until it isused to set the ball in motion;

– the association of the words “force” and “displacement”: only the forces associated with somedisplacement are of interest for the energy transfer. For example the force Fhs between the handand the stick is not transferring any energy;

– the “duration” of the energy transfer: energy is transferred while the force acts during thedisplacement. Once the energy is transferred, the effort is finished, although one might continueto apply a force;

– the “amount” of energy transfer: the larger is the force or the displacement, the larger is theenergy transfer.

The words used to describe the energy are thus quite different from those used to describe theforce. For the latter, it was important the aspect of the “interaction”: the force implies an interactionbetween two objects, thus it is important to understand which are the objects and how they interact.The energy is something different: it is something that “belongs” to the object, but not in the senseof the typical properties of the object, such as the weight or the volume. The energy of an objectdepends on the situation, on the previous interactions that the object underwent; it can havedifferent values; it can change because the object can transfer it to other objects, etc. For example:when the rubber is stretched, it gains energy; the amount of stored energy depends on the force andon the displacement; the energy stored in the rubber is then released to the ball, etc.By the analysis of other examples, other important properties of the energy are discovered, forexample the transformations of energy. Finally, at the end, also the mathematical expression of thework done to transfer energy is given, to complete the formalization process.

4. ConclusionsThe experience of two years of teaching physics to students of the university course of SFP hasshown that the association of a concrete object with an abstract concept by a basic formalizationhelps the learning process of aspects of physics generally considered difficult and obscure. We haveby now a rather large collection of toys and common objects, which we found to be very useful fordifferent physics concepts. They were partly developed by us, partly by the students (each studenthas to prepare, for the final exam of the course, a few objects or toys and discuss how he can usethem for the didactics of a given concept) and partly by the primary school teachers whocollaborate with us and host the SFP students during their practical training in the schools. Thedescriptions of the objects and of their educational use can be found in our web sitewww.iapht.unito.it/giocattoli/: the site is open to comments and collaboration of interested people!

References[1] Giorgio Hauesermann, Toys in the classroom, Proceedings of the GIREP International Conference, Duisburg,

August (1998), 208-209.[2] Seta Oblak, Toys as a part of a physics curriculum, Proceedings of the GIREP International Conference,

Duisburg, August (1998), 288-290.[3] see for example Pietro Cerretta, Lo scoglio didattico del principio d’inerzia e le questioni connesse, Proceedings

of the XXXVIII Congresso AIF, La Fisica Nella Scuola 34, Supplemento 1, (2001), 120-134


SOME PHYSICS TEACHING MYTHS

C.H.Wörner, Instituto de Física, Universidad Católica de Valparaíso, Chile

1. Introduction Informal learning (sometimes called experiential learning) has been recently recognized as a usefulmethod widely used for learning purposes. Some people asset that the use of botanical gardens,science fairs or exhibitions, museums, etc. may be used -and in fact they had been used- for theenhancement of science knowledge. Other learning situations described by this term refer to theside-by-side knowledge obtained for fellows in the same environmental situation, i.e. by high schoolor undergraduate students. A third situation came today in mind: it is not really necessary to havea physical neighborhood to access the information. The World Wide Web network is a source of analmost infinite quantity of informal data and facts. Furthermore, they are advocates for the thesisthat, because all the information is located in the web, the main purpose of instruction is to learnto use it and not to discuss the content of this information. Last but not least, I guess that almostall you (and me) know about PCs arises from informal learning.Appealing as it is, it is not the purpose of this note to discuss the utility and pertinence of theselearning resources.(People interested to be in touch with informal learning can consult [1]. I simplystress the fact that we, Physics teachers, are submitted to this type of influence since the pre-nettimes. Also, caution must be exerted as that this type of learning must be severely tested beforeacceptation. In order to discuss this fact, I will present some examples of commonly believed facts,that, although have been shown false in the literature, continue to be taught by a kind of fellow-to-fellow heritage. I will call these cases myths, in a different sense that other well know (proper)myths such as the Newton’s apple fall or the Galileo’s free fall experiment in the tower of Pisa.These later legends are taught as really unfounded or as unreal facts (although they can beprofitably used for pedagogical purposes), instead the facts we will dealing with, are false, but witha mode that is well reflected by the Italian proverb, “si non è vero, è bene trovato” [2].A recent nice account by Sawiki [3] on myths about gravity and tides triggered the present note.

2. Examplesa. The coincidence of Newton’s birth with Galileo’s deathTo be in tune with the last paragraph let us consider the common statement that the year Galileodies, Newton was born.The myth even consigns the date: 1642. It is easy to confirm this fact: Galileodies on January 8, 1642 and Newton was born on December 25, 1642.The fallacy stems from the fact that both dates belong to different calendars. The Newton birth’sdate was given in the old Julian system; instead, the Galileo’ death date is expressed in the (then)new Gregorian calendar [4]. This is a net example of the fact that equal numbers do not alwaysrepresent the same physical fact.

b. The flow of cathedral window glassesIt is a well-known fact that atoms in glasses and liquids present a disordered state or short-rangeorder as opposite to crystalline materials that present long-range order. Therefore, solid glasses (asthe ones belonging to cathedral windows) must possess liquid properties, that is, they must flow.Surely, solid glass viscosity must be greater than ordinary liquids, but for long exposure times, theconsequent deformation can be noted. Medieval glass windows seems to be the perfect target fortesting purposes and it is asserted that they are wider in its lower than in its upper edges.In fact, it has been shown [5] that this effect is not measurable –at room temperature- duringhistorical times (say, 5 OOO years), and therefore the thickness difference –if it exist- can beattributed to manufacturing defects. Furthermore, it is curious that older glass objects (ancient glassvases) do not seem to show this effect and do not appear in this myth.

c. The different rotation sense of water into a washbasin exhaust in the North and South hemispheresVortex movements are well known phenomena in domestic bathroom environments. Claims had


been whispered that –due to Coriolis non-inertial forces- liquid spins in opposite senses in differenthemispheres, in due account for the earth rotation around its North-South axis.This argument has been labeled an “old wives’ tale” by Crane [6], showing that this effect would beunnoticeable if the liquid is initially quiet. What the observer visually perceives is the increase inrotation (due to angular momentum conservation) of the liquid due to the fact that its initialangular momentum is different from zero. If fact, it is possible to whirl the liquid in either sense, bysimply selecting the initial angular moment of the system in a “clockwise” or “anti-clockwise” state.

d. The static friction force does not do workThe chapter on rotation dynamics in introductory Mechanics courses challenges Physics instructorsin various ways. One of this, is the treatment of the combined translation and rotation motion. Intreating this subject, common use of the example of a rolling body on the incline, permit the use ofenergy conservation, so avoiding the more complex settings of relationships between torque andthe rate of change of angular momentum. To be concrete, let us assume that we are talking aboutthe rolling without slipping of a cylinder on the incline. We can state the conservation of energyequations to calculate, in example, the final speed. The question is: why can we apply thisconservation law in presence of friction (indispensable for rolling without slipping)? , and theanswer is: because static friction does not do work. A further comment can follow: there is notrelative displacement between the two touching surfaces, and as work equals force times displacement,then with zero displacement, zero work.The fact is that –in general- static friction does work. Let us consider the combined motions of twoblocks with masses m1 and m2 under the action of a force acting on the upper body, as depicted in Fig.1.

Fig. 1:The joint movement of two blocks (masses m1 and m2), coupled by static friction forces on a frictionless surface.F is the external force acting on the upper block

There is (static) friction between the bodies and no friction with the table. The lower bodyaccelerates to the right due to the friction force f, and this force indeed do work, changing thekinetic energy of the body. Likewise, friction force does (negative) work on the upper body [7].Therefore, the real answer to the question is that static friction does not do dissipative work. In thecylinder-incline problem, the work done by the friction force equals the increase in rotationalkinetic energy and therefore, in including the energy rotation term in our equations, we are, defacto, taking due account for the work done by the friction force.

3. Final remarksA good pedagogical advice is do “not always believe written arguments” (by the only merit of beingwritten). A better would be do “not always believe whispered arguments” (although written on thenet). Informal learning is an inevitable and useful way of learning, but extreme care must beexerted on the examination of the offering evidence. Recourse to authoritative sources (althoughold fashioned) must be considered. Finally, the author does not assume to know all these myths andtherefore the reader would add his/her experiences on true/untrue explanations forknown/unknown phenomena.


m1

f

f m2

F

No friction

AcknowledgementsThe author acknowledges helpful discussions with G. Iommi Amunátegui and A. Romero.

References [1] S. Brailie, C.O’Hagan and G.McAleavy, http://www.mcb.co.uk/services/conferen/nov98/vuj/background_paper.htm[2] Quotations are untranslatable, but in a free version: “ if it is not true, it seems to be”; or in a more literal mode,

“if it is not true, it is worth to be discovered”.[3] M. Sawiki, TPT, 37(7), (1999), 438 .[4] S.E. Babb, Am.J.Phys,. 48, (1980), 421.[5] E. Dutra Zanotto, Am.J.Phys, 66, (1998), 392 .[6] H.R.Crane, TPT, 25(8), (1987), 516 .[7] For the interested reader: in order to assert that there is no slipping between the blocks, the applied force must

satisfy the inequality: F≤ µm1g(m1+m2)/m2, being µ the coefficient of static friction.



3.5 First Steps in Formalization

BUILDING FIRST EMPIRICAL CONCEPTS AND KNOWLEDGE BY ONESELF ANDOTHERS, IS THE WAY TO FORMAL THINKING FOR YOUNSGTER

Claudette Balpe, Institut universitaire de formation des maîtres d’Aquitaine (IUFM), France

1. A double question for teachersThe topic of the seminar is “developing formal thinking in physics”. I think that this problemsupposes two points of view because understanding physics in secondary school is a doublequestion for a teacher.Firstly, it depends on – What is implied in formal thinking. Secondly, it concerns – The way of teaching science.I’ll study these two cases one by one successively. I’ll illustrate my communication with someexamples of written or drawn products of children during physics activities in elementary or pre-elementary school.

2. About formal thinkingFormal thinking is one among the highest levels of thought, so, its building implies several kinds ofknowledge and needs to be progressive.Formal thinking means generally,– Knowing concepts in physics and their relations,– Linking data and equations, and,– To be able of coming and going between experiment and theorisation.In case of physics in secondary school, relations are often quickly introduced in a mathematicallanguage, which is very difficult for young people because using mathematical language in physicsrequires knowing what formalism means before using it.

3. Qualitative facts and formalismFormalism is, first, necessarily linked to qualitative facts: the problem of physics is to understandnature, therefore, to build abstract or mathematical models from concrete phenomena.Consequently, formalisation needs going in stages from qualitative situations. Moreover, in physics,mathematical expressions always imply logical relations between symbols of concepts relative tonature. Therefore, they imply on the one hand, knowledge of what is a concept, and on the otherhand, mastery of logical relations, first in a qualitative way, and further, in mathematical equations.Consequently, preparing children for formalisation is possible at elementary school, firstly becauselearning formalisation begins with a qualitative approach, and secondly because this approach isthe only form of studying which a child can reach. (examples: activities with water, ice, air, someliquids or solids, balances, light and shadows, calendar, nights and days, electromagnets,thermometer, etc…)

4. Tree levels of abstractionAccording to cognitive psychology, we can think about three levels of abstraction.– First, empirical abstraction to build concepts (according to “empirical abstraction” by Piaget)– Secondly setting of qualitative relations between concepts– And finally, generalisation (of a qualitative relation) with formal expressions and mathematical

symbols.

5. Concept and conceptual field to the meaning At elementary school, a concept is defined by an operative way. According to Gerard Vergnaud, aFrench searcher (CNRS) in cognitive psychology, a concept means a triple proposition:

– First, a concrete problem-situation to solve,– Secondly, mental logic operations about the situation (like classifying, comparing, ordering,

etc… things or quantities, or etc…). More often, the goal is to get the invariability (invariance)of the situation ; that is sufficient at elementary school.

– At last, a symbol — often, a word — which represents the concept and its meaning.Usually, the convention for symbol is a word which is easily accepted by children ; since it is aconvention, the pupils will be able further, to replace this word by an other symbol like amathematical symbol. For example, resistance in electricity will be replaced by “R”, surface by “S”,mass by “m”, distance by “d” and volume in geometry by “V”, etc… so as to be parts ofmathematical equations later.After this first qualitative elaboration, the concept is related to manydifferent contexts which give its several meanings, and therefore which form a sort of “conceptualfield” including all the meanings of the concept. So, the child knows all the qualitative sides of theconcept to which he has access at his age.This way of building concepts is very significant for the children, mostly if they do it by themselves:thus, they will get a sound and permanent knowledge of concepts and will be able to enrich themaccording to new contexts of learning. Therefore, it will be easy for them to use further a symbol ofconcept, either letters in qualitative relations, or mathematical symbols as parts of equations.

6. Process and logical, qualitative relationsSome concepts such as melting of ice or evaporating cannot be represented at elementary schoolby a measurable quantity but only by a linguistic symbol — that is a word. This sort of conceptimplies a process and logical qualitative relations between measurable quantities (for example:weight, height or surface area of water, time or temperature). It is of course, more difficult to study,and is tackled later in schooling (in the last years of elementary school) as said afterwards.These sorts of concepts — first symbolised by a word — will enable children to go further bysolving problems in experimental situations.

7. Solving experimental situations: a first introduction to logical thinking in a qualitative wayAfter having built some main concepts during the first years of elementary school, students are ableto solve experimental situations — for example: how to make water leaving out wet clothes ? For this, they have to conduct a research: what are the factors of this situation, how to foresee howit works, how to test hypotheses with experiments and end up to conclusions. (ex: evaporation goesquicker with a high temperature, a wide surface area and changing the air above water).In this way, logical relations are built between empirical factors: that is a first introduction to logicalthinking in a qualitative way.That way of thinking — by logical relations — is the most difficult partof learning in science.At the end of elementary school, youngsters can set up simple natural rules from calculating anddata — for example, the rule for balancing a mobile hung up on a thread.. If children know how tomultiply, the rule — expressed in words: heavier the object is, closer it has to be to the thread toequilibrate something else — can be translated first into a very simple equality between numbersfor each data (the product of multiplication = mass x length is egal on each side). Then, with themathematical symbols, this equality can form an equation. Thus, it will be easy to go throughequations, which are logical expressions with abstract symbols. At that time, it will be necessary foryoungsters to know about appropriate mathematics.To sum up, physics at elementary school are first building concepts, and then leading to logicalrelation firstly qualitative. Later, when youngsters have a good level in mathematical knowledge, insecondary school, they can do similar reasoning introducing to mathematical equations.In conclusion, it seems that if the symbolism of concepts and the functioning of logical thinking areacquired — even with empirical concepts — it will be easy to get formal thinking. Because, sincethe most important obstacle in the path to formal thinking is to give sense to equations, by buildingeach concept, its mathematical symbol will be easily mastered by children or youngsters. In the


same way, if relations between empirical concepts are firstly built, their mathematical expression willbe also mastered afterwards.And, little by little, when more and more equations are set up, young people become able to usethem, to come and go between experiments and theory through equations, to foresee, etc…

8. Some conclusionsThe first conclusion of this communication is that formal thinking cannot be taught or learned withlessons. It takes roots progressively, in the first learning, when children build their knowledge:otherwise, it makes no sense to youngsters. At elementary school, formal thinking is only anintroduction to logical thinking and to symbolisation.The second conclusion is that to build formal thinking requires to break with the pedagogy of speech,and to teach methods rather than results. In this new style of pedagogy, the youngster — and not thecontent — is central. That doesn’t mean that the content is not important, but, on the opposite, thatscientific activities in the class-room aim at building the meaning of official contents.

9. The second problem is not to make physics available to youngsters but to enable youngstersto build scientific knowledge?The first and most important results come from Jean Piaget who has analysed and searched aboutchild’s understanding. Actually, new considerations in cognitive psychology go further and provideus with the conviction that young people have to build their knowledge by themselves (and of course,with others and the teacher in the class-room): it is the only condition for them to get sound andstable knowledge. In physics, the progressive way described before, seems to be the only way toreach formal thinking in secondary school.The problem is not to make physics available to youngsters but to enable youngsters to buildscientific knowledge. If they can build concepts and relations — empirical, then, mathematical —young people will be able to come and go from experience to theory and thus, to develop fromelementary school, introduction to formal thinking.

10. How do teachers have to teach science at elementary school with this aim in view? First, they have to organise activities in the classroom for the children – Observe nature and phenomena,– Research how to solve problems – Answer questions about nature.Children are naturally curious and must satisfy this curiosity collectively and by themselves, ratherthan by listening to their teacher. Physics will begin by questions, interrogations, paradoxes, etc…raised by the teacher or by the children themselves, about nature and phenomena, as ancientpeople (Greek, Egyptian, Chinese…) did it; so, children become little researchers in the class-room. Thus they will learn by solving situations or answering problems in interaction with nature.I have proposed an indicative way for young teachers to teach elementary physics.– Didactic table (5-8) [proceeding table for teachers in cycle 2] – Didactic table (9-11) [proceeding table for teachers in cycle 3.] The teacher has to be an organiser of the activities, an organiser who knows about science but whodoesn’t teach it. He just leads activities and helps pupils in their research, their communication, insetting up conclusions and finally in writing about their activities.Generally, there are four parts in a scientific activity in classroom. First, children start with an open(or, half-open) situation or question, or with a challenge, to introduce the subject by observing ortrying to see what is at works. For example, if the aim of the lesson is the discovery of the rules ofequilibrium, the teacher proposes to the children to build a mobile. It is relatively easy for them andgives them a successful activity.

11. Example of solving concrete problem: building and equilibrating a mobile, a seesaw andknowing about their equilibrium

264 3. Topical Aspects 3.5 First Steps in Formalization

The teacher’s objective is to enable pupils to understand by themselves the qualitative rules ofequilibrium of an object moving only by rotation around a fixed point.This project will allow the children to build rules of equilibrium and understand about levers, balancesand further, to introduce measuring of masses.The pupils (6 - 8) have to equilibrate first a mobile with two different pieces of cardboard (one largeand heavy, the other small and lighter) which they have cut up.They have seen first, the mobile of theirteacher and afterwards, they can make their own mobile as they want. This need intuition and thepupils progress by trial and error. In French this is called “tâtonnement experimental”. Finally, theyachieve success and call their teacher to show him what they have made.While some of the pupils are finishing, the others have to draw on a sheet of paper a representationof their device and try to remember how they did it. When all the groups have finished, the teacherorganises a discussion and every group explains to the others how they did it.Two solutions appear from the discussion between them and the teacher who conducts the debate:first, moving only the hanging–thread and stopping it near the heavier object; secondly, maintainingthe main thread in the middle of the rod and moving the objects by trial and error and by intuition,until the mobile is equilibrated. In both cases, the teacher proposes to compare the lengths of thetwo parts of the rod on both sides of the main thread: first by foreseeing, and secondly bymeasuring. The results of all groups are similar: the heavy object (the larger) is nearer the threadthan the light one. This first result is valid for all the groups, so it is admitted as a truth.The conclusion is a qualitative relation: heavier the object is, closer it has to be to the thread toequilibrate something else.The transparent films I am showing you are some examples of writings from children I workedwith.To go further, another research can be proposed which permits to apply what they have builtbefore: how to equilibrate a seesaw with two children, one heavier than the other ? By groups,children have to research how to do with an apparatus (on a pivot, they put a ruler with two littleballs of clay or something else)Very quickly the pupils succeed in equilibrating the device and can say why they were successfulimmediately. They draw a representation of their trial (in classroom, with two balls in clay) and canpropose the right conclusion for lifting a heavy thing. (→ to lift up something heavy) For the second time the qualitative rule for equilibrating a rotating object is the same: heavier isthe object, closer it is to the thread. That becomes the general and scientific rule admittedqualitatively by the children about a balance. Then, to widen the field of the concept, the childreninvestigate the levers.Other researches are proposed:What happens when you use two little boxes with some little screwsinside it, (instead of balls of clay). First, the children try with the same number of screws and putthe boxes at the same distance from the main thread: the apparatus is equilibrated. The pupils haveunderstood that if the weight is the same, the distance is equal: that application is different than theprevious experiment. Then it is possible to go on studying balance. Then, it is proposed to add, oneby one a screw in one box. The children have to foresee what they will have to do. And then, theydraw a conclusion from their hypothesis: the heavier the container is, the closer it has to be to thepivot. It is possible to measure and draw up a table of data from which the pupils can foresee othercases. If pupils know how to multiply, the teacher can offer them to search the mathematicalconstant term of this situation. That is a possible beginning to approach the algebraic “theorem ofmovements” studied in secondary school.Such a way to build this content at elementary school enable the youngsters in secondary school todevelop quickly formal thinking about equations of equilibrium: they have already learnt aboutweight and distance, and their qualitative relations.Because the Principle is first tackled in a qualitative aspect and recalled when the subject ismathematically treated, there is no important obstacle for the youngsters to understand it, and allthe more since they gave sense to it when they were searching about it.


12. A fine perspectiveIn this example, pupils are like researchers. They have questions or phenomena to understand byobservations and experiences.They work together in little groups.The teacher plans their activities,organises debate, prepares apparatus and material. He knows about the topic but doesn’t teach itby lessons. He foresees the way the children will build their knowledge according to cognitivepsychology and pedagogy of groups. He thinks about how to organise final knowledge and aboutchildren’s writings: because structuring the presentation of the writing is a way of giving sense tothe contents and reminding the children of the scientific activities. This sort of scientific activitybuilds qualitative relations and conceptual notions about equilibrium of an object rotating arounda fixed point.This example is one among many others, all of which giving roots to the formal thinking.

ReferencesBalpe C., Les sciences physiques à l’école élémentaire, Paris, Colin, (1991).Balpe C., Enseigner la physique – Approche historique, Rennes, Presses universitaires de Rennes, (2001).Vergnaud G., (Ed) Apprentissages et Didactiques. Paris, Hachette, (l994).Vergnaud G., Les fonctions de l’action et de la symbolisation dans la formation des connaissances chez l’enfant, In

Piaget J., Mounoud P., Bronckart J.P., Psychologie, Encyclopédie de la Pléïade, Paris, Gallimard, (1987), 821-844.Weil Barais A., Vergnaud G., Students’conceptions in physics and mathematics: Biases and helps. In J.P. Caverni, J-

M. Fabre, M. Gonzalez (Eds), Cognitive biases. North Holland, Elsevier Science Publishers, (1990), 69-84.Vergnaud G., La théorie des champs conceptuels, In J. Brun (Ed), Didactique des Mathématiques. Delachaux et

Niestlé, Lausanne, (1996).Vergnaud G., La formation des concepts scientifiques. Relire Vygotski et débattre avec lui aujourd’hui. Enfance, 1-

2, (1989), 111-118.

INTRODUCING THERMAL PHENOMENA QUANTITIES

Domen Ferbar, Faculty of Education, University of Ljubljana, Slovenia

Big worldTemperature and heatIn the first step heat1 and temperature are differentiated. It is important to stress the differencebetween everyday experience and language connected with it and the refined meaning we aretrying to introduce.

1. Heat can be stored in warm bodiesPour some water in a glass. Heat it up. Draw a strip. Water in the right glass is warmer then waterin the left one. It makes sense to say that the second glass contains more heat. Changes intemperature and changes in heat go hand in hand. There is more heat in the glass of water of 70 °Cthan in the glass of water of 10 °C. There is more heat where there is higher temperature.

2. What is the difference between temperature and heat?1) Pour some water into a glass. Pour some more. The temperature of water in the glass stays the

same. For which picture it would make sense to say that water contains more heat?


1 Everyday meaning of the word heat is used. Heat in everyday language (amount of heat) differs form the meaning ofthe word in science courses only by the fact, that heat (in everyday language) can be stored.

Change of heat departs from temperature changes: the amount of heat goes up but the temperaturestays the same. There is more heat in the glass with more water.

2) When pouring the water out of the glass, the temperature of water in the glass stays the same.For which picture of glass A it would make sense to say that it contains more heat?

Heat in glass A diminishes, temperature stays the same. Change of heat again departs fromtemperature changes: heat goes down, temperature stays the same. There is less heat in the glasswith less water.

3. Can heat flow without flow of water?Hold a can of hot water with both hands. You can feel the flow of heat into your hands. Hold a canof cold water with both hands. You can feel the flow of heat out of your hands. Put a can of hotwater into a vessel of cold water.What is happening with temperature? Draw a strip using different shades of grey different fordifferent temperature. What is going on with heat?

Heat in hot water is decreasing. Heat of cold water is increasing. It makes sense to say that heatflows from hot water to cold water. Put an arrow in the picture to indicate the direction of heat flow.

Energy and entropy

1. Warming hands with heatIt is possible to heat hand by putting them into warm water.

2. Is heat the same as energy?It could be. It can be transferred from body to body and it can be stored in a body. Let us give someenergy to Domen and see, whether his hands warm up as in case when giving him heat.Both Janez and Domen are holding a rope. Domen is standing on a trolley. Janez pulls the ropewhile Domen holds it firmly. Clearly Domen receives energy as his speed increases and thus hiskinetic energy increases. But his hands do not warm up. So heat is not the same as energy.

3. Can one warm hands without heat?The previous experiment is repeated with a slight change: Domen holds the rope loosely so it slipsthrough his hands.Domen does not move. Domen’s hands warm up as when he was receiving heat from warm stove.Touch the rope at different places. You can feel that heat is not flowing along the rope to Domen,but the energy does. So something is added to energy on the contact surface between rope andhands. We will call this addition entropy. Thus energy flowing through the rope and entropy addedon the contact surface together make hands hot.


It is safe to say that energy and entropy together make heat. Energy flows through the rope,entropy does not. It makes sense to say that entropy is created while rubbing the rope against thepalm of Domen’s hand.So one can heat hands without heat. In the slipping rope experiment only energy was supplied toDomen’s hands and entropy was created.Revise the chapter ”Temperature and heat“ and exchange the word heat by word-pair ”energy andentropy“. Both energy and entropy inherit the properties of heat introduced in this chapter. Theyare both extensive attributes: they can be stored, they can flow and they are additive.

Entropy can be produced

1. Heating by rubbingRub your hands strongly against each other until it hurts. Count the number of hand-moves. Putsome drops of salad oil on your palms. Rub them again and count the number of hand-moves untilthey get as warm as before. Put some washing powder on the oiled palms. Do the counting again.The energy is already in the hands. By rubbing entropy is produced. Decreasing friction with oildiminishes entropy production rate. More hand-moves are needed for the same temperaturechange. Adding washing powder increases friction. Entropy production raises. Only a few scrubs isenough to heat up hands as previously.

2. Heating by reactionBreak a ”heating pad“. Hold it in one hand. What happens?Energy released by the substance combined by the entropy produced by the chemical reactionwarm up the substance in the same way as if it were receiving heat.

3. Heating by currentTake a piece of heating wire and connect it to the battery. You can feel the wire warming up.Energy transferred by electric current is combined by entropy produced by it. This produces thesame effect in wire as if it were warmed up by heat.

4. Heating by lightMake a patch of aluminum bronze on the back of one hand and a black soothe patch on the other.Put both hands on near strong (IR) light source.Light is reflected by bronze and is absorbed by soothe. Absorbed energy and entropy produced byabsorption will make skin unbearably hot.Aluminium bronze reflects and disperses light in all directions. Entropy produced by dispersion istaken away by reflected light.In all these processes entropy is created and “added” to energy. This produced the same feeling asheat would. In the first two examples the energy was already in the body and entropy was createdand in the third and forth energy was also supplied.

Micro worldLet’s create a representation that will help us to think about selected phenomena. Let’s pretendthat we are looking at what is happening through a microscope.

Temperature and energyWhat we see is a simple world of atoms2. Every atom can either have a small amount of extraenergy or it does not have it. In the computer model we simulate the two states by either colouringthe atom (atom has some extra energy) or by colouring it white (atom has no extra energy).Atoms randomly exchange extra energy among each other. In a computer model we make a rulethat simulates this process: a randomly picked up atom exchanges its colour with the colour of a


2 By atoms we mean only the basic particles of which this world is made of. Some, though very limited analogy can bemade with real-world atoms.

randomly picked up neighbour.In the “experiment” a hot body is placed in a contact with a cold body. A red block of atoms isplaced beside a white region.Redness spreads out from small red block into big white region until both are “equally red”. Wecan say that difference in “redness” between two blocks disappeared and interpret the redness astemperature.Because redness is passing all by itself from high “concentration” to low “concentration” we saythat the difference in concentration is the cause of change.

What is entropy in microscopic world?In the model, energy stops flowing when there is no difference in temperature. If this is the case in“big” world, heat (energy and entropy) stops flowing. It is sensible to connect the two quantitiesalso in the micro world.To introduce entropy, the number of ways in which atoms within a body can exchange extra energywithout the change of number of atoms with extra energy is counted.

The difference between temperature and heatIt makes sense to make a revision of the difference between temperature and heat (energy andentropy). The same argument makes sense also in the microscopic view.The temperature (redness) of the system does not depend on the size of the system.

Energy and entropy on the other hand are connected with the number of particles with extraenergy and thus depend on the size of the system.

Increasing entropy – spreading out energy to larger number of placesConsider a hot iron block in cold environment. The iron block is denoted as a block with a thickerline around it.

Time passes and iron block cools down and the environment warms up. Energy goes form the ironblock to the environment.In the iron block the number of arrangements lowers and increases in the environment. Theentropy of the iron block decreases and the entropy of environment increases.


Obviously energy spreads out. The entropy of the region, from which the energy flows, decreases,because there are less different arrangements. The entropy of the environment increases.A computer model can be used to compute the number of arrangements. It can be shown that theincrease of entropy in the environment is larger than the decrease in the iron block. So entropy isproduced in the process.

Disco type entropyBy now the model of the micro word was simple. It needs to be broadened.Entropy depended only on the number of arrangements of extra energy. We can think of it as discotype entropy because it can be connected with movement of molecules. In the process of spreadingout energy molecules bump into nearby molecules. “Fast molecules” slow down and “lazy” nearbymolecules speed up - exactly what happens in disco, if some good dancers start with a wild dance.They agitate the nearby neighbours but by this process the wild dancers slow down.From experience we know that fast moving and slow moving dancers will mix up. But also dancersthat differ in some other property might mix up in a similar way.

Lego type entropyTwo groups of dancers are on two halves of the dance floor. One is wearing red shirts and the otheris wearing white shirts. They represent two different substances in two separate regions. While timepasses they mix. Red ones spread into the white group and white ones spread into the red group.On the whole the process goes on until the red/white ratio is the same on both parts of the dancefloor.Mixing is the process that runs by itself. The particles are moving from regions of highconcentration to regions of low concentration in a similar way as energy is transferred from hot tocold region. Because of this similarity we expect that entropy be produced in such a mixing process.This forms a counterpart to the disco type entropy. Let’s call it lego type entropy3. This is the partof entropy that depends on the number of arrangements of different types of particles.

Increasing lego type entropy - spreading out particlesIn the “hot iron block” example a change that happens by itself was connected with a diminishingtemperature difference. Now we can say that by spreading out energy the disco type entropyincreases.Other changes that happen all by itself exist in which other differences are disappearing. Spreadingout of matter is an obvious example.

When spreading out of matter occurs differences of concentration of matter are disappearing. Thechanges in entropy are similar to changes in entropy that come about because of spreading out ofenergy.Examples:A small amount of perfume is sprayed into the air in a large room. The molecules of the perfumesoon fill the entire room. Matter spreads out.Let’s involve both, spreading out of energy and matter.A match is burned. Because the temperature of the match and its constituents and also a nearby


3 Lego type entropy is tightly connected with spatial distribution of particles

environment is increases, the energy spreads out. But also the molecules constituting of the matchmolecules and nearby oxygen molecules spread out.To ease the thinking, it is often possible to think of the lego type entropy increase as the increasein mess. Some precaution is in needed since the meaning of the word mess can be misleading. Messis created, when two substances mix, like in the Cinderella story, where ashes and millet grains weremixed. Entropy increases. But then, if the two particles bind somehow (like atoms bind inmolecules), the entropy decreases, since within a certain diameter of an ash particle a millet graincan be found.If water freezes, a crystal is formed out of fluid. The lego type entropy decreases, since one is quitesure where a certain particle in the crystal is to be found.A box of well-aligned matches is in perfectly good order until matches fall out of the box. If onecollects them in a hurry, not all the caps are on the same side. So entropy increases.Something similar happens if NaOH is dissolved in water, dipole water molecules surround Na+and OH- ions. The alignment of water molecules is now better, so the entropy of the watersurrounding the ions decreased.Two facets of spreading energyA model of solid made of Styrofoam balls connected with spring is left to fall on the floor.During the fall all the “atoms” move with the same velocity in the same direction. Kinetic energycan be ascribed to movement in one direction – towards the floor.After the solid hits the floor balls are moving more vigorously than before. They move withdifferent speeds in different directions. Energy has been spread from the movement of all atomstowards the floor to movements of atoms in all directions.Spreading of energy has two facets – energy can spread in two manners:• to larger number of places (atoms) or • to larger number of movements (movements in different directions with different speeds).In the Styrofoam example energy spread only to larger number of movements.

Connecting the micro world with the “big” world

Macro worldEnergy before the bump of block of solid on the floor is ascribed to movement of the body. We say,that the block of solid has kinetic energy. Energy of disordered movement of particles inside thebody is called internal energy.Kinetic energy of the block of solid diminishes in the process of breaking or collision. We say thatkinetic energy is transformed into internal energy. Because the energy of the solid does not change,we associate the increase of temperature with the increase of entropy.

In micro worldOrderly movement of all the particles of the body downwards is transformed into a large numberof disordered movements – movements of particles in all directions with various speeds. It lookslike disorder in movement increased. Increase of disorder in movement in micro world can beperceived as increase of entropy in the big world.

Both worldsDispersion of energy either to bigger space or to a larger number of different movements is aprocess that is difficult to reverse. We call it irreversible process. Of course this does not mean thatthe process cannot be reversed – it only means that this does not happens by itself. Reverseprocesses of irreversible processes must be driven from outside by some effort.So we can conclude microscopic picture by saying that in irreversible processes (dispersion ofenergy and matter) entropy is produced and this reflects in increased disorder of distribution ofenergy and matter in space and among movements.


In the big world it seems that there are many more types of processes that run by themselves onlyone way and have to be driven in reverse direction. These are irreversible processes: things slowdown, currents die out, waves are absorbed and chemical reactions proceed in one direction. Justfour irreversible processes in the micro world can explain irreversible processes in the big world:dispersing energy and particles in space and among movements. This is summarised in the abovetable.

ReferencesAtkins P. W., The second law, New York, W. H. Freeman and Company, (1984).Black P. J., Davies P., Ogborn J. M., ‘A quantum shuffling game for teaching statistical mechanics’ American Journal

of Physics, 39, 1154 – 1159.Boohan R., Energy and change Support materials (London: University of London Institute of Education), (1996).Boohan R., Ogborn J., (Trans. Ferbar J) Energija In Spremembe (Ljubljana: Modrijan), (1996).Ferbar D., Statistične igre in ireverzibilnost, (Ljubljana: Pedago_ka fakulteta Univerze v Ljubljani), (1995).Ferbar J., Irresistibility of irreversibility, in Oblak S. et al. (ed.), New ways of teaching physics - Proceedings of GIPEP-

ICPE International Conference, (Ljubljana: Board of Education of Slovenia), (1996).Ferbar J., Teaching wisdom, in Physics Teacher Education beyond 2000, Proceedings (Barcelona), (2000).Herrman F., Der Karlsruher Physikkurs, Teil 1 - Energie, Impuls, Entropie, (Karlsruhe: Abteilung für Didaktik der

Physik, Unversität Karlsruhe), (1995).Herrman F., Der Karlsruher Physikkurs, Teil 3 - Reaktionen, Wellen, Atome, (Karlsruhe: Abteilung für Didaktik der

Physik, Unversität Karlsruhe), (1995).Hribar M., Entropijski zakon - poskus zaključene obravnave, (Ljubljana: Pedagoška fakulteta Univerze v Ljubljani),

(1995).Keohane K. W., (co-ordinator), Change and chance, (published for the Nuffield Foundation by Penguin Books),

(1972).Kuščer I., Žumer S., Toplota, (Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, Zveza organizacija

za tehnično kulturo Slovenije), (1987).Mandl F., Statistical physics, (New York: John Wiley and Sons Ltd.), (1988).Marx G. (ed.), Disorder in the school, (Budapest: Educational Branch of the Roland Eötvös Physical Society).Reif F., Statistical physics, (New York: McGraw-Hill Book Company Inc.), (1967).Zemansky M. W., Heat and thermodynamics (New York: McGraw-Hill Book Company Inc.), (1951).

WHAT FRACTION OF PUPILS REALLY REACH THE STAGE OF FORMAL THINKERIN PHYSICS?

Rudolf Krsnik, Planinka Pećina, Maja Planinić, Ana Sušac, PMF, Physics Department,University of Zagreb, CroatiaIvica Buljan, Primary school ”Zaprude”, Zagreb, Croatia

According to Piagetian theory transition from preoperational stage of thought to the concreteoperational stage occurs between 7 and 11 years of age. Around age 11 child is concrete thinker; sheor he is capable to think causally and that is of crucial importance for physics teaching. Concreteoperational mental structures permit intuitive conservation reasoning, reversal thinking, multipleclassification, assimilation of data from concrete experience and their arrangement andrearrangement into serial ordering etc. However, thought operations of concrete thinker are strictlyrelated to objects and physical processes with which he is in direct physical contact and are directly


Dispersion of energy of matter

by position heat conduction diffusion

by direction

conversion of kinetic energy into internal energy dispersion of light

dispersion of beams dispersion of particle beams

tied with physical experience. Objects and environment themselves are not of crucial importance;what is important for cognitive development is the child’s activity in acting on those objects.However, concrete thinker is not able to think on the basis of verbally stated hypothesis andabstractions.With further cognitive development to the stage of formal thought operations these limitations areoutgrown. Formal thinker has no need for objects and direct physical experience; she or he is ableto think in abstractions using propositional logic. While the earlier stages were described in termsof domain specific descriptions, it was not so with the formal stage. Formal operational stage wasdescribed in terms of operational schemes, i.e. certain patterns of reasoning, for example:• focusing on the important variables what includes isolating relevant variables and controlling

variables in a given process;• formulation and stating of hypotheses to concepts and abstract properties and the use of

propositional logic;• combinatorial reasoning,• proportional reasoning,• stating functional relationships, etc.According to original Piagetian theory [1,2] transition from concrete operational stage of thoughtto formal one is a) universal and b) occurs between 11 and 15 years of age. Numerous results ofresearch in the light of Piagetian ideas have showed that the first statement does not hold. It turnedout that ability of formal thinking strongly depends on context inside which development hasoccurred. In the second statement Piaget was too optimistic. It was shown later that large fractionof adults still stay at the stage of concrete thinker. As school children are concerned it turned outthat less than 20% students of English comprehensive schools are in late formal stage at age 16 [3].There exists certain criticism of Piagetian stage theory, for instance that it is not explanatory butdescriptive. Nevertheless, Piagetian stage model is very useful and simple enough for the use inphysics teaching. One important reason for that is the fact that in majority of Piaget’s originalquestions in tests and clinical interviews are about physical objects and processes. There is more orless consensual agreement that dynamics of child’s cognitive growth is strongly dependent on thequality of teaching process. Stage model gives to curriculum developers and teachers importantinformations of two kinds. Knowing cognitive level of students (by their age) one can: a) chooseappropriate level of teaching contents, and b) structure and organize teaching contents in a mannerwhich can accelerate student’s cognitive development. Piaget himself claimed that physics is in thatrespect the most convenient of all school subjects.Our experiences with pupils of all ages and with university students (future physics teachers)demonstrate significant difference between those who has attended traditional teacher centeredteaching and those who attend pupil centered constructivist oriented teaching.The great majority of interviewed students attended some of traditional teacher centered teachingduring their schooling. The achievement of middle school students is in accordance with quotedresults of Shayer et al [3]. Their data, that not more than 20% students age 16 are formal thinkers,is particularly interesting. Namely, we have found that in teacher centered teaching process(lecture) only about 20% of students try to follow the lecture longer than 5-10 minutes; the othersswitch themselves off. Also, according to Solomon [4] percentage of English school students whoare able to think about physical problems in appropriate manner is about 20%. Coincidence ofthese data cannot be accidental.Students in higher classes up to age 19, and even university students (future physics teachers), incertain situations do not handle some operations which are associated with formal thinking. Itseems that these inadequacies are tied with weak qualitative understanding of concepts which havebeen acquired mostly via definitions and mathematical formalism. It is often difficult todiscriminate what is the reason of inadequacy in response: undeveloped formal thought operationsor undeveloped conceptual structure. We will consider in more details few typical examplesrestricted to thought operations of isolating and controlling variables.


• Definition of electrical resistance, R = U/I, student interprets in the following manner:“Electrical resistance is proportional to voltage and inversely proportional to current.”Obviously, student is not using control of variables, but he does not understand the concept ofelectrical resistance either. It is indicative that such mistakes are often made by future teachersof mathematics and physics (and not by future teachers of physics, physics and chemistry orphysics and technology); it probably reflects their way of thinking in mathematics.Similar and even more drastic case, but fortunately less frequent, is the interpretation of secondNewton law, F = m a, by stating: “Force is proportional to the mass of the body.”

• In answer to question how does centripetal force depend on radius r, most of students choose oneof the answers: “inverse proportional” (led by equation Fcp = mv2/r ), or “proportional”(equation Fcp = m 4π2ρ / T2 ). One could say they are just not applying control of variables,therefore they are not formal thinkers. However, the other possibility is that they have not atdisposal sufficient conceptual understanding of given situation, and that could be the reasonthey are not able to apply formal thinking.

• Isolating relevant variables is particularly important operation in qualitative reasoning onproperties of harmonic oscillator.a) In the first case system is simple harmonic oscillator, a weight suspended on elastic spring.Thetask is to find functional dependence of free oscillations frequency in dependence of relevant

parameters of the system (i. e. dependence ). In spite of the fact that there evidently

exist only two relevant parameters, k and m, student have lot of troubles with their isolation(these parameters were known to students, but were not mentioned in the formulation of theproblem). While isolating relevant parameters is a serious problem for appreciate part ofstudents, procedure of dimensional analysis is not a problem at all.b) Interesting case is analogous procedure for simple pendulum, i.e. to find out functional

dependence using dimensional analysis. Here is the operation of isolating relevant

parameters more complic5ated and more challenging. Namely, beside the length l of a thread asthe obvious parameter, immediately appears the mass m of the bob (which turns out not to berelevant parameter for frequency, and should be eliminated in the later procedure), andgravitational field of the Earth, g, which is hidden at least at the first glance. Majority of studentssucceed in isolation of g as a relevant parameter only after additional questions (why is the bobswinging at all?).

• In the course of our (constructivist oriented) variant of school curriculum in 11th class (age 17)the problems are stated to find functional dependence of speed of (sound) wave on relevantparameters in stretched wire, and in bulk specimen. Students, future physics teachers, workingon that curriculum have serious problems in autonomous isolating relevant parameters whichcould influence the speed of the wave. Probable reason for these difficulties lays in the way howthey were educated. They have learned physics mostly following traditional lectures, usingdefinitions and mathematical approach without appropriate qualitative reasoning.

In the case of stretched wire it should be easy to isolate the length. l. and the mass

m .of wire (which enter in the relation through linear density µ = m /l and tension T of thread;it is really difficult to anticipate some additional relevant parameter.

In the case of bulk specimen students have even more problems in

isolatingrelevant parameters because they are more abstract. Only a part of the students succeedin autonomous isolating of these parameters, but only when during discussion they are reminded

2743. Topical Aspects 3.5 First Steps in Formalization

m

k0

l

g0

( µ

Tv )

(E

v)

that analogy should be drawn with simple harmonic oscillator and parameters whichcharacterize inertia and elastic properties of the system respectively. Isolating of elastic constantE (or some other modulus) makes more difficulties than isolating of density, because it is moreabstract.

How to improve formal thinking of pupils in physics? We see the solution in interactive, studentcentered and constructivist oriented approach to the physics teaching. That is necessary forachievement of science literacy of all scholars; in the same time that is useful for the most able aswell because they have more opportunity to express their special abilities. For example, in 4th class(age 10) elaborating experimental theme on water waves, we asked pupils one inappropriatelydifficult question: One boat drives with speed 10 km/h, another one with speed 20 km/h. Both ofthem produce water waves. Compare the speed of waves produced by these two boats. On oursurprise there were pupils (4%) who answered that speed of both waves are equal, because wavespeed should depend only on properties of water.It seems that majority of scholars from age 11 to the end of middle and high school (age 18-19) aremixture of concrete and formal thinker. Certainly, the portion of formal thinking part is increasingwith age, but is far from being completed. Consequences for physics teaching could be that it isalways convenient to start with gaining physical experience, and through discussion move towardsgeneralizations (but not strictly by inductive method). The development of formal thinking isstrongly dependent on context, so in teaching practice (and curricula development) Piagetian ideasabout cognitive stages should be combined with perception on preconceptions and conceptualchange, and complete teaching process should be subjected to the ideas of educationalconstructivism. In such educational surrounding pupils are unbelievably able participants. Forexample, in 8th class (age 14, second year of physics teaching) during experimental treatment oflight dispersion, Newton’s experiment of light dispersion on prism was presented and historicalsituation in Newton time was described, particularly old ideas that white light is primitive andcolored light is a mixture of white light and of something else. Pupils were told that Newton createdtwo crucial experiments by which he succeeded to refute that old ideas. Pupils were asked to createthese two crucial experiments themselves (by the use of two identical prisms, light source and ascreen with a small hole).With the help of some discussion (answers to pupil’s additional questions)and a small assistance of teacher pupils succeed in creation of both crucial experiments. In spite ofthe fact that it was paradigmatic change in historical development of physics, which had manydifficulties with acceptance, pupils took it as a quite normal thing. Pupils obviously had no strongpreconceptions about that matter and this result was hardly conceptual change to them. This resultis on the line with important constructivist thesis that knowledge is dependent on socialsurrounding. By the way, such way of teaching process is not limited to very small number ofexclusively able teachers (what is occasional comment of uninformed persons, even physicists). Inour case the role of teacher was taken by two students on pre-service teaching practice.

References[1] J. Piaget, Les stades du développement intellectualle de l’enfant et de l’adolescent. In Symposium on Le Problème

des Stades en Psychologie de l’Enfant. Presses Universitaires de France, Paris, (1955).[2] B. Inhelder, J. Piaget, The Growth of Logical Thinking from Childhood to Adolescence, Basic Book, New York,

(1958).[3] M. Shayer, D. E. Kuchemann, and H. Wylam, The distribution of Piagetian stages of thinking in British middle and

secondary school children. British Journal of Educational Psychology, 46, (1976), 164-73.[4] J. Solomon, Teaching about the nature of science in the British National Curriculum, Science Education 75 (1),

(1991), 95-103.


COGNITIVE LABS IN AN INFORMAL CONTEXT TO DEVELOP FORMALTHINKING

M. Michelini, A. Stefanel, C. Moschetta, Research Unit in Physics Education, University ofUdine, Italy

1. IntroductionIt is necessary to start scientific education early, so that it is an integral part of the culture withwhich the individual grows and contributes to his education, with its ways of putting together andusing the work done by all, of recognizing as valid what can be shared, by defining and discussingproblems, by paying scrupulous attention to the heart of these problems, transcending the personaldimension to give value to the social dimension. In Italy this is still something which has to be done,due to the influence of Gentile on our school system [1,2].Science teaching in the compulsory school can construct scientific interpretation on every-day andcommon sense experience, joining with the experience of the senses, with which each one of usspontaneously explores the world from the first moments of life [3,4]. The connection betweenscientific knowledge and knowledge based on experience is one of the main problems of learningin the scientific field [5]. Individual operativity (practical and conceptual) appears to be importantin all didactic activity, so that the educational itinerary can set off and direct a progressive andexplicit connection between cognitive dynamics and disciplinary structures [6-12]. This connectioninvolves the methodological plane, when a fundamental role in the educational process isattributed to the peer’s cooperative learning [13].Conceptual change seems to require a cognitive crisis and a re-structuring of concepts [14] bymeans of dynamic mental models inextricably linked to the context [15] in order to interpret andforesee situations and facts.It is therefore necessary to explore the ways children learn in specific fields in order to obtaincompetence in strategies which will be effective in educational path. The preparation of materialsfor a coherent curriculum of scientific education from primary school to university is the mainobjective of the national research project SeCif [16], where the Udine Research Unit has, amongits specific tasks, the basic knowledge on thermal phenomena. Therefore, studies have been carriedout on two lines: educational path have been experimented in class and specific moments ofinteraction with the children have been monitored.To support the first type of activity, work groups were organized with teachers from the primaryschools [17], who for one year worked with us in weekly meetings. Educational paths wereproposed to these teachers, to be done in class as a type of action-research in order to identify workstrategies and to study the organization of knowledge, with particular attention to the wayschildren construct formal elements in relation to thermal phenomena. The path proposed are theresult of previous didactic studies in the same phenomenological field, documented in a hypertext[18] and used to make a multi-media prototype as a didactic support [19], which was also madeavailable to the teachers involved in this study.For the second type of activity three types of laboratory were organized, 2 hours long, in the contextof a yearly activity for the diffusion of scientific knowledge [20], where the teachers find connectingactivities between didactic research and school practice.For three weeks every year, the University is transformed into a science center [21,22] andperforms various activities, which are included with research already under way with theteachers [6-11]. The use of Explorative and Inquiring Cards (EIC) in the framework of theinteractive exhibit GEI [10] and the experience of constructing conceptual maps with thechildren have taught us a great deal on the role of operativity in the construction of concepts[23] and have shown us the limits of the instruments of enquiry used to analyze learningprocesses. Together with laboratories for experimental exploration and for constructingconceptual maps, redesigned in the light of our previous experiences, a cognitive laboratory was


proposed, centering on an operative discussion with the children: Cognitive Laboratory ofOperative Exploration (CLOE). Groups of children took part in this lab, participating in thefirst type of activity and also groups not involved with this activity. In this conference we shallreport on the research associated with this type of laboratory (CLOE) which has included theuse of thermal sensors connected to the computer for activities on which to center cognitiveinterviews.

2. Formulation and objectives of the researchA change of perspective, from the phenomenology of states to the thermodynamics of processes, inthe disciplinary framework of thermal phenomena, allowed us to show that learning problems inthis field are connected to a lack of recognition of the processes, the need to distinguish betweenstate sizes and process sizes [24,25], which only occasionally is manifested in the temperature-heatdichotomy, initially identified as central [27].Our long experience in this field, in class experiments [27,28], with the GEI exhibit [10,25,27,28],and indirectly through teacher education [29], provided us with the elements to define newmethods of working and approaches which cannot be structured in a short time, but which developover a number of years, constructing the thermo-dynamic concepts starting from the processes,using on-line sensors with the computer in order to collect data in exploratory activities [25,30,31].The research which we present here is intended to explore the children’s potential while it analyzesthe effectiveness of our proposals [18]. In particular, by using on-line sensors and by means ofspecific strategies of interaction with the children, we want to test, in the field of thermalphenomena, what other research has already highlighted on children’s ability to formalize[6,7,8,9,10,32]. Our intention is to understand how the recognition of formal features facilitates theformation of concepts even in primary-school children and how they organize their knowledge ofthe phenomenological world on the formal plane too.The tools we used for the investigation with the students in the Cognitive Laboratory of OperativeExploration (CLOE) consist of some Rogers-type dialogues [33] with small groups or individualstudents, preceded by an interview-discussion with an integrated experimental exploration activityto be done with groups of about ten children. For this last activity we perfected an interviewprotocol, following our experiences in the classes [18], with questions on the concepts of a cognitiveitinerary in which it is desired to stimulate the construction and evaluation of interpretativehypotheses by means of conceptual microsteps, which in previous studies [10] were found tocharacterize children’s reasoning.

3. Context of the research and characteristics of the sampleThe CLOE activities were carried out in laboratory/classrooms equipped with various copies oflow-cost materials, for experimental exploration, a system with temperature sensors [28] able todisplay in real time the temperature values of the sensors. With an OHP and a blackboard withsheets of paper it was possible to manage and document some collective phases, for example thediscussions.In each laboratory there were at least two researchers: one conducted the activity, managing thecollective interaction with the children. At the end he noted down his comments on the way theactivity developed. The other carried out systematic monitoring, by means of audio recording, freenotes on the attitudes and significant comments by the children or on situations which would bedifficult to reconstruct based on the recordings alone. In some (few) cases, the whole activity wasvideo-recorded.In the labs, lasting about 2 hours, phases involving the whole class were alternated with phases ofindividual work or work for small groups, taking care to involve all the students.All in all, 250 children from the primary and middle school participated in the cognitive laboratoryactivities. 170 of them were involved in CLOE. In the table 1 is summarised the sample composition(117 primary school pupils).


Eight classes were involved, working in groups of 12-20 students, between the ages of 6 and 11. Incolumns 2 and 3 of the Table the numbers in brackets refer to the classes and students who werecarrying out the research experiments in class as above.

4. The interview protocol and the organization of Lab AThe interview protocol used by the researchers in CLOE, of which some parts are reported below,was organized as a grid of problematic points on situations of everyday life or operative proposalson selected scenarios from familiar contexts.

PROTOCOL FOR THE RESEARCHERInterview on situations: predictions, actions, evaluations Q: Questions A: Actions

K1 Q: How can we establish if something (an object, air, … ) is hot or cold?K2 A: Distribute the drawing of a scenario made in an experimental class (Fig. 1)

Q: What are the hot objects shown in the drawing?A: Let the objects be ordered by groups: hot, lukewarm and cold, monitoring the criteriaof order and the way of presentation

K3 A: Identify objects which can be considered equally hot: list and analyze how it is done(If a minimum of grouping is done, ask)Q: The objects which are equally hot, are they all the same?

…..K5 A: Let the objects in the tank be ordered according to thermal sensation (scissors, grain

of brass and aluminum, rubber, playdo, similar cubes of aluminum-wood-plastic-polystyrene, pencil-sharpener)Q: The different bodies give different thermal sensations. What will their temperature be?

……K6 A: Get the student to hold a sensor in his hand and discuss the development T(t). Whose

T do I measure?Q: What is the temperature of the table?Q: What is the temperature of the hand?

K7 A: Measure the temperature of the objects previously put in order.Q: Does the thermometer work? (we make sure by holding it in our hand and wemeasure the temperature of the objects again)Q: Does the information obtained with the sensors coincide with the thermal sensation?Q: In what does it differ, if at all?

We explore the ways of formalizing and constructing cognitive structures through conceptualmicrosteps wherein the everyday experience of the children is compared and connected with thatof selected scenarios. We shall discuss some of them.With the initial question K1-Q, “how do we know if an object is hot or cold?”, we explore the waysin which the cognitive connection between “being hot” and “thermal state” is get explicit. Weexamine the ways in which thermal sensation and measurement of the temperature areassociated/distinguished/identified in personal mutual experience. We evaluate if and how thestudents recognize that those identified by the individuals as thermal conditions depend on thesituation (recognition of temperature as property of state).

2783. Topical Aspects 3.5 First Steps in Formalization

Table 1 Classes Number of classes Number of students

1 - primary (6-7 years old) 1 20 2 - primary (7-8 years old) 2 (2*) 36 (36*) 4 - primary (9-10 years old) 3 (2*) 47 (32*) 5 - primary (10-11 years old) 1 (1*) 14 (14*)

*: number of classes and students involved in experiments conducted by us.

The K2 request aims at recognizing criteria and methods with which the objects are put into order.The children experience the conceptual crisis necessary to overcome the identification betweenthermal sensation and temperature measurement both by a sensorial exploration of objects ofdifferent material, mass and shape (they touch different objects placed on the table), and also by acomparison with the graphic representation of the temperature data of a sensor which is takenfrom the table and held in the hand for a sufficiently long time.The representation of the data supplied by the sensors in real time is used to study the ways inwhich processes are recognized with respect to states, and those in which they are associated withtransformations or conditions of thermal balance.We investigate the recognition of the intrinsic interaction process in measuring temperature, theinvestigation being guided by proposals to examine variations in the temperature of the sensors.The following table reports the conceptual knots on which the interview-activity is centered.


K1: Ways to establish the thermal state of a system. K2: Classification of objects according to thermal sensation. K3: Comparison of sensation and temperature in defined situations and in what they imagine they mean K4: From representation to interpretation. Intensive and extensive sizes. Recognition of

describers and transformation producers. K5: Analysis of transformations through selected situations. K6: Reading and analysis of the temporal evolution of the temperature of the human body,

short- and long-term environmental data, on small and large scale, of the transformations in the kitchen and in everyday environments.

K7: The properties of materials in transformations: sizes of state and process.

5. Data and analysis The data were analyzed on both a quantity and quality plane.The qualitative analysis was carried out in an initial step of synthetic correlations between ourprior hypotheses [6bis,7,19] on mental representations concerning thermal phenomena and waysof organizing the relative knowledge in individual situations. From the interviews we madeorganizational paths of the concepts, identifying the emerging processes of formalization.Then, weidentified individual conceptual microsteps common to the different paths.In the quantitative analysis we evaluated the use of characteristic words or expressions.The correlation between the two types of analysis concerned the groups of children in relation tothe contexts of use of the quantitative elements.Here we report only some of the data monitored, concentrating particularly on those relating tothe conceptual difficulties which the children explored most in CLOE.The children’s attitudes. All the students (95%) showed a great involvement in the operativecomparison of their own ideas with reality and in remembering their everyday experiences. Thecollective discussion was a highly motivating element. Right from the first phases more than 90%became familiar with the use of the measurement system based on sensors, as shown in previousexperiments [22].K1 – Ways to establish the thermal state of a system. When asked the question “How can weestablish if an object is hot or cold?”, most of the children replied “with our hand”, “by touchingit” (in half the cases this was the most immediate answer), others with a thermometer, others againsaid “by looking at” the object and relying on one’s own experience. It is interesting to note thatabout a quarter of the children answered by indicating how to produce the thermal state assumedas known (”I put it on the gas”, “I heat it up”). The cognitive connection between “being hot” andthermal state emerged in more than 80% of the cases, during the discussions stimulated by the firstquestion. In some isolated cases the identification of heat and temperature emerged. More

frequently, especially in the younger children, it was the association of fever and temperature.K2 – Classification of objects according to thermal sensation. The first division into micro-classes(hot/lukewarm/cold) generally occurred by indicating only one of those objects subsequentlyidentified as “hotter”. Few children (less than 10%), indicating several objects simultaneously, showa first tendency to construct isothermal classes of systems for individual elements. Any furtherclassification occurs: referring to the thermal sensation “I felt it hot” (24%), recognizing a processthat a state is produced “The cup gets hot because of the coffee poured into it” (40%). 36% do notindicate any relations of order. 19% of the children employs the classes hot-lukewarm-cold toidentify the concept of thermal condition (thermal state) from the diagnostics of the sensationsproduced by the various objects in different conditions. In the same way, they recognize that it is aproperty which depends on the situation and the history of the object, overcoming the contingentlimit of sensorial information, which calls to mind other parameters, such as thermal conductivity,the material, the mass. 24% of the children need to pass through an intermediate step to limit thedifficulties, which step consists of identifying equally hot objects.The references to everyday sensorial experience show the children’s ability to distinguish thermalstates relating to a limited range (accessible to experience): beyond this, systems in different statesare considered in a single state of “very hot” or “very cold”.K3 – Comparison of sensation and temperature. The way of exploration used leads to anunderstanding that thermal sensation is connected to a process, whereas temperature is connectedto a state (70%). There were few students who limited themselves to recognizing differences inthermal states (30%), but they overcome the dimension of sensorial relationship and recognize thatthe systems have their own thermal condition. However, they don’t overcome the cognitive crisiswhich separates states from processes. Those who succeed often also know how tomotivate/interpret the reasons for the different sensations, sometimes recognizing the role of thedifferent materials in the processes.K4 – From representation to interpretation. In exploring the process of interaction between a sensorand the hand, a great majority (68%) distinguish the three phases of thermal balance of the sensorwith the table (initial), transition (intermediate), thermal balance of the sensor with the hand(final). The other children manage to read the value of the temperature of the hand correctly, butdo not reflect on the way it occurs and think that in any case the sensor always measures thetemperature of the object it interacts with from the moment it interacts with it.K5 - Transformation analysis for selected situations. The cursor proved to be decisive with regard tothe ability to identify quantitative aspects and in particular to associate the qualitative andphenomenological elements with the data. Identifying critical moments in the evolution of thetemperature, done by only 15% of the children, becomes mastered by 95%.The autonomous abilityto refer to differences in temperature and time intervals to recognize behavior is much rarer.Observing the graph, 60% of the children can predict the final thermal state of a process. It isinteresting how it is done by 20% of the children: by looking at the way the temperature varies intime. There is, as it were, an intuitive idea of the derivative both of the first order (speed of growthof the temperature) and also of the second order (curvature). The confirmation that 72% of thechildren tend to objectualize the graph (48%) or the temperature (24%) [21] is perhaps the answerto the apparent contradiction in their not knowing how to examine individual variations, but waysof varying.This is confirmed by the 78% of the children who are able to recognize the processes from thegraph and the limited number of children (22%) who make the connection between therepresentation shown in the graph and the numerical displays.Also the understanding that thermalbalance must be achieved in order to make a temperature measurement is connected to this wayof looking at data. While the use of the cursor helps 66% of the children to overcome the missingconnection between numerical values and the graph.Effectiveness of the strategies.In order to accustom the students to seeing the numerical values associated with the individual


points of the graph, it is important for the researcher to make focused questions, as described in theprotocol. Imitative attitudes will help the students to acquire this familiarity.Using the cursor makes the children’s point of view pass from a simple recognition that thedevelopment of the data has changed, to the recognition of the individual point where the datachanges. In the interviews it is the children themselves who explain how, and confirm ourhypothesis as set out above on their privileged way of looking at data.The conceptual difficulty remains of recognizing the temperature as an autonomous parameter andas the property of a system with respect to the general identification of the system with the variableconsidered (I measure the desk!… not the temperature of the desk).In the light of the above, we therefore developed a strategy, which revealed itself to be effective in80% of the cases, which uses objectualization precisely to overcome objectualization.We report thedocumentation here in synthesis:F1/Q; Where was the sensor? A: on the table;F2/Q: So whose line is that there? A: the table.F3/Q: What were we measuring then? A: the temperature;F4/Q: So what does that line represent then?The whole class replies enthusiastically: A: the temperature of the table!

6. Concluding remarkesThe recognition of the children’s attitudes is wide and complex. We can already draw someconclusions.The practical and conceptual operativity used in the Lab acts as a motor to connect everydayexperience and scientific knowledge. Dedicated systems reduce the need for abstract thought and fororganizing hypotheses, compared with open environments. Their limit is thus overcome by the conquestof familiarity which, thanks to real-time graphic representation, produces intuitive correlations betweenthe plane of action, the plane of the phenomenon and its formalized representation.With regard to the formalization processes, we find that the concept of thermal state is structuredthrough the recognition that it can be modified with processes of thermal interaction. Thisrecognition and consequent conceptual structuring are activated in ordering and classifying thethermal conditions of different objects. The ability to put the objects in order according to theirthermal conditions implies a recognition that they are in defined thermal states.This is the cognitivepassage from the sensorial “feeling hot” to overcoming the subjective reference in favor ofaccepting the existence of a “being hot”. This conquest of an outside reality, which can be observedand described with quantitative properties, is one of the steps of becoming an adult and of knowinghow to manage observation and interpretation of phenomena: it is a requirement for doing physics.We have seen that it is activated very early and cannot be ignored without paying the price of anempirical knowledge of the world, disconnected from the knowledge taught later in physics.The conceptual restructuring through which the identification of thermal sensation andtemperature is overcome is activated by comparing the results of sensorial explorations andmeasurements of temperature when it is possible to identify the process of “getting hot”. It passesthrough the recognition that measuring the temperature of a system requires conditions of balance,the result of an interaction between the measurer and the thing measured. The role played by theability to employ some formal elements, such as temperature developments or differences, in thisprocess of restructuring seems decisive and possible with processes of objectualization.Knowing how to obtain information from a graph does not generally occur spontaneously, but therecognition of what has been done (actions) and observed (graphic representation) and the sizewhich describes the phenomenon (temperature as a function of time) helps the students todistinguish the different planes (operative, descriptive, interpretative) for the conceptualconstruction. This is shown, for example, by the majority of the children, who knew how to makethe correlation, with certainty, between deeds-actions and graph in order to recognize states andprocesses taking place.


The interview protocol describes an effective strategy in encouraging the students to read and usethe graphs, recognize elements, sizes of state and process and in this way has some didactic value.The simple operative proposals which it provides involve important aspects on the level of a firstformalization. For example picking up and holding a sensor implies associating the action ofinteraction of the sensor with the hand, with the curve of the temperature over time, associating thedata with the temperature; it also implies recognizing the non-linear nature of the process. Theresearcher, by means of focused questions (“whose temperature am I measuring?”, “What is thetemperature of the table?” – “ of the hand?”), explores how the children recognize the variousaspects shown.We confirm the already known advantages for learning which derive from using on-line sensors.Moreover, a role emerges, which was not so evident before, of the cursor as a software instrumentcapable of constructing a conceptual bridge between actions and deeds. It appears to make theconnection between the qualitative dimension and the quantitative dimension more powerful.The conceptual restructuring already identified in literature as a unique step in the development ofgeneral and dynamic mental models and of powerful cognitive networks, against the provisionalnature of contingent interpretative models, is shown to be connected to the ways in whichformalization occurs. It is associated with the individual relationships between concepts and henceconnected to the formal aspect, which characterizes such relationships. It is precisely in theparticular link between the restructuring paths and the relative methods where we think we can seethe greater complexity and the need to make deeper analyses.The methods followed in this work seem to give the expected results and can constitute a referenceprotocol for investigation.

References[1] Croce’s and Gentile’s foundations applied in the post-war period school reform.[2] B. M. Dibilio, L’evoluzione dell’insegnamento della fisica in Italia, La Fisica nella Scuola, XXIX, 1 suppl., (1996).[3] J. Piaget, Ou va l’education, UNESCO, Paris, (1972).[4] J. S. Brunner, Savior faire savoir dire. Le développement de l’enfant, Presses Universitaires de France, Paris,

(1983).[5] H. Pfundt, R. Duit, Bibliography: students’ alternative frameworks and science education, IPN University of

Kiel, Germany, (1993).[6] D. H. Jonassen, Objectivism versus constructivism: do we need a new philosophical paradigm?, Educational

Technology Research and Development, 39, 3, (1991), 5.[7] T. M. Duffy, D. H. Jonassen, Constructivism and the technology of instruction, Hillsdale, New Jersey, Erlbaum,

(1992).[8] B. M. Varisco, Paradigmi psicologici e pratiche didattiche con il computer, TD7, (1995), 57.[9] D. P. Ausbel, Educational psychology: a cognitive view, New York, Holt, Rinehart and Winston, (1968).[10] Bosio, V. Capocchiani, M. Michelini, S. Pugliese Jona, C. Sartori, M.L. Scillia, A. Stefanel, Playing, experimenting,

thinking: exploring informal learning within an exhibit of simple experiments, in New Way for Teaching, Girepbook, Ljubljana, (1997).

[11] S. Bosio, A Di Pierro, G Meneghin, M Michelini, P Parmeggiani, L Santi, A multimedial proposal for informaleducation in the scientific field: a contribution to the bridge between everyday life and scientific knowledge,European Multimedia Workshop, Lille, (1998); International Conference on Science Education for the 21stCentury - SciEd21 Book, K Papp, Z Varga, I Csiszar, P Sik eds, Szeged University, Hungary (1999).

[12] P. Guidoni, private communication; P Guidoni, Il calcolatore come strumento cognitivo: esempi e riflessionisulla didattica possibile, TD7, (1995), 33.

[13] C. Pontecorvo, La condivisione della conoscenza, La Nuova Italia, Firenze, (1993); C. Pontecorvo, A.M.Ajello,C. Zucchermaglio, Discutendo si impara. Interazione sociale e conoscenza a scuola, NIS, Roma, (1991); M.Santi, Ragionare con il discorso. Il pensiero argomentativo nelle discussioni in classe, La Nuova Italia, Firenze,(1995).

[14] S. Caravita, O. Hallden, Reframing the problem of conceptual change, Learning and Instruction, 4, (1995), 89; S.Caravita, Costruzione collaborativa di prodotti e tecnologie della comunicazione, TD7, (1995), 6.

[15] S. Vosniadou, Capturing and modelling the process of conceptual change, Learning and Instruction, 22 (1),(1994b), 45-69.

[16] SeCiF is the acronym for Studiare E Capire In Fisica, (Studying and Understanding Physics), a national researchproject financed by MURST (Ministry for the Universities) and coordinated by Paolo Guidoni. It involves thePhysics Teaching Research Units of Milan, Naples Palermo, Pavia, Turin, Udine to study materials able toactuate vertical curricula for scientific education from primary school to university; P.Guidoni et al., Explaining


and understanding in physics SeCiF: a project for correlated teachers’ formation and curriculum innovation,http://pctidifi.mi.infn.it/SeCiF/.

[17] It involved three groups from a primary school in Udine (50 teachers), in Fogliano - Gorizia (8 teachers) and inManiago (10 teachers).It also involved three groups of primary schools of about 20 teachers in the three schoolsindicated and a further group of 10 teachers in San Vito al Tagliamento – Pordenone.

[18] G. P. Meneghin, M. Michelini, CD-Stati e processi termici, (Udine, 1999); G.P. Meneghin, M. Michelini, Unipertesto per studiare e contribuire all’apprendimento dei concetti di termologia a 11-16 anni, La Fisica nellaScuola, XXXI, 1, sup., (1998), 56; S. Bosio, G. Calogero, M. Michelini , Un software a supporto dell’esplorazionesperimentale in un contesto di educazione informale, La Fisica nella Scuola, XXXI, 1, sup., (1998), 76.

[19] M. Michelini, L. Santi, A.Di Pierro, G.P. Meneghin, CD-GEIWEB, Udine, (2000); S. Bosio, M. Michelini, P.Parmeggiani, L. Santi, GEIWEB - Una proposta multimediale per l’educazione informale in campo scientifico,La Fisica nella Scuola, XXXII, 3, sup., 1999, 46; www.uniud.it/CIRD/ or www.fisica.uniud.it/GEI/GEIweb.

[20] M. Michelini, A. Stefanel, Diffusione della cultura scientifica attraverso la scuola: un’esperienza come proposta ,La Fisica nella Scuola, XXX, 3 Suppl., (1997), 48 (2 IR, 1997).

[21] AAVV, Games, Experiments, Ideas - from low cost materials to computer on-line, 120 simple experiments to doand not only to see, Exhibit Booklet, Udine, Forum ed., (1996); G Bosatta, M Bosia, S Bosio, G Candussio, VCapocchiani, D Ceccolin, L Marcolini, M C Mazzadi, M Michelini, S Pugliese Jona, L Santi, C Sartori, M LScillia, A Stefanel, Games, Experiments, Ideas from low-cost material to the computer on-line: 120 simpleexperiments to do and not only to see, in Research in Science Education in Europe: the picture expands, MBandiera, S Caravita, E Torracca, M Vicentini eds, Roma, (2000).

[22] B. N. Honeyman, Science centres: building bridges with teachers, Science Education International, 7, 3, (1996),30.

[23] S. Bosio, M. Michelini, S. Pugliese Jona, C. Sartori, A. Stefanel, A research on conceptual change processes in thecontext of an informal educational exhibit, in Research in Science Education in Europe: the picture expands, MBandiera, S Caravita, E Torracca, M Vicentini eds, Roma, (2000).

[24] M.R. Sciarretta, R. Stilli, M. Vicentini Missoni, Le proprietà termiche della materia. I- nozioni di senso comunedi studenti ed insegnanti, La Fisica nella Scuola, XXIII, 1, (1990); Le proprietà termiche dei materiali. II -Schemi di conoscenza di studenti ed insegnanti, La Fisica nella Scuola, XXIII, 2, (1990), 99; M. Vicentini, IlContesto dei contenuti: il caso della termodinamica, in Didattica della Fisica, La Nuova Italia, Firenze, cap. IV,(1996), 69.

[25] S. Bosio, V. Capocchiani, M. Michleini, L. Santi, Computer on-line to explore thermal properties, in Teaching theScience of Condensed Matter and new materials, Girep-Icpe Conference, Udine ed Forum (1996), 351.

[26] A. Tiberghien, Revue critique sur les recherches visant a éluder le sens des notations de temperature et dechaleur, Proceed. La Londe les Maures, (1983), 55.

[27] A. Loria, M. Michelini, Technological Activities in the Teaching of Mechanics - Experimenting a MethodologicalProposal: Low Cost Experiments done by Students, Proceedings of the GIREP International Conference onPhysics Education ‘The Many Faces of Teaching and Learning Mechanics’ Utrecht, (1985), 439.

[28] E. Mazzega, M. Michelini, Termografo: a computer on-line acquisition system for physics education, in Teachingthe Science of Condensed Matter and New Materials, GIREP-ICPE Book, Forum , 1996, 239.

[29] M. Michelini, A. Mossenta, The EPC Project - Exploring, Planning, Communicating; M Michelini, Thecontribution of institutions to improvement of physics teaching: Supporting scientific knowledge by structuresand curricula which integrate research into teaching; G Marucci, M Michelini, L Santi, The Italian Pilot ProjectLabTec of the Ministry of Education; R Martongelli, M Michelini, L Santi, A Stefanel, Educational proposalsusing new technologies and telematic net for physics, in “Physics Teacher Education Beyond 2000”, Girep Book(2000) and in Essevier Ed.

[30] E. Mazzega, M. Michelini, Termografo: a computer on-line acquisition system for temperature measurements inphysics education, in Teaching the Science of Condensed Matter and new materials, Girep-Icpe Conference, (edForum, Udine (1996), 223.

[31] Thermal conduction in solids: an on-line experiment for secondary school and undergraduate students, i nTeaching the Science of Condensed Matter and new materials, Girep-Icpe Conference, ed.Forum Udine (1996),239.

[32] P. Guidoni, private communication.[33] L. Lumbelli, Gestalt theory and C. Rogers’ definition of subject-centered interview, (1997).


MAKING PHYSICS FASCINATING TO….. ALL !?

Grazia Zini, Lab.di Did. della Fisica, Dipartimento di Fisica, Università di Ferrara, ItaliaAngela Turricchia, Aula Didattica Planetario, Comune di Bologna, ItaliaLeopoldo Benacchio, Osservatorio Astronomico, Padova, Italia

1. IntroductionMaking Physics interesting to people is a real challenge but worth trying for a number of reasons:many people judge Science and Physics on the basis of information given by media (TV, Internet),information often incorrect, incomplete or just wrong and on these basis they are often requestedto judge on Physics aspects that have social impact such as electrosmog, nuclear plants etc.; Physicsis perceived as too difficult to be approached by a “standard” student so the number of Physicsstudents is going down steadely everywhere; ……..and also because just trying is rewarding!Our work regards LIGHT. Why light? It is one of the most fascinating physics phenomenon forpeople of all ages, it is basically interdisciplinary: from arts to all the experimental sciences suchas geology, natural science, etc. where Physics of light play a basic role, moreover the visualapproach is today of paramount importance (TV, Internet, advertisements....)’.We havedeveloped a complete curriculum [1], based on students activities, which goes from elementarylevel to university level.The logic of the whole physics proposal is: light as perceived by our eyes vs light as revealed byman-made instruments (spectroscopes, various type of sensors, etc.) in order to present thefascinating world beyond what we see or touch, such as the stars or the electromagnetic radiationsbeyond the visible range. Here only a section is of the whole work is reported.It regards the use we are making, of a hand-held diffraction spectroscope, produced forGemmological analysis [2], to introduce visible spectra and to present light as information carrier.Hand-held spectroscopes are not new in Physics education, but this instrument is so simple that itcan be effectively handled by pupils about ten years old, and accurate enough to interest Universitystudents. Obviously goal, objective, language and Physics level should be suitable to the age of thestudents. Examples of activities made in classes of ten, thirteen, eighteen year old (and over)students respectively, are here reported. The educational validity of the reported activities wereexperimented in the last two years with primary school students, with University students, and isalso applied to the pre-service and in-service teacher formation. Obviously the activities areintegrated in different curricula. We present also the use of the same nice instrument in theinteractive section of a museum of scientific instruments [3].

2. The starting activitiesThe starting activities are almost the same for all ages and typeof audience, a step which takes 2 lessons at least, for pupilsabout ten years old and just half an hour for high schoolstudents.This first step is conceived to stir curiosity, in order to triggerinterest.After announcing that the lesson will begin with the study of

everyday-life light sources, but without giving furtherexplanation, we put in our students hands a mysterious, niceobject :“What is it ? ““Try it, look at the white light of the ceiling lamp” “Wait a

moment ! White?”.Only at this point we give the instrument name and, when suitable, explain how it works, otherwisewe present it only from an operational point of view. The procedure follows :“Use the little spectroscope to look at the incandescent white lamp, at the fluorescent white lamp,


Fig. 1: The hand spectroscope, rulerinserted for length comparison

at a burning table salt, at street yellow lamps, at the sunlight.Is white light really white?” “And colours? Look at the light transmitted by a red glass, a blue glass,an incandescent blue-light lamp, a mineral, a gem”.Here the second step: surprise which give rise to intentional learning [4].These qualitative activitiesgive operational concepts: of light source, of emission, absorption of light, of continuous and linesspectra. They give the concepts that:• our eyes are limited and might be deceiving: man-made instruments are needed to enlarge our

knowledge (they too might be deceiving!);• light carries information: of the source and of the materials it interacted with, so we have

information on substances which are e.g. too far for reaching (astronomy), too deep inside atransparent solid to be touched (defects in crystals, material science), too scarce to be perceivedby our senses (e.g. elements traces in a mixture Chemistry)........

The teacher role is to foster intentional observation and discussions from which to attain firstqualitative report, then quantitative measurement, and to introduce theory at the level suitable tothe age and the previous knowledge of audience.For young pupils the proposed experimental activities are intended to make them observe, report,discuss, make assumptions and validate (or not) them by experiment; for higher students theactivities are also the basis to main parts of optics: diffraction and the photon model of light. Thisis the beginning of a learning/teaching process which has been developed in various contexts thatuse different level of formalisation and of mathematics. We present here examples 1 in astronomy teaching2 in a lecture to Physics students 3 in the interactive section of a scientific instruments museum,

3. Examples 3.1 An application to the teaching of astronomyThe educational procedure for the younger pupils (8-13 ys old students) is the one presented in theProject “Cielo! Astronomia e Fisica per la scuola dell’obbligo”, (Heavens!, Astronomy andPhysics” [1]. The title contains a little joke, in fact “Cielo!” is used in Italy sometimes in case ofsurprise).The spectra activities takes about 2-3 hours, and it comes after other simple experiments whichenlighten the concepts of source and light sensor, and show that the light intensity transmitted bya number of exposed photographic slides diminishes increasing the number of slides.Important astronomy concepts comes out directly from the “discovery” of the difference betweenthe spectra of the light from an incandescent lamp and the sun, followed by the observation of theflame produced by a burning table salt and/or other substances.


Fig. 2: A report in the lab-book of a tenyear old child (“Elementari” school,Bologna, Italy)

Fig. 3: Observing the flame of cooking salt

In fact each pupil (without support from teacher) is able to see, with the little spectroscope manyFraunhofer lines superposed on the continuous spectrum and a very bright line on second case(only one, elas!). Discussion on these effects with the whole class and the use in class of an applet[5] which displays the emission and absorption spectra of the periodic table elements, makestudents understand how astronomers are able to deduce the composition of the stars “withoutgoing to them”.The Sun emission spectrum induces also understanding of the Earth atmosphere effect on the lightwhich reaches the Earth. [Other experiments, not in Cielo! Project, are now available to deepen theunderstanding of the effect on light by other stars atmosphere[6]].Experimentation on this activities were made during 2000/2001, in “Elementari” schools (about 650pupils, 9-10 year old) and “Medie” schools (about 150 students, 12-13 year old) [7]. Activities werein part made by the class teachers working alone, in other cases researchers were active in classduring the lessons and other times the students do the activities during a visit to the “Aula DidatticaPlanetario” of Bologna.Spectroscopes and written material about the lesson procedure were given as support to theteacher. In-service teacher formation was adopted in some cases, other teachers were able toproceed by themselves with little help, by e-mail or phone, from Physics education researchers.Evaluation of the proposal were obviously made in the usual way by written tests, open-end-question to students and discussions with students and teachers. The results were very good. Herewe report the phrasing of one of the teacher: “The results are clearly positive for my students whotook part with enthusiasm to activities and discussions. From my point of view I liked very much toteach Physics in this way; it is perhaps the first time I liked it so much”

3.2 Application in Physics lessons at High school, University and Teacher trainingThe activities outlined above were proposed also to students at the beginning of their Universitycarrier (17-20 ys old students), both Physics and non-Physics students and also to in-service andpre-service teacher training [8].The approach was almost the same as above, only the playful procedure takes less time (about halfan hour) and it is followed by the theory with the appropriate mathematical level.


Fig 4: High school student using the spectro-scope during a visit to the Physics Dept. ofFerrara

Fig 5: The “blackboard”at the end of a lessonto Physics students (Dept. of Physics- Ferrara)

The spectroscope was first used to display the spectrum of helium-neon laser light, and to discussdiffraction phenomena and related theory. Then the observation of line spectra made easy tointroduce (or revise) the photon theory. For non-Physics students an applet of a photoninteracting with an Hydrogen atom [9] proved to be very useful to the understanding of theemission-absorption model.The experimentation of the Physics educational proposal here presented, took place during the

last two years at the Department of Physics, University of Ferrara. It involved students of “Diplomadi biotecnologie agro-industriali” (non-Physics students), Physics students of the first year courseand students specializing in Physics teaching (a two year, post degree course which must befollowed by all future teachers). The qualitative approach revealed to be particularly importantsince very few student in Italy have Physics in their pre-University studies and so lack of thequalitative appraisal of the Physics phenomena, which in my opinion, is necessary when studying aPhysics phenomenon for the first time.

3.3 Application in an interactive section of a scientific instruments museum

Obviously here the spectroscopes and the educational strategies above explained can bestraightforward applied. The Scientific museum visitors are of all ages and scientific knowledgelevel so the experiments for the younger children are proposed for a public without previousknowledge in Physics, while the higher level approach is for a public with deeper knowledge ofscience. Working sheets suited for an exhibition were prepared and, obviously, proposal forsecuring the hand spectroscopes are advanced.

4. ConclusionsThe educational strategy here presented links common knowledge to science and shows oneexample of the role of man-made instruments in Science. Moreover the playful simple practicalactivities provide motivation for theory, discussion and modelling from a very early age.We can saythat the main points of the learning process are:• Light carries information• The decoding of light carried information (and its integration in our “Mind”) depends on

instruments used Our eyes gives us the first, basic idea of the visual world, that necessary for our life, Using different instruments we “see” different aspects of “objects” or different “objects”

Eyes, used to observe and analyze, and man-made instruments are for intentional learning [10] (oureyes most of the time are used for not-intentional learning). Intentional learning requires efforts,by efforts we enlarge our knowledge of world from the basic ideas and, doing so, we develop alsoour Mind. In this way instruments are seen as part of the intentional learning mind.(Mind= brain with its neural web and knowledge acquired)

References[1] L. Benacchio, G. Mistrello, M.G. Pancaldi, M.Sasso, M.G. Somenzi, A. Turricchia, G. Zini, “Cielo! Un percorso di

Astronomia e Fisica per la Nuova Scuola dell’Obbligo”, Giornale di Astronomia, 4, (2000), 31 (andwww.polare.it)

[2] http://www.gagtl.ac.uk/sinst.htm#spectroscopes Giornale di Astronomia, 3, (2000), 31“ The Project “Cielo!” isfunded by M.P.I. Project S.E.T.: http://www.istruzione.it/argomenti/autonomia/progetti/set.htm


La Sezione interattiva:(esperienze ed attività pratiche di laboratorio)

12 attività interattive con schede per il pubblico

4 esperienze per scuole superiori con schede insegnanti

La sezione museale:9 oggetti : prismi, reticoli, banco di

Melloni, spettroscopio, spettrogoniometro.

Schede di presentazione per visitatori con informazioni storiche

Fig. 6. Scheme of the museum project composed of history and interactive sections


3.6 Strategies: Methods and Tools

SWITCHING FROM EVERYDAY FACTS TO SCIENTIFIC THINKING

Pilar León, Universidad Simón Bolívar, VenezuelaMarina Castells, Universitat de Barcelona, Catalonia, Spain

1. IntroductionFirst year physics is not the same for all freshmen. Certainly each curriculum assigns a relativeimportance to this subject, and the physics content is selected because of its relevance to attain thedesired goals. In engineering courses and other technological careers physics plays a subordinaterole. Physics knowledge is directed to practical applications, often without time enough to foster adeep understanding. Therefore, the teacher is usually under time pressure to cover a great amountof content and to provide the students with a repertoire of skills. Besides it, lectures prevail inuniversities with large enrolment; despite the fact that many other teaching strategies areconsidered more effective. So, plain lectures to large groups are programmed, but there is still littleresearch on the communicative skills used by teachers in their lectures.The aim of this pilot study was to investigate the processes by which the scientific knowledge ispresented to students attending lectures, and how formalisation is intended using a multi-representational discourse. In short, our focus was on communicative processes in lectures.

2. BackgroundLanguage has been seen as a full medium to give adequate expression for everything that needs tobe expressed. Some efforts have been made to explain how written and spoken language are usedto give accurate descriptions of physical phenomena (Talmy, 2000). But written and spokenlanguage is not enough in the classroom. A better approach to language awareness has forced tolook at language as a multiplicity of quite distinct semiotic resources (Kress et al, 1998). Therefore,when teachers try to communicate scientific concepts and processes to their students, they need touse other resources as gestures, drawings, graphs and equations that contribute to give adequateexpression to their explanations (Lemke, 1998; Jewitt et al, 2000). The use of this multi-representational discourse (van Dijk, 1985) in scientific explanations must be directed to make thesubject matter more accessible, understandable and manageable to students.There are differences between the language used by the scientific community and the languageused in teaching physics. Thematic patterns to be taught are not original because they must beconstrued with reference to accepted science knowledge. In consequence physics teaching requiresa careful analysis of the subject matter, to produce a versioned form adapted for teaching in a givenlevel (Ogborn et al, 1996). That is, a didactic transposition. As a result of such transformation,linguistic and other symbolic resources used by physicists to construct entities, processes, states,relations and outcomes (Ochs et al, 1996) are reviewed, restructured, and fashioned in a suitableform to promote learning. When teachers teach an inverse process is used: everyday facts andprevious knowledge are invoked to re-build scientific knowledge in a coherent way from thestudents’ point of view.In this work we focussed on the meaning-making process supported by the teacher in a currentlecture on physics. Using the results of the analysis, we tried to identify both semantic relationshipsestablished along the lecture and summarise the thematic patterns built by the teacher through hismulti-representational discourse.

3. MethodFrom the ontological point of view we were interested in teaching as it occurs in natural settings.So we adopted a qualitative approach to carry out a case study addressed to a deep understandingof the process under study. Three different sources provided the information gathered by the

researcher: 1) Video-recording and field notes from direct observation of the whole lecture. 2) Thephysics syllabus currently used, and the textbooks as a valuable way to establish the didacticsequence in which the lecture was inserted. 3) An informal interview with the teacher to inquireinto his teaching approach and what he considered the most important features in his lectures fromthe point of view of the subject matter content.The research was carried out within the usual lectures in the middle of a four-month termaddressed to architecture students. The topic of the lecture belonged to first year Physics Programspecially designed to reach the specific aims of the technical focus given by the institution. Thelecturer had been previously contacted and informed about the purpose of the study and he hadagreed to collaborate with us.After a brief talking with the teacher about the arrangements needed to record the lecture, weproceeded to the observational stage of the research. A ninety-minute lecture was video recordedusing one fixed camera, settled in front of the blackboard and among the students to appreciate thelecture as the audience did.The researcher, as a non-participant observer, took field notes to recorddrawings, graphs and equations not clearly visible in the video. A few months later the informalinterview with the lecturer was held. The purpose behind it was to inquire on how he faces his ownteaching in this particular context, while avoiding a probable bias created by the proximity of theobservational stage.Subsequently, the lecturer’s speech was integrally transcribed jointly with the other communicativeresources used by the speaker. In this way the interactions between the different communicativeresources were accurately registered.

4. Results and discussionWe centred our analysis on the development of the thematic patterns of physics, that students mustmaster and use when reasoning about specific contents (Lemke, 1990). So, we went through thevideo and its transcription searching for the main strategies used by the teacher to build patternsof semantic relationships.Repetition of key terms with different emphasis each time prevailed, and the main features of thecontent were highlighted by a reiterative use of naturalistic drawings (Kress and van Leeuwen,1996).A basic drawing, a horizontal rod supporting two loads near its extremes, was used every timeeither to expose a new facet of the phenomenon or to make direct linking among real facts, theirconceptual interpretation within physical knowledge, or their mathematical formalisation.This variety in presenting the main thematic concepts in the lecture was carried over, not only fromone part to another in the lecture but from previous and forthcoming knowledge, both in physicsand in other subjects. Besides, semantic relationships were articulated around three main topics:elasticity of materials, internal effect of loading and local deformations.At the beginning the teacher made an introductory link to the thematic sequence and, at the sametime, he began to talk about the discourse itself (metadiscourse):«now we are going to continue with the theme we began the other day».This linking to previous activities gave thematic continuity to the teaching going on. But metadiscoursewas also used as a mark or signal in the discourse, putting forward relevant points that would be theelements to be taking in account for the students in their process of meaning-making:«Our working methodology is going to search for normal stresses (...) we will connect them with theflexion moment (...) we will use the geometry (...) to construct a function that is able to describe thecurvature of the rod».A core term, «flexion model», was a constant along the speech acting as a pivot in the teacher’sdiscourse. Rhetoric and semantic connections were made using analogies from ancient and modernarchitecture provoking implicit relationships: a strategy directed to transferring semanticrelationships and thematic patterns, already known by the students, to new relationships andpatterns the teacher was trying to build up.


Meaning-making and formalisationRelevant elements of the transcript were extracted to analyse the process of meaning making. Wefocused mainly on how premises were stated through a narrative, and on how the argument wasdeveloped1 to achieve the formalisation.The process of meaning making began when the teacher put in context the topic to be taught:«After our talking about the moment of inertia we are going to talk about flexion (...) using a modelrestricted to avoiding additional stresses as shear stresses».In this way, the teacher engaged the topic in a didactic sequence, giving to the students a sense ofcontinuity and a useful linking to the forthcoming argument. On the other hand he was implicitlyanticipating that `moment of inertia’ would be an important part in his explanation.Further he made more drawings, pieces of the curved rod, that he used to build new explanations,giving material support to the premises of the argument: the assumed restrictions of the model. Atthis moment a closing was done by the teacher, who passed to discuss similar situations in ancientarchitecture.Here gestures were the most predominant mode used to communicate the behaviourof stones, beams and rods supporting weights.A coming back to the main line of the argument was done, marked by a change in the intonationand rhythm of the voice. A new and condensed version of the effect produced by loads was made,emphasising the inner behaviour of the fibres into the rod:«...once we had applied the weights, the beam was deformed, its fibres made a rotational movementaround an axis (...), perpendicular to its length (...), in this way the plane, initially a vertical one, hasbecome an oblique plane».This speech would be senseless without the gestures and drawings made at the same time. Really,in this part of the explanation, verbal language had a subsidiary role because the argument reliedon visual resources. Figure 1 shows the original transcription matrix where the speech, drawingsand gestures used at the same time were registered.Then the teacher began to build a set of mathematical hypotheses:

290 3. Topical Aspects 3.6 Strategies: Methods and Tools

SPEECH GESTURES DRAWINGS COMMENTS

Then, once we have applied theload, the rafter was deformed andit has a rotational movement.

And this rotational movementtakes place around an axe,perpendicular to the rafter, that,we will draw later,...whichcoincides with the neutral fibre.

This plane, initially vertical, hasbecome an inclined one.

Points to a transversalsection in a previousdrawing that representsthe deformed rafter in 3Dand shows how thevertical plane was rotated.

He draws an inclinedplane, points to the Z-axisand uses one hand toindicate the inclination ofthe plane

.

Y

Z

X

Visual language prevails:

The information isorganised by the interplay oftwo drawings (a previous aand a new ones), and themovement of his hand..

Here, verbal speech, gestureand drawings are combined,but verbal language is usedas a complement of thevisual language (gesture anddrawings).

Figure 1. Beginning of the formalisation process in this lecture

Record of the speech in the original transcription matrix. In the «gestures» column, the observed gestures aredescribed in the same order as they were made; the inclined plane in the «drawings» was the main visual resourceto which the teacher came back many times in the subsequent explanations; «comments» reflect field notes.

1 We are using the terms ‘narrative’ and ‘argumentation’ in the Aristotelian sense. That is, for us narrative refers to theformulation of the premises for the argument, which in turn is the process by which the transference from premises toconclusions (in this case, key concepts to be learned) occurs.

«...given any curve -of any size- I can always find a circumference that approaches the curve asclosely as we wish around a given point (...) if that point is on the neutral fibre, we would assign aradius of curvature to that fibre».This piece of speech was the announcement of another kind of discourse, in which mathematicallanguage prevailed over verbal, gestual and graphical forms that dominated the communicativeprocess in the first part of the lectureThen the lecture was developed almost exclusively on mathematical language. After that, he tookadvantage of the relationship between length variation and the properties of materials, to speakabout the behaviour of materials subjected to tension and compression.Then he construed a general equation describing the curvature. At this point, a new change in thediscourse was done to show how the found equation would be used to solve specific problems inarchitectonic design.

5. Conclusions In this analysis and discussion we could identify how new meanings were construed using adynamic interplay among multiple resources.We have grouped our conclusions under the followingheaders:

General features of the lectureThe general strategies most frequently used for the development of thematic patterns were: (a)metadiscourse to introduce and to guide the process of meaning-making; (b) drawings and gesturesas visual support to move from real facts to its scientific understanding.• The general strategies most frequently used for the development of thematic patterns were: (a)

metadiscourse to introduce and to guide the process of meaning-making; (b) drawings andgestures as visual support to move from real facts to its scientific understanding.

• The meaning-making process was carried on as a back and forward process, proving links tocomplementary explanations, and coming back to the main line of the argument.At the same time,such complementary explanations were kept, incorporating them in the conceptual framework.

• The lecturer assigned various specific purposes to the new meanings, but purposes appearedmixed because of their overlapping. For this reason, we will only say that some of them take partof further explanations or were used to get a better understanding of the world around us.Sometimes new meanings were grouped to support formalisation, and other times they wereused to illustrate practical problems.

• Communicative resources interchanged their roles along the lecture in a complementary way.According to their semantic power verbal, written, graphic or mathematical resourcesfunctioned sometimes as the main expressive resource, and other times as support for the others.

Characteristics of the lecturerAnalysing the whole performance of the lecturer we could identify some aspects of didacticinterest. In this case the lecturer is a physicist with theoretical physics background, teaching formany years in the same faculty. Despite an observed tendency to give canonical explanations to thestudents, he frequently went back in his discourse linking different concepts. His efforts is centredaround the elaboration of the ideas he considers fundamental to achieve didactic objectives So heattempted to trace a clear pathway from real life to conceptual knowledge. In the interview it wasclear that his focus was to teach a kind of physics that could be useful to the students.The teacher’s whole performance was coherent with his ideas of being explicit about which wouldbe key ideas in his teaching, and the analysis of his lecture shows that indeed he made this essentialfor the ultimate goal of his lecture.

Implications for educational practiceWe feel that this kind of research may be important to show the possibilities of lectures to fosterdidactic objectives consistent with current theories on effective teaching. In the observed sequence,


scientific and everyday facts were repetitively described by means of different communicationalmodes, a strategy consistent with a constructive approach to teaching, since giving multi-representational descriptions attends to individual differences in meaning-making. This is animportant result because for a long time it has been assumed that teaching through lectures isopposed to a constructivist point of view, dismissing its potentiality to do good teaching despite itsmassive audience.Communicative effectiveness is an important step in teaching. Not in order to transmit knowledgebecause knowledge is a personal construction of each learner, but to communicate the importanceof the subject to be learned and its main features.It is well known that teachers’ approaches to teaching are influenced by contextual factors such asclass size and faculty efficiency criteria (Prosser and Trigwell, 1997). This is especially true inuniversities where staff conceptions about teaching are centred more in their own academiccapabilities and the institutional frameworks requirements of their instruction (Bos & Tarnay,1999).We believe that now it’s the time to promote a new approach to the teaching of large groups,and faculties’ staff should be aware of the necessity to develop communicative skills as animportant facet in their qualification as academic lecturers.

Future researchThe analysis of the gathered data gave us some insights about future research on communicativeprocesses in lectures:• Thematic analysis is a useful tool to identify relevant features of different teaching styles in

lectures. It isn’t an easy challenge in high level teaching due to the complexity of the contents tobe developed in a short time. Therefore, we need to search for new tools or improve the existentones in order to deal with the complexity of teaching in natural settings.

• It is necessary to do a broad study on lecturers’ individual differences in their explanations, andthe way in which students perceive such explanations. This kind of study should lead to a betterunderstanding of the effectiveness of the different communicative strategies currently used inhigher education.

6. Final commentPlain lecturing is very popular but at the same time it is heavily criticised. One critical commentsays that lecturing consists in a transference from teacher’s notebook to students’ notebooks withoutpassing through the brain of neither of them. Others point to some well-known facts, e.g. that some ofthe most learned faculty members are bad teachers. And so forth.However, the very fact is that in a high number of cases, teaching consists of broadcasting with nocommunication, which makes imperative more investigation to improve communication in teaching.

ReferencesBos W. & Tarnay Ch., University faculty members and students perceptions of university academic systems. Int. J. ofEducational Researc., 31, (1999), 699-715.Jewitt C.; Kress G.; Ogborn J. & Tsatsarelis C., Multimodal teaching and learning. Lecture for the Research Seminarat the University of Barcelona, February, 4-5, (2000).Kress G., Ogborn J. & Martins I., A satellite view of language: some lessons from science classroom, LanguageAwareness, 7, (2/3), (1998), 69-89.Kress G. & Van Leeuwen T., Reading images. The grammar of visual design, London: Routledge, (1996).Lemke J., Teaching all languages of science: words, symbols, images and actions, (1998).http://academic.brooklin.cuny.edu/education/jlemke/papers/barcelona.htmLemke J., Talking science: language, learning and values. Norwood, NJ: Ablex Publishing, (1990).Ochs E., Gonzalez P. & Jacoby S., «When I come down I’m in the domain state»: Grammar and graphicrepresentation in the interpretive activity of physicists. In Elinor Ochs, Emanuel Schegloff & Sandra Thompson(eds.), Interaction and grammar, Cambridge, Cambridge University Press, (1996).Ogborn J.; Kress G. Martins, I. & Mcgillicuddy K., Explaining science in the classroom, Buckingham, Open UniversityPress, (1996).Prosser M. & Trigwell K., Relations between perceptions of the teaching environment and approaches to teaching,British Journal of Educational Psychology, 67, (1997), 25-35.Talmy L., Toward a cognitive semantic, Vol. I, Concept structuring system,. Cambridge, The MIT Press, (2000).Van Dijk T., (Comp.), Handbook on discourse analysis, London, Academic Press, (1989).


ENGLISH AS A MEDIUM OF INSTRUCTION IN SCIENCE-TEACHING

Claudia Haagen-Schützenhöfer, Leopold Mathelitsch, Institute for Theoretical Physics,University of Graz, Austria

1. What is EMI?English as a Medium of Instruction is a recently developed bilingual teaching-method. The mainidea of EMI is to combine the conventional instruction of content-area subjects with foreignlanguage-learning: instead of the mother-tongue, a foreign language is used as a ‘tool’ forcommunication in different subjects.The concept of EMI was promoted by the Austrian Ministry of Education to improve and intensifyforeign language education at schools. During the last decades powerful communicationtechnologies have been developed and improved. This quick and easy way of getting in touch witheach other has led to the establishment of numerous international co-operations in different fields.However, one prerequisite for the functioning of global communication is a common languageunderstood by everyone. English has become this ‘lingua franca’ in science, as well as in technologyand in economics.EMI was designed as a flexible concept which can be used in all subjects as well as in all forms andtypes of school at the level of secondary education. Varieties of EMI in use range from sequentialinstruction through the medium of a foreign language in one subject to continuous instruction withEMI in a majority of subjects [1].So far this bilingual concept has been successfully introduced in several Austrian schools. Thenumber of schools with EMI in their program is not even able to meet the rush for places in EMI-classes [2].

2. Physics taught in English – a means to “horrify” pupils even more?Recent international developments increase the need for workers with a well-founded expertknowledge and a profound competence in English. Pupils instructed with EMI fulfil theserequirements, since they show an enhanced flexibility in foreign-language communication. Incontrast to pupils with conventional foreign-language instruction, EMI-pupils are able to talkabout a large variety of subject-specific topics [1].Next to its linguistic advantages EMI also influences pupils’ subject-specific skills and abilitiespositively. First of all, the instruction of Physics through the medium of a foreign language requiresthat complicated subject-specific topics are explained more slowly and on a simpler linguistic level,which often contributes to a deeper subject understanding. As information given by teachers isfiltered through the use of a foreign language, it takes pupils longer to process information.Therefore compact units of information need to be split up in smaller sub-units. Furthermore thesesmaller chunks of information need to be encoded into simple language exponents in order not toovertax pupils. Finally maximum input is achieved by using additional non-verbal representationssuch as pictures, graphs, animations and hands-on.Secondly, the concept of EMI favours the use of various authentic teaching materials, which oftengive different access to a topic [1] and thus broaden the repertoire of adequate problem-solvingtools. Foreign language materials often show a topic from another point of view since they arebased on a different scientific and epistemological tradition due to a different historicaldevelopment.Thirdly, the use of a foreign language frequently avoids the confusion between everyday conceptsand scientific concepts. One major problem in the instruction of Physics is, that pupils can hardlydistinguish between the meaning of a term denoting a concept acquired through everydayexperience and that denoting a scientific concept. Since pupils acquire their mother-tongue longbefore they are instructed in Physics it causes big problems for them to relate a later acquiredscientific concept to an already familiar term.As a result this mixing up of meaning frequently leadsto inadequate ideas about physical phenomena. The EMI method however makes it easier for


pupils to acquire scientific concepts as there is usually no temporal gap between the acquisition ofa term and the matching scientific concept.Finally the instruction of Physics through the medium of a foreign language often increases pupils’motivation. On the one hand, pupils find the subject Physics much more interesting since differentteaching-methods like team-teaching are used [3]. On the other hand especially pupils who like thesubject English better than Physics find the EMI method motivating. In EMI lessons English isused as a medium for the exchange of subject-relevant information in contrast to the often artificialcommunicative situations in the language-classroom which are dominated by pseudo-communication.

3. Physics taught in English – the “ultimate cure” for the Physics-crises?EMI may be one method of making Physics more attractive to teenagers but it is surely not the solecure for the negative image of the subject. Since EMI is a quite new method there are still somemajor problems which must be faced.First of all the successful application of EMI depends on the ability and motivation of pupils.Experience shows that especially pupils with bad language abilities - in the foreign-language as wellas in their mother tongue - become easily overtaxed and frustrated.The shortage of science teachers who have a profound knowledge of English poses anotherproblem. EMI is frequently used in subjects like History or Geography. Subjects like Physics orChemistry however are seldom taught through the medium of a foreign language [2]. The mainreason for this is the frequency of a certain subject-combinations. In addition there is a lack ofcontinued education [4], which would prepare science teachers for the instruction with EMI.Another reason why many teachers refuse to use the EMI-method during their lessons is that thereis a shortage of adequate teaching-materials [2]. American or British school books can usually notbe applied for teaching with EMI as either their language is too difficult and thus frustrates pupilsor, taking the appropriate language-level, their scientific content is far under pupils’ cognitive leveland thus bores them.

4. Physics taught in English – examples of EMI-lessonsThe lack of teaching-materials is a main reason why teachers hesitate to use EMI. In order tosupport Physics-teachers using English as a Medium of Instruction we have developed a collectionof teaching materials suitable for Physics classes at the secondary level. These materials could beeasily adapted for mother-tongues other than German.Each of these thematic units, whether designed for lower or upper secondary forms, is similarlystructured. A working-text dealing with a certain physical topic is the basis for every thematic unit.As far as the working-texts are concerned we decided to choose various text-types which differfrom conventional school-book texts. Historical texts by Newton or Faraday were included in ourcollection of teaching materials as well as comics, texts of children’s books and recent popular-science publications.The first step of every EMI-unit is the pre-reading stage. The function of pre-reading activities isthe arousal of interest and motivation as well as the introduction of the topic [5]. In the course ofthese introductory exercises pupils are asked to brainstorm their ideas about a given text.The mainaims of this phase are to get an overview of pupils’ knowledge of topic-related vocabulary and tomake them aware of their pre-concepts acquired through everyday experience.This can be done bybrainstorming activities which use pictures or titles of working-texts as stimuli. Pre-reading is a firststep to promote student’s conceptual understanding of a physical content through activelyengaging them into the process of creating knowledge.In the second step pupils are supposed to read through the text on their own. During this stage theyare asked to highlight words and passages of the text which they do not understand. Each textcontains also a few explanations of words which pupils might not know but which play a centralrole in understanding the text.


The next step is concerned with Vocabulary-Work.The aim of this stage is to broaden pupils’ activeknowledge of vocabulary [5] and to construct new knowledge. In different types of exercises pupilsare asked to guess the meaning of given words or phrases from the context and to use themadequately. During this phase pupils work on their own or in pairs. Finally their results arecompared in class. In the end the meaning of all remaining passages which are highlighted isdiscussed. In the course of this stage students build up a net of semantic relations in which physicalconcepts are represented by terms in the foreign language. The main advantage is that these termsare newly introduced and that they do not denote already familiar non-scientific concepts acquiredthrough everyday experience like terms in the mother-tongue often do.The aim of the next stage is to check whether pupils have understood what they were reading [5].For pupils on the upper secondary level this step is subdivided into one part focusing on linguisticcharacteristics of the text and a second part which checks their understanding of the underlyingphysical phenomena. At this stage pupils work again in pairs or in groups of three. Finally theresults of the exercises are discussed in class.The last part of every thematic unit contains a German parallel-text which summarises the physicalphenomena discussed in the working-text. These parallel-texts provide additional backgroundinformation about the topic treated on an adequate linguistic level. At this stage the Englishdenotation of physical concepts is related to its equivalent in the mother-tongue. Finally pupils’recently acquired physical concepts are compared with their ideas stated during the pre-readingstage. The function of this part is to make pupils aware of how their concepts have changed andhow their pre-concepts differ from scientific valid concepts.Units for classes on the lower secondary level are completed with simple students’ experiments.These practical exercises show the effects mentioned in the working-text in the context of everydaylife.

5. Empirical study - how does EMI influence practical lab-work?This empirical study was carried out at a Grammar School in Graz where pupils were alreadyfamiliar with bilingual instruction in content-area subjects [3]. When this survey was done in thesummer-term 2000 the pupils were in the seventh form of Grammar School, this is one year beforeA-level. Their Physics lessons on the topic ‘Electricity and Magnetism’ were divided into two parts,into theoretical instruction and practical lab-work. Divided in groups of three, pupils had to readthe instruction of an experiment, arrange the listed components as shown in a circuit diagram, carryout the experiment and make lab-notes of their observations.Although pupils were used to bilingual lessons they worked with instructions in German due to thelack of adequate authentic material. This situation provided the opportunity to investigate howtheir performance changes if they worked with experiment-instructions in English. In the course ofthis empirical study the lab-teams were divided into two categories: in a group who continuedworking with instructions in German, called E— group, and into a group who worked withinstructions in English, called E+-group. In order to find out how the language of instructionsinfluences the practical lab-work, pupils of both groups were asked to fill in a questionnaire. Theaim of this questionnaire was to collect pupils’ personal opinions and experiences. In addition bothgroups’ subject-performance was compared with the help of their individual lab-notes. Finallypupils’ behaviour was observed to get an external impression of how the language of instructionsinfluences practical lab-work.The first item-complex of the questionnaire asked which steps of the workshop were made moredifficult because of an instruction in English/German: the identification of components and deviceslisted in the instructions, a quick and correct arrangement of the experiment or the understandingof the theoretical background of the experiment. All E+-pupils agreed that none of the mentionedsteps was more difficult because of the instruction in English. Most E—pupils claimed that instructionsin German did not make a big difference. A minority of E—pupils however thought that it made theidentification of components and the quick, correct arrangement more or less difficult.


In the second question-complex pupils were asked if the use of instructions in English/Germancaused general language problems. The majority of the E+-group answered that there were no oronly small language problems which were easily solved. E—pupils stated that instructions inGerman did not cause serious language problems.The comparison of individual lab-notes confirmed the results obtained by questionnaires. As far asPhysics related results are concerned, there was no significant difference between the E+-group andthe E—group. However, as far as language is concerned, the E+group clearly outperformed the E—group.In addition pupils were observed during their practical lab-work in order to get an externalimpression of their performance. First of all their reactions to the new class-room situation was asubject of investigation. At the beginning most E+-pupils reacted negatively. But soon E+-pupilsperceived that the use of instructions in English did not influence their work negatively. Finallythey even asked for instructions in English. Secondly their task-performance was observed. E+-pupils did not have noticeable difficulties in carrying out experiments.On the whole, the use of English instructions in practical lab-work did not lead to significantdisadvantages for the E+-pupils; as far as language is concerned they had an even better knowledgeof technical terms in both languages.

6. ConclusionEnglish as a Medium of Instruction is an innovative bilingual concept which influences foreignlanguage abilities and cultural awareness positively. In the future, the method of EMI will spreadeven more and prepare young people for a cosmopolitan social and professional life. As EMI is aquite new method, scientific research concerning the instruction of Physics through the medium ofEnglish is at the very beginning. There are several questions which cannot be answered withscientific certainty so far. Therefore it is necessary to carry out further studies to ground thisbilingual concept on a scientific solid base.

References[1] G. Abuja, D. Heindler (eds.), Englisch als Arbeitssprache – Fachbezogenes Lernen von Fremdsprachen. Graz:

Zentrum für Schulentwicklung, BMUK, (1993).[2] K. Oestreich, G. Grogger, Der Einsatz einer Fremdsprache als Arbeitssprache in nichtsprachlichen Gegenständen

– Ergebnisse einer bundesweiten Direktorenbefragung an Schulen der Sekundarstufe. Graz, Zentrum fürSchulentwicklung, BMUK, (1997).

[3] E. Svenik, Zur Evaluation neuer Lernstrukturen an der Graz International Bilingual School, Graz, Zentrum fürSchulentwicklung, bm:bwk, (2000).

[4] S. Grangl, “Survival of the fittest ...? Lehrerfortbildung für EAA“, In: English als Arbeitssprache – Modelle,Erfahrungen und Lehrerfortbildung, Graz, Zentrum für Schulentwicklung, BMUK, (1998).

[5] F. Grellet, Developing Reading Skills, Cambridge, Cambridge University Press, (1981).

DEVELOPMENT OF FORMAL THINKING ON KINEMATICAL ASPECTS OFMOTION FROM CHILDREN KNOWLEDGE TO EARLY MATHEMATISATION

Gagliardi Marta, Grimellini Tomasini Nella, Pecori Barbara, Physics Department, University ofBologna, Italy

1. Physics, knowledge and culture: the basic assumptionsPhysics can be looked at as the «cognitive game» of building links among three elements: naturaland technological systems (what physics explains), laboratory experiments and data reduction (thetools for physics investigation), a set of laws, models, principles and theories (the «shapes» physicsuses to represent reality). Like other more trivial games one cannot really understand how thegame works without playing it. Besides, the most effective way to understand the «specializedgame» of physics is to play it consciously as a specialisation of the more general game -peculiar to


the human being- of constructing intellectual tools for understanding and interacting with reality.In this contribution we shall try to exemplify how school teaching can contribute to thedevelopment of such intellectual tools by describing an ideal path from children knowledge aboutmotion to early mathematisation of kinematical concepts. The path is based on the assumption thatthe aim of compulsory education should be the construction of individual culture by every student.We agree with the statement that culture is what remains for the life when «school knowledge» hasbeen forgotten and we think that knowledge becomes individual culture when it is consciouslyacquired, it can be communicated and defended and it is used as a tool for looking at one self andthe world around in order to make judgements and choices and to take decisions in variousmoments of one’s life.In order to analyse what this assumption implies for teaching physics at school, we must thereforelook not only at the pupils who attend school lessons but also, and mainly, at the adults they willbecome, and, in particular, at those who will not make of science their professional activity. Fromthis perspective, rather than to accumulate a stock of scientific notions - that will be soon forgotten- it is most important for them to understand the «game» of constructing inter-subjectiverepresentations of some aspects of reality on which physics knowledge development is based, to beable to identify similarities and differences between this game and those peculiar to otherdisciplines, to understand why human beings felt the need to start and pursue this game, to beaware of its limits and to be able to use this kind of conscious knowledge when necessary.

2. Physics knowledge as culture: the research problemResearch in cognitive psychology has pointed out the crucial role of two factors in the spontaneousprocess of knowledge construction peculiar to every person: interaction with objects andphenomena (Piaget, 1955) and social interaction (Vygotsky, 1966). Our research hypothesis is that,in order to enhance the process of construction of one’s own culture, school learning must fit inwith the spontaneous process of knowledge development, mediated by common culture, bypromoting the process of conceptual change towards scientific knowledge (Grimellini et al., 1992).School learning has to be integrated into the system of cognitive interaction with reality that eachindividual develops during his/her life (Arcà et al., 1983,1984; Guidoni, 1985). Besides, this processmust be explicitly recognised by the pupil and scientific learning must appear step by stepmeaningful in itself to the learner and not only to the teacher (Bonelli et al., 1999).Someone might ask: Is all this really feasible at school? What does it imply in practice?Several long-term research projects, based on these assumptions, investigating a range of pupils’ages from 4 to 14 years and concerning different content areas, have shown that the researchhypotheses can be put into practice.In order to document how this process can be started from the very first years of schooling and whywe believe it can be most efficient if started so early, we have chosen to analyse the transition frompatterns of common sense knowledge (Piaget, 1946a, 1946b) to patterns of scientific knowledge(Gagliardi et al., 1989; Gagliardi et al., 1999; Gagliardi and Giordano, 1993) concerning theinvestigation of motion and in particular the description of motion.How impressive the transition appears to be is pointed out in figure 1 where we tried to sketch acomparison between the two alternative perspectives in looking at motion. In the followingsections we shall illustrate possible paths of development of pupils’ ideas based on research resultsdocumented in the literature. The development of pupils’ ideas will be described following twoaspects: the patterns of motion and the rate of motion.

3. From ways of moving to elementary patterns of motionHow can children be involved in the study of motion from 4-5 to 6-7 years of age? According toour assumptions we need to exploit the spontaneous activities of children of that age. Therefore wecan put forward all kinds of play that involve looking at/generating/reproducing different kinds ofmotion (of the human body, of animals, of toys and all sorts of objects). Also the fun of building


things can be used by encouraging the children to build objects (e.g.: with Lego materials) that canbe put in motion by hand, by loaded springs, by battery supply.… By asking the pupils to observe and discuss the different motions and to produce/re-producedifferent representations of them (by words, drawings, cartoons.., see fig 2), we can encourage theconstruction of an early socialised knowledge of motion. Also virtual animals like the Logo turtlecan be very useful at this stage for starting to think about the link between motions and motionrepresentations.What can children gain from these activities? They can begin to identify, separate and correlatethree kinds of patterns: «of the objects», «of their motion», «of the causes/conditions that makethem move». These are elements hard to be separated at this age level whereas their identificationis fundamental for the development of a description of motion. At the same time a process ofidentifying elementary aspects of motion within a global view of the motion phenomena can alsobe started by encouraging the pupils to separate and correlate different ways of moving of differentparts in a system (the wheels turn; the car moves forward).At 7-8 years of age, unlike before, activities can start to be perceived as part of a global project of«study of motion». Balls, screws, rings and all sorts of toys can be set in motion in all the ways one


Fig. 1: Comparing common sense knowledge and scientific knowledge.

MOTION vs. MECHANICS

Of a child, of a snake, of the sun, of a ball, of the river, of a flag, of leaves, of the train, of blood, …

Of a point mass, of a rigid body, of a system of points, of a fluid, …

To walk, run, fly, roll, crawl, fall down, bounce, swing, swim, jump, wave, …

Translation, rotation, compounded motions, stationary motion, oscillations…

Forward/backward, to the right/to the left, up/down, straight, bent, curved…

In a Cartesian reference frame (in 1, 2 or 3 dimensions), in a polar reference, on a fixed trajectory, point-by-point; x(t), r(t), (t), s(t), …

Slow, fast, normal, very fast, faster and faster, at snail’s gallop, …

V(t), V(t), V(s), V(s), a(t), a(t), a(s), a(s), d /dt, d2 /dt2, …

On the ground, in water, on a waxed floor, on tracks, …

With/without friction (contact, rolling), in vacuum, in a medium, …

By yourself, engine powered, electrically powered, with a push, by its weight, uphill/downhill, …

With no force acting, in a force field, with torques, involving transformation of mechanical energy , with energy dissipation, with momentum variations, …

Fig. 2: Drawings by infant school pupils with their comments written by the teacher.

can think of, but now we can also just think about objects that move or can be moved in wayssimilar or different from the ones observed in the classroom. Pupils can begin to identifyelementary patterns of motion (rotation, translation) and compounded ones, movements thatalways keep the same pattern and movements that change while they happen.The pupils’ drawings and descriptions become now very rich (see fig. 3) if compared with those ofinfant school pupils.


Fig. 3: Drawings and comments by a 4th grade pupil.

But even more can be done: one can start to distinguish between the motion of an object and thatof different points on it: a top, a screw, a ball all turn and move forward but they do not move in thesame way. In order to describe in what they differ one needs to find out the «right» things to lookat.All along the path the pupils recognize the necessity of a language (e. g., like that of geometry)suited to describe what they see. Sometimes it will be constructed on purpose, sometimes it will besimply used. In any case, the power of specialized languages becomes evident and can beappreciated.

4. From races to speed comparisonRaces on pre-definite tracks, according to rules that may change from race to race, are theprivileged context where to realise the need for finding tools to describe the «rate of motion».At infant school level it is important to criticise the stereotype statement «the faster one arrives firstand wins» which is taken as true by children of this age whatever the context of the race might be.It can be discovered that what happens depends on the type of path (its shape and length) and bothlength and time must be taken into account to decide about speed. Ways of comparing the lengthof different paths and the time to run along them can be found, still non quantitative methods butadequate to judge in terms of equal/different, more/less. These methods do not provide a solutionin all situations: pupils learn to identify when we can judge whose speed is greater and when wecannot.

A way to bounce is goingforward, rolling andbouncing forward. Theother is to bounce on its selfwithout going forward. Theheight of the ball gets lowerand lower, so it is as if thespeed grew, but it gets slowinstead. This happensbecause the ball gets lowerin height, so it is as if itgrew but it doesn’t.

Perhaps we are not allowed to call it «physics learning» but certainly we must recognize it is«culture» that supports the comments made by a group of children after one year of work aboutraces (see figure 4). They clearly show to be able to use their knowledge as a personal tool forinterpreting reality.

«The smallest wheel is faster because it makes more turns.»«No, it’s smaller, it isn’t faster. It is only smaller. When we made runs if I made a smaller bit Iarrived first, but I wasn’t always the faster»«I move my hand [he is turning the crank] and it takes the same time to make a turn, but thewheels move in different ways: one makes more turns, another less. Then one goes faster and theother slower.»«In runs while I make a longer bit you make a shorter one two or three times, it takes the sametime, but you are faster because you made the bit two or three times.»«You didn’t get it at all! You must look at the difference in length between the long bit and the shortone. If three times the short is longer than the long bit and you make it in the same time then you arefaster. If I put myself on the top [the rim of the wheel] it’s different from putting myself near near here[the centre of the wheel]. Because if we make a row of three, being in the middle I turn much slower.The one who stays at the end runs faster and both of us make a single turn.»

Fig. 4: Excerpt from a conversation among 5-6 year old pupils looking at a machine model by Leonardo Da Vinci.

In primary school we can go beyond mere comparison. Eight-nine year old children can makeactual measurements of length and time, in standard and not-standard units, following the need,now made explicit, of relating the three quantities space, time and speed. The pupils know thatthere are «numbers» also for speed and an intuitive meaning of «space covered in a given time»can be given to measures of speed. So, the pupils can look for a solution also in some not trivialcases by intuitively using proportional reasoning.

5. First steps in mathematisationObjects in motion are the starting point also at higher age levels. If we ask to reproduce as preciselyas possible the motion of a radio controlled toy-car, pupils realise that in order to reproduce itstrajectory they need to be able to describe it. If the floor is covered with tiles we can assign a nameto each of them, like in maps... but we may find that one tile is too large... On this basis the teachercan introduce the idea of frame of reference and of co-ordinate system providing the pupils withtwo powerful and everlasting tools of investigation.At this level races need not to be performed, they can simply be imagined. Even by the end ofprimary school pupils can appreciate that a proportion between spaces and time intervals is validonly when motion is uniform (always «at the same rate», as they say). Now it is possible tointroduce the algebraic relation among s, t, and v that requires looking at the problem from anabstract perspective, independent from the hundreds of examples one can think of. But giving aphysical meaning to the relation is not at all trivial: a ratio or a product between numbers thatcorrespond to different quantities is an entirely new idea that cannot be justified intuitively butonly on the ground of the fact that the numbers we get are confirmed by what happens in reality.Pupils should be aware of that in order to understand the power of the mathematical tool (seefig. 5).


Eli: I had found this too [the relation v=s:t ]. I shall take the simplest example, given that it has towork for all, with all numbers.Adult: OK, this is an important point.Eli: If the distance is 2 km and the time 2 hours, if I ... that is I have proceeded by exclusion,because if I make distance times time it cannot... it cannot be right, because distance 2 km, time 2hours, the speed would be 4 km per hour, but the speed should come out... cannot come outhigher, it has to be always the same, I say...(…)Eli: given that multiplying doesn’t work, it must be...Adult: Ha, you say: first I try by categories of operations... but there are two divisions...Eli: Indeed I tried the first that came to me, of those two, then as I found one and saw that itworked, I thought... since these are two different ways, if this works, the other cannot work too, soI haven’t even tried.

Fig.5: Excerpt from a discussion in a 5th grade class about the relation among space, time and speed

By the end of primary school the pupils are able not only to understand the meaning of averagevelocity but also to get an idea of instantaneous velocity. The motion of balls of different densityfalling in a liquid can be studied and the limit velocity regime discovered (see fig.6).

«The speed of the ball, as soon as we leave it, becomes maximum and then doesn’t growanymore.»«The speed of the ball in the tube starts to grow little by little, then grows less and less, but alwaysgrows.»«The speed of the ball grows down to half of the tube then stays the same.»«We would need to measure speed every centimetre!»

Fig. 6: Comments of 5th grade pupils about the motion of a ball in a liquid.

The motion of a person can be investigated by means of a sensor of motion interfaced to acomputer: the pupils can learn to «read» their own motion on the graphs produced on the screenin terms of instantaneous position, of versus of motion, of rate of change of position, etc. So far, thelearning path consists in constructing a scientific way of looking at those kinematical variables thatcan be directly perceived (length, time, velocity) and that are essential in common knowledge too.Next step is the introduction of the concept of acceleration, which is a purely disciplinary concept,and furthermore the introduction of vectorial aspects of motion. But here we are going ahead inthe process of mathematisation …

6. Final remarksThe examples of activities briefly described in this paper somehow represent an ideal path from infantto lower secondary school. Research results collected in the last decades seem to point out that it is afeasible and effective path of knowledge construction and that a child who could follow it frombeginning to end would probably achieve even better results than those documented in literature.Looking at the research works of groups that in recent years have dealt with the problem ofeducation in science from a long term perspective one can find many other examples of pathsconcerning different topics. They all appear to rely on three basic features of the activities putforward and of the learning environment they advocate. We shall summarise them here briefly:- dealing with «facts», from everyday experiences to lab experiments, because the ultimate aim ofphysics is to describe/interpret reality;-dealing with the others, since discussion among peers and with the teacher appear to befundamental in starting/encouraging/supporting the process of knowledge construction;-dealing with one’s own process of knowledge construction, as metacognitive activities can be


started very early in education and produce an increasing level of awareness in the pupils (see fig.7), a fundamental component of the process that makes scientific knowledge become part of one’sown culture.

Al: Because before…I couldn’t…I mean.. describe motion very well: for example there was a carpassing by, I couldn’t tell its speed, so I said: a car passed by… and nothing more, it was goingmore or less fast… Now instead I can tell, I can tell which was its speed.Adult: Tell me more precisely: what do you do now?Al: That is I look more or less… I say for instance, if it made 1 km I say at what speed, I can sayin how much time, at what speed and what distance.Adult: Wheras before, what would you say before?Al: I would… look and say: this car was fast or slow.

Fig. 7: Comments of a 5th grade pupil about her own transition from common sense to disciplinary knowledge

ReferencesArcà M., Guidoni P. and Mazzoli P., Structures of understanding at the root of Science Education (part I), European

Journal of Science Education, 5, (1983), 367-375.Arcà M., Guidoni P. and Mazzoli P., Structures of understanding at the root of Science Education (part II), European

Journal of Science Education, 6, (1984), 311-319.Bonelli Maiorino P., Gagliardi M. and Giordano E., Métacognition et éducation scientifique in O. Albanese et al.

(Eds.) Métacognition et éducation, Peter Lang, (1999), 243-264.Gagliardi M., Gallina G., Guidoni P. and Piscitelli S., Forze, deformazioni, movimento, Emme Edizioni, Petrini

Junior, Torino, cap. IV, (1989).Gagliardi M. and Giordano E., Pupils’ Representations of Physical Transformations and their Evolution in

Classroom Situations, Cahiers de la fondation Archives Jean Piaget, 13, (1993), 259-277.Gagliardi M., Grimellini Tomasini N. and Pecori B., A challenge for lifelong science understanding. The role of «lab

work» in primary school science in J. Leach and A.C. Paulsen (Eds.) Practical Work in Science Education,Roskilde University Press, (1999), 210-228.

Grimellini Tomasini N., Pecori Balandi B. and Gagliardi M., Reasoning, development and deep restructuring,Eric/Resources in Education, ED (1992), 347-183.

Guidoni P., On natural thinking, European Journal of Science Education, 7, (1985), 133-140.Piaget J., 1946a Les notions de mouvement et de vitesse chez l’enfant, P.U.F., Paris.Piaget J., 1946b Le développement de la notion de temps chez l’enfant, P.U.F., Paris.Piaget J., De la logique de l’enfant à la logique de l’adolescen,t P.U.F., Paris, (1955).Vygotskij L. S., Pensiero e linguaggio, Giunti-Barbera, Firenze, (1966).

WHY DO WE RUN, WHEN WE WANT TO MOVE FASTER?

Erich Reichel, Bundesgymnasium und Bundesrealgymnasium Graz, Austria

1. IntroductionIn some physics textbook you can still read, that physics is the science of “non- living” matter. If weassume this to be true, then the human beings are excluded from physics and we are only“independent” observers or experimentors and not part of the world of physics. This might be onereason for making some students believe, that they are not included and there is no need to deal withphysics. Therefore interdisciplinary topics combining physics with e.g. biology or sports must becomean intrinsic part of our lessons. My experience shows, that these topics interest the students much morethan calculus based examples, which only pretend to deepen the understanding of physics sometimes.This unit should show a simple experiment with a variety of interesting consequences, althoughthere is a little bit of formal thinking and basic understanding necessary. But this was accepted bythe students, because they were interested. The unit deals with walking and running and thetransition between both, when we increase our speed. Background is the understanding of circularmotion or oscillations (depending on the model used). A further interesting aspect is the use of


scaling laws, which help to reduce the number of physicalquantities, which have to be measured in the experiment.

2. BasicsThe understanding of moving interests scientists since along time. Very interesting are the results of EadweardMuybridge from 1887 [1]. He took serial pictures of themotion of humans and animals. Nowadays such studiescan be performed computer- supported and are of greatimportance for robotics.

There are different models suggested for the understandingof walking and running. [2] Simplification of the modelsfor this unit leads to the same results. As it deals mainlywith the transition from walking to running we canreduce the model to circular motion of the feet. Fig. 1shows this simplified model and indicates the basicquantities. Walking means that one foot must have contact to the ground always, whereas inrunning there is a phase with no contact of both feet to the ground.From Fig. 1 one can see, that the important parameters are the velocity (v), the length of the leg (h)and the stride length (λ). Each point of the foot moves along a circle with the hip joint as center.The maximum radius is given by the leg length. If we move forward with the velocity v, the leg mustmove backwards at the same velocity. Therefore a centripetal acceleration must act on the foot,which amounts to

On the other hand the leg “falls down” with the gravitational acceleration g. It is also possible touse forces for this description, but it is not necessary.If we consider the real problem, we see that there will appear a lot of different leg lengths and stridelengths due to the different sizes of persons. This all will result in many different values of thevelocity. To achieve useful results from the experiment it is necessary to measure with a largenumber of different test people leading to a huge number of data. This is the reason for the use ofscaling laws and dimensionless numbers for data reduction and analysis.The law of Buckingham tells us the number of independent dimensionless numbers for a givenproblem. It says, that every problem which needs n quantities for description, consisting of mdimensions has n - m dimensionless numbers. In our case we have four independent quantities: _,h, displacement and time (the latter give v). For the measurement of these quantities twodimensions are necessary: length and time [1].Thus leading to two dimensionless numbers: the relative stride length and the dimensionless

speed , which is also known as the Froude- number (g is the free- fall acceleration). This Froude-

number has further interesting meanings independent of walking and is as important forengineering as the Reynolds- number.These two dimensionless numbers can be derived by a calculus suggested by the theory ofdimensionless numbers. But it is not necessary to do it on this level.The dimensionless numbers allow to gather enough relevant data without the need of having testpersons with varying height or leg length in a wide range.

3. The experimentAfter an introduction of the problem the students get a worksheet (see Appendix 1), which helpsthem to go through the experiment. The worksheet is just a suggestion, which should be adapted tothe particular class and the student’s age.


Figure 1: Model of walking (the red arrowsindicate the velocity v, l means length ofstride, h is the leg length)

h

v 2

(1)

h

hg

v 2

The only materials used are measuring tapes with a length of about 10 m and stopwatches. For theevaluation a computer with a spreadsheet program can be helpful.During the measurement the number of steps needed for walking along this distance must becounted and then the stride length can be calculated by division of the distance’s length by thenumber of steps.It always appeared that the performance of the experiments and the graphical representation ofthe data were reproducible for students with an age of 16 and older.It is important to emphasize with the students that this experiment is not a competition! Somestudents believe, that it is important to be the quickest walker and cheat in the walkingmeasurement by exceeding the limits of walking.

4. Discussion of the results and applicationsI repeated this unit under different conditions with students and in-service training teachers. Theexperiment showed to be very insensitive to measuring errors, because the measurements are verysimple. The only problem is the cheating (see above).The result is a diagram like that shown in Fig. 2. The measurements were performed not only fornormal walking and running, but also for rapid walking and slow running. The diagram shows somekind of clusters for the particular movements.An interesting limit is shown by the Froude- number 1. No data for running lies below this limitand no value for walking lies above. Only slow running makes an exception. This results from thedifficult distinction between running and rapid walking by the test person.The limit at the Froude- number 1 will be found in every experiment of this kind.


Figure 2: Froude- number vs. Relative stride length for different kinds of movement.

Also the fitting of experimental data to mathematical functions can be discussed. In this case linearregression was applied automatically by the used spreadsheet program.And now for more physics. Why do we find this limit at Froude 1?A little bit of analysis of the data will help. A little bit of algebra which means multiplication bythe free- fall acceleration leads to the following result:

-0,5

0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,5 1 1,5 2 2,5

Relative stride length

Fro

ude-

num

ber

Slow walking Rapid walking Slow running Running Linear ( )

ghg

v= 1

2

gh

v=

2

If we remember on expression (1), the left side of (3) shows the centripetal acceleration. And thisequals g. Walking means always contact with one foot to the ground, which means g is larger thanthe centripetal acceleration, which lifts the leg in direction to the hip joint. Thus contact to theground is ensured. While running the centripetal acceleration becomes larger than gravitation andthe foot is raised from the ground. So the results show the expected picture.What can be done with this result?For different persons the maximum walking speed can be calculated. This is done by calculatingtheir relative stride lengths, go into the diagram and find out the corresponding Froude- number.With the help of the leg length the speed can be calculated from that Froude- number. Theknowledge about this can be helpful for basic considerations in sports.

5. ExtensionsThe knowledge about theFroude- number and walking orrunning can be expanded tofollowing examples.Similar evaluations have beenperformed for the gait of horses(see Fig. 3). The limit betweengallop and walk is also given byFroude 1. The interpretation isthe same.R. McNeill Alexander calculatedthe speed of dinosaurs in thesame way [3]. He compared theskeletons and traces of dinosaurswith similar today living animals.In that way he calculated themaximum speed of aTyrannosaurus to about 40km/h.Ships displace water by their hulls. The longer a ship is the quicker it can sail. This is also describedby the Froude- number.

References[1] Thomas A. McMahon, On size and life, Scientific American Books, Inc., New York, (1983).[2] Leopold Mathelitsch, Sport und Physik, Verlag Hölder- Pichler- Tempsky, Wien, (1991).[3] R. Alexander McNeill, Dynamics of Dinosaurs & other extinct giants, Columbia University Press, New York,

(1989)[email protected]

Appendix 1: worksheet for the studentsWhy do we run?

Humans and other animals change their way of movings to speed up. Horses begin to gallop and menstart running. Why is it not possible to walk at any speed? (Walking means, that one foot is alwaystouching the ground.)

ProblemThe transition from walking to running will be investigated. For that walk and run in succession alonga certain distance. While doing so measure the time and count the number of steps. For useable resultswe need a certain amount of data, because people are different. For data reduction we usedimensionless numbers.Perform each experiment in a group of three. One is walking, the other measures the time and the thirdrecords the data.


Figure 3: Movement of horses (The quantities on the axes areinterchanged compared to figure 2) [1]. (Translation of the captions:Schrittlänge = stride length, Hüfthöhe = leg length, Geschwindigkeit =velocitiy, Gravitation = free- fall acceleration, Gehen = walking,Galopp = gallop, Laufen, Traben, Paßgang = trot)

Procedure

TIME TRAVEL – MORE THAN A PHYSICAL CONCEPT?

Tanja Tajmel, Leopold Mathelitsch, Institute of Theoretical Physics, University of Graz, Austria

People of every age are fascinated by the idea of travelling through time. The science-fictionliterature dealing with this topic is enormous and many TV-series and movies on this subject havebeen produced (a comprehensive overview is given in Ref. 1.) So, the teacher can try to take up thisinterest in time travel and transfer it to the class. Or it may even happen that the physics teacher isasked whether time travel is possible – either forward or backward in time.Teaching should build on the experience, on the preconceptions of the students. But what is theknowledge of students about time travel? Since our experience with time is based on our senses, inthe first instance, and on objective instruments like clocks later, one should not be surprised to findout that also the concept of time travel is not always a physical one. In order to gain moreinformation about the ideas of people on time travel, about 80 persons of different nationalitieswere questioned. This survey, the main results thereof, and possible implications for teachingphysics are presented in this contribution. We start with some examples of famous time travel.

1. Time travel in fictionTime travel started to become a point of interest when H.G. Wells published his novel ”TimeMachine” [2]. How did the time traveller travel through time? He used a so called time machine.


1. A distance (L) with a length of about 10 m or longer will be marked out. Actual length of distance L: _________________ m.

2. Measure the leg length (h) of each person (this is the distance between ground and hip joint) My leg length h: ________________ m.

Every test person is first walking and then running along the distance. To get a uniform speed start before the actual beginning of the distance. While walking or running count the number of steps (N) and measure the transition time (T).

3. Calculate the speed: s

m

T

Lv ===

.

4. Calculate the stride length: ===N

Lm

5. Calculate the relative stride length: =h

6. Calculate the dimensionless speed (= Froude- number): =h

v

819

2

,Results

DurationT (s)

Speed v (m/s)

Number of steps N

Stridelength

Rel. stride length

Dimensionless speed

Walking Running

The results of every person are collected and put into the computer to get a diagram with relative stride length vs. dimensionless speed.

How did he know in which time period he was? He had some kind of a clock or counter which gavehim information about his orientation in time.This time travel is quite a technical one, which is verysignificant for the beginning of the 20th century.With the physical (e.g. theory of relativity) and technological (e.g. space flights) progress also timetravels got new impetus. The famous science fiction author Carl Sagan wrote a novel entitled“Contact” [3]. In this novel a person travels to a completely other point in space and time by passingthrough a tunnel which connects these two regions of space-time. Carl Sagan always wanted hisnovels to be close to physical feasibility.Therefore he asked his friend Kip Thorne, whether a tunnelthrough space-time could be possible from the physical point of view, that means do there existsuch solutions of the basic equations of general relativity. Kip Thorne found out that it is notimpossible, and he and his colleagueses Morris and Yurtsever published their finding in PhysicalReview Letters in 1988 (“Whormholes, Time Machines and the Weak Energy Condition”) [4]. Withthat, wormholes became the new time machines of our fantasy and they are used in many sciencefiction movies or novels.But there exist other time-travel stories which are much older and which have nothing to do withtechnology or physics. For example the story of “Sleeping Beauty”. In this story the princess sleepsfor hundred years until a prince comes up and wakes her up with a kiss. This is also a time travel,not by technology but by sleeping. We can say that it is a kind of psychological time travel. Howdoes Sleeping Beauty know that she slept for hundred years? Because of the roses:They had grownso much that they covered the whole castle. This cannot happen within one night. It is logicallyimpossible.These were three examples of time travel, a machine without physical explanation, a physicalpossibility and a psychological one. What kind of time travel is usually in the mind of people?

2. The common sense of time travelWe tried to find out what the common conception of time travel is, what ideas different people haveand where these ideas came from. For this purpose we prepared a questionnaire which was filledout by eighty persons of age between 12 and 60 years and different nationalities (Austria, Italy,Germany, Portugal, Great Britain and Spain) [5].The questions were:1) Which idea do you have about time travel? Please give a short example!2) Do you think time travel is possible?3) Do you know novels, movies, etc. about time travels? Which ones?4) Please describe an experience which was like a time travel for you!

ResultsQuestion 1: The major part answered that you need a time machine for time travels, but also otherexplanations were given.Here some examples:Time travel is ....... when you travel by a time machine into the Middle Ages.... to sit into a machine and travel mentally into another time, that means: with all your senses butnot physically.... under hypnosis or in meditation, when you speak about experiences which you had in yourchildhood.… we are always on a time travel because we cannot stop time.Question 2: The main answers were:Time travel will be possible because of technological progress (44 %).Time travel is impossible because there does not exist any time machine (38%).Maybe time travel is possible (18 %).Question 3: The most frequently mentioned novels and movies were: Back to the Future I, II, III;


Stargate, Time Cop, Terminator; Star Trek; Sliders; The Time Machine. In each of these novels ormovies time travel happens by time machines.Question 4: More than half of the respondents indicated that he/she had experiences which werelike time travels!... when I travelled to Pompei and imagined how the people lived.... when I saw on TV the transmission of the first landing of human beings on the moon.... when I dream.... when I had an accident I saw situations of my past in the position of an observer.... every journey where I loose the contact to the environment which I am used to.The interesting point is that even those, who said that time travel is absolutely impossible,mentioned experiences which they would call “time travel”.

Conclusion• Time travel is not seen only as a physical phenomenon. Many persons put forward some

example of a psychological time-travel experience.• Both aspects of time travel are connected to certain characteristics, which result from logical

thinking. That means, if some situation is not logically compatible with a person´s consciousnessin time, the person feels a kind of dislocation in time. Such characteristics can be1) The cultural and biological environmentExamples: a journey to India, the roses in Sleeping Beauty2) Historical settingHistorical persons or happenings show up which are definitely connected with another time, e.g.World War II, Einstein, ...3) The personal backgroundA person meets her/his own past or future.4) An objective clock A clock shows a completely different time from what one would think.

• The possibility of time travel is mainly associated with technology and physics. The experience oftime travel is connected with consciousness and feeling of time.

But there is a discrepancy in the answers: About forty percent of the respondents said that timetravel is not possible, more than forty percent believe that it is possible. But 65 % of therespondents indicated that they had time travel-experiences. Actually there shouldn’t be morerespondents with time travel experiences than those who said that time travel is possible.Apparently these respondents have problems to connect their physical and psychologicalrepresentations of the same object and they are also not aware of this mismatch.• Finally, we could not find any differences depending on the nationality.

3. Implications for teaching The persons questioned have shown notions of different aspects of time travels, but not in a verywell coordinated way, physical aspects and psychological components are separated. Therefore adiscussion of time travel in a physics class has to be based on both possibilities of time travel, notonly on the physical one. The starting point could be to ask whether one agrees on a definition likethe following:Time travel is a transfer of someone´s body or consciousness in time. One becomes aware of thisdislocation because there are characteristics of the surrounding which are logically not compatiblewith the background one is used to.This definition implies that there are many possibilities of time travel, the students can certainly putforward several of them, the teacher can add some more. It is natural to distinguish betweenphysical and psychological time travels, but one should also try to work out common features. Thephysical view is just one side of the coin.

Psychological time travel: To experience a time travel, it is enough in many cases not to be awake, that means not to be aware


of the happenings and changes around you.Sleeping: During our sleep we loose our consciousness of time. But we are so familiar witheveryday´s sleep that we do not have the feeling of disorientation when we awake.Similar to this is being in narcosis or falling unconscious. This does not happen so often, thereforemany people have problems in orientation in time afterwards.Dreaming: In dreams one does not only lose consciousness of time but one sees and feels situationsof the past, or one dreams something which could happen in the future.Similar to dreaming are experiences with drugs.Circadian clock: The biorhythm of human beings does not have a period of 24 hours but of 25hours. If a human being is at a place where he does not have any contact with his environment, aftersome days his biological rhythm will have a periodicity of (in average) 25 hours. When this personleaves the isolation there will not have passed as many days for him/her as the objective clocks say.Journeys to other cultures or different seasons can also give the feeling of time travel.A very new kind of time travel is virtual reality. Different surroundings can be created virtuallywhich give you the feeling not to be in the present any more. It depends on the technical progresshow “real” the simulations are.

Physical time travel:The physical possibilities for time travel are very few, and they are very expensive and dangerous.Twin paradox: When the brother who comes back and is younger than his twin, he may have thefeeling that he travelled into his future. But he never left his system of reference, and within thissystem the twin paradox is actually a psychological time travel, not a physical one.Wormholes: Theoretically wormholes are possible. They are connections of two distant parts ofspace-time allowed by the laws of general relativity. But they are very instable. A wormhole couldbe transformed into a time machine because the time evolutions of the two openings of a wormholeare different. One opening is accelerated nearly up to the speed of light and returns later at theoriginal place. So, considered from outside, there is a time difference between the two openingsaccording to the twin paradox [6].Beaming: Also teleportation, that is transport of information at least at the speed of light, could becalled time travel. When a person gets teleported in Enterprise, he/she is immediately at anotherplace [7]. Teleportation is realized already, but just with the information-content of a photon [8].Tachyons: Tachyons are particles which travel faster than light. According to the laws of specialrelativity they can therefore travel into the past The main reason is the difference of proper timein two systems moving relatively to each other. Let us assume that a tachyon moves to a rocketleaving the earth, which also has a source of tachyons. Immediately after a tachyon has reached therocket the source in the rocket should send a tachyon back to the earth at the same speed. Thistachyon would be measured before the first tachyon was emitted [9].To develop both aspects of time travel, the physical and the psychological one, with students openspossibilities and chances.• Interdisciplinary aspects are (too) seldom included in physics teaching, although demanded by

many curricula. The topic of time travel gives the possibility to address different school subjects.It would be advantageous if the teachers of language/psychology/biology/physics couldcooperate.

• Physics appears to the students often as a very closed field, where there is always a definiteanswer – right or wrong. Bringing in literature and psychology should change this opinion. Agiven question – does time travel exist? – is discussed and answered from very different pointsof view and also within different ways of thinking.

• To be aware of – and maybe even enjoy – these two different ways of thinking can lead to ahigher level of thinking, also of formal thinking.

References:[1] P.J. Nahine, Time Machines. Time Travel in Physics, Metaphysics and Science Fiction, AIP, New York, (1993).


[2] H.G. Wells, Time Machine, in The Definite Time Machine, H.M. Geduld (Ed.), Indiana University Press,Bloomington, (1987).

[3] C. Sagan, Contact, Pocket Books, (1997).[4] M.S. Morris, K.S. Thorne, U. Yurtsever, Whormholes, Time Machines, and the Weak Energy Condition, Physical

Review Letters 61 (1988), 1446.[5] T. Tajmel, Zeitreisen, Diploma Thesis, Univ. Graz, (1999).[6] K.S. Thorne, Black Holes and Time Warps. Einstein´s Outrageous Legacy, Norton, (1995).[7] L. M. Krauss, The Physics of Star Trek, Basic Books, New York, (1995).[8] A. Zeilinger, Quantum Teleportation, Scientific American 282 (2000), 515.[9] R. Ehrlich, Nine Crazy Ideas in Science, Princeton Univ. Press, Princeton, (2001).

THINKING ON VECTORS AND FORMAL DESCRIPTION OF THE LIGHTPOLARIZATION FOR A NEW EDUCATIONAL APPROACH

M. Cobal, M. Michelini, Physics Department, University. of Udine, ItalyF. Corni, Physics Department, University. of Modena and Reggio Emilia, Italy

1. IntroductionPolarization is a very common and relevant phenomenon in nature and in everyday life, which has relevant applications. It is important for what concern physics as a discipline since it represents afundamental property of the light. From the learning point of view, the polarization has a greatpotentiality and is an important conceptual resource, since it offers the possibility to be reviewed –at several analysis levels – as a property of a wave, as well as a property of a physical state. Inaddition, it can give a contribution on how to built and manage the process of formalization, tohandle the physics interpretation of the phenomena using mathematics.Looking at the methodological point of view, polarization allows to see how a given representation,can find different meanings according to our interpretation of the polarization phenomenon understudy.As it will be shown later, we can use the same formalism (the vector) to describe the lightpolarization. According to the interpretative frame chosen, the vector will have three differentmeanings. Therefore, polarization offers a useful bridge between electromagnetism and optics, aswell as between classical physics and the physics of quanta.Some recent didactical proposals have built an approach to the quantum mechanics, which usessimple experiments on the polarization [1,2]. Ferguson showed in his paper [3] a method todemonstrate the interference of polarized light and the Fresnel-Arago laws. He concluded bysaying that explanations of the observed phenomena are possible in terms of classical wave opticsor in terms of superposition of quantum states, so that it is possible to discuss it in both optics andquantum mechanics courses.However, light polarization is normally a neglected topic, during the teaching of the secondaryschool.This is due to the assumption that it requires a deep background of mathematical and physicalknowledge. Working experience with teachers, clearly showed that polarization is alwaysintroduced as a property of the electromagnetic wave, and correlated to the Maxwell’s equations.However, as shown in many contributions [4-11], there are a large number of simple experimentswhich one can use to introduce the light polarization.Strangely enough, is not even selected an activity from the phenomenology of this phenomenon –maybe at a more elementary and experimental level – as it is normally done for other topics.For other subjects in the physics field, a more experimental approach finalized to the understandingthe physical process and to build its phenomenological characteristics is tried.This does not happenfor the polarization.The contribution of this paper is to show that polarization requires just a vector to be described,

3103. Topical Aspects 3.6 Strategies: Methods and Tools

and the vector that is used to describe the same process (polarization), acquires completelydifferent meanings, depending on the interpretative frame that is assumed.In literature there are already some efforts in this direction: in [12] it has been shown how theprocess of light scattering and polarization can be described using matrices.Describing a vector by means of geometry or matrices is completely equivalent. The potential ofthe geometrical representation offers a bridge between the phenomenological context and itsformalisation: an imaginative reduction of the formal entity and the observable of the processunder study.In discussing our proposal we will make a brief excursus on the main methods and kinds ofpolarization. Our idea is to introduce the study of the polarization phenomena very early in thecurriculum. A phenomenological approach allows underlining that the light in several differentsituations (e.g., when it is transmitted through a Polaroid or through a crystal which producesbirefringence, when it is diffused in a direction normal to the incident light), it acquires a property– called polarization – which can be interpreted with a vector P.

2. Polarization: a rich learning context of everyday phenomena and applicationsThe missed connection between everyday experience and schoolwork in the scientific field hasbeen identified as the main cause of difficulties in learning [13]. Therefore, experimentalexploration and personal involvement are important components for the construction ofknowledge as an individual interpretation of the world [14]. The phenomenology of polarization isso present in our everyday life, that we get used to not interpret it. However, it is a rich context forthe link between experience and scientific knowledge. In the site www.uniud.it/cird/secif/ we offera pool of proposals for school activities, with examples of everyday phenomena, applications andexperiments. The material to which we refer has been developed within the SeCiF project1. Manystrange optical effects, seen by watching the ski during day or night, can be identified consideringthe type of light polarization present, and its orientation [15]. The polarization of an ordinaryrainbow can be shown, by comparing two photographs of it, onetaken with a polarizing filter, and one without.When the filter hasa certain orientation, the appearance of the rainbow is enhanced.At another orientation of the filter the scene remains the same,but the rainbow is not visible anymore. Many of us had the chanceto observe how a ray of light interacts with a long crystal thatpresents birefringence, i.e. has two indexes of refractiondepending on the direction of the light polarization inside thecrystal. In Fig. 1 two crystals are put on top of a book page, and itis possible to see two images of what is written.Another example: we know that in a sunny day is better to useglasses with Polaroid lenses.By rotating of 90o a Polaroid, around the direction of observation,we can see that the intensity of the transmitted light changes withthe angle of rotation, if the floor reflects the light, or if the skydiffuses the light in a direction orthogonal to the sun. It does notchange instead in the case of light coming directly from the sun,from a lamp, or reflected by a mirror. In the first case, theanalyzed light is polarized, in the second case is not polarized.Lights from liquid crystal displays (LCD) of digital watches andcalculators are polarized, and provide a very convenient source of


1 SeCiF means: “Studiare e Capire in Fisica”. It is a national project, which involves nine universities, with 1999-2001MUR financial support.

Fig. 1: Two crystals are put on top ofa book page: birefringence is visible.

polarized light in a classroom situation. By placing a small Polaroid sheet in front of the display, theappearance of alternating dark and clear views of the background after every quarter turn of thesheet can be observed [16].In addition, several tools and techniques used to understand material properties are based on thepolarization phenomenon. Here only few examples are given [17].Glasses with imperfections occurred during the cooking process, develop internal stresses that canbe spotted out by analyzing the polarization of light passing through them. Similarly, it is possibleto draw a map of the superficial stresses of opaque objects undergoing external solicitations, bycovering these last ones with a film of optically active material. Ellipsometry, is a technique basedon the variation of the polarization state of light incident on a sample. It allows measuring theoptical parameters of the material that composes the sample, as well as the thickness of possibledifferent layers.The presence of organics materials in a solution can be identified and their concentration measuredusing the so-called “rotational dispersion” technique, which is a phenomenon related to thepolarization. And many other examples could be quoted as well to show how often we meet thisproperty of the light.A research on the use of simple equipment inside the school, in an organized and open way [18-22]has confirmed that operability (manual and conceptual) determines the involvement of the learner,the activation of cognitive resources and the separation of the descriptive and interpretative plans[19,20].Links have been pointed out between the sensorial information, the experimental exploration andthe building of formal thinking [23,24].For the classroom work, it is possible to realize many simple experiments that can blow up the mainproperties of polarization, as well as show when it is relevant, using very common materials.For example, scotch tape presents a large birefringence, more than 10 times that of quartz, withoptic axis parallel to the plane of the tape itself (not necessarily along the length). Plexiglas sheetsexhibit slight birefringence, 300 times smaller than that of quartz, with the optic axis almostperpendicular to the plane: with Plexiglas it is then possible to do low cost, macroscopically thick,interference devices based on polarization, such as quarter or half wave plates and compensators.Tapes and Plexiglas are suitable for overhead projection experiments. All the above describedphenomena and applications are elements which can stimulate the curiosity for an explanation andto motivate for an interpretation. This involves the understanding of the possibilities ofinterpretation, and therefore of the role of the different formal tools used.

3. Our proposal on the formal descriptionSeveral studies indicates that the cognitive development, has to be considered a didacticalobjective, rather than the result of a spontaneous evolutionary process [25,26].The construction of the formal thinking is a fundamental aspect in the process of learning physics.It has to start gradually, and must not appear suddenly, while a certain physics topic is treated.Two are the elements that can activate the learning processes [27]:1) A cognitive crisis, which allows the students to abandon any misconception they had in

precedence about the topic under study (this crisis can be started by showing them someexperiences which are in contradiction with their previous believes);

2) Anchoring conceptions or bridging analogies [28], for grounding instruction on students’intuitions.

These two elements guarantee a change from a provisional and partial spontaneousinterpretational model, to the scientific one. Polarization can be studied at three different levels: afirst interpretation of the phenomenology with vectors which describe the properties of theobservable, a second one, which is a complete interpretation of the polarization supported by theelectromagnetic theory, and a third one where the polarization is a quantum state of light. The firstone includes the potential to prepare the background for the second and third descriptions, and in


particular for the third one where the vector that represents the state is described in a 2-dimensional space. It is possible to use the same formal tool: a vector, in order to describe thepolarization in the three above-mentioned interpretative contexts.Our proposal has to be seen in the frame of a more general idea that sees as a relevant point forthe formation, the formalization already at the level of the description of the phenomenology, in asort of “generalized kinematics” of the phenomenon [29].This proposal includes a first representative level of the polarization as a property of the light,through the use of a vector P. The P vector is just a geometrical entity that accounts for thephenomenological observable. In addition, it prepares the way to look at the polarization in otherinterpretative contexts, offering the opportunity to recognize that the same formal entity canassume different meanings.This is important in the didactic of physics where often there is a naturaltendency to look to the physics contents, keeping well separated the conceptual and the formallevels.In a classical view, the vector is related with the electrical field vector E (which represents theamplitude of the electromagnetic wave)2. In quantum-mechanics the polarization, which is aproperty of a physical state of light, imposes exactly such a mathematical description with vectors,and the vector, in a sort of formal conceptual continuity, becomes –from the photon “microscopic”point of view – the vector of the quantum state (in this case we will assume the language of thequantum-mechanics, and call this vector S) [2].The P vector represents for us the anchoring conception on which the various interpretations of thepolarization can be built.At the same time, it is also the element which produces the cognitive crisis,posing the question of which is its physics meaning in an organic theoretical frame. This point is thecentral problem that the didactic of physics has to face, coming from a tradition in which theunderstanding of physics was based on a classifying reductionism, fragmented with respect to theinterpretative models and to the whole theory.The underlying idea is that vectors can be the best formalism to describe polarization. They alsohave the advantage that they can be associated to a geometrical representation (which always helpsin the learning process). Their link with the waves representation it is once more offered bymathematics, through the Euler equations. According to the needs (i.e., solving a new equation), itis always possible to make an “extension” of a class of numbers. For example, from the naturalnumbers, one passes to integers, than to the rational and to the real numbers.A further class is given

by the complex numbers which – with the imaginary unity i - allows the operation: . Andwhat comes after the complex class? Is it conceivable an additional extension? Mathematicians areable to demonstrate – with the so-called “Main Law of algebra” that the class of the complexnumber is the ultimate one, and no additional extension is possible. From the point of view ofphysics, this means that all phenomena in nature, can be formalized by using complex numbers.In addition, it is possible to demonstrate that there are functions that can be used to build all theother functions. For example, with a proper sum of the sinus or co sinus functions of differentfrequencies, any other function can be obtained. Therefore, we can say that – as for numbers –alsofor the functions it exists an ultimate function, which, for example can be the sinus.Euler demonstrated that the complex number eiφ, where i is the imaginary unity and φ a phase can be written as eiφ = cos φ + i sen φ. But the complex number eiφ can be represented as a 2-d vector ina plane (Argand diagram).As well known, we can use this result to say that the quantum mechanicscan be expressed with both the complex numbers - therefore with vectors - and with waves, whichcan be written as sum of co sinus functions. Also in this view, the concept of vector allows to createan easy link between the description of polarization in a classical and in a quantum-mechanicalframe.

i=1


2 In fact, now it is preferred to link the polarization direction with that of the magnetic field B.

The three vectors E, P and S can play a role in describing and interpreting the polarization in manydidactical paths, at different levels, without loosing the possibility to make a complete explanationof the subject. In the following, we will illustrate and support the description of the polarizationusing the vector P: this is the intended original contribution to the topic in this paper. This is donewith an excursus on the main phenomena of polarization and their properties, to show in parallelthe wave and vector description. The study of polarization can be exploited both through theanalysis of the polarization properties of light refracted, reflected, transmitted as well as throughthe study of the production of polarized light.

4. Polarization of light: an excursusBy means of quantitative studies performed with ordinary filters, Polaroid filters and crystals withbirefringence, it can be recognized that all filters absorb part of the light that passes through them,and the absorption increases with the number of crossed filters. In the case of Polaroids or crystals,one can see that the light is transmitted differently, according to their orientation. This is a firstcharacterization of the polarization. We can examine a setup like the one depicted in Fig. 1, wheretwo filters (one called “Polarizer” and the second one, tilted by an angle θ called “Analyzer”) aremounted between a source of natural (non-polarized) light and a detector. If the light transmittedby the Polarizer is observed after having crossed the Analyzer, it will be found that the lightintensity changes by rotating the Analyzer with respect to the Polarizer. The second filter transmitsa fraction of the intensity of the incident light, which, in turn, is a fraction of the light which crossedthe first filter, in agreement with the Malus’ law.


Natural light

Polarizer

Analyzer

Polarized light

E0

E0cos

Figure 2: Optical setup with a Polarizer and an Analyzer.The first Polaroid selects the incident light component whichis linearly polarized along the direction of the transmission axis. The second Polaroid is used to recognize thepolarization status of the incident radiation, through the Malus’law.

In a classical picture, light is a transversal electromagnetic wave. The wave equation is given by theformula:

(1)

Which is solution of the Maxwell’s equations. This is not the most general one, but it representswhat it is called a linearly polarized wave (for the sake of simplicity, our arguments will be limitedto this case): the electrical field E oscillates always on the same plane. This plane contains the Eand k (propagation direction) vectors. Polarization can be linked to the way the electrical field isoriented in space. By superimposing two waves with same frequency but linearly polarized inorthogonal planes, any direction for the resulting oscillating field can be obtained. If there are twowaves in phase, or out of phase by an odd number of π, of equal frequency and linearly polarizedin two orthogonal planes:

(2)

(3)

( )tkzEiE xx = cosˆ0

r

( )tkzEiE xx = cosˆ0

r

( )tkzEjE yy = cosˆ0

r

The resulting wave is given by their superposition:

(4)

Which is linearly polarized, has a constant amplitude equal to E0x + E0y and oscillates on a planetilted by an angle with respect to the two original normal planes (as shown in Fig. 2).A source of natural light is composed by a great number of atomic emitters that are randomlyoriented. Each excited atom emits a wave train of different frequencies for about 10-8 s.The so-called natural light it is therefore composed by rapid changes of different polarizationstates. Natural light can be represented in mathematical form, with the superposition of twopolarized waves in normal planes, with equal amplitude and being out of phase by a value whichchange rapidly and randomly in time.If we represent the light wave as the superposition of twonormal components, the polarizer works as a tool that allowsone of the components to pass through, and absorbs theother one. If the electrical field E transmitted by the firstpolarizer is E0, after the analyzer it will be reduced to E0 cosθ. Since the light intensity it is proportional to the squaredamplitude, we will have that the transmitted intensity will beproportional to cos2θ (this relation is the so-called Malus’law: I(θ) + I(0)cos2 θ. What happens with this setup in termsof the polarization vector P? The polarizer transformsnatural (non-polarized) light in polarized light. Aftercrossing the first filter, the light can be characterized by aspecific vector P: a particular polarization state (vector) hasbeen selected, which we can associate with the direction ofthe electrical field E. We can see this vector P as the sum oftwo components Px and Py, being the x-axis chosen in thedirection of the Analyzer transmission axis. Our second filterwill allow only the x component to pass, absorbing thesecond one.As an example of link with the quantum mechanics, we can give a quantum-mechanical descriptionof what happens with our setup, by assigning a polarization vector S (in this case a vector whichdescribe a physics state) to the low intensity light, property of a single photon from another pointof view.The two polarizers have directions allowed along two orthogonal axes represented by two versors,u and v. In particular, we will concentrate on the light that is filtered by the first polarizer.This lightis “polarized”. Polarized, means that after the first polarizer, the photon polarization vector S - hasnow a well-defined direction, along the u versor and gives a complete description (of course interms of polarization) of the light which passed the first polarizer. As for every vector in the two-dimensional space, any S vector can be written as a linear combination of two vectors normal toeach other, which can be named H and V:

(3)

where the component ψ1 and ψ2 are the so-called “amplitudes”. Of course, they have to obey to thenormalization condition:

(4)

Since H and V are unit vectors, they also represent two possible states for a linearly polarizedphoton. One can choose H and V exactly along the allowed direction u and v of the first and secondpolarizers. The concept of orthogonality between physical states can then be introduced: two statesare orthogonal when their physical properties are mutually exclusive. The two polarization states


( ) ( ) ( )tzEtzEtzE yx ,,,rrr

+=

Ex

EyE

Figure 3: Example of linearly polarizedlight.

VHSrrr

21 +=

122

21 =+

H and V are an example of orthogonal states, since, when light has crossed a polarizer orientedalong u, we can certainly exclude that it can cross a second polarizer with allowed direction v.The superposition principle allows therefore describing physical systems in terms of vectors andtheir scalar products. As additional examples of our proposal, we will now examine the mainmethods to obtain polarized light (namely: dichroism, reflection, scattering and birefringence),using both the descriptions given by the field E vector and the polarization P vector.

DichroismThe word dichroism refers to the selective absorption of one of the two orthogonal states ofpolarization that composes an incident non-polarized beam. The simplest setup is a grid of parallelconducting wires. If a non-polarized radiation crosses this grid (see Fig. 3), the component polarizedwith the electrical field parallel to the wires will be absorbed since it will generate electrical currents(and therefore waste energy by the Joule effect) within the wires. Instead, the other orthogonalcomponent, will not be able to interact in a relevant way with the wires, and therefore it will becompletely transmitted. This phenomenon can be easily shown in the lab, using microwaves and amacroscopic grid made by electrical wires at a distance of few centimeters. Some materials are sourceof dichroism by nature, since they present an anisotropy in the crystal structure (e.g.: the tormaline).The phenomenological observation can be interpreted as the selective absorption by the grid ofwires of one of the twoperpendicular states (Px or Py) oflinearly polarized light, whichcompose the incident light.

Polarization via scatteringIf we send light into a saline solution(sufficiently concentrated) andobserve the light diffused atdifferent directions using a Polaroid,we will see that this light is notpolarized in the direction ofincidence, and is only partiallypolarized at intermediate angles,with a higher degree if observed perpendicularly to the direction of incidence.If the incident light is polarized, it will be predominantly diffused in a well-defined direction,perpendicular to the direction of incidence.In terms of classical optics, we say that if an electromagnetic radiation hits an atom and itsfrequency is far away from those typical of the atom absorption, the electrons that belong to the atomare not excited, but are put in vibration by the electromagnetic field. Such electrons can be consideredas dipoles, which absorb and re-emit the electromagnetic energy. Let’s imagine that a wave linearlypolarized on the vertical plane is hitting a molecule of air. The electrons of the molecule will vibrate

along the polarization direction. They can beimagined as electrical oscillating dipoles. Thelight that will be diffused will not be emitted in alldirections but only in the orthogonal directionswith respect to the dipole axes, and with linearand vertical polarization.In terms of vectors we can imagine to have anincident light with a certain polarization P,which can be decomposed –as usual- in twocomponents, as shown in Fig. 4: Px and Py.The component with Px will be diffused mainlyin a direction orthogonal to the plane defined


Figure 4: Setup with a grid of parallel conducting wires to observe thedichroism phenomenon.

E

Figure 5: Polarization vectors in the diffusion.

Diffused light Py

Px

Incident light

Diffused light

by the incident light direction and by Px itself. On the contrary, the component with Py will bediffused mainly in a direction orthogonal to the plane defined by the incident light direction andby Py itself.

Polarization via reflectionWhen the light hits a surface that separates two different materials, reflection and refraction takeplace. These phenomena can be analyzed from the point of view of polarization, applying theconcepts introduced above.The reflected light turns out to be non-polarized for angles of incidencenear zero (almost normal incidence), and polarized for an angle of incidence near 60o. In general,it will be partly polarized.In analogy to what happens with the polarization via diffusion, the model of the electronicoscillator gives a classical description of what happens. Let’s consider the case of a wave that travelin vacuum and hits the surface of a dielectric material. Let’s assume, in addition, that the wave ispolarized in a plane perpendicular to the dielectric surface (see Fig. 5). The wave is partly reflectedand partly transmitted and refracted inside the material. The electrical dipoles of the material willoscillate in a direction parallel to the electrical field of the refracted radiation, and therefore willre-emit the radiation in a direction normal to theoscillation axis. A part of the radiation will emergefrom the surface as reflected radiation, and thisradiation will be weaker as its direction will beparallel to the dipole oscillation axis. For differentangles of incidence, the refracted light turns out to bealways polarized in the same direction. At theBrewster’s angle, the reflected and refracted wavesare linearly polarized in orthogonal directions: thepolarization component of the incident light parallelto the incidence plane is completely refracted. Thereflected light is therefore completely polarized in adirection orthogonal to the incidence plane. In termsof vectors we can imagine to have an incident lightwith a certain polarization P, which can be imaginedas made by two components, one along the directionperpendicular to the incidence plane (as defined above), Port , and one parallel to this plane, PparReflection at the surface will act as a “projection” of the initial polarization vector P on thedirection of Port.

BirefringenceA case of anisotropy in the propagation of light in matter is given by the birefringencephenomenon.If we put a crystal on a book and we look through it, we can see two images of what is written onthe book.The first image (ordinary ray) is produced following the usual Snell’s law.The second oneinstead (straordinary ray) is based on a different process. If we send on the crystal a linearlypolarized light, two parallel beams come out, whose separation is a function of the crystal thickness.Both the beams are characterized by mutually orthogonal polarization, as can be checked by usingPolaroid filters.The beam intensities are different according to the incident light polarization. If thiscoincides with that of the ordinary (straordinary) beam, only the ordinary (straordinary) beam willpropagate in the crystal. In a classical picture, the index of refraction of a transparent material isintroduced by considering the atoms as composed by a heavy nucleus practically at rest, and by anelectronic cloud that can oscillate thanks to the force of an external electrical field. When anelectromagnetic radiation hits a material, the electrons absorb the energy, which is thereafter re-emitted in all directions. This introduces a reduction of the propagation speed. If the atoms in amaterial are arranged in such a way to present asymmetries in the direction of oscillation there will


Figure 6: Reflection of light

Ei Er

BB

be different speeds of propagation allowed for the light, and therefore different indexes ofrefraction.

Fig. 7: Birefringence.

The relevant physics observable in our case is the direction of the light polarization. Which effectswe would expect if polarized light is sent through a plate of a substance that presents birefringence?If the polarization vector P is parallel to the optical axis, the light will go through with one velocity;if the polarization is perpendicular to the axis, the light is transmitted with a different velocity. Ifthe light is polarized at a certain angle (see Fig. 6) to the optic axis (i.e.: P has a certain angle withrespect to the optic axis), since P can be decomposed as a sum of two orthogonal vectors (oneparallel and the other normal to the optic axis), the two components will travel at differentvelocities. We can check the output light polarization along a certain direction v, for exampleinserting two polarizers in our setup, and looking whether the two beams have crossed them or not.

5. ConclusionsThe missed connection between everyday experience and schoolwork in the scientific field hasbeen identified as the main cause of difficulties in learning. Therefore, experimental explorationand personal involvement are important components for the construction of knowledge as anindividual interpretation of the world.Polarization is a very common and relevant phenomenon in nature and in everyday life, which isimportant from many points of view: disciplinary, for learning of formal thinking, methodological,etc.It is therefore a rich learning context of everyday phenomena and applications.However, light polarization is normally a neglected topic, during the teaching of the secondaryschool, since it is believed that a solid background of mathematical and physical knowledge isrequired.Our proposal is to introduce the study of the polarization phenomena very early in the curriculumA phenomenological approach allows underlining that the light in several different situations (e.g.,when it is transmitted through a Polaroid or through a crystal, when it is diffused in a directionnormal to the incidence one), it acquires a property – called polarization – that can be formalizedwith a vector.The idea shown in this paper is that the vector that is used to describe the same process(polarization), acquires completely different meanings, depending on the interpretative frame thatis assumed (whether it describes the electrical field direction –as in the classical optics–, thedirection of light polarization, or a quantum state of light, as in quantum mechanics). Therefore, itoffers a useful bridge between electromagnetism and optics, as well as between classical physics andthe physics of quanta.The P vector represents for us the anchoring conception on which the various interpretations of thepolarization can be built. At the same time, it is also the element that produces in the students acognitive crisis that allows replacing their misconceptions with a proper scientific knowledge.In this view, we illustrated and supported the description of the polarization using this vector,presenting an excursus on the main phenomena of polarization and their properties, to show inparallel the wave and vector description.


S

E

Optic axis

References[1] G. C. Ghirardi, R. Grassi, M. Michelini, A Fundamental Concept in Quantum Theory: The Superposition

Principle, in Thinking Physics for Teaching, Aster, Plenum Publishing Corporation, (1996), 329.[2] R. Ragazzon, L Santi, A Stefanel, Proposal for quantum physics in secondary school, Phys. Educ. 35, 6, (2000),

406.[3] J. L. Ferguson, Am. J. Phys., 52, 12, (1984), 1141.[4] A .J. Cox, Am. J. Phys., 46, 3, (1978), 302.[5] E. Fortin, Am. J. Phys., 47, 3, (1979), 239,[6] W. Herreman and H. Notebaert, Am. J. Phys, 51, 1, (1983), 91.[7] J. L. Ferguson, Am. J. Phys, 52, 12, (1984), 1141.[8] F. Behroozi and S. Luzader, Am. J. Phys., 55, 3, (1987), 279.[9] J. F. Borges da Silvia and J.A. Brandao-Faria, Am. J. Phys., 59, 8, (1991), 757.[10] W. K. Koo, C. S. Chong and B. K. Merican, Am. J. Phys., 63, 2, (1995), 184.[11] P. R. Camp, Am. J. Phys., 65, 5, (1997), 449.[12] W. S. Bickel and W.M. Bailey, Am. J. Phys., 53, 5, (1985), 468.[13] M.D.Merrill, Constructivism and instructional design in T.M. Duffy, D.H.Jonassen ed., Constructivism and the

technology of instruction, Hillsdale, New Jersey, Erlbaum (1992).[14] D. Hestenes, A modeling theory of physics instruction, American Journal of Physics, vol. 55, (1987), 440-454.[15] G. P. Konnen, Polarized Light in Nature, Cambridge U.P. New York, (1985), 172 .[16] H.R. Crane, “The Quartz Watch with Digital Read-out”, Phys. Teach. 32, (1994), 298.[17] F. Corni, http://www.uniud.it/cird/secif/ottica/corni1.htm [18] M Michelini, A Mossenta, L Benciolini, Teachers answer to new integrated proposals in physics education: a case

study in NE Italy, in Information and Communication Technology in Education, Intern. Conf. Proceedings, E.Mechlova ed., University of Ostrava, (2000), 149; M Michelini, A Mossenta, The EPC Project – ExploratingPlanning, Communicating, in Physics Teacher Education Beyond 2000 (Phyteb 2000), R. Pinto, S. Surinach Eds.,Girep book - Selected contributions of the Phyteb 2000 International Conference, Elsevier, (2001), 457.

[19] A. Loria, C. Malagodi, M. Michelini, Teacher’s Attitudes: on undating curriculum in particular, Proceedings of theInternational Conference on Education for Physics Teaching - ICPE, (1981), 272-273.

[20] S Pugliese Jona, M Michelini, A M Mancini, Physics teachers at secondary schools in Italy, in The Training Needsof Physics Teachers in Five European Countries: An Inquiry, H Ferdinande, S Pugliese Jona, H Latal eds., vol 4,Eupen Consortium, European Physical Society, (1999), 63-89.

[21] L. Benciolini, M. Michelini,A. Odorico, Formalizing thermal phenomena at 3-6 year: action research in a teachertraining activity, in Developing Formal Thinking in Physics, Girep Book of selected papers, in press.

[22] A Stefanel, C. Moschetta, M. Michelini, Cognitive Labs in an informal context to develop formal thinking inchildren, in Developing Formal Thinking in Physics, Girep Book of selected papers, in press.

[23] The main research projects carried out by the cooperation of the physics education reasearch units of Milan,Modena, Naples, Palermo, Pavia,Turin, Udine are: 1) National Project financed by Ministerium_1996_In-servicesecondary school teacher education for new curricula based on ITC experimented in school, 2) National Projectfinanced by CNR_1996-1997-1998__ITC in physics education and teacher education l, 3) National Projectfinanced by CNR_1999_ ITC in physics and in teacher education l, 4) Relevant National Project financed byMinisterium_1999-2000_Spiegare e Capire in Fisica (SeCiF) – Explaining and understanding in Physics.

[24] See the documentation of the school experience in www.uniud.it/cird/SeCiF, Section: Fenomeni termici –Attività in classe.

[25] P. Violino, B. Di Giacomo: An investigation of piagetian Stages in Italian Secondary School Students, Journal ofChemical Education, 58, (1981), 639.

[26] U. Besson, Rappresentazioni mentali e strutture logiche, Schola Europaea, n. 114, (1992).[27] U. Besson, Une approche mesoscopique pour l’einsegnement de la statique des fluides, Ph.D. Thesis, Université

de Paris 7 “Denis Diderot”, Paris.[28] J. Clement, D. Brown, A. Zietsman, Not all preconceptions are misconceptions: finding anchoring

conceptions”for grounding instruction on students’intuition, International Journal of Science Education, 11,(1989), 554-565.

[29] M. Vicentini, http://griaf.fisica.unipa.it/FORM/pag1.htm.


HIGHER ORDER THINKING IN PHYSICS EDUCATION (HOT-PHYSICS)

Jens D. Holbech, Poul V., Thomsen, Centre for Studies in Science Education, University ofAarhus, Aarhus C, Denmark

1. IntroductionDenmark has 9 years of compulsory and comprehensive schooling (the “folk-school”, 7-16 years ofage). Integrated Science is taught in grades 1-6 for 1-3 lessons per week (each lasting 45 minutes)and integrated Physics/Chemistry is taught for 2 lessons per week in grades 7-9.At the age of 16 the pupils have a number of possibilities for continued schooling and about 20%of the population choose the mathematics line of a general upper secondary schooling lasting 3years (the “gymnasium”). In the mathematics line, physics is obligatory with 3 lessons per week fortwo years and optional in the third year (A-level: 5 lessons per week).For a student the change from the folk-school to the gymnasium is a very real change since not onlythe physical surroundings change - the pupils also meet quite a different kind of teaching and ofteacher: in the folk-school the teacher is trained at a teacher education college and familiar with(theoretical) pedagogy, but not so much with the subject. In the gymnasium the teacher has amaster’s degree from a university and is very well versed in his subjects (e.g. mathematics andphysics) but knows little about pedagogy! Some of the problems and perspectives - especiallystudents’ interests and attitudes - connected to the transition from folk-school to gymnasium aredemonstrated in Krogh & Thomsen (2001).In this paper we address a quite different problem: the problem of missing thinking skills and howto treat it. In the folk-school, practical work and calculations of physical quantities are done on aregular basis but at a rather low level of abstraction (15 years ago this was different, but(unpublished) research showed the teaching to go far beyond students’ possibilities forunderstanding and the curriculum was changed accordingly). From the first day in the gymnasium,however, the students are required to do formal thinking and manipulate formulas. This evidentlycan only succeed if the new students are carefully selected - but this is not the case. As a result,physics teachers have been complaining for years about falling standards.

2. Inspiration from CASEOur inspiration for a project aimed at remedying the above problem came from the work by Shayerand Adey on the Cognitive Acceleration through Science Education (CASE)-project (e.g. Adey etal. 1995, Adey & Shayer 1994, Shayer 1999). Their work is aimed at students about 12 years old andbased theoretically on a combination of Piaget’s ideas of developmental stages (especially thetransition from concrete to formal thinking) and Vygotsky’s concepts of the ‘zone of proximaldevelopment’ and socially mediated learning.Most students at that age are ‘concrete’ thinkers (level 2A-2B in Piagetian terminology) and thehope was through carefully planned instruction to induce students to progress to early formalthinking (3A) and beyond. For this purpose, the CASE team developed a teaching schemeconsisting of a concrete preparation ideally leading to cognitive conflict and ensuing construction ofmeaning. This construction is stabilized by metacognition. The teaching scheme runs decoupledfrom the science lessons (intervention lessons) as independent ‘thinking lessons’, but during thefollowing science lessons ample and explicit use is made of the (Piagetian) reasoning patternsintroduced, e.g. proportionality and control of variables. Shayer and Adey claim striking successesand are especially proud of long term and far transfer effects as e.g. far better results in theexperimental classes than in control groups at the nationwide GCSE examinations at age 16, i.e. 4years after the intervention - not only in science, but also in mathematics and English.

3. The HOT-projectAs mentioned above, about 20% of a Danish cohort enters the mathematical line of the uppersecondary school and there are widespread problems with thinking skills. Our starting point was a


reserved acceptance of Shayer and Adey’s ideas - something like: ‘it may work, so let’s try. If itworks: very good; if it doesn’t: we got that wiser and we’ll try something else’.According to Shayer and Adey’s early results (Adey & Shayer 1994, see also Shayer and Adey1981), only about 5% of students around 16 should be late formal thinkers (3B) and another 25%early formal thinkers (3A). Even in the best case our target population would be dominated by 3A-students. This implies that the students may handle formulas but without a deeper understandingof what they are doing, e.g. considering U = R ≅ I, R = U / I and I = U / R as three different formulas(or prescriptions for use) instead of one; they may work with problems with two independentvariables and control for factors explicitly named but not deal freely with multivariable problems;and they may do simple problems with constant ratios but are not able to do proportionalreasoning - in short: they are not able to understand the concepts, the reasoning and the modelsrequired in the physics curriculum.In order to find out the distribution of ourstudents on Piagetian stages we used atranslated version of the ‘pendulum task’from the Science Reasoning Tasks (1992)which is a test of students’ abilities to dealwith control and exclusion of variables. Inthe very beginning of the HOT-project(Feb. 2000) we tested 4 classes, i.e. about100 students, and convinced ourselvesthat most students actually handledcontrol of variables at early formal levelonly. The teaching material was thereforedesigned to start at that level (or a littlebelow) and aimed at lifting the studentstowards late formal thinking.Before starting the actual teaching inAugust 2000 we did the pendulum test onall 22 classes participating and on 15control classes from the same schools.As described later, the actual course of the project was rathermessy and only 7 of the classes tried more than half of the teaching material. We have only usedthese classes for statistics together with the control classes from the same schools (8), and Figure 1shows the initial distribution of these 330 students on Piagetian levels. There are rather differentdistributions for each single class but no significant difference between experimental and controlgroups.We had expected to do piloting of the material (to be described later) in just a few classes in ourlocal area, but 10 teachers offered to try it out and the newly started Danish project ‘World-ClassMaths and Science’ (www.matnatverdensklasse.dk) asked us to take a further 12 teachers from theCopenhagen area. The pilot phase thus comprised 22 classes with about 600 students.The teachers were brought together at Aarhus University for two half-days during which they gotan introduction to the thinking behind HOT-physics and the pendulum test. Most of the time was,however, spent on an introduction to the preliminary teaching material and discussions on its actualuse and how to do the bridging to later physics lessons. One of us (JDH) taught a class himself usingthe HOT material and used the experiences to make a first adjustment. The adjusted material wasimmediately distributed via an Internet based conference where the participating teachers couldalso give their suggestions for changes, ask questions and initiate discussions.Our original idea was that the HOT-lessons should be over before Christmas, but unfortunately thiswas not the case. Some of the teachers from the Copenhagen area only started in Novemberbecause of simultaneous participation in another project under ‘World-Class Maths and Science’.It also turned out to be more difficult to get feedback from the teachers about the working of the


Figure 1: Initial percentages of students in Piagetian levels ofreasoning.

0

0,1

0,2

0,3

0,4

0,5

0,6

2B 2B* 3A 3A/3B 3BLevel of Reasoning

Science Reasoning TaskCopenhagen & Aarhus, Autumn 2000 N=330

project.The Internet conference was little used and we could only manage to have one or two shortmeetings (1-2 hours each) with the teachers. At the end of the school year JDH was the only oneto have done all lessons and the most arduous teachers had taught about 75%, the least only about25%.At the end of the school year most classes - experimental as well as control - were tested again withthe pendulum test. Due to the difficulties described above we were very modest in our expectationsand we decided to look only at classes having trialled more than about 50% of the material and thecontrol classes from the same schools. This gave us 7 experimental classes and 8 control classes.Also, we only used data from students having taken the test at the beginning as well as at the endof the school year and the net number of students then turned out to be 154 in experimental classesand 176 in control classes (these are the students behind Figure 1).As mentioned, there were no significant differences between the two groups in the beginning of theschool year, but there did turn out to be striking differences at the end of the year: the experimentalclasses had a significantly larger gain in reasoning level than had the control classes. To bring theresults into a quantitative form we used the same numerical scale as Adey and Shayer (1994), i.e.2B = 5, 2B* = 6, 3A = 7, 3A/3B = 8, 3B = 9, and it turned out that the initial value for Piagetian levelwas 7.07 _ .24 (corresponding to early formal). The control classes ended at 7.60 _ .27, i.e. anaverage gain of .52 (‘half a level’) while the experimental classes ended at a mean of 7.96 _ .38, i.e.an average gain of .91 (‘nearly a whole level’). The difference between the gains is .36corresponding to the experimental classes being 1.3 (control) standard deviations above thecontrol classes. This is highly satisfactory (especially in view of our difficulties) but of course it isnot clear whether it is due to the HOT-material or to the HOT-teachers (who joined voluntarily).The touchstone for this requires a much larger scope with participation of ‘non-voluntary’ teachers,but at least our initial hope is not falsified!

4. The HOT teaching sequencesIn the CASE-project there are 32 intervention lessons distributed over 2 years. The lessons aredesigned to last for about 70 minutes each, i.e. a total of ~ 40 hours. The total amount of time forphysics teaching in the first grade of Danish upper secondary school is 79 hours and it is quite clearthat our ambitions about time consumption had to be at a much lower level than in CASE - maybe15-20 hours in the best case. It was also clear to us that the HOT-lessons should be closer coupledto the physics teaching than were the CASE-lessons - partly in order for the teachers to feelcomfortable about fulfilling their duties towards the syllabus, partly to have an opportunity tointroduce the thinking skills before the students need them. Taking the speed of delivery in physicstextbooks into account, this means that the HOT-lessons should be given during the autumn.The thinking skills needed in the physics curriculum pertain mostly to using variables and formulas.We therefore chose – much along the same lines as CASE – to start with the concept of variables,relationship between variables and control of variables. Later on this was supplemented withcompound variables (e.g. pressure, density), but in the meantime we did ratios and proportionalityand inverse proportionality (‘compensation’) as an introduction to the use of formulas. Finally, wemade lessons on equilibrium where connections and control of up to 4 independent variables comeinto play (a 3B-activity). We wanted also to include modelling – both as an independent lesson andas continuous input - but due to lack of time this never came into play.A typical module follows the CASE teaching scheme with a concrete preparation where the teacherintroduces materials and terminology and makes sure the students understand ‘what it’s all about’.This is followed by hands-on and mind-on activities where students work alone or in small groupswith concrete material and/or worksheets (hopefully) establishing cognitive conflicts and a need forconstruction of meaning. The teacher has a central role as mediator and guide into the zone ofproximal development6 and peers have important roles too as sparring partners in discussions andas collaborators in trying out theoretical as well as practical suggestions for explanations andsolutions. The newly acquired thinking patterns are used in a number of thought experiments and


problems in order to consolidate them and further consolidation is aimed at through metacognitiveactivities, e.g. students summing up and clarifying the new terminology, what they learned in themodule, and how they learned it.The teaching material consists of worksheets for students, concrete material for demonstrationsand hands-on activities, and a teacher’s guide with introduction to the theory and the aims behindthe module, advice on actual teaching, and suggestions for bridging during the physics lessons tofollow.Space does not permit description of all these modules and we have to restrict ourselves to just oneexample: control of variables:The module about control of variables followed a module on variables and relationship betweenvariables: what is a variable, a variable may have qualitative or quantitative values, one maydistinguish between independent and dependent variables, and finally some examples ofqualitative relationships between dependent and independent variables.The concrete preparation was a practical work with tubes (adapted partly from CASE, Activity 3:The ‘fair test’). The tubes are of 3 different lengths, 3 different widths and 2 different materials. Theteacher demonstrates the possibility of making sounds by tapping the end of a tube in a hand (inthe first edition we asked for blowing across the tube, but one of the teachers informed us thattapping was much better than blowing and gave no problems with hygiene) and introduced thehighness of the tone as the dependent variable wanted. Students were asked to try the tapping andafterwards identify the independent variables (length; width; material) and their values (long,medium, short; wide, medium, narrow; metal, plastic. Some students wanted to measure the lengthsand widths to get quantitative variables and were allowed to do so).The activity in the construction zone was for each student to pick pairs of tubes, compare the notesand try to find the relationship between highness of note and the independent variables. Thestudents have access to all the tubes (the materials box contained 72 tubes of 18 different kinds)and get a worksheet asking the student to write down the combinations used and the conclusionsdrawn from the experiments: ‘What is the influence of the width?’, ‘Which experiment tells youthat?’ etc. The teacher walks around and mediates cognitive conflict by asking for conclusions andchallenging incorrect as well as correct suggestions: ‘How can you be sure that width does notmatter?’, ‘How can you be sure that only length matters?’ ‘Is one pair of tubes enough to draw aconclusion?’ etc.Some simple problems are given and the teacher initiates metacognition by asking the students toanswer worksheet questions like ‘What have you learned about control of variables when you haveto find relationships between variables?’ and ‘How many independent variables must there bebefore needing control?’. Afterwards, a class discussion is organized about the topic and thestudents are asked to do a number of problems/thought experiments of increasing difficulty.Suggested bridging during the physics lessons following the module is to explicitly ask students forcontrol of variables in practical work, e.g. by planning experiments on Joule’s law (P = R ≅ I2)instead of giving them ‘cookery book recipes’. Correspondingly, they may be challenged to‘construct’ the equation of state for an ideal gas or at least to design experiments showing the effectof each variable on the other.

5. The futureAt the brief meetings with the teachers we got very positive feedback about the teaching materialand the modules they had tried out. They clearly saw HOT-physics as kind of an answer to at leastsome of their problems and they reported also about most students being very engaged in theactivities which were at the right level for them to understand: considered as a teaching material(and looking apart from our ambitious aims) the HOT-modules were successful! The teachersstrongly suggested us to continue.In view of this we decided to run a second generation of HOT-physics - despite the messy way ithad worked and despite the fact that we did not know at that time whether there were any positive


effects. It was clear to us, however, that for methodological reasons we should prepare the teachersbetter and have a closer monitoring of the whole process.We therefore advertised HOT-physics as an in-service teacher training course consisting of a 2-dayintroductory meeting in August 2001 (instead of 2 half-days) followed by one whole day lateSeptember and another late November.The response was overwhelming: about 60 teachers wanted to participate, but we had to restrictourselves to a doubling the course and could only accommodate 40 teachers (neverthelesscorresponding to 10% of the first grade physics teachers in Denmark). A few of them hadparticipated the year before, had been assigned to a new first grade class and wanted to try againin a more serious way. This of course is encouraging, but at least as encouraging is that most of ournew teachers come from schools with teachers who participated the year before and hadrecommended the course.At the time of writing we have just finished the two 2-day courses. As we hoped, they turned outto be much better and far more reflective than the stressful course last year and afterwards theparticipants were very keen to go back to their schools and do HOT-physics.Whether the project will fulfill its aims is of course uncertain at the moment, but we think we havegood reasons for optimism. But it is quite certain that about 60 physics teachers have beenchallenged to reflect on their former teaching habits, have experienced cognitive conflicts andmade explicit decisions to change their practice. For the better, we believe, but time will show!

ReferencesAdey P, and Shayer M, Really Raising Standards, Routledge, London, (1994).Adey P, and Shayer M, Yates C., Thinking Science Thomas Nelson and Sons Ltd UK, (1995).Krogh L and Thomsen P V, Teaching Style and Learning Outcomes, Proceedings of GIREP-conference: “Physics

Teacher Education Beyond 2000”, Elsevier, (2001), 257-260. Science Reasoning Task III: The Pendulum ScienceReasoning, 16 Fen End, OVER, Cambridge CB4 5NEShayer M Cognitive acceleration through science educationII: its effects and scope Int. J. Sci. Educ. 21, (1999), 883-902.

Shayer M and Adey P, Towards a Science of Science Teaching, Heinemann Educational, (1981).Vygotsky L S, Mind in Society, Harvard University Press, (1978).

COMPREHENSION AND TEST RESULTS AFTER INTRODUCTION OF WORKSHOPPHYSICS

Thomas Lundstrom, Mats D. Lyberg, Alf Svensson, School of Mathematics and Systems Eng.,Dept of Physics, Växjö University, Sweden

1. IntroductionHere we report on results of introducing Workshop Physics as a complementary education tool inthe introductory course in mechanics. The course corresponds to the first thirteen chapters ofcommon introductory textbooks such as Benson, Giancoli, Halliday-Resnick-Walker, Serway,Tipler, Young and Freedman, Wolfson and Pasachoff, etc. Our traditional course comprises fiveweeks of full-time study with twenty hours of lectures, twenty hours of problem solving and twentyhours of laboratory work. Two reference groups, one physics student class of about twenty studentsand one mechanical engineering class of about forty students followed this curriculum.The test group consisted of a physics student class of about twenty students.This group was for partof the course introduced to Workshops Physics [1-4], tutorials [5] and interactive lecturedemonstrations [6] to enhance the understanding of some important concepts. The items selectedfor Workshop Physics have been motion of a particle subjected to a force, gravitation and the actionand reaction of forces.Our laboratories are equipped with The PASCO Physics Systems which include physics apparatus,computer interface 750, sensors and the data collection system DATA STUDIO


2. VariablesIn this report we have studied the interdependence of three variables, the student’s result from theexamination of the mechanics course, and the student’s result from a conceptual test [7] before (pre-test) and after (post-test) the mechanics course. All variables have been reworked to a scale from 0 to100. Examination results are given in a scale where 0 corresponds to 0 points and 100 corresponds tothe maximal number of points. The force test is a multiple-choice test of 29 questions each having 5alternatives. A test group knowing nothing about mechanics would thus score an average of about sixcorrect answers. However [8] ,completely non-newtonian thinkers may tend to score below therandom guessing level because of the very powerful interview-generated distractors so we have notmade any correction for this.

3. Test result versus examination resultThe test is intended to measure theunderstanding of the force concept andNewton’s laws. The examination consists ofsolving problems. Obviously, they must notmeasure the same thing. In Fig. 1 is displayedthe relation between post-test result andexamination result for students of physics andmechanical engineering, respectively. Thecovariance is for physics students 64% and formechanical engineering students 51%. There issome connection between the two variablestest result and examination result, but not avery impressive one.

4. Pre-test result versus post-test result.Presumably, most students have improved theirunderstanding and knowledge after a course inmechanics. This assumption may be investigatedby comparing the pre-test result to that of thepost-test. The result is displayed in Fig. 2. As canbe seen, there is a very good correlation (91%).The improvement is of the order of 25 %.Taking70% correct test answers as a measure of anacceptable understanding of newtonianmechanics, 10% of the students are newtonitesalready before the mechanics course, whileanother 15% achieve this level after the course.In the same manner, taking a test result of 30%correct answers as a measure of a decidedlyaristotelian view of mechanics, 40% of thestudents are aristotelians before the course and15% stubbornly remain so. Applying the samecriteria to the mechanical engineering studentsafter the course, about 20% are newtonites andequally many are aristotelians.For analysis purposes we have used the gainfactor g defined by Hake [8]


20 40 60 80 100Posttest

20

40

60

80

100

Exam Examination vs Posttest

Fig. 1: Plot of the post-test result versus the examinationresult for physics students (squares) and for mechanicalengineering students (triangles).The covariance is 64%for physics students and 51% for engineering students.

20 40 60 80 100Pretest

20

40

60

80

100Posttest Covariance 91%

Fig. 2: A plot of the post-test result versus the pre-testresult for physics students. Data include students whohave worked with WP as well as those who not.

g =posttest(%) pretest(%)

100 pretest(%)

If we apply this to our student group we achieve <g> = 0.53 for the WP students and <g> = 0.36 forthe traditional group. The first group falls in the region between Medium-g and High-g [8].If we instead use the Gaussian fit to histogram presented by Redish [9,10] this group falls in theWP-group from Dickinson College. In table 1 we give the gain factor with standard deviation as acomparison.

Table 1: A comparison of gain factor for different groups

5. Results from working with Workshop PhysicsIn this section we compare in more detail the two groups of physics students who have, respectivelyhave not, worked with Workshop Physics. In Table 2 we give some results from a comparisonbetween these two groups.The concepts selected comprised the force concept (questions 15, 18, 22 and 28, one-dimensionalmotion of a particle subjected to a force (questions 3, 5, 16 and 23), and the concept of action andreaction for forces (questions 2,11,13 and 14)

Table 2: Effects of working with Workshop Physics


14 trad courses from Hake 0.23±0.04 Traditional Vaxjo 0.36±0.20

WP Dickinson 0.42±0.04 WP Vaxjo 0.53±0.29

Physics students Physics students not Engineering working with WP working with WP students, no WP

Examination result [%] 66 64 54 Pre-test result [%] 41 35 - Post-test result [%] 66 57 42 Force concept, pre-test 32 36 -

post-test 59 40 21 Linear motion, pre-test 45 38 -

post-test 61 42 37 Newton III, pre-test 39 34 -

post-test 83 79 39

Comparing the two groups of physics students, the introduction of WP has no significant effect onthe examination result. The group working with WP has a higher post-test score, but this was thecase also in the pre-test. However, looking at the parts of the course where WP was applied, thereis a markedly higher improvement regarding the force concept and linear motion for the WPgroup. Regarding the force action and reaction both groups have improved substantially. Thegroup of engineering students has a markedly lower score than physics students in the post-test.

6. Results from a gymnasiumWorkshop Physics has been introduced into many a curriculum at the college or university level.Students starting studies at the gymnasium level have at best some rudimentary knowledge ofphysics from the compulsory school, while students starting at the university are obliged to havetaken physics courses at the gymnasium.The Swedish gymnasium (school years 10 to 12) has two Physics courses for students orientedtowards natural sciences.The first course is in the first year and is compulsory for all students. It comprises mainlymechanics.The second course of the last two years is voluntary. In many cases the interest in taking

the second course of physics has been very low. In an attempt to improve this situation, WorkshopPhysics [1-4] was introduced in some classes. We have studied the effect of this introduction.Workshop Physics was introduced to two classes out of three. The number of students continuingwith the second course of Physics is the main interest of this study. The two classes receivingWorkshop Physics (WP) during the first year did so also during the two consecutive years to studya possible influence on the final examination in physics.The main results are presented in Table 3. We assume that three factors may influence the choiceof taking the second physics course.These are the perceived success in studying mathematics and physics during the first year, adecision made already before the first year not to choose further studies in physics, and anincreased understanding of physics due to the introduction of WP. In practice, we do not believe itpossible to disentangle the first two factors, so we have taken the examination result inmathematics and physics during the first year of studies as a measure of both factors.

Table 3: First year Physics students

In the two classes with WP a much larger number of students selected the second course in physics.However, students not continuing with physics had a lower grade in mathematics and physicsduring the first year. This is probably the most important factor, the covariance between the firstyear examination result and the decision to continue with physics is 75%, while for the exposure toWP the first year and continued studies in Physics the covariance is 35%.The application of WP during the last two years does not seem too greatly influence theexamination result in Physics.This study is continued by a follow-up of how many students continue to study natural sciences orengineering at university. There are no definite results yet, but there is a clear indication thatstudents having taken part in WP do so to a much larger extent.

7. ConclusionsThere is a comparatively weak correlation between examination result and understanding, asmeasured by the Force Concept Inventory, for an introductory course in Mechanics, the co-varianceis of the order 50%. The introduction of Workshop Physics for part of the course markedlyimproves the understanding for this part of the course, but does not seem to greatly effect theexamination result. The introduction of Workshop Physics at the gymnasium level resulted in alarger of students selecting more than one physics course.


Student category Number of students

WP course, Class 1 24 choosing Physics B 20 no more Physics 4

WP course, Class 2 26 choosing Physics B 20 no more Physics 6

Ordinary course 22 choosing Physics B 10 no more Physics 12

1st year grade in math- physics (average)

75 76 67

62 68 44

52 70 38

Students continuing with physics [%]

83

77

45

3rd year physics grade

63

72

70

References[1] P. Laws, “Calculus-based physics without lectures”, Physics Today, (1991).[2] P. Laws, “Millikan lecture 1996: Promoting active learning based on physics education research in introductory

physics courses”, Am.J.Phys. 65, (1997).[3] Sokoloff, Thornton, Laws, “Real time physics” Wiley & Sons[4] P. Laws, “Workshop Physics”, Wiley & Sons[5] L. McDermott et al, “Tutorials in Introductory Physics”, Prentice Hall[6] R. Thornton and D. Sokoloff, “Interactive Lecture Demonstrations”, Physics Teacher, (1997).[7] D. Hestenes, M. Wells, G. Swackhamer, “Force Concept Inventory” The Physics Teacher, 30, (1992).[8] R. Hake, “Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test

data for introductory courses”, Am. J. Phys. 66 (1), (1998).[9] E. Redish, J. Saul, R. Steinberg, “On the Effectiveness of Active-Engagement Microcomputer-Based

Laboratories”, Am. J. Phys, 65, 45-54, (1997).[10] E. Redish, “Millikan Lecture 1998: Building a Science of Teaching Physics”, Am. J Phys, 67, (1999).



3.7 Mathematisation

ON HOW TO BEST INTRODUCE THE CONCEPT OF DIFFERENTIAL IN PHYSICS

R. López-Gay, J. Martínez-Torregrosa, Dpto. de Didáctica General y Didácticas Específicas,Alacant, SpainA., Gras-Martí, Dpto. de Física Aplicada, Universitat d’Alacant, Alacant, SpainG. Torregrosa, Dpto. de Análisis Matemático y Didáctica de las Matemáticas, Alacant, Spain

1. IntroductionThe almost unanimous opinion of 103 high school Physics teachers, and the results of the analysisof 38 Physics textbooks, indicate clearly that Differential Calculus is necessary to teach Physics inthe last years of high school, and it becomes indispensable at university level (Lopez-Gay et al.,2001). This necessity is obvious since Differential Calculus allows to study physical situations witha higher degree of complexity than the ones dealt with in elementary courses.In agreement with this importance, it should be expected that the use with comprehension ofDifferential Calculus would be a concern of Physics teachers and a result of the learning process;and that it should be perceived as an indispensable tool to progress in the comprehension ofPhysics. In spite of this, our teaching experience has showed us that the reality of the customaryuse of Differential Calculus in Physics lectures is very different: textbooks and lecturers use it inan operational and mechanical way, the students don’t understand what is being done and why isit done when it is used even in simple situations, both teachers and students have low expectationsof it being used with comprehension and autonomy, its use is perceived as an obstacle and a sourceof reject against Physics. This impression is confirmed by our analysis of the usual teachingpractice.A clear uneasiness among Physics teachers exists, for they feel obligated to use DifferentialCalculus in their lectures, but are aware that their students don’t understand the reason and themeaning of what is being done. Even worse, there is a diffuse uneasiness as a result of theirincapability to identify clearly the cause of this situation and to find an adequate way to overcomeit.In order to confront this diffuse uneasiness, we have carried out a historical and epistemologicalstudy (Martínez-Torregrosa et al., 2001) looking for answers to the following questions: in whatkind of situations is Differential Calculus used?, which is the global strategy of Calculus to dealwith those situations?, which is the meaning of the different concepts used?, which is therelationship among them? These questions have remained concealed in the customary educationpractice under a stream of rules and algorithms, which shows a clear disdain towards the moreunderstandable and conceptual aspects (Artigue and Viennot, 1987; Ferrini-Mundy and Gaudard,1982, NCTM, 2000).Taking into account that differential expressions constitute the starting point in the reasoning andmathematisation of the physical situations, we have centred our study in the concept of differential,although without losing sight of the global strategy of Calculus. The conclusions we have obtainedhave enabled us to understand better the present deficiencies and to introduce proposals toovercome them.

2. Towards a better understanding of the differential in physicsThe concept of differential has not always answered to the same definition, nor has played thesame part in the whole of Calculus. In its origin, in the 17th century, the differential was inventedto undertake the step from approximation to exactness. For this, the differential of a quantity (dy)was considered as an approximation of the increment of that same quantity (∆y), but an incrementso small that they tended to be the same (dy = ∆y).This way one could ignore the error made when

the (very small) increments were substituted by differentials, or when the terms containingdifferentials were neglected (which L’Hôpital resumed in the equation: y+dy = y).An analysis of 38 Physics textbooks shows that this concept of the differential -and the argumentsthat accompany itis the only one that appears in Physics lectures. The same criticisms that weremade more than 300 years ago about the use of infinitely small quantities, can be formulatednowadays almost in the same terms: how small is the differential?, is it a fixed quantity, or a variablequantity?, is it really possible to obtain an exact result by neglecting terms that are not zero?, whatcriteria should be used to determine the differential expression in each specific case?In general, in order to write the differential expression, an approximate expression of the incrementis written, with the certainty that, when that increment is very small, it can already be consideredan exact expression. For instance, it is said that the variation of the intensity of a plane wave whencrossing a material medium of very small thickness is: dI = -α·I·dx. But this is not true, no matterhow small dx or dI are. It is true, however, that when dx is very small, that expression is veryapproximately –although it never coincides with- ∆I, but the same happens to other expressions(for example: dI = -α·I2·dx, dI = -α·dx/I, dI = -α·I·dx2...). If dI never gets to coincide with ∆I, an errorwill always exist when substituting one for the other. How then get to the exact result?Furthermore, what criteria do we use to select which starting expression is adequate?In usual teaching these questions are not dealt with. The wrong belief is implied that, whatever theform of the approximate increment, as it is so small, the error will always be practically zero andwill not affect the result. This is wrong: there is only one correct differential expression for eachsituation1.Therefore, the criticisms made to the use of the differential as an infinitely small quantitydo not only affect formal or philosophical questions, but have practical consequences.Why are not wrong results obtained more frequently then? On the contrary, the use of DifferentialCalculus is considered almost as a guarantee that the result is more correct than when it is notused? It is known that by recalling past examples one selects the correct results, and the absurdresults that have historically been obtained are soon forgotten (Orton, 1983; Schneider, 1991). Inteaching, the starting differential expression is accepted as a dogma, without arguments noranalysis that justify that particular choice and not a lot of other possible ones. This gives rise to thefeeling that the simple idea of differential as a very small increment, where everything is allowed,is enough to sustain the whole of Calculus in the physical applications.The weight of these criticisms and the adverse results lead to the creators of Calculus to doubt ofthe identification between the differential and an infinitely small quantity [Kline, 1980, p. 480 and511]. Leibniz left the differential without a rigorous meaning but, convinced of its utility, did notgive up the use of the differential: “the differentials may be used as an instrument, in the same waythat the algebraists use the imaginary roots with great benefit” [Kline, 1980, p. 509]. We haveconfirmed this same ambiguity when we study in depth the significance that teachers and studentsassign to the differential expressions they constantly use in the Physics lectures.The following 150 years showed the enormous power of Calculus, but no substantial progress wasmade in understanding correctly what was being done: one worked “in an almost blind way, oftenguided by a naïve intuition that what they did had to be valid” (Eves, 1981). Early in the 19thcentury Cauchy laid out the foundations of Calculus; based on a precise definition of the conceptof limit, he transformed the concepts of derivative and integral into the pillars of Calculus, leavingthe concept of differential relegated to a marginal role, useful to abbreviate certain formalexpansions. Even the definition of differential (dy = y’·dx) depended on the previous definition ofderivative (y’). This work, nevertheless, barely affected the use of Calculus in physical applications,where the differential expressions continued to play an essential role, as starting points to tackle agreat number of problems and situations; not in vain, the Mathematics of the 19th century marks

330 3. Topical Aspects 3.7 Mathematisation

1 For example, approximating the portion of a sphere by the corresponding cylinder, the volume of the sphere is correctlyobtained, but not the area of its surface [Artigue and Viennot, 1987].

the break up between Physics and Mathematics, considered by some as a divorce (Gonzalez-Urbaneja, 1991), and by others as a decolonization (Aghadiuno, 1992). It is not strange, then, thatthe differential has kept in Physics the original meaning of Leibniz, without overcoming itsinconsistence, in spite of the precise definition of the concept of limit and of an infinitesimalquantity provided by Cauchy2.

3. A definition which improves its use with comprehension in physicsAt the beginning of the 20th century, in the context of Functional Analysis, the Frenchmathematician Fréchet formulated a new definition of differential, which “resembles the olddefinition ... and offers all its advantages, but escaping all the objections of rigour that very justlyhad been made to it”, according to a textual quote of the author [Artigue, 1989, p. 34]. Thisdefinition is independent of the derivative, it is centred in the original idea of an approximation,and provides a precise meaning to any differential expression; for this reason, we have used it–adapted to functions of one variable- to rebuild the use of Differential Calculus in a wide varietyof physical situations, giving a precise answer to all the enigmas related with its comprehension(López-Gay et al., 2001).In short, dy (the differential with respect to x) is an estimate of the ∆y produced by an ∆x; such anestimate is linear with respect to ∆x. Among all the possible linear estimates, the differential is theone whose gradient (dy/dx) coincides with the derivative (y’); this guarantees the possibility ofobtaining the exact result by integration, that is, that the limit of the total error accumulated whendifferentials are added is zero. This conception allows us to overcome the deficiencies related withthe original concept:- It clearly shows that it is necessary to resort to the differential in non-linear situations and, for

this reason, the differential will never coincide with the increment (it can be bigger, or smaller),no matter how small it may be.

- The differential is a new function of two variables: x, dx. Its value may be big or small, dependingon the value assigned to the variables x, dx. What is infinitely small, to a first order, is the

difference between differential and increment:

- The differential expressions acquire a precise meaning. For example, the expression: dp = F·dt isused when the force is variable in the interval of time ∆t o dt, and is an estimate of how muchwould the linear momentum change if the force remained constant during ∆t.

- Sometimes, it is easy to determine the differential expression: when the uniform behaviour isknown beforehand (for example, when the force is constant we know that: ∆p = F·∆t, thereforethe corresponding differential expression when it is not constant will be: dp = F·dt). But, in mostphysical problems, that behaviour is not known beforehand, and the starting differentialexpression must be advanced as a hypothesis, based on the physical analysis of the situation, andwith the only formal requirement of it being linear with respect to the change of variable. Theconfirmation of that hypothesis can only be done through the result to which it leads.

This clarification not only leads to a better understanding of the nature and the significance of thedifferential expressions, but also of the general strategy of Calculus to tackle a physical problem,the reason for the steps that are taken when applying that strategy, and the meaning of other basicconcepts (derivative and integral) and the relationships among them. With this clarification wehave identified some indicators of what is an adequate comprehension of the differential in Physics:


2 Cauchy defines an infinitely small quantity as “a variable whose numerical value decreases indefinitely so that itconverges towards the limit zero” [Cauchy, 1998, pp. 26-27]. It is obvious that the increment of any continuous functionobeys this definition, and therefore it is not useful to characterize the differential expression. Nor can one say that it isthe limit of the increment when it tends to zero as, if it is continuous, the differential would always be zero.

00

=x

dyylímx

1. To know when and why its use becomes necessary, that is, to know which is the problem thatmakes ordinary calculus insufficient (i.e., it is necessary to resort to the differential when weneed the ∆y produced in an ∆x, and the relationship between both is not linear).

2. To understand the strategy that Calculus offers to solve that problem and comprehend thereason of the different steps that are taken, i.e.:a. To be able to explain with accuracy and physical sense the meaning of the expressions.b. To know and justify the relationship that exists between the differential and the derivative y’= dy/dx, and accept without ambiguity the reasoning in which this relationship is used.c. To know the meaning of the integral and to know how to justify the so called FundamentalTheorem, i.e., why the definite integral requires the calculation of antiderivatives or primitivefunctions.d. To use that strategy with full knowledge of the physical content in situations and problems.

3. To be aware of the hypothetical and exploratory nature, in almost all physical situations, of thestarting differential expression, and to know that the validity of that hypothesis cannot bechecked directly but through the result to which it leads.

4. To value positively the role of the differential in learning Physics. This axiological componentshould be a natural consequence when the crucial role that the differential plays in thetreatment of physical situations of interest is understood.

These indicators have been of use to us as a guide to analyse the use of Calculus in the commonteaching practices, and to elaborate an alternative proposal.

4. Improving the use comprehension: first resultsAlthough we are aware that it is arguable which is the adequate school year to introduce the useof Calculus, we are convinced that, once that decision is made, it is necessary to do it right, withoutexpecting the comprehension to be acquired in the future, out of the blue. In our case, we begin theuse of Calculus in the 3rd year of BUP (16 years old) and complet it in the year of COU (17 yearsold). We have designed programmes of activities for those two years based on the idea of thedifferential as a linear estimation of the increment, paying attention to the indicators of anadequate comprehension as listed above. Only in the 2nd year the concept of integral is introducedto all the students, therefore we are not taking into account the indicators 2c and 2d in the 1st year.We have carried out a detailed experimental design to confirm that the incorporation, from thebeginning, of the new proposal on the differential, improves the teaching and learning. Thefollowing diagram shows the results of the analysis of a problem solved by students of COU, inwhich the use of Differential Calculus was necessary. The students of the experimental group havehad one of us (RLG) as their teacher during two years.We may mention also some results in relation with the attitude that students adopt. 63% of thestudents of control groups (COU) do not pay attention when Differential Calculus is used in Physics forthey know they are not going to understand it and only pay attention to the final formula. This clearlyreflects the preponderance of the mechanicism in standard teaching practice. On the other hand, 62%of the students of experimental groups (3rd of BUP) taught by teachers trained by us, and 85.4% ofthe students of one us (RLG) (3rd of BUP), plainly reject this idea, which we interpret as an indicationof the use with comprehension which is promoted in their Physics lectures.In general, the comparative analysis of the results obtained by experimental and control groupsshow clearly that significant differences exist among them in all the indicators of an adequatecomprehension, always in favour of the experimental groups, whether of one of us or of teacherstrained by us. The differences become bigger with students who have used the new teachingmaterials for two years. We find ourselves, then, in an adequate direction to improve the use withcomprehension of Differential Calculus in the teaching of Physics.


ReferencesAghadiuno M.C.K., Mathematics: history, philosophy and applications to science, International Journal for

Mathematical Education in Science and Technology, 23, (5), (1992), 683-690.Artigue M., and Viennot L., Some aspects of students’ conceptions and difficulties about differentials, Misconceptions

and Edu. Strategies in Sci. & Math. Cornell, Ithaca, USA, (1987).Artigue M., Le passage de la différentielle totale à la notion d’application linéaire tangente, en: Procedures

différentielles ... (Annexe I). Université Paris 7: IREM et LDPES, (1989).Cauchy A.L., Cours d’Analyse de L’École Royale Polytechnique, Facsímil de la 1ª ed., Sevilla: SAEM Thales, (1998),

1821 Eves H., Great moments in Mathematics (After 1690), Washington, D.C.: The Mathematical Association of America,

Dolciani Mathematical Expositions, 7, (1981).Ferrini-Mundy J., and Gaudard M., Secondary school calculus: preparation or pitfall in the study of college calculs?,

Journal for Research in Mathematics Education, 23 (1), (1992), 56-71.Gonzalez-Urbaneja P.M.,. Historia de la matemática: génesis de los conceptos y orientación de su enseñanza,

Enseñanza de las Ciencias, 9 (3), (1991), 281-289.Kline M., Math. Thought from Ancient to Modern Times, New York,Oxford, Univ. Press, (1980).López-Gay R., Martínez-Torregrosa J., and Gras-Martí A, What is the meaning and use of this expresion: dN=-

α·N·t2·dt?, GIREP 2000. (2001) Selected Contributions. R. Pintó & S. Surinach (eds.). Paris: Elsevier editions; andEnseñanza de las Ciencias, (to be published)

NCTM, (2000) (National Council of Teachers of Mathematics). Principle and Standards for School Mathematics,http://standards.nctm.org/document.

Orton A., Students’ understanding of differentiation, Educ. Stud. in Math., 14, (1983), 235-250.Schneider M., Un obstacle épistémologique soulevé par des “découpages infinis” des surfaces et des solides,

Recherches en Didactique des Mathématiques, (1991).

TEACHING QUANTUM THEORY

Hans Grassmann, Department of Physics, University of Udine, Italy

1. IntroductionMathematics is important for the understanding of Quantum Theory. Often students haveproblems with some aspect of mathematics, without being aware of this problem. As a result thesestudents will for the first time realize to have a problem, when confronted with Quantum Theory.They will therefore wrongly conclude to have a problem with Quantum Theory. Having wronglyidentified their problems, they will have difficulties to solve them: they fail to understand that theyneed to study again some particular aspects of mathematics, in spite of the fact that they are havingproblems in their physics course, not in the mathematics course.


0

20

40

60

80

100

Uses DifferentialCalculus

He/She justifies it Some meaning tothe differential

Writes integrals Addition of a lotof terms

JustifiesFundamental

Theorem

Per

cen

tage

Control group COU (57) Experimental group COU (18)

In this paper we discuss a possible strategy of teaching quantum theory in such a way, that thestudent more easily can identify his problems: while it is not possible to do quantum theory withoutmathematics, it should be possible to formally disentangle them to a point, where the presentationof the subject to the student enables him to see, for instance, that his problems with quantum theoryreally are problems in his understanding of the mathematics of waves.Quantum theory can be expressed either using complex vectors and matrices or waves. There is nofundamental difference between these representations, as is discussed in the presentation of Cobalet al. at this conference.For our presentation we make use of the wave formalism. This choice is motivated by theobservation that many students are even less familiar with complex numbers than with waves.

2. Prerequisits Ideally, the student should have knowledge of the following:He should be familiar with the basic concept of what a field is, in order to better understand whata wave is, and why and how disturbances in fields are propagating as waves. From a formal point ofview, the wave equation can be introduced simply as a generalization of Newton’s second law. Forthe discussion of the kinematics of a point, which can be described in terms of some function intime, f(t), the second law is sufficient, F=md2f/dt2, but for a multitude of points, which are“connected to each other” (concept of physics field), one needs for the description a function bothin space and time f(x,t), there must then be a force F’= αd2f/dx2, too, and these two forces must beequal (3rd law of Newton), that is βd2f/dx2 = d2f/dt2, which is the wave equation.The student ideally should learn about the superposition of waves and the part of mathematicalwave theory, which in physics is called “uncertainty principle”.Where a deeper study of the Fourierformalism is not possible for that, a phenomenological understanding would be totally sufficient(for examples refer to H.Grassmann, “Alles Quark?”, Rowohlt Berlin, 1999).And the student should understand, that one can construct each and any function from asuperposition of waves.The ideal student should know that in special relativity no signal can travel faster than c, and somephenomenological evidence should be given for the atom structure of matter and/or some greekphilosophy. As definition for what is “elementary”, one would define “a particle is elementary, if itcannot be made smaller, without destroying it”. (For which reason one can describe a proton by asimple wave function, without taking into account, that it is composed of quarks.)While these prerequisits seem to be rather demanding, it should be noted, that they are in the endrequired anyhow, no matter how the subject is tought. Sometimes they are in one way or othertought as a part of Quantum Theory - while we rather suggest to identify them clearly asprerequisits.

3. Foundations of quantum theoryThe discussion of the physics of quantum theory could for instance start with a challenge to thestudent: “Go and build a world. You are in charge and you can do whatever you want. You onlyneed not to come into contradiction with yourself, be consistent.” First, this kind of approach israising the student’s curiosity and should increase his motivation, since he is now personallyinvolved. Secondly, it helps to make the student understand that quantum theory results as anunavoidable consequence from a total (though temporary) lack of alternatives. Referring to theprerequisits, a hint is given to the student “try a world, which is made of discrete units, “elementaryparticles”, like the Greeks wanted to do (atomos).”What are the mathematical entities from which one could build the world ? Points would be most easy, physics also knows the concept of “point mass”, so the points couldindeed have mass. But points do not have any dimensions, we cannot create extended objects frompoints only, which do not have any spatial extension. (And this is the more true, if we want to limitourselves to a finite number of point objects.)


One-dimensional objects (strings) could be considered, or two-dimensional objects (branes). Thatindeed is being done presently (string theories), however, there are no conclusive results yet.Theorists continue working, up to now building a world from strings seems in principle possible, butit seems at least very difficult, theorists have not yet succeeded.Can we use three-dimensional massive objects, like the Greeks imagined ? They imagined theatoms to be little spheres, very hard, which never change form or properties (=are elementary).Theanswer again is negative: a body made of hard little spheres would - like the spheres from which itis made of -, be totally hard too. If one moves it at one end, the other end would need to move, too,at the same time. One could transmit messages at the speed of light.The student should be encouraged to come up with suggestions of his own. So that he realises, thatat the present status of science, only one possibility remains (though it can be realised in differentrepresentations, as mentioned before, as complex vectors or as wave formalism).We need to use fields: Fields are three-dimensional, and they are not hard : if one disturbs the fieldat some point (“hits it”), this disturbance propagates as a wave at some certain velocity. Actually,we are not interested in static fields, since in a static world nothing ever would happen, but we areinterested in the field changes, and those are the waves.We therefore need to use waves, from whichto build the world.We need to describe the particles as waves. Therefore we need to identify what property of theparticle is described by the characteristic parameters of the wave, which are amplitude, frequency,wavelength.Obviously, the amplitude of a wave somehow (to be clarified later) tells, “where the wave is”. Fora classic wave, which is composed of very many elementary waves, the amplitude then also tells usabout how much energy the wave has at a given point: in a region, where there is “much of theclassical wave”, there must be correspondingly “many of the elementary waves”, and therefore acorrespondingly large fraction of the total energy of the classical wave.Instead, for an elementary wave, the amplitude of the wave cannot have anything to do with itsenergy. The amplitude of a wave in general depends on space and time, while the energy of aparticle may very well have a well defined and constant value. This discussion was qualitative. Aquantitative measure, of “where the wave is” is given by its intensity, which is the energytransported by the wave per area radiated and per time. The intensity of the classical wave isproportional to the square of the amplitude. Since the intensity of the classical wave in a certainvolume is equal to the probability to find one of the particles from which the wave is composed inthis volume times the total number of particles, it follows, that for the elementary particle theintensity gives the probability to find the particle in this volume.If a wave is not of infinite length, but has a certain finite length (expressed in number of maxima,for instance), then we find that the energy of the wave is proportional to its frequency. Since ingeneral particles are presented by waves of finite length, one would expect, that in quantum theorythe frequency gives a measure for the energy of the wave.For the energy of the elementary wave we therefore would expect a law of the form E=hν, with hto be determined in experiment.The wave length of the wave can be associated to its momentum, when we consider the process ofabsorption of the wave : the wave shall be absorbed by a force F which acts over a time t and adistance l (by which the wave is moving in this time t). We can express t in terms of the period ofthe wave T, as t=αT, in which case l = αλ. A can have any suited value, and the argument also holdsif ν and λ are functions of time. The force F then absorbs the momentum p=F*αT and the energyE=F*αλ. p can be written as p = (E/αλ)*αT = hν*T/λ.= h/λ = p.At this point the student should understand, that one can express elementary particles in terms ofwaves, where the square of the amplitude of the wave gives the probability to find the particle, thefrequency gives the energy and the wave length the momentum of the particle. The studentsunderstanding of the implications of quantum theory, which need to be discussed next, is limitednow primarily by the extent to which he understands the prerequisits.


For instance, an elementary wave in a potential will automatically lead to • Stable states with certain eigenvalues. In this way one can build up for instance the atoms, or

molecules.• The uncertainty principle follows directly from the mathematics of waves.• Interference experiments with, for instance, electron beams show the wave behavior of the

particles directly. Etc.These and similar well known observations and experiments are very important in order to makethe student familiar with Quantum Theory, and they obviously need to be presented with great careand effort also in the kind of teaching which is suggested here. But since they can be used in ourcontext in the same way as in any other context, it is not necessary to describe them here separately.

4. Is the wave real?In some sense, even in classical physics waves are an idealization : on the one hand the waveequation is a differential equation, it has its roots in the differential calculus, that is, in the infinitesmall. On the other hand one needs to average over large distances and large numbers of particles,to observe wave–phenomena. For instance, sound is described as a wave, but the behavior of theatoms or molecules of the gas in which the sound wave travels, individually do not show this wavebehavior.Even if the discussion presented in this paper, or any other kind of discussion, has shown to thestudent, that it is rational and unavoidable (at our present status of knowledge) to build the worldfrom waves, still often a vague feeling of uneasiness will remain with the student, which might bephrased as “it is weird to build a world of waves”. The student wrongly may interpret this “feelingof uneasiness” as evidence for not having understood Quantum Theory, even if he has. Thisunnecessarily will undermine his enthusiasm for physics and his motivation to proceed with hisstudies. Concluding this paper, we therefore would like to suggest, to make it clear to the student,that Quantum Theory is simply the best we can offer at the time being, not more and not less :Quantum Theory as well as the Theory of Relativity are nowadays believed to be incomplete orpartial pictures of a more fundamental point of view, a more fundamental theory of time-space.Depending on how advanced the students is, one or some of the following arguments should begiven to the student :No single elementary wave has ever been observed and it cannot be observed. This is, because thewave collapses everywhere, when observed at one point, otherwise one and the same particle couldbe observed several times. Having been observed at one point, the wave disappears immediatelyand at infinite velocity. Nowhere else in physics is there anything “infinite”.Furthermore, the discussion presented above uses space and time as if something like an “absolute”time and space would exist at least in a given inertial system. While the electromagnetic and thestrong force are known to result precisely from the contrary of this assumption, namely from therequirement, that the elementary waves need to be invariant under global and even under localgauge transformations, respectively.But if the electromagnetic and the strong force in this sense can be seen merely as properties ofspace-time, then the particles, which mediated these forces, should be allowed to be seen asproperties of space-time, too. And one needs to ask, whether the particles in general can be seensimply as properties of space-time:Due to E=hν and p=h/λ. not only the position, but all of the kinematics of a particle can beexpressed in terms of space (λ.) and time (ν), and the mass of the particle can be seen merely as thedispersion relation between ν and λ.The existence of such a dispersion relation in turn indicates a discrete space-time structure.And instead of describing a particle in terms of E =F*l and p=F*t, one can equally well describe itby its action H=F*t*l. But the action of particle waves is described by the formula E= hν, whichsays, that the action is always the same : it is h. It is easy to understand, why the action of a particlenever changes, that is because it never senses any effective force (due to not being extended, as


explained above). But that would not yet explain, why the action of all particles is the same. Ifhowever particles are nothing but space-time properties (like oscillations on a solid body areproperties of this body), then one could understand h as a fundamental property of space-time, andh naturally would need to be the same for all particles. From this point of view it becomes clear,why matter particles are fermions and force carrying particles are bosons : it is a consequence ofthe fact, that particles never can change their action. For an electron of action h to emit a photonof action h, without changing its action, the only way is to flip the electron spin by 1 (from –1/2 to1/2 h, for instance), emitting a photon of spin 1.These or similar arguments should help to communicate to the student, that Quantum Theory -being for the time being the best and only way of describing elementary particles - is very wellcompatible with a possible existence of a more simple picture on a more fundamental level. Underthis point of view, the initial “uneasiness” of the student with the idea to build the world from wavesshould from the beginning not be undermined or questioned. Rather it should be confirmed at theend of the discussion, by telling the student “yes, you have got the right feeling, there is a greattheory but it still may be improved. Come and help us. Or simply stay tuned.”

ANALYSING DIFFERENT FORMS OF PRESENTING NEWTON’S LAWS EMPHASYSINGTHE RELATED CONCEPTS

Josè Luis, Jiménez, Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa,MéxicoIgnacio Campos, Departamento de Física, Facultad de Cienci,s, Universidad Nacional Autónomade México, MéxicoGabriela Del Valle, Área de Física Atómica y Molecular Aplicada, Universidad AutónomaMetropolitana-Azcapotzalco, México

1. IntroductionOur scientific knowledge is an organized conceptual structure that intends to capture somestructural aspects of the world. In other words, we try to mirror a material system with a conceptualsystem.In our scientific tradition we have as a model of conceptual systematization, Euclidian Geometry(300 B.C.); that is the first example of axiomatic systematization.This particular conceptual system encompasses the geometrical knowledge accumulated throughthousands of years of experience in land surveying, constructing and trading. Thus theaxiomatization of a particular domain of knowledge is a late stage in the development of thatdomain. In such development the first step is the establishment of the pertinent concepts, that areusually decanted through years of controversy.We have, for example, that the concepts of constant velocity and constant acceleration wereestablished only with N. de Oresme and J. Buridan in the fourteenth century during a longcontroversy with the aristotelians. These thinkers also introduced through the theory of impetus,the concept of inertia that later was refined by Descartes, Galileo and Newton (H. Butterfield,1957).Once the pertinent concepts of a domain of knowledge are developed, these are used in statementsabout certain aspects of the world. In this way Galileo began to use the concepts of velocity,constant acceleration and inertia, to describe the motion of projectiles and falling bodies.On these grounds, Newton structured the accumulated knowledge of the motion of bodies andtried to give this conceptual structure the format of Euclidian geometry, that is, the format of anaxiomatized system.


2. Elements of an axiomatized systemSince Newton the idea is to give our physical theories the structure of an axiomatized system ofpropositions.In such a system we distinguish:1) Primitive concepts; these are concepts that are not explicitly defined within the conceptual

system, as “point” or “line” in Euclidian geometry, or space and time in mechanics.2) Definitions. Some concepts are defined in terms of primitive concepts or other definitions, as

velocity and acceleration are defined in terms of space and time.Definitions are a matter of convenience, “but theoretically, all definitions are superflous”, (B.

Russell, 1910).3) Primitive propositions or axioms. These are propositions that are not proved within the system;

they are rather the initial assumptions from which all other propositions follow by logicalentailment.

4) Theorems. These are the propositions that can be proved starting from the axioms or from othertheorems already proved; according to their importance, sometimes theorems are called lemmasor corolaries.

In the case of Newtonian Mechanics, as exposed in The Principia, we find some definitions, as thatof mass, quantity of motion, and others, but there is not an explicit list of primitive concepts.However, Newton does state his three axioms or laws of motion. Hundreds of years later Poincaréwould complain that Newton did not distinguish clearly what is experimental fact, what is adefinition, or what is an assumption. Indeed, this complain is just an indication of the difficulties inaxiomatizing a physical theory. Until now there is not a completely satisfactory axiomatization ofclassical mechanics, and this task is included in the fundamental problems listed by Hilbert.A succesful axiomatic system must satisfy the following conditions:I. It must be consistent, that is, a proposition and its negation can not be logical consequences of

the system.II. It must be complete, that is, any known true statement of the domain of knowledge that the

system intends to mirror must appear either as a theorem or as an axiom.III. The axioms must be independent, that is, an axiom cannot be a logical consequence of the

other axioms.Of course these conditions are not easy to establish and therefore most axiomatic systems aresubject to permanent investigation. It was precisely the investigation of the independence ofEuclid’s axioms what lead to non-Euclidian geometries. These conditions are still harder toevaluate in physical theories.

3. Mach’s Criticism to Newtonian MechanicsAs an example we consider Mach´s criticism to Newtonian mechanics. In this case we can see theinfluence of empiricism in Mach, who tried to reduce dynamics to kinematics, eliminating theconcepts of mass and force because he saw in them traces of “metaphysical” concepts as matter(mass) and cause (force). A great deal of Mach´s view has passed to modern textbooks onmechanics. We give some examples in the appendix.In his program Mach criticizes Newton’s definition of mass in terms of density and volume.However, since Mach does not list primitive concepts, his criticism is useless.It is possible in principle to take density as a primitive concept and define mass in terms of it.Thereis some freedom to choose our primitive concepts as well as our axioms. The aim is to obtain aconsistent and functional conceptual structure. Mach instead proposes to define mass in terms ofaccelerations. Here we find some problems since accelerations are vectors, while masses are scalars,however, it is commom to find this “definition” of mass in textbooks.Besides Mach, following Kirchhoff, defines force as mass times acceleration. Thus Newton’s secondlaw is regarded as superfluous and replaceable by a definition.This is again a common view in manycontemporary texts. However, if force is really a definition then, following B. Russell, it issuperfluous!


If the second law is not a proposition about the behavior of the world, but a definition, then whenwe find in a statement “force”, we can substitute “ma”, or vice versa. This cannot give us anyinformation about the possible motions of a body.Poincaré (1902) gives us some indications about this dilemma. He proposes that the acceleration ofa body depends only of the position of the body and the positions and velocities of near bodies.This, he says, implies that the motions of any body are ruled by second order differential equations.If the world were such that velocities instead of acelerations depend on other bodies, as inAristolian physics, then the equations would be first order, and if it were the case that change inacceleration depends on other bodies, then the equations would be third order. In this way we cansee that Newton’s second law certainly is a statement about how the material bodies move underthe influence of other bodies.Hence our view is that it is much more sound to consider mass and force as primitive concepts thanto pass them as useless pseudo definitions. Then the study of classical mechanics amounts toinvestigating the possible motions implied by the differential equation

for pertinent force functions F. In the study of these models of force, that the theory does notprovide, lies the pragmatic aspect of classical mechanics (Moulines, 1979).These force models may be suggested by experiment, like Hooke’s law, or may be conjectured, asthe law of universal gravitation, but certainly they are additional hypotheses that must be fed intothe general theory.Therefore the role of experience is to suggest and test our theories, but certainlyit does not logically implies them.Another point that deserves attention is the common statement that the first law of motion, the lawof inertia, is a particular case of the second law: no force no acceleration. Sometimes it is said thatthe first law defines implicitly inertial frames of reference.Here again Poincaré’s observations may clarify the point. We can express the content of the firstlaw as a further characterization of force: far away bodies cannot accelerate a given body. Hencethe relation to inertial frames: the farther a body the better inertial frame it is. This is the reasonwhy Newton proposed the “fixed stars” as the best inertial frame.

4. ConclusionsWe can conclude that in teaching classical mechanics we must take into account that the involvedconcepts are “free creations of the spirit”, as often remarked Einstein. Then the students must beacquainted with enough mechanical experiences in order to draw the pertinent concepts from thecontext of those experiences.On the other hand, the laws relate these concepts and therein lies their usefulness to understandsome aspects of the world. Thus our students must see that studying physics enables them tounderstand many phenomena and the technological advances familiar to us. In this respect it maybe helpful to study the evolution of our knowledge in order to see how the concepts are createdand refined. For example, Galileo saw the need to substitute the Aristotelian concept of resistanceto motion by the concepts of inertia and friction, identifying this as a force.This historical perspective can also warn us about the tentative character of our concepts. AsEinstein once wrote: “And yet in the interest of science it is necessary over and over again toengage in the critique of these fundamental concepts, in order that we may not unconsciously beruled by them.” A. Einstein in the Foreword to “Concepts of Space” by , Max Jammer (1993).Thus, in our view, the study of mechanics must begin with a familiarization with mechanicalphenomena, as rotational inertia, motion of projectils, etc. Then we study how these phenomenahave been understood in terms of different conceptual structures, as Aristotelian physics andNewtonian physics. The student must appreciate why Newtonian mechanics is a betterapproximation to mechanical phenomena than Aristotelian physics. This may be a beginning to a


dpdt

= F (r, r, t)→ → .

critical attitude that Einstein considers so necessary to the development of scientific knowledge.As they advance in the study of mechanics, they can be exposed to the most elementary model offorce, for example the almost constant gravitational force near the Earth. In advanced courses theymay be put in contact with the mathematical structures of differential equations, variationalprinciples and differential geometry, without forgetting that these structures give us anapproximation to the motions that real bodies in interaction do have.

ReferencesRussell Bertrand, Principia Mathematica, Cambridge U. P., 1910, reprint (1962).Butterfield Herbert, The Origin of Modern Science, McMillan Publishing Co, 1957, reprint (1965).Poincaré Henri, La science et l’hypothése, (1902), Flammarion, Paris, (1968).Ernst Mach, Desarrollo Histórico-Crítico de la Mecánica, Espasa-Calpe Argentina, (1949).Moulines C Ulises., Theory-nets and the evolution of theories: The example of Newtonian Mechanics, Synthese, 41,

(1979).Bunge Mario, Foundations of Physics, Springer-Verlag, New York, (1967).Bunge Mario Epistemología, Siglo Veintiuno, México, D. F., (1997).Jammer Max, The Concept of Space, Third Edition, Dover, New York, (1993).

Some textbooks that follow Mach’s view:M. Alonso and E. Finn, Physics, Addison-Wesley, Reading Massachusetts (1992).M. Alonso and E. Finn, Fundamental University Physics, Mechanics Vol. I, Addison-Wesley, Reading, Massachusetts

(1967).K. R. Atkins, Physics, Wiley, New York (1967).V.D. Barger and M.G. Olson, Classical Mechanics. A Modern Perspective 2nd edition, Mc Graw Hill, New York

(1995).H.C. Corben and P. Sthele, Classical Mechanics 2nd Edition, Wiley, New York, (1960).R.P. Feynman, R.B.Leighton and M.Sands, The Feynman Lectures on Physics Vol.I, Addison-Wesley, Reading,

Massachusetts, (1963).W. Hauser, Introduction to the Principles of Mechanics, Addison-Wesley Reading, Massachusetts (1966).U. Ingard and W.L. Kraushaar, Introduction to Mechanics Matter and Waves, Addison-Wesley, Reading,

Massachusetts, (1960).T.W.B. Kibble, Classical Mechanics, Mc Graw Hill, London, (1966).J.B. Marion and S.T. Thurnton, Classical Dynamics of Particles and Systems 4th Edition, Saunders, Philadelphia

(1995).K.R. Symon, Mechanics 2nd Edition, Addison-Wesley, Reading, Massachusetts, (1965).P.A. Tipler, Physics for Scientist and Engineers 3rd Edition, Worth, New York, (1994).

HOW TO “AVOID” WORK. UNDERSTANDING THE WAYS IN WHICH PHYSICS USESMATHEMATICS TO RECOGNIZE THE CONSTANT OF MOTION OF MECHANICALENERGY

Marisa Michelini, Gian Luigi Michelutti, Department of Physics, University of Udine, Italy

1. IntroductionResearchers and teachers often find themselves faced by the difficulties which students have withthe mathematical instruments which physics uses both on the descriptive level and also on theinterpretative level; almost as a logical, unquestionable consequence, they look for ways to avoidor reduce involvement on the formal level [1,2,3]. For example, they try to give greater weight toexperimental activities (the educational role of which is beyond discussion for an experimentalsubject) [4,5,6], or they try to entrust to operativity and/or informal education the connectionbetween the perception and observation of phenomenology and the physical description [7,8,9].Didactic projects done in the 60s and 70s throughout the Western world [10] are examples of atranslation of the various pedagogical theories into operational strategies for effective teaching.These projects mainly relied on experimental activity to construct a gradual awareness of theformal relationships between the significant variables in selected experiments (PSSC, IPS, PS2),


even when the formulation was of a historic type (PPC). Such experiments were conducted over awide scale, and established that it is not sufficient to optimize teaching in order to achieve goodlearning, and that other types of difficulties occur [11], such as those linked to the lack ofconnection between common-sense interpretation and physical interpretation [12,13], or thoselinked to the ability to use ways of representing things which physics uses, for example graphs [14].The difficulties in mechanics are particularly well-known [15-21]. If we examine them we noticethat the difficulties are of a conceptual type, both with regard to the significance of the elements offormulas which physics introduces to describe and interpret, and also with regard to the styles offormalization which physics assumes.Studies on learning processes [22,23], and theories on conceptual change [24] have given usefulindications for involving students, for the processes of building knowledge, for ways to encouragethe contextualization of concepts, and for the ways in which to help students look at the world froma physical point of view. Research into the use of the computer in physics teaching has given animportant contribution to the ability of looking at processes from a physical point of view, toreading and using graphs [25-29]. Such research has contributed decisively to putting into students’hands the process of constructing formalized physical models starting from qualitative hypotheses[30-34].With regard on how to make students aware of the ways physics uses mathematics to deal withdescriptive and interpretative problems in various circumstances: this problem is still open. It seemsto us that this cannot be considered a secondary problem for a subject like ours, which has assumed,as a work style, a predictive capacity based on the description of phenomena by means ofmathematical tools. It is a style which is a part of the epistemic roots of physics and we do not thinkit is possible to give this up, if we want to give young people the opportunity to develop a passionfor this discipline [35]. Therefore, we need contributions which show the ways physics usesmathematics, which familiarize students with these ways, and which give young people theopportunity to operate on this level without inhibitions, overcoming the prejudice that the symboliclanguage is impossible for them to manage.This work wants to give a contribution to this end, and offer a new way of recognizing mechanicalenergy as a constant of motion.

2. Definition of the proposalThe point of view which directs this proposal favors the principles of energy conservation in theknowledge of physics and associates them with the existence of one or more constants of motion.In this conference we shall consider the conservation of mechanical energy, which allows to identifyinteresting characteristics of the motion of a material point, establishing a relationship betweenposition and velocity.In the teaching of physics, the principle of conservation of mechanical energy is traditionallyintroduced by using the concept of work. In reality, in many basic physics problems, it is this verysame concept which takes on significance and usefulness from the formulation of the principle ofenergy conservation.We therefore propose a didactic definition which sees the principle of energy conservation as thepivot of a mechanism which holds the concept of work as marginal and reserved only for dissipativeforces.This approach develops from the second law of dynamics to a search for quantities which areconstants of motion, that is, which re-write the law in terms of temporal derivatives of quantity,which express it in the direction of the motion.In order to do this, we consider the scalar products with the velocity v of both terms of Newton’ssecond law.On the didactic level, in carrying out this first step, we point out to students that the scalar productof two vectorial sizes selects the contribution of one in the direction of the other. Moreover, we getthe students used to exploring the significance of the temporal variation of one size with respect to


its value: this usually helps us to know new properties, as happens in the description of phases inspace. By representing the quantity of motion as a function of the position for a mass-springoscillator, we see immediately if the motion of the system is dissipative or not.The expression of the kinetic energy emerges from the search for a quantity whose derivative givesthe scalar product of the quantity of motion with the velocity. The expression of the potentialenergy is defined as that quantity whose temporal derivative gives the scalar product of the forcewith the velocity and it is recognized that this is possible only if the force has constant componentsin all directions. Since the two scalar products are equal, students recognize the conservation ofmechanical energy both in classical dynamics and also in relativistic dynamics. The student findshimself face-to-face with a new way of looking at the characteristics of force. It must be subjectedto verification and the general characteristics must be examined. From the expressions alreadyfound it is easy to obtain the known expressions of mechanical energy in the case of weight force,elastic force, gravitational force, Coulomb force and Lorenz force. The definition of conservativeforce emerges as the consequence of the fact that the scalar product of force times the velocity isequal to the temporal derivative of the potential energy. Work is obtained by examining thevariation of mechanical energy when forces of friction are in play.This approach uses elementary mathematical tools, without losing the formal elegance andgenerality of the ordinary treatment based on infinitesimal calculus. It can therefore be given tofirst year students on no-calculus courses or as more detailed work for secondary school studentsspecializing in science.


3. The formal itinerary proposed

Newton's second law

pp

F &==dt

d, (1)

compares the resulting force F acting on the material point with mass m and velocity v with the derivative of its quantity of motion p = m v. To recognize the contribution in the direction of motion, we consider the scalar products with the velocity v of the two members of the law.

vF = vp& (2)

3.1 The classic case of the scalar product vp&

It is known that the second member of (1) in the classic case of constant mass can be written as follows:

p& =dp

dt=

d

dtmv( ) = m

dv

dt= ma (1.1)

and hence the scalar product vp& which is the second member of (2) can be written as

)(dt

dvv

dt

dvv

dt

dvvmm z

zy

yx

x ++=va . (3)

Each term of (3) contains the component of the velocity vector in that direction and its temporal derivative, we shall therefore look at it as the temporal derivative of a single quantity. It is recognized that:

+= xxx

x Cvdt

d

dt

dvv 2

2

1, vy

dvy

dt=

d

dt

1

2vy

2 + Cy , vz

dvz

dt=

d

dt

1

2vz

2 + Cz ,

+ + = +

+ + = = + +

( ) =

= =

( ) = ( )

= +

= ( ) = =

= + = +

where Cx , Cy and Cz are arbitrary constants, from the second member of (3) we get

+ + = +

+ + = = + +

( ) =

= =

( ) = ( )

= +


= ( ) = =

= + = +

vx

dvx

dt+ vy

dvy

dt+ vz

dvz

dt=

d

dt

1

2v2 + C ,

Due to the principle of composition of motions we can put vx

2 + vy2 + vz

2 = v2 and C = Cx + Cy + Cz .

Thus we obtain

+== '2

1 2 Cmvdt

dm

dt

dvav

p. (3.1)

putting C'=0 so that the kinetic energy Ec v( ) =1

2mv2 , is cancelled when velocity is nil, we obtain

dt

dE

dt

d c=vp

(4)

3.2 The relativistic case of the scalar product vp&

In relativistic dynamics the expression of the quantity of motion vector is

p = m v = m 1v2

c2

1 2

v , (5)

where m is constant and v( ) = 1 v2 c2( )1 2

. Deriving (5), therefore, we find

dp

dt= m 1

v2

c2

3 2v

c2

dv

dtv + m 1

v2

c2

1 2dv

dt. (6)

By definition, therefore,

We want to show that there are four basic cases where the law of force, acting on a free material point, allows us to find the constant of motion mechanical energy, using elementary rules of derivation

= + +

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

avF

21

2

2

2

23

2

2

11 +=c

vm

c

vv

c

vm

&. (7)

4. The law of force and mechanical energy in free motion

= + +

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

If the force acting on the material point is constant and can be written as

= + +

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

F = F0 = F0,x x + F0, yy + F0,z z . (4.1)

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

= + +

The scalar product of the force times the velocity gives

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

= + +

( )CzFyFxFdt

d

dt

dzF

dt

dyF

dt

dxF zyxzyx +++=++= ,0,0,0,0,0,0vF (4.2)

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

-1/2

-3/2 -1/2


= + +

Thus we can introduce the function of the coordinates of potential energy

Ep x,y,z( ) = F0, xx + F0,y y + F0,z z + C , (4.3)

= ( )=

( )= ( )

+( )=

= +

= + +

( ) = + + +

where C usually represents an arbitrary constant

= ( )=

( )= ( )

+( )=

= +

= + +

( ) = + + +

By choosing C = 0 , so that Ep 0,0,0( )= 0 we can write

( )= ( )

+( )=

= +

= + +

( ) = + + +

= ( )=

( )pEdt

d=vF . (4.4)

( )= ( )

+( )=

= +

= + +

( ) = + + +

= ( )=

Remembering vpvF = & .

Replacing in (4), we find

d

dtEp( )=

d

dtEc( ), that is to say,

d

dtEc + Ep( )= 0 . (4.5)

= +

= + +

( ) = + + +

= ( )=

( )= ( )

+( )=

We can interpret (5) saying that mechanical energy

Em = Ec + Ep

= + +

( ) = + + +

= ( )=

( )= ( )

+( )=

= +

remains constant during the motion of the material point. We point out that the expression of kinetic energy is always given by (2.4) or by (2.9), whereas the expression of potential energy changes,

case by case, and is deduced from the equation pE&=vF .

Replacing (4.3) in (3.1) we find, in classic dynamics,

Em =1

2mv2 F0, xx F0, yy F0,z z (4.4)

= ( )

=

= +

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

or, in relativistic dynamics,

= ( )

=

= +

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

Em = mc2 1( ) F0,x x F0, yy F0,z z . (4.5)

=

= +

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

= ( )

Let us examine the case of weight force and take, to fix our ideas, F0 = mgz . (4.4) becomes

= +

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

= ( )

=

Em =1

2mv2 + mgz (4.6)

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

= ( )

=

= +

that is

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

=

= ( )

=

= +

1

2mvx

2 +1

2mvy

2 +1

2mvz

2 + mgz =1

2mv0, x

2 +1

2mv0, y

2 +1

2mv0,z

2 + mgz0 . (4.7)

From (4.6), since the Cartesian components of acceleration ax e ay are nil, we deduce the relations

between the components of the position vector and the velocity vector in the motion of free fall of bodies

= = = ( )

= +

= ( )=

= +

= ( ) +

=

= ( )

=

= +

+ + + = + + +

vx = v0, x , vy = v0,y , vz2 v0, z

2 = g z0 z( ). (4.8)

= +

= ( )=

= +

= ( ) +


=

= ( )

=

= +

+ + + = + + +

= = = ( )

Let us consider a material point subject to a force, which obeys Hooke’s law,

F = -k xx . (5.1)

By calculating the scalar product of force times the velocity, we find

+== Cxkdt

d

dt

dxxk 2

2

1--vF . (5.2)

By comparing (5.2) with (3.1), we can put

Ep =1

2k x2 + C . (5.3)

If we take C = 0 , so that Ep 0( )= 0 , we find

Em =1

2mv2 +

1

2k x2 (5.4)

or, distinguishing the relativistic case from the classic case,

Em = mc2 1( ) +1

2k x2 . (5.5)

=

= ( )

=

= +

+ + + = + + +

= = = ( )

= +

= ( )=

= +

= ( ) +

6. Gravitational force and Coulomb force

=

= ( )

=

= +

+ + + = + + +

= = = ( )

5. Elastic force

= +

= ( )=

= +

= ( ) +

On the level of macroscopic objects, motion is governed by two basic forces only, gravitational force and Coulomb or electromagnetic force. Mathematically, these two forces can be represented by the same formula,

F = k r 3r = k x2 + y2 + z 2( )

3 2xx + yy + zz( ). (6.1)

If we put k = GMm , then (6.1) identifies the gravitational force which the material mass point M ,located at the origin of the reference system, exerts on the material mass point m , located in the point identified by the position vector r . On the contrary, if we put k = Qq 4 r 0 then (6.1) identifies the Coulomb force which the punctiform load Q exerts on the punctiform load q .If we calculate the scalar product of the force times the velocity, from (6.1) we directly obtain

( ) ( )zyx zvyvxvzyxk ++++=23222vF . (6.2)

On the other hand, if we take u = r2 = x2 + y2 + z2 e u 1 2 = r 1 = x2 + y2 + z2( )1 2

, we find d

duu 1 2 =

1

2u 3 2 =

1

2x2 + y2 + z 2( )

3 2, (6.3)

andd

dtu =

d

dtx2 + y2 + z 2( )= 2 xvx + yvy + zvz( ). (6.4)

= = + +( ) + +(

= +

=

= + = +

= + = ( ) +


= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

Consequently, by combining (6.3) and (6.4), we have

= = + +( ) + +(

= +

=

= + = +

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

d

dtu 1 2 =

d

duu 1 2 d

dtu = x 2 + y2 + z2( )

3 2xvx + yvy + zvz( ). (6.5)

= +

=

= + = +

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

Therefore, by replacing (6.5) in (6.2), we can write

= +

=

= + = +

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

( ) +=+= Cr

k

dt

dCuk

dt

d 21vF . (6.6)

= +

=

= + = +

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

By comparing (6.6) with (3.1), which provides the definition of potential energy, we can put

Ep =k

r+ C . (6.7)

=

= + = +

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

= +

If we choose C = 0 , so that the potential energy is annulled at an infinite distance from the origin, the formula for the constant of motion mechanical energy is

Em = Ec + Ep =1

2mv2 +

k

r (6.8)

in classical dynamics, while in relativistic dynamics it becomes

= + = ( ) +

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

= +

=

= + = +

Em = Ec + Ep = mc2 1( ) +k

r. (6.9)

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

= +

=

= + = +

= + = ( ) +

Lorentz force acts on a punctiform electric load moving in a magnetic field

= = + +( ) + +( )=

=

= = + + = = + +( )= = + +( )

= + +( )= + +(

= = + +( ) + +(

= +

=

= + = +

= + = ( ) +

7. Lorentz force

FL = qv × B , (7.1)

where q is the algebraic value of the movable load, v is the velocity of the load, B is the magnetic induction of the field where the load moves. The scalar product of the Lorentz force for the velocity is nil,

=

= ( )= ( ) =

= ×

( ) 0=×= vBvvF qL , (7.2)because in the double mixed product the same velocity vector appears twice. In this case, therefore, the motion constant of mechanical energy coincides with the kinetic energy,

=

= ( )= ( ) =

= ×

=

8. Conservative forces

= ( )= ( ) =

= ×

=

We underline the fact that the mechanical energy of a material point on which any force F acts, is a motion constant if pE&=vF . This consideration also suggests the way to define the concept of

conservative force.

= ( )= ( ) =

= ×

=

9. The exact differential pdEd =rF

= ( )= ( ) =

= ×

=

Let us take a free material point, immersed in a force field F = F r( ) , which describes the trajectory

identified by r = r t( ), with velocity v = Ýr . If the hypothesis ( )rvF pE&= is valid, we find

( )dtEdt p rvF &= , that is

= ×

=

= ( )= ( ) =

( )rrF pdEd = . (8.1)

= ×

Em = Ec . (7.3)

= ( )= ( ) =


= ×

=

= ( )= ( ) =

(8.1) expresses the variation in potential energy along an elementary segment of trajectory. If we take a finished arc of the trajectory, with ends rA ed rB , the variation in the potential energy is given by

( ) ( )BpApp EEdEdB

A

B

A

rrrFr

r

r

r== . (8.2)

If we consider all the possible trajectories, with ends rA and rB , then (8.2) states that the integral B

A

dr

rrF , although calculated along different integration paths, always assumes the same value

expressed by Ep . Therefore (8.2) suggests the well known definition of conservative force; if it

is an exact differential, the force F is conservative, that is, ( ) ( )BpAp EEdB

A

rrrFr

r== , whatever

may be the path which has rA ed rB as its ends.

10. Constrained motion, friction forces and the concept of work

Let us consider the material point on which not only active forces but also constraining reactions act. We shall indicate these constraining reactions with the symbol . The law governing these forces is not generally known in advance, that is, before finding the law of motion. However, if the

= ×

=

= ( )= ( ) =

10. Constrained motion, friction forces and the concept of work

Let us assume that the resultant force acts on the material point

R = F + , (9.1)

where F represents the active force, while represents the constraining reaction. If

0=v , (9.2)

the mechanical energy is always a motion constant. In fact, we can write

( ) vp

vFvF ==+dt

d. (9.3)

Consequently, if pE&=vF , we can conclude that the principle of conservation of mechanical

energy Em = Ec + Ep remains valid.

We observe that can represent the tension of the unstretchable wire, T , in the motion of the mathematical pendulum, or the normal constraining reaction of the supporting plane, N , in the case of motion along an inclined plane without friction.

= + +

= ( )=

[ ]

constraining reactions are normal at the trajectory of the material point, that is, normal to the velocity vector, the mechanical energy is always a motion constant. But if there are constraining reactions acting on the material point which possess an anti-parallel component to the velocity vector, the mechanical energy is not preserved. In may cases which occur in practice the anti-parallel component is due to the friction or resistant forces. If we calculate the quantity of mechanical energy dissipated due to the friction forces, this leads to the natural definition of work done by a force.

11. Constraining reactions normal to the velocity vector

= +

= +

= + +

= ( )=

[ ]


= +

= +

Let us now suppose that on the material point a resultant force is acting, given by R = F + + Fr , (10.1)

= ( )=

[ ]

= +

= +

12. Friction forces

= + +

= ( )=

[ ]

= +

= +

= + +

where F represents the active force, the constraining reaction perpendicular to the velocity vector and Fr = Fr v( ) a friction force, anti-parallel to the velocity vector. In the case of dynamic

grazing friction, for example, we have ( )vF ˆ= Ndr µ , in the case of viscous friction Fr = bv v( ).

Remembering that pR &= , from (10.1) we find

[ ]

= +

= +

= + +

= ( )=

vFvFvp

= rdt

d, (10.2)

[ ]

= +

= +

= + +

= ( )=

that is, thanks to (2.5) and to (3.1),

[ ]

= +

= +

= + +

= ( )=

vF= rmEdt

d. (10.3)

[ ]

= +

= +

= + +

= ( )=

Therefore the mechanical energy is not preserved, in fact the product vrr FvF = expresses the

rapidity with which it is dissipated. The variation of energy in a time interval tA , tB[ ] is defined by means of the integral of the first and the second member (10.3), that is,

=B

A

t

t rm dtE vF . (10.4)

=

[ ]

= ( )

=

Since vdt = dr , from (10.3) we obtain rF ddE rm = and we can re-write (10.4) in the form

=B

A

dE rm

r

rrF , (10.5)

[ ]

= ( )

=

=

[ ]

= ( )

=

Traditionally, the integral B

A

dr

r

rrF is called work done by the resistant force Fr and is indicated by

the symbol

=

where rA and rB are the ends of the trajectory arc described by the material point in the time interval tA , tB[ ].We should point out that, in general, (10.4) or (10.5) do not allow us to calculate Em andtherefore of the integral which appears as the second member, if the dependency of the velocity on the time v = v t( ) is not explicitly known. Therefore, for the actual calculation of the dissipated energy, we must first solve the differential equation

TFTFTp ˆˆˆ += rdt

d, (10.6)

obtained from (10.1) by multiplying in scalar fashion the first and second member by the versor T ,

tangent in every point of the trajectory, replacing R = Ýp and remembering that 0ˆ =T .

13. Work

=

[ ]

= ( )

=

=B

A

dL r

r

rrF . (11.1)

The physical meaning of (11.1) is supplied by (10.4) or by (10.6). We would add that this definition of work also clarifies the etymological meaning of the term , which in Italian derives from the Latin labor and, apart from work, action, enterprise can also denote labor, toil, worry.

14. ConclusionsThe teaching of physics to young people must include activities suitable to develop awareness ofthe role of mathematics in physics and the ways in which it is used in different circumstances. Inphysics, it is the integration of the two planes, phenomenal and formal, which creates the meanings;it builds those concepts which allow us to describe and interpret the world, using models able toexplain classes of phenomena through theory. The well-known limits in managing mathematics aremainly of a conceptual character and due to the fact that students are not used to looking for thesignificance of the calculus which they have learnt to do. Attention must therefore be paid to thisproblem and time must be spent on didactic research in order to construct proposals able toencourage the acquisition of a thorough knowledge of quantitative features of physics.Starting from the following considerations:- The principles of conservation are part of the deepest roots of physics and, in particular, are

decisive in the development of mechanics.- Every principle of conservation entails the existence of one or more constants of motion.- Among these, mechanical energy is particularly important, which allows us to identify

interesting characteristics of motion of a material point, establishing a relationship betweenposition and velocity.

We have developed a proposal which leads to identify mechanical energy and its conservationstarting from Newton’s second law, without the prior introduction of the concept of work. Itscoherence and completeness are recognized by the possibility of obtaining the expressions ofmechanical energy in the same way when the forces in play are of various types: from gravitationalforces to Coulomb forces, from weight to elastic force, from Lorenz force to restraining reactions.The definition of conservative force can be recognized in terms of a corollary with respect to theprior introduction of a potential energy. The case of forces of friction introduces the non-conservation of mechanical energy and the concept of work. The itinerary proposed, which doesnot require mathematical knowledge and/or ability above those which every student learns in asecondary school specializing in scientific subjects, allows the student to familiarize himself withsome important ways of looking at physics through mathematics, such as using the scalar product,vectorial de-composition and the properties of temporal derivatives with respect thereto, thesearch for primitive functions, the identification of quantities which are conserved.If we do not choose it as an alternative to the traditional way of introducing mechanical energy andits conservation, it represents a powerful opportunity for reflecting on concepts which connect theway of looking at the world through forces with the way of looking at the world through energy.

References[1] J. Piaget, R. Garcia, Les explications causales, Etudes d’épistèmologie génètique, vol. XXVI, P.U.F. (1972).[2] E. Guesne, A. Tiberghien & G. Delacote, Methodes et resultats concernant 1’analyse des conceptions des hleves

dans differents domaines de la physique, Rev. Fran. Pbda., 45, (1978), 25-32.[3] S. Strauss & R. Stavy, U-Shaped behavorial growth: Implications for theories of development, in «Review of

child development research», W.W. Hartup Ed., Chicago: University of Chicago Press, (1985).[4] P. Black, B. E. Woolnough, The role of the Laboratory in Physics Education, Girep International Conference,

Oxford, University Press, (1986).[5] La Fisica nella Scuola, Special issue on strategies of physics teaching: the role of the problem and the role of the

laboratory , in Italian, (Speciale sulle strategie di insegnamento della fisica : il ruolo del problema e il ruolo dellaboratorio), XXVII, 4, (1994).

[6] M. Michelini, Quale laboratorio per la formazione degli insegnanti: un contributo sul problema del laboratorionella didattica della fisica, Quaderni MPI, Roma, (1998).

[7] Physics Education special issue on hands-on science, vol.25, N.5, (1990).[8] S. Chaikin, J. Lave, eds., Understanding practice. Perspectives on activity and context, Cambridge, University

Press, (1993).[9] S. Bosio, V. Capocchiani, M. C. Mazzadi, M. Michelini, S. Pugliese, C. Sartori, M. L. Scillia, A. Stefanel, Playing,

experimenting, thinking: exploring informal learning within an exhibit of simple experiments, in Proceedings ofGIREP-ICPE International Conference – Ljubljana, (1996).

[10] Some of the well known projects translate in different language are the following: PSSC - Physical Science StudyCurriculum, developed in MIT, USA; PPC : Project Physcs Course, developed in Harvard University, USA;


Nuffield developed by Nuffield Foundation, UK; IPS & PS2 developed in MIT, USA for introduction of sciencein 14-16 age; Spoerg Nature developed in Denmark; IPN Curriculum science, developed in Kiel, Germany

[11] J. Novak, D. Gowin, Learning how to learn, Cambridge Univ. Press. Cambridge, UK, (1978).[12] H. Pfundt & R. Duit Students’ alternative Frameworks and Sci Educ Inst for Sci Educ (1995)-Kiel (D).[13] M. Vicentini M. Mayer, Didattica della fisica, La Nuova Italia, Roma, (1996).[14] L.C. McDermott, M.L Rosenquist and E.H Van Zee, Students difficulties in connecting graphs and physics:

Examples from kinematics, Am. J. Phys., 55, (1987), 503.[15] Speciale rappresentazioni mentali, LFNS, 1986.[16] L. Viennot, “Le raisonnement spontané en dinamique élémentaire”, Hermann, Paris, (1979).[17] E. Saltiel, J.L. Malgrange, “Les raisonnements naturels en cinématique élémentaire”, BUP, 616, (1979), 1326.[18] D.E. Trouwbridge and L.C. McDermott, Investigation of students understanding of the concept of velocity in

one dimension, Am. J. Phys., 48, (1980), 1020.[19] D.E. Trouwbridge and L.C. McDermott, Investigation of students understanding of the concept of acceleration

in one dimension, Am. J. Phys., 49, (1981), 242.[20] L. Viennot, L. “Bilan des forces et loi des actions réciproques. Analyse des difficultés des élèves et les enjeux

didactiques”, BUP, 716, (1989), 951.[21] M. Gagliardi, G. Gallina, P. Guidoni, S. Piscitelli, “Forze, deformazioni, movimento” Emme Edizioni, Torino,

(1989).[22] K. Bednar, D. Cunningam, T. M. Duffy, J. D. Perry, Teory into practice: How do we link? in Instructional

technology. Past, present and future, J. C. Angelin ed., Englewood, Colorado, Libraries Unlimited, (1991).[23] M. D. Merrill, Constructivism and the technology of instruction, Hillsdale, New Jersey, Erlbaum (1992).[24] G.L. Posner, K.A. Strike, P.W. Hewson and W.A. Gertzog, Accomodation of scientific conceptions: toward a

theory of conceptual change. Sci. Educ., 66, (1982), 211.[25] R.K.Thornton,Tools for scientific thinking - microcomputer based laboratories for physics teaching, Phys. Educ.

22, (1987), 230.[26] K. Swan, M. Miltrani, The changing nature of teaching and learning in computer-based classrooms, Journal of

Research in Computing in Education, 25, (1993), 121-127.[27] M. Riel, Educational Change in a technology-rich environment. Journal of Research in Computing in Education,

26, (1994), 31-43.[28] L.T. Rogers, “The computer-assisted laboratory”, Phys. Educ. 22, (1987), 219.[29] R.F. Tinker, “Computer-aided student investigations”, Computers in Physics, Jan/Feb (1988), 46.[30] R. K. Thornton, Changing the physics teaching lab:Using tech and new approach to learning physics concepts,

AJP, 58, (1990), 858.[31] M. Michelini, L’elaboratore nel laboratorio didattico di fisica: nuove opportunita’ per l’apprendimento, Giornale

di Fisica, XXXIII, 4, (1992), 269.[32] S. Hennessy, D. Twigger, R Driver, T. O’Shea, C. E. O’Malley, M. Byard, S. Draper, R. Hartley, R. Mohamed, E.

Scanlon, A classroom intervention using a computer-argumented curriculum for mechanics, InternationalJournal of Science Education, 17 (2), (1995), 189.

[33] M. L. Aiello Nicosia et al., Teaching mechanical oscillations using an integrate curriculum, Int. Journ. in researchin Science Education, 19, 8, 1997, p. 981.

ONE-DIMENSIONAL QUANTUM SYSTEMS AND SQUEEZED STATES

Carlos A. Vargas, Área de Física, Departamento de Ciencias Básicas, UAM-A,Azcapotzalco, MéxicoArturo Zúñiga-Segundo, Departamento de Física, Escuela Superior de Física y Matemáticas-IPN, Zacatenc., México

1. IntroductionOne hundred and two years ago quantum mechanics begun and at the beginning of other centuryit is necessary to ask a question on the understanding of this theory by the students on several levelsand careers. At the present there are many books, articles and web sites, directed to show theproperties, experiments, paradoxes and mathematical foundations of quantum mechanics. Manygroups of teachers in the world try to instruct the students on this important theory, mainly duringthe last decade motivated by technological advances. The Planck constant was a consequence of arevolutionary postulate to solve the blackbody radiation law [1]. Each atom into the radiationcavity oscillating with frequency ν can only gain a multiple of an energy E=hν, called the quantumof energy. Einstein was able to treat the radiation from the blackbody as a gas of photons. Einstein


postulated the maximum kinetic energy of the liberated electron must be KEmax =hf-φ, in this case,the electromagnetic radiation consists of a stream of photons of energy hf and φ is the workfunction. Bohr extended the notion of quantization to the hydrogen atom. Bohr postulated thatelectrons confined to certain stationary states. The radiation only will be emitted when theelectrons make transitions from one stationary state to other: the atom emits a single photon ofenergy equal to the difference between orbital energies. As the transition must be happeninginstantaneously, is common refer to this type of transitions as quantum jumps [2]. The earlyquantum theory was supported on classical mechanics and supplied Newton’s laws withquantization properties, then was possible to obtain the selection of the stationary states in theBorh model. The matter waves proposed by de Broglie move away from the Newtonianpropositions. He postulates that the frequency of the wave associated with a particle is related totheir energy by the same equation for the electrons, it is E=hf. De Broglie established that λ=h/p,in this way he proposed the wavefunction as follows:

The confirmation of de Broglie ideas were due to the Davisson-Gremer and Thomson experimentson electrons scattered by crystals and metallic films respectively. These results secured the waveproperties of the electron but also that the particle waves satisfy the principle of superposition. Bythis time, Schrödinger formulated the now generally called Schrödinger wave equation:

This equation applied to the hydrogen atom generated the quantization of both angular moment Land energy E. The Schrödinger absolute wave function |ψ|2 represent the probability distributionfor the position of the electron. This Schrödinger formulation or wave mechanics is the newquantum mechanics that is included in most of the textbooks. However, another alternativeformulation was developed by Heisenberg and is known as the matrix mechanics.In this scenario the classical entities, position and momentum had no meaning. Matrix mechanics

replaces these classical quantities with ones directly related with the stationary levels. Bothformulations are equivalents and produce the same results.

2. Strategies to Teach Quantum MechanicsThe state of art in science embraces from research to divulgation. However, as we know, in somecases the physical theories have some centuries, by example, the classical mechanics, while othersas by example the cosmological or particle theories have only some decades.The first one has manyexperiments that corroborate its propositions; the new theories are waiting for experimentalreplication. The quantum mechanics for example has nowadays several disagreements in the areaof the observations. This type of problems are of philosophical nature and falls into the territory ofquantum interpretations [3,4]. We consider these types of questions are more appropriate toadvanced students because of possibly they require many tools to explain the phenomena.Moreover quantum mechanics in many cases is contrary to the intuition. This is a difficulty whenwe try to teach the quantum mechanics. But this challenge can be surmounted with fine scientificliterature. By example, at the present there exists much interest in the physical societies to putpublic web sites with the more recent advances of the physics. In this sense, we remember byexample the outstanding efforts of the European Society of Physics and the American PhysicalSociety. Other institutions like Nobel foundation provide a series of descriptive posters on differenttopics of the physics directed to all people; they also have a public web site. As we know anycontemporary physics topic securely have in its structure the quantum theory. The other source ofquantum physics information comes from the regular articles published in specialized journals. Wecan observe now the great interest to focus the attention to secondary level by several Europeanphysics groups, where is possible to distinguish works on physics history, physical concepts,experimental and curricular physical research [5]. For the secondary level we think the concrete


)].)(/2sin[()]/(2sin[ pxEthxft ==

.0),())((2

),(22

2

=+ txxVEh

mtx

dx

d

activities on quantum mechanics are very important. It is preferable that the students of this levelhave a first hand experience. It is not necessary to be an experiment; it can be for example, anexercise with matrices. Also, the thought (gedanken) experiments are beautiful “quantum”recourse when we have not other alternatives [6,7]. The single and double slit diffraction and theStern-Gerlach experiments are the most ubiquitous in order to explain the quantum mechanics[8,9]. Fortunately there is a vast and brilliant literature on them. However, we try to use otherexamples, in our case not only to understand quantum mechanics but also to clarify certain type ofstrategies we suggest here. We will go from the old to the present quantum mechanics in order tofix ideas that possibly can be included for teaching quantum theory. Also, we will go from theapparent simple to more complex systems, and also from the one to more dimensions. We believethe problems here suggested not only suitable as a pedagogical tool for introduce basic conceptsbut also ideal for communicating some aspects of the physics behind quantum mechanics foradvanced undergraduates.

3. Electron-Atom ScatteringSometimes, we see the footnotes accompanying the principal text or simply a little mention, andpossibly we have no time to reflect on this point. The problem on electron-atom scattering to lowenergy, so called Ramsauer-Townsend effect, we describe here, appears mentioned in manytextbooks as footnote or it has been included as a little mention [10]. When we reflect on itscharacteristics, it involves many interesting features. It is possible that the authors consider it as aproblem that is only applicable in a few cases. But when we are introducing problems on barriersand wells in a course of quantum mechanics, it is a very simple and funny problem; moreover, it ismore interesting if we can do the experimental demonstration [11]. The strategy is as follows: firstwe ask for a classical mechanics analogy, later we proceed to think on the constellation electron-atom and their relative velocities in order to understanding the undistinguished character of theelectron, that is we cannot put a label to the incoming particle in order to distinguish it of thestream of electrons. Therefore, we ask for kinetic theory in order to calculate the cross section.Finally, the square well is proposed and resolved. With them we can put in confrontation thetheoretical computation with experimental results. The student has a first quantum approximationto the problem, which is sufficient to adopt this point of view. In this stage the student has moreinformation on the theoretical structure and is able to recognize that the spherical potential is amore convenient explanation of the problem. In this point the student will need to know the partialwave treatment. As we see the student get an insight in the problem with the consequent skills toattempt to resolve similar problems.The noble-gas atoms can be considered to present an attractivepotential, the Schrödinger equation for slow electrons indicates that the cross section will be havea minimum to electron energies near 1 eV. In the kinetic theory the cross section is independent ofthe energy and thus gives nearly four times the quantum result. The experiment exhibits an energyminimum approximately to 1 eV.

4. Hydrogen Atom in One-DimensionThe Hydrogen atom in one dimension (1DHA) has interesting features, and a possible didacticvalue, and diverse applications for example: atoms in high magnetic fields or models of theinteraction of electrons with the surface of liquid helium [12-14]. If we consider the 1DHA withCoulomb interaction then V(x)=-e2/|x|. Its Schrödinger equation in momentum representation hasthe form


)()(||2

2

2

222

xExx

e

dx

d

m=

h

Transforming the previous equation to momentum space and rewriting the regions x>0 and x<0, weobtain the Schrödinger equation to a pair of Volterra integral equations one of them with ϕ+ andother with ϕ- thus:


)(')'()(2

22

pEdppie

pm

pp

+++ =+h

+

±± dppipxx )()/exp()2)(( 2/1hh

Notice that + ( -) are defined only for x 0 (x 0) while the functions + ( -) are defined for all values of p from - to + . The real and imaginary parts of the two momentum-space eigenfunctions + and

- do not describe completely the states of the 1H, + describes an electron moving to the right of the singularity (the origin) whereas - describes its movement to the left. It means that if the electron is moving to the left (right) it will remain confined to this region for all time. If we go on to obtain the eigenfunctions n

+ and n it is found that both correspond to the same energy spectrum, the energy levels appearing twofold degenerated; so that the right and left regions turn out to be independents. It is possible to say that we are dealing with two-dimensional problems simultaneously. Effectively the degeneracy is explained by the operation of a superselection rule between the bound states of the 1D

system. This system impose restrictions to the Hermeticity of their Hamiltonian ||

1

2

1 2

xpH = in

units m=e=h =1. This superselection rule forbids the superposition of states on one side of the singularity of the potential with those on the other side. Allow the superposition of two arbitrary states

+ += nm ba1 and + += nm cb2 , with |a|2+|b|2=1 and |b|2+|c|2=1. The condition that both of

these states vanish in x=0 implies that their Wronskian determinant must vanish too: W ( 1, 2) =0.Then the functions 1 and 2 cannot be regarded as describing independent states. Therefore we must conclude that there is a superselection rule between the states + and the states -. An immediate consequence is that the parity operator becomes an unobservable. Moreover the absence of a non-degenerate ground state implies the spontaneous breaking of supersymmetry in the 1DH atom.

The wave functions in the momentum space ϕ+ and ϕ- are related to the coordinate-spacewavefunctions as

5. Squeezed States

The squeezed states are relevant to optical interferometry and optical communications [15]. The relevant element for the development of this subject is the harmonic oscillator, which form a central part in any quantum physics course. The squeezed states of the electromagnetic field can be established with elementary quantum mechanics concepts and statistical properties of several harmonic oscillator states that are linear superpositions of its energy eigenfunctions [16]. The radiation in each standing wave mode in a cavity is analog to the harmonic oscillator. The oscillator displacements x correspond to the radiation electric field mode’s while the momentum p correspondsto the magnetic field mode’s. As a consequence the energy interchange of the oscillator is shared between potential and kinetic energy. The oscillator total energy according to quantum theory corresponds only to an integral number of photons into their radiation mode. The coherent states, a type of linear combinations, have constant the variances of the position and momentum in any time and their product equals the minimum allowed by the Heisenberg uncertainty principle. The variance is the squares of the uncertainties. A squeezed state is produced when the variances of position and momentum are oscillating in time 180° out or phase one with other and with the two times the oscillator frequency. The coherent and the squeezed states constitute the backbone of the theoretical framework of modern optics [17]. The Wigner function directly allows the calculation of expectation values in the quantum phase-space, with the advantage that the parameters q and p of these quasi-probability functions are c-numbers [17,18]. Is possible to make quantitative analysis in phase-space without use the Wigner function? The answer is yes. The coherent-state representation of quantum mechanics [19], allows analyzing completely the dynamics of the quantum systems in the phase-space


in the same way that in the coordinate representation. Here the projection < | > of the abstract ket | >into the basis vector | > =|p,q>, gives us a complex wave function in phase-space, i.e., a wave function with independent variables p and q, and the quantity | ( )|2 ( ) ( ) represents a probability density ( where ( )=< | >=< | > ). This definition ensures that the quantum density | ( )|2 is a nonnegative quantity in phase space and fulfills all the requirements of a probability density. The

closure relation for the basis vectors ˆˆ ( ) ( )I d A= (where the integration is carried out over the

whole phase space) can be used in the calculation of the expectation value of the operator Â, as the usual definition

ˆ ˆ( ) ( ).|A d A=

The actions of the operators P and Q on the arbitrary ket | > are given by

2

2

ˆ| | ( ) ( );

ˆ| | ( ) ( ).

pq

qp

P i

Q i

=

= +

h

h

Fig. 1: Square magnitude of the squeezed state in phase-space.

These operators are Hermitians and do not commute with each other, in fact ˆ ˆ ˆ, .Q P i I= h

Based on these operators, the phase-space Schrödinger equation is given by 21

2 2 2[ ( ) ( )] ,p qm q pi i V i

t= + +h h h

where 2( )qpV i+ h indicates that the potential function V(q) is evaluated in the operator 2( )q

pi+ h . For

simplicity in the following, we will use 1m= = =h .The phase-space squeezed wave function , 0 is obtained by squeezing the initially prepared

harmonic oscillator ground state 0 (vacuum state). The squeezed state with complex squeezing

parameter exp( )i= , corresponding to the vacuum state, is known as squeezed vacuum, and is

written as 2 21 1 1 1

2 2 2 2, exp[ ( ) ( ) ],N q p i pq= +

with 0 exp( )i= . The photon number N and the squared photon number 2N are the most

important operators in quantum optics. According to the harmonic oscillator Hamiltonian 12

ˆ ˆN H= ,we can write

References[1] A.J. Makowski, ‘A century of the Planck constant’, Phys. Educ. 35, (2000), 49-53.[2] G. Ireson, ‘A brief history of quantum phenomena’, Phys. Educ. 35, (2000), 381-385.[3] G. Pospiech, ‘Uncertainty and complementarity: the heart of quantum physics’, Phys. Educ. 35, (2000), 393-399.[4] P. Mittelstaedt, ‘Interpretation of Quantum Mechanics and the Measurement Process’, Cambridge University

Press, Cambridge, (1988).[5] M. Michelini, R. Ragazzon, L. Santi and A. Stefanel, ‘Proposal for quantum physics in secondary level’, Phys.

Educ. 35, (2000), 407.[6] H. Helm and J. Gilbert, ‘Thought experiments and physics education-part 1’, Phys. Educ. 20, (1985), 125-131.[7] H. Helm, J. Gilbert and D.M. Watts, ‘Thought experiments and physics education-part 2’, Phys. Educ. 20, (1985),

211-217.[8] R. Feynman, ‘The Character of Physical Law’, MIT Press, Cambridge, (1967).[9] J.C. Polkinghorne , ‘The Quantum World’, Princeton University Press, Princeton, (1984).[10] S. Gasiorowicz, ‘Quantum Physics’, John Wiley, New York, (1974).[11] S. Kukolich, ‘Demonstration of the Ramsauer-Townsend effect in a Xenon thyratron’, Am. J. Phys, 36, (1968),

701-703.[12] H.N. Núñez-Yépez, C.A. Vargas and A.L. Salas-Brito, ‘The one-dimensional hydrogen atom in momentum

representation’ , Eur. J. Phys., 8, (1987), 189-193.[13] H.N. Núñez-Yépez, C.A. Vargas and A. L. Salas-Brito, ‘Superselection rule in the one-dimensional hydrogen

atom’, J. Phys A , 21, (1988), L651-l653.[14] R. Martínez y Romero, C.A. Vargas, A.L. Salas Brito and H.N. Núñez Yépez, ‘Note on supersymmetry in

quantum mechanics and the hydrogen atom in one dimension’, Rev. Mex. Fis, 35, (1989), 617-622.[15] R.W. Henry, S. C. Glotzer, ‘A squeezed-state primer’, Am. J. Phys, (1988), 318-328.[16] A.K. Ekert, P.L. Knight ,‘Correlations and squeezing of two-mode oscillations’, Am. J. Phys., 57, (1989), 692-697.[17] Y.S. Kim and M.E. Noz, Phase Space Picture of Quantum Mechanics. Group Theoretical Approach, Lecture

Notes in Physics Series,. 40, (1991) (World Scientific, Singapore)[18] E.P. Wigner, ‘On the Quantum Correction For Thermodynamic Equilibrium’, Phys. Rev, 40, (1932),. 749-; M.

Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner, ‘Distribution Functions in Physics: Fundamentals’, Phys.Rep., 106, (1984), 121-167.

[19] Go. Torres-Vega and J.H. Frederick, ‘Quantum Mechanics in phase space: New approaches to the


( )( )

20

2104

20

22104

120

221 1

2 20

ˆ (cosh 4 1),

ˆ sinh 4 ,

N

N

= =

= =

agrees with the standard squeezed-state result in the Wigner representation [17]. In Fig. 1 the shape of the square magnitude of the squeezed state in phase space, with 20.25exp( ).i= is shown by means of density contours. We can see that the probability density from the squeezed state, is rotated by an angle / 2 in a counterclockwise sense, where is the argument of . In this figure, we show also the quantum probability flux vector corresponding to a squeezed state. These vectors assemble a non-

It is a pleasure to thank the referee for his or her comments and clarifying remarks.

In summary, we have described a series of problems that we consider with possible didactic value with the aim to motivating the teaching of the quantum mechanics. The Ramsauer-Townsend effect would be an interesting problem to contrast the kinetic theory with the quantum mechanics. The one- dimensional Hydrogen atom has illustrating aspects from elemental to high mathematical level. Finally, we suggest the application of the properties of the harmonic oscillator with the purpose of illustrating the analysis of squeezed states.

Aknowledgment

stationary vortex, allowing the squeezed density to rotate in a clockwise sense around the origin in phase space, according the Wigner density time evolution [20].

6. Conclusions


correspondence principle’, J. Chem. Phys., 93, (1990), 8862-8874; Go.Torres-Vega and J.H. Frederick, ‘A quantummechanical representation in phase space’, J. Chem. Phys., 98, (1993), 3103-3120; Go. Torres-Vega, ‘Lanczosmethod for the numerical propagation of quantum densities in phase space with an application to the kickedharmonic oscillator’, J. Chem. Phys., 98, (1993), 7040-7045; Go. Torres-Vega, ‘Chebyshev scheme for thepropagation of quantum wave functions in phase space’, J. Chem. Phys., 99, (1993), 1824-1827; Go. Torres-Vegaand J.H. Frederick, ‘Numerical Method for the Propagation of Quantum-Mechanical Wave Functions in PhaseSpace’, Phys. Rev. Lett, 67, (1991), 2601-2604; K.B. M∅ller, T.G. Jorgensen and Go. Torres-Vega, ‘On coherent-state representation of quantum mechanics: Wave mechanics in phase space’, J. Chem. Phys., 106, (1997), 7228-7242

[20] A. Zúñiga-Segundo, ‘Squeezed states obtained by means of canonic transformations in quantum phase space’,Rev. Mex. Fis., 47, (2001), (in Spanish).


GRAPHS AS BRIDGES BETWEEN MATHEMATICAL DESCRIPTION ANDEXPERIMENTAL DATA

Laurence Rogers, School of Education, University of Leicester

1. Why do physicists use graphs? When physicists accumulate numerical data from practical experiments they need to store the data insome form and this is traditionally done in a table of results.When it comes to analysing results, althougha table has a certain amount of use for comparing items of data and deriving further information, thegraph is a far more informative tool for this purpose. The visual impact of graphs and their relationalproperties makes them extremely valuable for analysing and appreciating the properties of data. It isimplicit that a typical graph stores a large quantity of numbers, the data pairs associated with each plottedpoint, but more importantly the graph contains valuable information about the relationship between thevariables represented by the data. Describing the relationship between variables is an important butsophisticated skill for students to acquire and an understanding of graphs, how they work and how theycan be used, can be a key factor in helping them develop this skill.

2. Information from graphsThe most obvious feature of a graph is its shape. When experimental data is presented as a graph,the shape of the graph immediately conveys information in a qualitative manner withoutconcerning the observer with unnecessary, numerical detail. The shape of a graph can give pupilsa quick overview of what may be going on in an experiment; pupils can ‘see’ gradual or suddenchanges, continuity or discontinuity, the difference between a complex sequence of events and aprogressive trend. A range of features give valuable information about the variables: a gentle orsteep gradient indicates a slow or rapid change, the curvature of a line indicates a varying rate ofchange, peaks and troughs indicate maxima and minima, and so on. Let us consider some commontypes of graph shape and consider what scientific information pupils might be expected to inferfrom their observation of the graph.The first is a graph whose dominant feature is an upward trend. The upward trend indicates that asone variable increases the other variable also increases.The classic example of this is a graph of currentagainst voltage for an electrical resistor. If the data is gathered using a data logger and plotted usingsoftware, it is very easy to vary the plotting format to emphasise the relationship between changes inthe variables. A useful variant is to plot both voltage and current against time. Inspection of the graphin Figure 1b leads to the simple observation that, as one variable increases, so the other also increases.Thus the graph shape provides basic information about how the variables are related.The second graph shows a general downward trend. This shows a relationship between twovariables which is the inverse of the first case, namely, as one variable increases the other variabledecreases.The example of pressure plotted against the volume of a fixed mass of gas illustrates thistype of relationship.Again, the use of software to plot the data provides a useful alternative formatfor plotting the graph. Figure 2b makes it abundantly clear that the two variables vary in theopposite sense.

3. Understanding the properties of graphsThe graphs in Figures 1a and 2a both illustrate progressive trends, without irregularities ordiscontinuities. It is reasonable to expect that such smooth trends can be described by fairly simplemathematical formulae. Clearly the straight line graph in Figure 1a can be described by theformula y = mx + c and the constants m and c are readily evaluated by taking measurements fromthe graph. Software provides useful tools for conducting these measurements speedily, using acurve fitting facility or by adjusting a trial function to match the graph under observation.

3.8 Software Packages-Multimedia

358 3. Topical Aspects 3.8 Software Packages-Multimedia

However, even more usefully,software allows pupils to explorethe properties of the graph toteach them the significance ofthe linearity for preciselydescribing the relationshipbetween the variables. The maintools for this type of explorationare the cursors which provideautomatic reading of data pointson the graph and which cancalculate changes of both sets ofvariable simultaneously. Fromsuch explorations a variety ofstatements may be made aboutthe properties of a straight linegraph:1. Changes in the variables

occur at a constant rate.2. For a given increment in one

variable, the other variablealways increases or decreasesin equal steps. (For the case ofa variable plotted againsttime, the size of the step for agiven time interval is always thesame.)

3. This is independent of themagnitude of either variable.

4. The ratio between the increasesor decreases in either variable isconstant.

5. When this ratio is not unity, onevariable changes more rapidlythan the other.

6. The gradient is the same at allplaces on the graph. i.e. it isconstant.

7. When the gradient is negative,an increase in one variable isaccompanied by a decrease inthe other.

To the tutored eye, thesedescriptions are clearly equi-valent to or follow from each other. Their significance here is that they are individually testableusing software tools: when a cursor is moved across the graph, changes in the variables may be readautomatically and the rate of change calculated; measurements may taken from any selected partof the graph; ‘x’ and ‘y’ cursors may be locked together, easily showing the relative changes in twovariables; the gradient at a cursor may be read automatically. Through a variety of explorations,pupils may learn to associate a characteristic set of properties with the linear graph so that whenexperimental data yields a straight line, they will have a certain understanding of how the variables

Figure 1a. Graph of current against voltage

Figure 1b. Graph of current and voltage against time

relate to each other, how changesin one variable are associatedwith predictable changes in theother. Thus for ohmic resistors,the straight line indicatesbehaviour which may bedescribed by phrases like “equalincreases in voltage lead to equalincreases in current”, or “the ratiobetween the voltage and currentdoes not vary”.The straight line graph is acommon occurrence in theanalysis of experimental data andits unique properties are easilydescribed informally andmathematically. However, it isonly one shape of many which candescribe variables which increasesimultaneously. The example ofthe ohmic resistor points toanother relationship which iscertainly not linear, that betweenthe power dissipation and thecurrent flowing. Figure 3 showsthe curve which is characteristicof this relationship.Although power increases whencurrent increases, the ratiobetween the increases is clearlynot invariant as it was for thestraight line graph. Software toolsallow the properties of this shapeof curve to be explored:Cursors are used to calculatesuccessive increases in power for agiven increase in current. For anupward curve it can be expectedthat the increase in power for agiven increase in current is largeraccording to the value of thecurrent. For this particular curve,the difference between successive

increases in power is always the same. This is a unique characteristic of the quadratic curve. The useof a software curve fitting facility identifies the formula and indeed confirms that this curve is aparabola described by y = ax2. Further use of cursors shows that the power increases according tothe square of the current.Again, through a variety of explorations, pupils may learn to associate a characteristic set ofproperties with this particular graph shape which can in principle be applied to any experimentaldata yielding a parabola. For example the braking distance of a motor car can be related to thesquare of its velocity, demonstrating that a required braking distance does not increase in simple


Figure 2a. Graph of pressure against volume

Figure 2b. Graph of pressure and volume against order of recording

proportion to the velocity of acar.The previous example of aninverse relationship betweenthe pressure and volume of agas can be analysed with asimilar range of software tools.Informally the downward slopeof the graph (Figure 2a) showsthat increases in one variableare associated with decreases inthe other.The use of cursors canconfirm that there is a definitepattern to this trend whichallows predictions to be made.For example when the volumeis reduced by half its value, thepressure doubles in value. pupilscan gain a feeling for thisrelationship by looking out for

this pattern. As for the previous graph shape, a software fitted curve identifies the characteristicformula for inverse proportionality, in this case expressing Boyle’s Law.The inverse square law is another common relationship with a downward curve graph which at firstsight appears to be very similar to the previous example. Exploration with cursors soon reveals thedistinctive properties of this curve which has much more rapid changes to the gradient. A casualview of a curve for exponential decay might also suggest a similar relationship, but curve fitting andcursor reading techniques soon reveal its unique ‘constant ratio’ properties.

4. Graph shape as an indicator of a relationshipThe discussion in this paper has argued that, when experimental data is represented by a graph, thisfacilitates three levels of description of the data:• informal qualitative description based on observations of graph shape e.g.“the temperature falls

quickly at first and then slowly rises.”• informal quantitative description based on numerical exploration of the data using software

cursor tools e.g. “at double the speed the distance is four times greater”• mathematical description using formulae evaluated by curve fitting and curve matching

techniques.Pupils should be encouraged to develop all these levels of description and build up a notion of thecharacteristic properties of each graph shape so that when graphs of experimental data areobserved these properties can be applied to the variables concerned.

ReferencesBarton R., Computer-aided graphing: a comparative study’, Journal of Information Technology for Teacher Education, 6, (1),

(1997), 59-72.Newton L.R., Graph talk: some observations and reflections on students data-logging, School Science Review, 79, (287),

(1997), 49-54.Rogers L.T., Probing the Hidden Secrets of Graphs, Hands-on Experiments in Physics Education - Proceedings of the

GIREP, Conference in Duisburg (1998).


Figure 3. Graph of power vs. current for an ohmic resistor

THE USING OF MULTIMEDIA COURSEWARE FOR COLLEGE PHYSICS RELATIVITYTEACHING AT HARBIN NORMAL UNIVERSITY

Zhang Changbin, Song Guilian, Mu Hongchen Department of Physics, Harbin NormalUniversity, PRC

1. IntroductionThe theory of relativity is a very important foundation stone for modern physics. The founding ofthis theory is also one of the greatest discoveries in the 20th century. The content of the theory ofrelativity has made strong influences on physics, astronomy, as well as philosophy. In this theory, aseries of fundamental conclusion and deductions such as the principle of relativity, the Lorentztransformation, the Length Contraction and the Time Dilation of an object in motion etc., hadchanged our notions of space and time as well as of matter and energy. As the theory of relativityhas made a revolutionary break through to the classical physics, it is not so easy for college studentsto understand about. In traditional physics class, teaching methods was usually of white chalk andblack board, teaching information was mostly transformed through blackboard drawing and orallanguage speaking from the teacher. The inefficient of such kind of teaching is obvious.Nowadays, as the development of modern technology, especially the using of computer andmultimedia technology in class room, has changed our physics teaching greatly. Of course, themaking of a suitable and powerful courseware for physics education is a very important step forcomputer assist physics teaching. Our group has done several such works since 1996, and theRelativity Multimedia Courseware is one of our works.

2. The idea of design of the Relativity Multimedia CoursewareThe greatest advantage of multimedia technology using in classroom is that through thistechnology the transformation of teaching information could be a way of all-round insounds(voices), movies(3D simulations), images(pictures), and texts(words). Physics phenomenonon-the-spot simulation is another important advantage for physics teaching. So, the main purposeof the designation of the Relativity Multimedia Courseware was to bring the advantages ofmultimedia technology into full play in our college physics education. That is to present theories,concepts, as well as events of the theory of relativity by vivid and formed texts, images, movies, andvoices in front of college students.Contents of the Courseware (Fig.1):• Galilean principle of relativity and the time-space notion in classical mechanics.• The Michelson-Morley experiment.• Einstein’s assumptions and the time-

space notion in the theory ofrelativity.

• The Lorentz transformation.• Fundamental theory of relativity

dynamics.• The verification of relativity theory.• The general theory of relativity.• A city of relativity—cartoon.• A brief account of Einstein’s life.• Old pictures related to relativity.The detail of the above contents areformed from the following four data-base:(1)Text data-base. (2)Image data-base.(3)Voice data-base. (4)Movie data-base.


Fig.1. Major manual of the courseware

3. The way of achieve to the Relativity Multimedia CoursewareAs we have mentioned before, the main purpose of the courseware is to take the advantage ofmultimedia technology to improve our physics teaching in college classroom. So, the processingsoftware should be popular and easy to use by physics teachers. Then they may modify thecourseware by himself during physics teaching. The processing tools we chose as follows:• Platform of composition: Office 97, Powerpoint.(Fig.2)• 3D simulation developer: 3D Studio MAX4.0 and 3D F/X• Image processor: Photoshop5.0• Movie processor: Premier5.0

4. The characteristics of the RelativityMultimedia Courseware• Contents full and accurate.• Operating and modifying easy and simple.• Teaching material vivid and form.• Utilizing environment moderate.

5. The teaching practice of the RelativityMultimedia CoursewareTo make is for to use. We have used therelativity multimedia courseware in ourcollege physics course since 1998. Thefollowing points should mentionedaccording to our teaching practice:– Teacher should always remember thatcourseware is only an assistant to ourphysics teaching. Teacher plays a leading

role in the classroom and courseware is the second.– The courseware is not a lifeless dogma. Teacher should reconstruct or modify the courseware to

suit the real teaching circumstances.– Pay more attention to the advantage of the courseware, fully and correctly use the materials in

image, movie, sound, and text provided by the courseware.


Fi 2 C fFig.2. Courseware front page

Fig.3. Example slides in the courseware

6. Concluding remarksThe designing and teaching practice on Relativity Multimedia Courseware has given us a strongconfidence on the reform in college physics education by using modern technologies. It is a finefeeling that the students heighten their learning interests and motivations when we are usingmultimedia courseware in our physics teaching and the educational results is growing up obviously.Being optimistic, we are sure that college physics education, together with the development of

modern multimedia and information technologies, will greet a new flourishing era in the 21th

century.

References R.P. Feynman, R.B.Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Publishing Company,

(1964).M. Alonso, E.J.Finn, Physics, Addison-Wesley Publishing Company, (1992).L. Fan, Multimedia and Education, Higher Education Press, (1998) (in chinese).e-mail [email protected]

DEVELOPING STUDENTS COMPETENCES BY MEANS OF SIMULATION: USE OF ARESEARCH TOOL FOR UNDERSTANDING ION-MATTER INTERACTION IN SOLIDSTATE PHYSICS

Federico Corni, Dipartimento di Fisica, University of Modena and Reggio Emilia, Modena, Italy

1. IntroductionScience and technology define the degree of development and potential progress of a modernsociety and researchers and engineers are stimulated to search for new materials and methods toimprove and renew technological artifacts. Physics education plays a fundamental role in theeducation system and it cannot undervalue the importance and urgency of integrating and updatingthe curricula with modern physics and technology.The introduction of modern physics in secondary school and university must be relevant and needscorrelation to the actual technological world. However, the understanding of the laws and the rulesthat govern the phenomena in the modern fields of physics often requires a background knowledgeof quantum mechanics, statistical physics and mathematical methods that make very hard orimpossible an approach to students.In this paper a possible way to introduce fundamental aspects of ion-matter interaction ispresented.The aim is pursued through the use of a program, SRIM: the Stopping and Range of Ionsin Matter, for the simulation of ion implantation, a widely used technique, in particular, in materialsresearch and industry.The background knowledge can be very essential and coincides with subjectswhich are normally introduced in secondary school, such as the atomic structure of matter, theperiodic table of the elements, neutral and ionized atoms, the charge interactions and the collisionphysics. This activity is then suitable for first level university students as well as for students ofsecondary school especially with materials science specialization. School teachers could not havethe needed attainments and a materials science training course is recommended.SRIM calculates the stopping and ranges of ions in the kinetic energy range 10 eV – 2 GeV per amuinto matter and simulates an ion implantation using full quantum-mechanical treatment of ion-atom collisions. It supplies as output complete numerical tables as well as 3D plot distributions ofthe ions with target damage, sputtering, ionization and phonon production. Each ion track andtarget atom cascade are followed in detail and presented in animated plots. Students experimentwith the use of an advanced research instrument and are allowed to understand and learn newphysical phenomena such ion diffusion, target atom recoiling, atom cascade and surface sputtering,as well as statistical concepts such projected range, straggling, electron and nuclear stopping powerand radiation damage.

2. SRIM simulation for physics educationIon implantation technology is widely used for electronic materials and recently is more and moreoften mentioned and treated in books or in scientific documentary films. Besides, the phenomenainvolved in this process such ion production, acceleration and interaction with matter are common


to many scientific fields. Few emblematic examples: ion beams are employed in medicine forirradiation of cancerous cells with high efficiency and spatial resolution; natural radioactivity andfast charged particles coming from space, in some cases, can be thought of as ionized atomsinteracting with the atmosphere, with the earth or the skin and the organs; the problem of thenuclear waste storage since the walls of the container are affected by alpha particles (helium ions)implantation coming from the nuclear decays.It is very important that a physics course gives students the opportunity of moments of directcontact with the research and technology world, just treating particular topics which involvearguments transversal to many fields of science [1].The problems often encountered introducing modern technology in physics teaching are: i) hightechnical instrumentation required to obtain reliable and significant experimental data; ii)prohibitive and expensive costs; iii) particular environmental conditions needed (e.g. powderparticle concentration, humidity and temperature); iv) health hazard for the use of high voltages,toxic gases and substances; v) long period of specialization training to acquire competencies in theuse of instrumentation, machines and processes.The use of a simulator, a technique also adopted in formation courses for technician andresearchers, overcomes all these technical difficulties making easier the students reaching theresults and learning physics. Moreover students are not required to know the numerical andanalytical methods, often overcomplicated, which are necessary to solve the physical equations.An other important advantage coming from the use of a simulator in a learning activity is thepossibility of trying situations not easily realizable or absolutely unrealistic and purely theoreticalto investigate the influence of the input parameters. This is perhaps the major motivation thatjustifies an activity exclusively employing simulations as this.SRIM simulates the ion implantation process starting from few input parameters about the beamand the target and makes Montecarlo calculations of the ion trajectories which can be displayedsingularly with details and synthesized into distribution graphs and synthetic statistic quantities.Simulation and modeling in physics education are didactic strategies widely experimented andadopted, and various research papers have been published on these topics [2,3].An important pointis that students could have difficulties to distinguish model and reality [3]. Working with SRIM,students are precluded to have any direct contact with the phenomena under investigation and theteacher has to guard the students (and himself) against confusion. The activity with SRIM shouldbe preceded by an introduction about the role of modeling in the knowledge of the physical world.An important topic to be submitted to the students and discussed is the Montecarlo methodemployed by SRIM. The simulation is made by solving the equation of motion of each ionsingularly. The calculation of a particular ion trajectory is obtained using random numbers toquantify the stochastic quantities involved in the collisions, e.g. the collision probability, thecollision parameter and the degree of energy loss. The final product, as it happens in an actualimplantation process, is the result of the superimposition of all the single ions events.Taking into account the above observations, we propose an activity employing SRIM designedaccording to the following points:i) qualitative exploration of the ion motion into matter;ii) analysis of the influence of the input parameters;iii) search for physical phenomena occurring in the implantation process;iv) identification of quantities typical of the ion-matter interaction physics;v) problem solving activities aimed to the evaluation of these quantities by elaboration of the

simulated data.

3. What is SRIM?SRIM is a free simulation program that can be downloaded from the internet addresswww.srim.org and runs on personal computers. It results from the original work by J.P.Biersack [4]on range algorithms and the work by J.F.Ziegler on stopping theory [5].


SRIM is used in research to plan ion implantation processes to fit the desired ion depth profile andtarget damage. SRIM calculates the stopping and range of ions (10 eV – 2 GeV/amu) into matterand simulates an ion implantation using a full quantum mechanical treatment of ion-atomcollisions. This calculation is made very efficient by the use of statistical algorithms which allow theion to make jumps between calculated collisions and then averaging the collision results over theintervening gap. During the collisions, the ion and atom have a screened Coulomb collision,including exchange and correlation interactions between the overlapping electron shells. The ionhas long range interactions creating electron excitations and plasmons within the target. Theseare described by including a description of the target collective electronic structure andinteratomic bond structure when the calculation is setup. The charge state of the ion within thetarget is described using the concept of effective charge, which includes a velocity dependentcharge state and long range screening due to the collective electron sea of the target. A fulldescription of the calculation is found in the tutorial book [6]. This book presents the physics ofion penetration of solids in a tutorial manner, then presents the source code for SRIM with afull explanation of its physics. Further chapters document the accuracy of SRIM and showvarious applications.SRIM accepts complex targets made of compound materials with up to eight layers, each ofdifferent materials. It calculates both the final 3D distribution of the ions and also all kineticphenomena associated with the ion energy loss: target damage, sputtering, ionization, and phononproduction. All target atom cascades in the target are followed in detail. Plots of the calculation ismade in real time, moreover they can be saved and displayed when needed. On request, numericaltables of the stopping of ions as well as of the ions and displaced target atoms trajectories andenergies can be compiled and saved for elaboration purpose.

Figure 1 shows the window of input parameters for stopping calculation of hydrogen ions intosilicon. The parameters to be supplied are: the ion type and its mass, the energy range of stoppingpower calculation, the target composition, the units. Table for frequently used materials areavailable.Figure 2 shows the window of input parameters for implantation simulation of hydrogen ions intosilicon at 100 keV. The parameters to be supplied are: the ion type and its mass, the initial kineticenergy, the angle of incidence, the target in terms of layer thicknesses and compositions, thenumber of ions to be calculated, the plotting window and the output table files to be saved. Thedamage calculation option and the type of live plots to be displayed during calculation can be setwith the choice windows.The large number of information supplied by SRIM can constitute matter of study at differentschool levels and for different degrees of specialization. In this initial work, only the ionstrajectories and the target atoms cascades will be treated. Physical background needed are thefundamentals of classical mechanics and electrodynamics.The particular choices of the target material, of the ions and their energy could limit the generalityof the study. However the aim of this paper is the suggestion of guidelines for the use of suchresearch tool to teach ion-matter interaction. The SRIM version adopted is the 2000.39.

4.1 Qualitative exploration4.1.1 Ion trajectoriesFigure 3 shows the simulations of the trajectories of four ions with same initial kinetic energy(100 keV) impinging on silicon. The ions are H (Fig. 3a), C (Fig. 3b), Ge (Fig. 3c) and U (Fig. 3d).The horizontal axes report the depths and the vertical axes report the lateral displacements of theions. It can be observed that the trajectories are finite in length and decrease with increasing theion atomic number and mass. The trajectories appear not rectilinear and of irregular forms. Ions ofthe same kind produce almost similar shapes, while different ions present different features in theirmotion. For example, light ions as H travel in a straight way at the beginning and tracing hook-



Figure 1. SRIM window of input parameters for stopping calculation.

Figure 2. SRIM window of input parameters for implantation simulation.

shaped trajectories at the end of their motion. Heavy ions as U travel in a more regular way, withlight curves distributed within the whole track.The statistics of the simulations can be improved by increasing the number of calculated ions.Figure 4 reports the simulations of 100 ions. The trajectories are contained within pear-shapedvolumes of different dimensions and proportions. The region where the ions come at rest appearsof cylindrical symmetry with the axis perpendicular to the target surface through the beamincidence point. Occasionally insulated trajectories, very long or with strong deflections, occur.4.1.2 Recoils.Figure 5 shows the simulations of one trajectory per ion type including the detailed calculation offull damage cascades. Green traces are the target atoms (recoils) tracks. For a more extensiveexploration it is possible to do multiple ion simulations with the caution of clearing the screen toprevent confusion. The simulator allows the formation of the collisional cascades to be followed byswitching the animation button.Observing Figure 5, very few recoils are produced along light ion trajectories (H: 11, C: 518) andconcentrated at the end of the track in correspondence of the hook. As ion mass and atomicnumber increase, the number of recoil increases (Ge: 1750, U: 2200) almost uniformly distributedalong the track. Occasionally long trajectory recoils are produced.

4.2 Influence of the input parametersIn this section the input parameters will be varied. For a matter of simplicity the target compositionwill be kept constant, since having considered ions either lighter either heavier than silicon, theinvestigation does not loose generality.4.2.1 Ion mass and atomic numberAll the above qualitative observations exhibit dependence on the ion type. A question rises: is thecontrol parameter the ion mass or the atomic number?4.2.1.1 Trajectory length. Figure 6 reports the simulations for H and He ions with their correct masses(Fig. 6a and d) and with the inverted ones (Fig. 6b: H with mass 4 uma, Fig. 6c: He with mass 1 uma).Comparison between Fig. 6a and b and between Fig. 6c and d reveals that the mass increases thetrajectory lengths, while comparison between Fig. 6a and c and between Fig. 6b and d reveals thatthe atomic number shortens the tracks. Same conclusions are deduced from similar simulations ofSn and U ions reported in Figure 7. Ion mass and atomic number have opposite effects and thisleads to the conclusion that the atomic number prevails on the ion mass in determining the tracklengths.4.2.1.2 Trajectory shapes. To investigate the influence of the two parameters on the trajectory shape,simulations of H ions with the U mass and of U ions with the H mass can be used (Figure 8). Fig. 8aappears very similar to Fig. 3d and Fig. 8b appears very similar to Fig. 3a indicating that, in this case,the ion mass is the control parameter.4.2.1.3 Recoil production. The simulations of Figure 8 are recalculated with full damage cascades andreported in Figure 9. It can be deduced that silicon atoms are displaced more efficiently by highatomic mass ions, either in terms of number (H: 2170 recoils/ion, U: 128 recoils/ion), either in termsof displacement from the ion track and dimension of the collisional cascade.4.2.2 Ion energy.The kinetic energy of the impinging ions is expected to play a role in the recoil production (thedisplaced silicon atoms appear at first in the ending part of the ion track, see Fig. 5c, where the ionenergy is low) other than, obviously, in the ion trajectory length. Figure 10 reports the simulationsfor C and U ions at different initial energies.4.2.2.1 Trajectory length and shape. The trajectory lengths increase with ion initial energy (note thedifferent depth axes). However the two physical quantities are not proportional, especially in thecase of U. The shapes seem to be unaffected.4.2.2.2 Recoil production. The number of recoils increases with ion initial energy (C at 10 keV: 108, Cat 30 keV: 225, C at 60 keV: 325, C at 100 keV: 518; U at 10 keV: 262, U at 30 keV: 714, U at 60 keV:


1390, U at 100 keV: 2200). Moreover, observing Figure 10a, the displaced silicon atoms are roughlydistributed along the whole ion track, while in Figure 10c they are less dense in the initial part. Inthe case of U ions the behavior appears the opposite: many recoils are produced along the trackand fewer at the ending part of it.It is useful for students to design and compile a table which summarizes the observations made. Asuggestion is the following:

The previous two activities, qualitative exploration and input parameter variation, constitutefundamental moments of the present didactic activity with SRIM, because they play the role of theexperimental stage in a usual laboratory activity, with the advantage of offering large space topersonal choices. In this sense, these activities are occasions for the students of measuringthemselves in a research activity and of acquiring project skills and scientific method.

4.3 Identification of physical phenomenaFrom the above observations, physical phenomena can be recognized. In the following, some ofthem will be identified and described in a schematic way, leaving to the teacher the job ofcompleting the treatment and the strategy with students.4.3.1 FrictionArguments: the ion trajectories are finite in length; the ion trajectories are not rectilinearProperties: it depends on the ion atomic number; it depends on the instantaneous ion

energy; it is not constant (exhibits different regimes within the same track)Conclusion: it is of coulombian origin; there are two principal ways of loosing energy: as in a

viscous medium and by collision with target atomsTopics motion in a viscous medium; elastic and inelastic collisions4.3.2 Energy loss as in a viscous mediumArguments: trajectories present rectilinear partsProperties: typical of light ions; more efficient at high instantaneous energyConclusion: model of coulombian collision of the ionized projectile with the electrons of the

targetTopics: energy transfer from a projectile to mass orders of magnitude lighter;

dependence of the charge of an ion on its velocity in matter4.3.3 Energy loss due to target atom collisionArguments: the trajectories present deflections; target atoms displacement (recoil

production)Properties: typical of heavy ions; more efficient at high energyConclusion: model of nuclear collision of the ion and the target atomsTopics: equations of classical collisions; energy transfer from a projectile to target mass

of comparable order of magnitude [7] Rutherford cross section [8]; nuclearcharge screening due to electrons; interatomic potentials

4.4 Typical quantities of ion-matter interactionThe stochastic character of an ion implantation process requires the identification of measurablestatistical quantities. Here will be mentioned the principal ones.An implantation is performed in order to locate impurities at a desired depth and with a suitableprofile in a material.


Trajectory length Trajectory shape Recoil numberIon mass Y/N Y/N Y/N

Ion atomic number Y/N Y/N Y/NIon initial energy Y/N Y/N Y/N

Ion instantaneous energy Y/N Y/N Y/N

4.4.1. The principal quantity is then the projected range, defined as the average depth of theimplanted ions. Moreover, a crucial point is the statistical width of the impurity distribution due tothe so called straggling. Such quantity can be represented by the distribution standard deviation orby more sophisticated statistical parameters, such skewness and kurtosis, taking into accountasymmetries in the ion distribution. SRIM supplies live graphs during simulation of the iondistribution with the corresponding statistical parameters which can be used to give the students agraphical representation of the ion implantation process.4.4.2. An other important point in implantation technology is the damage caused in the targetmaterial. The damage can be represented in first approximation by the distribution of the targetatoms taking part to the collisional cascade or the so called interstitials. Also in this case theprincipal quantity is the average depth of recoils, and, consequently, the standard deviation of theirdistribution.4.4.3. The sites originally occupied by the displaced atoms are called vacancies and also contributeto the damage. The vacancy depth distribution is in most cases practically superimposed to that ofthe interstitials, however, for light target elements the interstitial distribution can result deeper.SRIM supplies live plots both for interstitial and for vacancy distributions.4.4.4. The energy loss of ions during their motion into matter is called stopping. Such phenomenonis roughly distinguished in the two contributions of the electronic friction and of the nuclearcollisions.4.4.4.1. The electronic stopping power is defined as the energy loss per unit depth due to the

interaction with target electrons. Such quantity depends on the velocity (so the instantaneousenergy) of the ion since, in the typical energy range of ion implantation, it can be comparable withthe ion core electron velocities with a consequent stripping and increase of interaction. Theelectronic stopping power is energy dependent and initially increases with energy; after amaximum, roughly corresponding to the ion velocity similar to that of its core electrons (totalelectron stripping), the stopping decreases due to less and less interaction time spent by the ionnear the electron.

4.4.4.2. The nuclear stopping power is defined as the energy loss per unit depth due to the

collisions with the target atoms. It depends on ion energy since at low velocity the ion does notpenetrate the electron cloud of the target atom with a consequent decrease of nuclear coulombianinteraction. Also the nuclear stopping power reaches a maximum with increasing energy, then itdecreases due to the decrease of interaction time.Figure 11 reports the calculated stopping powers for the considered ions. Their dependence on ionatomic number and energy reflects the observations and the deductions reported above.

4.5 Elaboration of calculated dataAn useful activity for students is now the numerical evaluation of the identified physical quantities.A first ability they can acquire is the use of statistical functions and methods of data elaborationand representation. SRIM supplies all the needed data in numerical tables for depth distributionsof ions (RANGE.TXT), interstitials (RANGE.TXT) and vacancies (VACANCY.TXT).A more complicated and not univocal job is the evaluation of the stopping powers (both electronicand nuclear) employing the data of calculated ions. Numerical tables do not supply these quantities,but give many informations about the ion motion and the collisional cascades (E2REC.TXT,COLLISON.TXT). This activity can be proposed to student as problem solving and then theoutputs of the various adopted solutions can be compared with the stopping powers calculated(and employed) by SRIM as those plotted in Figure 11.

5. ConclusionsA possible didactic activity using SRIM has been proposed. The fundamentals of ion-matterinteraction can be easily evidenced, recognized and investigated by means of simulations of the ion


edx

dE

ndx

dE


H C

Ge

U

a) b)

c) d)

H C

Ge U

d)

b)

a)

c)

Figure 3. SRIM simulation of the trajectories of 5 a) H, b) C, c) Ge, d) U ions implanted in a Sitarget with energy of 100 keV.

Figure 4. SRIM simulation of 100 ions of a) H, b) C, c) Ge, d) U with energy of 100 keV in Si.


Fi 5 SRIM i l i i l di f ll d d ( k d i ) f ) H b) C ) G d) U i

H C

Ge

U

a) b)

c) d)

H H

HeHe

Z = 1 M = 1

Z = 1M = 4

Z = 2 M = 1

Z = 2 M = 4

d)

a) b)

c)

Figure 5. SRIM simulation including full damage cascade (marked in green) of a) H, b) C, c)Ge, d) U ion implanted in a Si target with energy of 100 keV.

Figure 6. SRIM simulation of H and He ions into Si with the right masses (a and d) and withthe inverted ones (b and c).


U U

Sn Sn

Z = 50 M = 119

Z = 50 M = 238

Z = 92 M = 119

Z = 92 M = 238

a) b)

c) d)

H UZ = 1 M = 238

Z = 92M = 1

a) b)

Figure 7. SRIM simulation of Sn and U ions into Si with the right masses (a and d) and withthe inverted ones (b and c).

Figure 8. SRIM simulation of a) H ions with the U mass, and b) of U ions with H mass.


H Z = 1 M = 238

U Z = 92M = 1

a) b)

C 10 keV C 30 keV C 60 keV

U 10 keV

U 30 keV

U 60 keV

a)

b) c)

d)

e) f)

Figure 9. SRIM simulation including full damage cascade (marked in green) of a) H ions withthe U mass, and b) of U ions with H mass.

Figure 10. SRIM simulation including full damage cascade (marked in green) of C and U ions implanted in Si atdifferent initial kinetic energies.

d)

a)

implantation process, overcoming all the constrains that an experimental activity imposes. Theactivity has been projected in a similar way to an experimental investigation, allowing the studentsto acquire method and ability as well as competencies in data handling and elaboration.

References[1] Teaching the science of condensed matter and new materials GIREP Book, ed. M. Michelini, S. Pugliese Jona, D.

Cobai, Udine (1995).[2] D. Hestenes, Toward a modeling theory of physics instruction, American Journal of Physics, 55, (1987), 455-462.

E. Sassi, L’uso dell’elaboratore nella didattica della fisica: potenzialità e problemi, Giornale di Fisica, 28, (1987),109-123. G. Marx, Il microelaboratore nella scuola, La Fisica nella Scuola XXIII, (1990), 27-31. J. Ogborn,Modellizzazione con l’elaboratore: possibilità e prospettive, La Fisica nella Scuola, XXIII, (1990), 32-43. D.Hestenes, Modeling game in the newtonian world, American Journal of Physic, 60, (1992), 732-748. S. PuglieseJona, Modellizzazione dinamica nel biennio di scuola secondaria, La Fisica nella Scuola, XXVI, S1 Q1, (1993),62-73. M.L. Aiello-Nicosia et al., Teaching mechanical oscillations using an integrated curriculum, InternationalJ. of Sci. Educ., 19, (1997), 981-995.

[3] R.M. Sperandeo-Mineo, Il calcolatore nella didattica della fisica, in M.Vicentini and M. Mayer ed, Didattica dellaFisica, ed La Nuova Italia, (1996), 251-286.


0 20 40 60 80 100

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Ge U

Figure 11. SRIM calculated stopping powers for H, C, Ge, and U ions in Si.


[4] J.P. Biersack and L. Haggmark, Nucl. Instr. and Meth., 174, (1980), 257-267.[5] The Stopping and Range of Ions in Matter 2-6, Pergamon Press, (1977-1985).[6] J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Range of Ions in Solids (New York: Pergamon Press,

(1985).[7] F. Corni, M. Michelini, L. Santi and A. Stefanel, Rutherford Backscattering Spectrometry: a technique worth

introducing into pedagogy GIREP Book, (Udine), (1995), 266-272.[8] F. Corni, M. Michelini, L. Santi, F. Soramel and A. Stefanel, The concept of the cross section GIREP Book,

(Udine), (1995), 192-198.

MULTIMEDIA PROGRAM - ELECTRIC CURRENT

Libor Koníček, Erika Mechlová, University of Ostrava, Faculty of Science, Department ofPhysics, Czech republic

1. IntroductionThe concept of electric current is difficult for pupils. There are many reasons of these difficultiesbut the very important one is that pupils cannot manage with electric current in the same way aswith the flow of liquid.Many articles were published on the subject of how pupils understand the concept of electriccurrent. The understanding of this concept depends on curriculum, on textbooks of physics, on thecontrol of pupils by teacher and on the opportunity of pupils to “handle” the electric current in thesame way as the flow of liquid.The conclusions on the investigations on the understanding of electric current, depend on differentcircumstances in different countries, however, there is an element which is common.We give the results on the understanding of the concept of electric current by pupils at a grammarschool in the Czech republic. On these results, the multimedia program was developed to improvethe children understanding of the concept of electric current.

2. Results of the investigation on the concept electric currentThe investigation was focused on the relations of the concept of electric current with otherconcepts.The electric current is a phenomenon, and a quantity that describes the characteristics of thephenomenon.The electric circuit may be explained as a system where energy is transformed or as a system wherecurrents flow and voltages are measured. The second way of looking at the electric circuit uses theconcepts of voltage V, current I and resistance R. These two views of an electric circuit areconnected with each other through the energy flow dW/dt into resistor: dW/dt = VI, which meansvoltage and current determine the energy flow. The two flows, the flow of energy and the flow ofparticles with charge, should be distinguished - while the energy flow is unidirectional, the flow ofparticles circulates [1].The test “Electric current” was constructed on these ideas and was given to 17-year-old-pupils. Theresults of the test gave answers on how pupils adopt the electric current (phenomenon andquantity) in relation to the other concepts.The results [2]: Correct or partially correct answers were given by 23 % of pupils on two basicconditions why electric current flows through a body. They gave electrons only as carriers of theelectric charge.

2.1 Conduction of electric current in metalsThe best results in thinking about electric current by pupils were in the area of electric current inmetals.There were 63 % of the correct answers that referred to the free carriers of charge in metals.Although the validity of Ohm’s law was recognised by 76 % of pupils, nobody gave the condition

of the validity of Ohm’s law. Ohm’s law expressed with words was given by only 4 % of pupils, andwith formula by 33 % of the pupils. Pupils gave incorrect formulae, that are not the functionalrelations R= U/I (41 %), U= RI (18 %). Correct answers in all items were given by only 11 % ofthe pupils. Only these pupils demonstrated mastery of Ohm’s law and could decide in case frompractice.For what concerns the Kirchhoff’s laws, pupils failed.The pupils did not adopt Kirchhoff’s laws.Thereason of this is the misunderstanding of a simple circuit on the basis of measuring the electriccurrent in its different parts. Only 9 % of pupils adopted a single circuit and the same current in allplaces of this circuit. Pupils think that the current is consumed in the circuit as energy, the biggestcurrent is near the positive pole, the smallest near the negative pole of the battery.

2.2 Conduction of electric current in liquidsThe results given by the pupils in thinking of an electric current in liquids were bad. The freecarriers of charge in electrolytes were correctly given by 45 % of the pupils. The condition ofvalidity of Ohm’s law was not given by anybody. The total current in electrolytes was not given by80 % of pupils.

2.3 Conduction of electric current in vacuum and gasesThe worst results were given in thinking about the electric current in vacuum and gases. The freecarriers of charge in gases were correctly given by 1 % of pupils, in vacuum by 4 %. Some of thepupils gave the incorrect answers that the carriers of electric charge are protons and neutrons.The condition of validity of the Ohm’s law was not given by anybody. The total current was notgiven.

3. Multimedia program ‘Electric current’3.1 Part one: Conduction of electric current in metals

a) Ohm’s lawOhm’s Law is the basic law for conduction of electric current in metals. It links the electric currentand the electric voltage.The real experiment is a very useful method to gain the understanding of the Ohm’s law. Theexperiment must be a convincing experiment for pupils. The set of instruments for demonstrationhas to give evidence that the Ohm’s law is a proportional relation I=V/R, where the resistance R isconstant during the measurement. For this experiment a bulb is inconvenient because it is a non-linear element since its resistance changes with temperature. The bulb is totally unsuitableinstrument as resistor.The Ohm’s Law is correct only if the resistor stays at a constant temperature.Multimedia program includes two types of instructions. The first are for classical real experimentswith battery, ammeter and voltmeter while the second one for computer-aided experiments withthe system ISES (Czech system). The recommendation is given to do the classical experiment atfirst to understand the electric circuit and the principle of connection of the various elements of thecircuit. The computer-aided experiments have the big advantage that they run very quickly. Pupilscan receive many V-A characteristics of different elements, and compare them.

b) Kirchhoff’s lawsA good understanding of the Ohm’s law is the condition for a good understanding of theKirchhoff’s laws.Multimedia program includes instructions for real experiments and measurements in single electriccircuit (Fig. 1) and afterwards electric circuit with sub-circuits (Fig. 2, Fig. 3). Later, inn the sameexperiments, pupils measure electric energy, electric power and compare current and energyconsumption in the same electric circuits.

3.2 Part two: Conduction of electric current in liquids Multimedia program includes instructions for investigations of liquids. Pupils investigate what


liquids are good conductors and bad conductor of electric current (distilled water, solution of sugar,solutions of acids and solutions of salts).Conduction of current in liquids is very complicated; the aim of the program is to give anunderstanding of the main principles and of the Faraday laws, only. Instructions lead to theconclusions that the electric current in a liquid has a similar effects as an electric current in a metal(magnetic field near the electrolyte with current, increase of temperature of electrolyte withcurrent, electric current as a function of the distance of the electrodes and of the immerse surfaceof electrodes).The special characteristics of the electric current in liquid follow: dependence on theconcentration of the electrolyte, chemical composition, moveability of positive and negative ions.


Fig.2: Example of electric circuit with sub-circuitsFig. 1: Example of a simple electric circuit.

Fig. 3: Example of electric circuit with sub circuits and voltmeters

Fig. 4: As an example, there is dissociation of NaCl molecule to Na+ and Cl- ions in water.

3.3 Part three: Conduction of electric current in vacuum and gasesMultimedia program ends with experiments on the creation of ions and free electrons by ionisation(thermal ionisation, ionisation by electric field) on video. Instruction for pupil’s point out that theconduction of current is visible as an ionised gas (many videos with different gas discharge). Themagnetic properties of gas discharge in tube are demonstrated on videos.


4. ConclusionThe Multimedia program Electric current comes from an investigation of the pupil’sunderstanding, and of their permanent knowledge. The aim of the program is to unify theunderstanding, and afterwards the knowledge, of pupils on electric current in all matters – metals,liquids, gases and vacuum. The program involved pupils in real experiments and contains manyvideos on the place where real experiments are difficult.The program will be verified in a grammarschool in the Autumn 2001.

Remark: Multimedia program ‘Electric Current’ is developed in the frame of the programLeonardo da Vinci No SI 143008 “Computerised laboratory in science and technology teaching”(ComLab-SciTech).

References[1] R. Duit, W. Jung, Ch. Rhoneck, Aspects of understanding electricity, Proceedings of an International Workshop,

Ludwigsburg. Kiel, IPN, (1985).[2] E. Mechlová, The investigation of concept electric current at gymnasium, In Acta Facultatis Rerum Naturalium,

Universitas Ostraviensis, Ostrava, Ostravská universita, (1993), 39-46.[3] L. Koníček, M. Svobodová, Multimediální výukový program – Vedení proudu v kapalinách, In Scientific

Pedagogical Publishing, České Budějovice. Pedagogický software 2000, (2000), s. 103. ISBN 80-85645-40-8.[4] L. Koníček, E. Mechlová, Preparation of Multimedia Modules of Physics by Students, International conference

Phyteb 2000, Barcelona ISBN 84-699-4416-9.

Fig. 5: Demonstration device for electrical discharge – principle of lightning conductor

DO YOU HEAR THE SEA FROM A SHELL?

Verovnik Ivo, National Education Institute of Slovenia, Ljubljana, SloveniaMathelitsch Leopold, Institut für Theoretische Physik, Universität Graz, Austria

Holding a sea shell to the ear produces a definite, reproducible sound.Where does this sound comefrom? It is clear that it is not connected to the sea as the saying goes. A widespread explanation isthat the seashell amplifies the sound of the blood. One can perform two simple experiments inorder to falsify this opinion:- When the outside is quiet, one should hear the sound clearer, but the opposite is true.- When one does some exercising (for example running), the blood streams faster; therefore the

effect should be larger – in fact it is not [1].The explanation of this sound phenomenon is the following: In our surroundings, there existsalways a certain level of noise. The shell acts as a resonator which amplifies the sound at specificfrequencies, which are given by the dimension and the shape of the shell. The question arises whetherit is possible to include this interesting example in a more detailed, maybe also quantitative way, in theteaching of acoustics at school. We propose an approach where the teacher/student can perform aseries of experiments accompanied by a computer assisted analysis.PCs are nowadays usually already equipped with a powerful sound card. In addition there existsoftware programs which allow comfortable on-line data taking by microphones as well as a fastanalysis of the input.The following results have been obtained with the shareware program Cool Edit[2]. The data can be visualized in form of the time-dependent pressure level; they can be Fouriertransformed and presented as a frequency spectrum (the intensities of the sound at differentfrequencies are calculated at a given time) or as a sonagram (the time development of the spectralcomponents is shown, where the respective intensities are indicated by different gray shading, Fig. 1).One can start the exploration of the sound of sea shells by the very simple experiment that oneholds a hand (formed as a cup) close to the ear. One can hear some sound. The effect is much morepronounced when one takes a larger resonator, for example a cylindrical tube. Holding the tube veryclose to the ear changes the sound to lower frequencies. This can easily be understood and is alsoexplained in every textbook on acoustics. Standing waves build up in the tube, and the frequencies of thefundamental and the higher modes (overtones) are related to the length L of the tube, and are given by

for a tube open at both ends (c is the speed of sound), and

for a tube closed at one end.


L

cnfn 2.=

L

cnfn 4

).12(=

Fig. 1: Sonagram of theeigenresonances of acylindrical tube. Twosides of the tube areopen (left and right)and one side is open(middle) giving rise tohigher/lowerfrequencies offundamental andovertones.

We have used a tube with a length of 0.48 m which would theoretically give fundamentals of f1 =350 Hz (open at both ends) and f1 = 180 Hz (closed at one end). The measured values were f1 = 340Hz and f1 = 170 Hz, respectively. Also the difference can be understood: The “acoustical length” ofa pipe is a bit longer than the real length giving rise to a slightly deeper tone [3].The analogy of a sea shell with a cylindrical tube looks a bit crude.As a next approximation we usedsome plastic bottles and cups for our experiments. Fig. 2 gives the result with a plastic can. Againone can see the fundamental and higher partials, but some more comments are in order regardingthese results.Acoustical resonating systems are often associated with Helmholtz resonators. Helmholtz firstdescribed these resonators in 1860, and he used them for the spectral analysis of complex sounds.They are hollow metallic bodies in various shapes (spherical or cylindrical), some of them eventunable. There is a small hole at one end so that the sound from outside can enter the resonator.On the other side there is some small nipple which should be placed into the ear canal. Thetheoretical model of a Helmholtz resonator is that of a large oscillating air volume acting like aspring on the small air volume in the nipple (being the oscillating mass of the system). Thecorresponding eigenfrequency is given by [3,4]:


LV

Acf

.2= ,

Fig. 2: Sonagram of thefrequency spectrum ofa plastic can. The can isopened and closedsuccessively. When it isclosed, no noise canenter from the outsideand theeigenresonancesdisappear.

where A and L are the cross section and length of the nipple, respectively, and V stands for thelarge air volume. Our examples, bottles or cans, should act in between these two models, namelyideal Helmholtz resonators and ideal cylindrical tubes, and the measured resonance frequenciesconfirm this statement.Fig. 2 exhibits also another sound detected at very low frequencies, i.e. around and below 20 Hz.Of course, this sound cannot be heard, it is infrasound. Nevertheless the question arises about theorigin of this phenomenon. A direct hint to the explanation can be given by the followingexperiments: One holds the can with one or two hands; the can lies on the floor or on some table;one holds the can firmer and firmer. When the can is not held by hands, the effect disappears, andthe firmer one holds the can, the larger the amplitude of the oscillation is. It is the trembling of themuscles which can be seen by these experiments: The microvibration of muscles have been foundin 1943 by the Austrian neuropathologist Rohrbacher. The average frequency of a relaxed muscleis between 7 and 10 Hz, a maximally contracted muscle can have a frequency as high as 30 Hz [5].Let us finally come to experiments with real sea shells.We recorded and analyzed the sound of twoshells, Murex and Cassis Fig. 3). The fundamentals are clearly visible, and, naturally, the frequencyof the larger shell is lower (400 Hz) than that of the smaller shell (640 Hz). The higher partials are


not so clearly developed in Cassis as in Murex. Due to the complicated geometry of the shells, theovertones are not multiples of the fundamental. From the frequencies of Murex (640, 1350, 2090,2870 Hz) one could be tempted to read off a similarity to the law of a tube. This mixture ofovertones which are not perfect multiples of the fundamentals and the appearance of rather broadresonances give rise to the impression of a non-technical, natural sound, as from the sea; thus wefinally come back to our headline.

Fig. 3: Sonagram of thesound of two shells,Murex (left) and Cassis(right).

We have tried to present an example, where one starts with a question which might be of interestto the students. Guided by the teacher, students can approach the final answer by someexperiments accompanied by theoretical considerations.The computer assisted analysis should notbe a barrier but an additional motivational component for the students, and for the teachers aswell.

References[1] G. Rosenberg, American Conchologist, March (1995), 21[2] Cool Edit 2000 by Syntrillium Software Corporation; http://www.syntrillium.com.[3] M.P. Silverman, E.R. Worthy, The Physics Teacher 36, (1998), 70[4] T.B. Greenslade, Jr., The Physics Teacher 34 (1996), 228[5] http://www.studio32.net/WWW/Portfolio/Optimalife/TheoreticalBasis.html.


3.9 Text Books

DISCUSSING THE PROBLEMS IN TEXTBOOKS

Consuelo Escudero, Margarita García, Physics Dept., Fac. de Ingeniería, Universidad Nacionalde San Juan, San Juan, ArgentinaSonia González, Physics and Chemistry Dept., Facultad de Filosofía, Humanidades y Artes,Universidad Nacional de San Juan, San Juan, ArgentinaMarta Massa, Physics and Chemistry Dept., Facultad de Ciencias Exactas, Ingeniería yAgrimensura, Universidad Nacional de Rosario, Rosario, Argentina

1. Introduction Problem solving is a basic recourse in Science teaching at secondary school, through whichstudents work with the analysis of events and associated data, and reason in order to design therequired strategy to solve problems. Many activities, such as written texts comprehension,interpretation of data (literal, numeric and graphic), production of inferences, planning strategies,calculus, argumentation, application of concepts and communication, need to be developed duringa class (Pozo Municio & Gómez Crespo, 1998). Solved problems presented in textbooks are usuallymain models that students use in their apprenticeship. Therefore, the perspective of the authordefines implicitly the role, goals and possibilities that students will attribute to problem solving intheir own apprenticeship.Previous researches have shown serious difficulties in a significant number of college students of thefirst Physics course (Escudero, 1996, Escudero et al., 1998; Sánchez et al., 1998) and that features oftheir secondary studies influence their performances. The inabilities to recognize the relevantinformation and the lack of procedural skills to organize it reflect a degree of scientific illiteracy.Therefore, it is important to analyze the teaching-learning strategies concerning with problemsolving, the didactic material, the criteria for its choice, the internal coherence and theoreticalconsistency of the textbooks, the relationships established between the conceptual contents andthose related to procedures.We present a study about problem solving that appears in secondary school textbooks. We intendto develop a model for the analysis of problem statements provided by secondary textbooks. Weassume that textbooks´ problems serve as efficient prototypes to elucidate useful didactic modelsfor the development of criteria which may be applied in teachers’ training.

2. TheoryWe assume that scientific literacy implies an optimal usage of a set of actions to arrive to a goal,but not with a sense of a sequence but as a global instruction to decide and select opportunelyduring action. It demands a sufficient declarative knowledge (Rumelhart, 1980), but it is alsonecessary that the subject knows how to activate and apply it to solve problematic situations.Therefore, learning Science requires the construction of the meaning of the experiences withlanguage (Lemke, 1997). Discourses, written or spoken, are used not only to inform but also toguide observation, to persuade, to convince and to structure thinking (Vygotski, 1988). Teachinglies on discursive activities that mediate apprenticeship and support strategies to relate concepts,organise explanations and validate or deny statements. Dialogic interchange between subjectsprovides instances to negotiate and to arrive to a consensus of meanings, by extending or refutingcommon sense (Vygotski, 1988; Ausubel, 1978).The teacher’s attitude towards science and, specifically, towards scientific procedures and activitiesdefines the orientation that is prescribed in the didactic model employed in classes. Therefore,teacher’s actions and the resources selected to support the discourse are relevant forapprenticeship. In this sense, it is important to provide some reflections on the didactic models andmaterials in which sustain classes planning.


The word model has been used with several senses in science education and in published papers.Colinvaux (1998) has detected five categories within them, that she characterizes as: mental model,consensus model, meta-model, educational modelling and didactic model.

3. DesignThe purpose of this article is to organize didactic models of problem solving which can be usefulfor the preparation of teachers, taking into account the set of problems that usually appear insecondary textbooks as introductions, examples or routine exercises, within explanations related toconcepts and laws, or, at the end of a chapter, as a base for workshops and laboratory activities,discussions on environmental issues, analysis of everyday experiences and cross-disciplinary topics.We analyse problems offered in textbooks commonly used by secondary students. These problemsare shown by authors and teachers as examples to orient problem solving processes. We haveconsidered all the problematic proposals stated in textbooks, presented with different title formats,such as activities, workshops, debates, problems, applications, researches, opinions, questionnaires andtests. Each problem, including its solution, is assumes as the analysis unit.We have assumed the following set of 8 categories for the analysis, with their corresponding modalities.Some of them have emerged from results of researches on problem solving, misconceptions and/orevaluation. Others have been incorporated pragmatically during the research.

Analysis categories1- Formulation level • Textual language: written problems exclusively. Questions are also included.• Textual language with graphics: written problems that include graphics.• Graphic with questions: Problems with a graphic predominance in their formulation.

2- Transformation level• Closed: the statement proposes an explicit or implicit model and the student only has to identify it.• Open: the student organizes the model. The problem can have one or more solutions and the

subject can generate a variety of plausible and fruitful outcomes.

3- Language for expressing the solution• Qualitative: descriptive, attractive, with logic and extra logic aspects (Stinner, 1990, cit. Lópes &

Costa, 1996). Measurements are no required.• Quantitative: characterized by the existence of precise relations among magnitudes and concepts

(Ibid). Includes possible comparisons with scales or tabulated values.• Formal: includes a reference to a physical – mathematical model as a semantic dimension.

Modelling is stated not only in a logic and mathematical sense for the interpretation of calculusbut also as a simplified representation, with a complete and relevant set of variables, strictlyrelated to the analysed fact. Predictions are based on modeling (Greca & Moreira, 1998);measurements are not implied.

4 - Alternative conceptions: includes texts that orient students to reflect or discuss specifically on misconceptions that havebeen detected as obstacles for conceptualization (Driver el al., 1989).

5 - Solving Method• Graphic: refers to a strictly graphic solution, without consideration of the presence of a

representation as a support to the solver.• Analytic: includes solutions done with explicit equations and algebraic transformations.• Verbal: includes solutions stated in a propositional level, without explicit calculus. Inferences and

analogies are included.• Numeric: includes solutions based on a direct probe and on substitution of numeric values in

algebraic expressions.

6 - Contents• Pure: considers topics of the highest specificity within a conceptual framework (e.g., kinetics

energy).• Theme: exclusively centered on a single content (e.g., energetic processes) • Integrated: problems that only involve general Physics contents.• Interdisciplinary: problems that include contents concerning different disciplines.

7- Statement context• Natural: refers to an everyday problem situation.• Academic: offers a model situation with a high level of abstraction, such as a particle with mass

m, a rigid body or an infinite charged plane. Problems only provide strictly relevant data.• Technologic: involves human-developed devices, processes and tools.

8 - Possibility of functional analysis (used to determine tendencies to variations anddependences of a variable on others):• Graphic: by means of representations.• Algorithmic: by means of analytic searching.• Experimental: by means of controls carried out with instruments and lab equipments.

SampleWe analysed 312 problems that were extracted from 11 textbooks commonly used at secondaryschools. Each problem was considered as an individual and each dimension previously mentionedas a variable. Therefore each individual was characterized by a set of eight values (modalities)associated with the category identified for each dimension. Results were organized in a data matrixof 312 files (problem) × 8 columns (variables). Data were processed using multivariate statisticaltechniques (Multiple Correspondence Analysis and Factorial Cluster) (Lebart et al, 1985), usingthe software SPAD. N (CISIA - CERESTA, 1998). The obtained clusters were used as initialmodels to develop didactic strategies.

4. Results The classification analysis allowed recognition of six groups (classes) of problems with similarcharacteristics, as shown in Figure 1:

384 3. Topical Aspects 3.9 Text Books

Figure 1: Affinity classification of the problems


Class 1: includes 136 problems, with an academic content. They are written with a quantitativelanguage since they include literal expressions but all the relevant data are expressed with numbers.They require an analytical solution. They are oriented to an efficient calculus and to a coherentnumeric result. Problems that are explicitly solved in the textbooks allowed to say that the authorassumed that the reader is able to comprehend the text adequately. It is also implied that studentsdominate algebraic strategies.Class 2: includes 95 problems related to everyday situations. They are written with a qualitativelanguage. They also demand a verbal solution. They focus on misconceptions or on specificconcepts, in order to support the construction of the new concepts previously treated in thetextbook.Class 3: includes 17 open problems that are stated with an academic language. They arecharacterized to use a graphic solving method. They emphasized the development of alternativestrategies to arrive to the goal.Class 4: is constituted by only 7 problems. They are stated through a graphic, accompanied byquestions. They require an image processing, related to some numerical data. The authoremphasizes the comprehension of representations.Class 5: includes 26 problems that deal with technological situations. They take into account theapplication of scientific concepts to comprehend or to explain how devices work.Class 6: is formed by 31 closed problems, stated with a formal language. They demand theintegration of different contents. Therefore, the solver ought to relate concepts and laws during thesolving process. An analytical, and not always direct, solution is required. These problemsemphasize the construction of conceptual structures within the discipline.

5. ConclusionThe six categories recognized in textbooks may be used to model different strategies for teachingproblem solving. Class 1 offers prototypes whose purpose is the calculus, based on the correctcomprehension of numeric data. These problems can be used to attract teachers´ attention on thelack of discussion about data and are an opportunity to discriminate between relevant andirrelevant data. Problems of class 2 may be employed as introductory ones when the purpose is tofill the gap between abstractions and natural contexts. The scarce percentage of problemsbelonging to classes 3 and 4, many of them found in a same textbook, reveals that graphic solvingstrategies as well as graphic interpretations are considered by few authors. The relevance ofrepresentation and the interesting opportunity to install the discussion in the codification betweenconcepts, models and geometric symbolism are not taken into account. Classes 5 and 6 reflect theintention of the author to produce an “actionable” knowledge through problems that demand theapplication of concepts and laws to interpret devices or to integrate them to previous knowledge.From a didactic perspective, these results provide tools of analysis for the discussion of theusefulness of textbooks’ problems not as scaffolding examples that orient students in theirindependent study, but as a means to enlarge the criteria for the design and selection of problemsin the different stages of teaching and learning processes.

ReferencesAusubel D. P., Psicología educativa: un punto de vista cognoscitivo, México, Trillas, (1978).Colinvaux D., Modelos e educação em ciencias, Rio de Janeiro, Ravil, (1998).Driver R., Guesne E., Tiberghien A., Ideas científicas en la infancia y la adolescencia, Ediciones Morata, Madrid,

(1989).Escudero C., Los procedimientos en resolución de problemas de alumnos de 3er año: caracterización a través de

entrevistas. Investigaciones em Ensino de Ciencias, 1, (3), (1996).Escudero C., González S., García M. y Massa M., Las leyes de Newton: problemas y ejercicios propuestos en libros

de enseñanza media, Memorias del II Simposio: La Docencia de las Ciencias Experimentales en la Enseñanzasecundaria, Madrid, (1998).

C.I.S.I.A. CERESTA, SPAD. N integrado versión 4. París, (1998).Greca I., Moreira M. A., Modelos mentales, modelos conceptuales y modelización,. Caderno Catarinense de Ensino

de Fisica, 15, (2), (1998), 107-120.

Lebart L., Morineau A., Fenelon J., Tratamiento Estadístico de Datos, Barcelona, Marcombo, (1985).Lemke J., Aprender a hablar ciencia, Madrid, Piados, (1997).Lopes B., Costa N., Modelo de enseñanza-aprendizaje centrado en la resolución de problemas: fundamentación,

presentación e implicaciones educativas, Enseñanza de las Ciencias, 14, (1), (1996).Llonch E., Sánchez P., Massa M., La activación representación ⇔ situación en los procesos de comprensión y

resolución de problemas, publicación en CD, Actas del V Simposio de Investigadores en Educación en Física, SantaFe, (2000).

Rumelhart D. E., La representación del conocimiento en la memoria, Infancia y Aprendizaje, (1980), 19-20.Pozo Municio J. I., Gómez Crespo M. A., Aprender y Enseñar Ciencia, Ediciones Morata, (1998)Vygotski L., El desarrollo de los procesos psicológicos superiores, México, Grijalbo, (1988).e-mail: [email protected]

DO PRIMARY AND SECONDARY TEXTBOOKS CONTRIBUTE TO SCIENTIFICREASONING?

Marta Massa, Hilda D’Amico, Physics and Chemistry Dept., Facultad de Ciencias Exactas,Ingeniería y Agrimensura, Universidad Nacional de Rosario, Rosario, Argentina

1. The discourse and knowledge representation in science textbooksTheory and practice of science, as knowledge, are transmitted, explored and re-constructed throughthe semiotic function of language in classes and textbooks. Lemke (1997) has stated that teachers’discourse in classes may frequently alienate students as it presents an opposition between a worldof impersonal, authoritative and boring facts and their own world of uncertainties, beliefs andinterests. Unfamiliar discourses with technical words and mathematical expressions will either boreor attract students, mainly in secondary school. These features may also be attributed to textbooks’discourse and may act as a model for teachers when they prepare their classes (D’Amico & Massa,2000).Textbooks have traditionally helped Physics teaching, by introducing a “scaffolding” discourse toconstruct a conceptual framework and to develop reasoning skills. Therefore, textbooks inducelearning situations by means of the provided information, the questions to orient the reader, theproposed activities, the graphic design and the language.Discursive styles of the authors externalise their reasoning and influence readers’ thinking, throughthe way ideas are supported or contrasted; experiments are designed or beliefs are denied orreinforced (Billig cited in Kuhn, 1991; D’Amico & Massa, 2000). The logical structure andcoherence of written discourse may be considered as a good expression of the reasoning that theauthor develops when he organizes the contents, applies them to certain facts and brings orstrengthens the conclusions. The argumentation of the authors, as a way to support or to refusestatements, defines a position towards the construction of knowledge.The language of science has evolved in the construction of a special kind of knowledge – a scientifictheory of experience. Science textbooks deal with information and explanations related to facts,processes and devices. Therefore the language is a basic unit to introduce concepts, principles andlaws, whose meanings are negotiated by specific figures and expressions provided by the writer ina communicative and constructive process. In this sense, language is viewed as a meaning-makingsystem within a context rather than a meaning-expressing one (Halliday et al., 1993).Theories dealing with information processing assume that external information is registered,codified and processed by subjects before being expressed by actions. Written discourses, drawings,photographs, diagrams are “inputs” for learning processes. In fact, when a subject reads, heintegrates the meaning of each sentence in order to construct the global sense of the text within acontext. Language comprehension lies on the processing of propositions (assertions, arguments,questions), images and mental models (Johnson-Laird, 1983) as individual representations. Thelatter are organised dynamically and their meanings are developed in different levels as a result ofthe comprehension of the text associated with the construction of a mental representation of the



stated situation. This process is known as the “resolution of the reference”. The links between thenew information and previous knowledge and beliefs, and the inferential processes that derivedfrom them, characterize the style of the discourse, which may be descriptive, informative orargumentative (de Vega et al., 1990).Researchers have identified many problems related to the organization of science textbooks, suchas: a “cookbook” approach to experimental activities, scarce attention paid to the development ofscientific processes, the introduction of terms preceding the exploration of examples. In addition,the style is descriptive rather than argumentative, with a frequent lack of justification or refutation.Graphic design and structural devices are oriented to attract the attention of the reader rather thanpromoting reasoning skills (Musheno & Lawson, 1999).Assuming that textbooks are the main source of models of written scientific language for students,they contribute, together with formalized productions of teachers, to construct an alternativeinterpretation of the world (science versus common-sense). Therefore four questions emerged: (a)How do textbooks’ discourses contribute to a progressive development of scientific concepts?; (b)What propositional and procedural knowledge is introduced for constructing a formal thinking?;(c) How do authors use the argumentation as a mechanism for conceptual changes?; (d) How arelanguage and language activities (discussion of concepts, questions, formalization, linguisticresources) used to promote understanding?This exploratory study focuses on the development of the concept of ENERGY in sciencetextbooks. Assuming that textbooks provide a relevant discourse that defines an attitude towardsScience, we research on the use of argumentation as a resource for a meaningful understanding ofthese concepts.

2. MethodThirteen science textbooks, mainly adopted by teachers of the 3rd cycle of the Basic GeneralEducation (for students aged 11-14 years) and Physics textbooks for Polimodal courses (studentsaged 15-17) were analysed (see Table I). The texts contain prose, pictures, diagrams, mathematicalexpressions and suggested activities throughout the book. The study was focused on the chaptersdealing with Energy.Two stages were followed in order to identify textbooks’ structure and strategies to promotelearning,: (a) a general approach with four analytical dimensions: conceptual development,structure, language, and proposal of activities; (b) an analysis of the evolution of the argumentation.The latter was done using Toulmin’s scheme (Alvarez Pérez, 1997). It considers the existence of six

Code Title Author- Editor-Edition

Level Grades

A Natural Sciences Santillana - 1997 General Basic Education 8th

B Physics Plus Ultra General Basic Education 7th, 8th,9th

C7, C8, C9 Natural Sciences Santillana 2000 General Basic Education 7th, 8th,9th

D Natural Sciences 9 Kapelusz 2001 General Basic Education 9th

E Natural Sciences 7 Kapelusz 2000 General Basic Education 7th

F7, F8, F9 Natural Sciences andTechnology

Aique 2000 General Basic Education 7th, 8th,9th

G Physics I Aique 1998 Polimodal 1st

H The Universe of Physics El Ateneo 1997 Polimodal 1st

I Physics I Santillana - 1999 Polimodal 1st

Table I. Analysed textbooks

elements in an argument, and the relationships among them. Three basic elements forcomprehensive reasoning are: data [D] (all relevant information related to the events, subjects andobjects), conclusion [C] (derived proposition for a general acceptance) and justification [J](statement that gives continuity to the gap between [D] and [C]). The other three elements, thatenrich the reasoning, are: modal qualifiers [M] (intensity with which the data support theconclusion), refutation conditions [Rf] (particular circumstances in which the justification is notvalid) and support conditions [Rs]. Refutation [Rf] and support conditions [Rs] sustain theacceptability of the justifications.

3. ResultsThe first stage of the study involved the identification of the features employed by the author totry to enable the comprehension of the text.The different dimensions for the analysis of the textbooks, the associated variables and therespective modalities are described in the following paragraphs.

Conceptual development I. Central topics: problems related to the use of energy (1) - energy in nature and/or energy

resources (2) - forms of energy, energy transformations (3) - conservation of energy (4) - heatand nuclear energy (5) - electrical energy (6) - physical and chemical changes (7).

II. Energy concept: A concept impossible to define. Energy is transformed and transferred (8) -ability to do work (9) - associated to movement and substance (10) - the total energy of a systemis the sum of the macroscopic energy and microscopic internal energy (11) - power to makechanges (12) - heat as a form of energy that travels from one body to another due to ∆T (13) -identifiable by its forms (14) - social concepts (bills, home devices) (15).

III.Misconceptions (towards a conceptual evolution): existence (through initial questionnaire) (16)– omission (17) - existence of perpetual mobile (18).

StructureI. Introduction: energy conservation (1) - energy sources (2) - historical reference (3) - questions

related to misconceptions (4) - lab experiments (5) - physical and chemical transformations (6)- scheme as previous organizer (7) - electric energy: generation system, distribution andconsumption; conceptual activities (8).

II. Body: work, heat, internal, nuclear, mechanical energy (9) - analysis of forms and principlesrelated to devices and nature (10) - energy: forms and transformations in industries (11) - workand impulse related to kinetic energy and linear momentum (12) - work and kinetic, potentialand mechanical energy (13)

III.Conclusion: historical references of applications, reflective and argumentative activities basedon papers and Internet sites (14) - historical references of applications, reflective questionnaires(15) - energy transformations in industries and technological devices (16) - work equal to thevariation of kinetic energy (17) - reflection on energetic self-sufficiency (18) - environmentaleffects (19) - W–∆Ek and W–∆Ep , energy conservation and Wfnc - ∆E; work done bygravitational forces along a closed trajectory; analysis of several cases (20).

LanguageI. Approach: Declarative and descriptive (giving information) (1) - interrogative (demanding

information) (2) - imperative (demanding services or providing instructions) (3).II. Format: photographs with explanation (4) - schemes and related questions (5) - maps (6) - new

concepts in vignette without explanations (7) - caricatured drawings (8) - tables withinformative data (9).

III.Formal structure: scientific terminology; definition (without units); principle and laws withoutformalization (10) - scientific terminology; definition with mathematical expressions withoutdeductions and units (11) - scientific terminology; definition stating dependences between



variables (12) - scientific terminology; definition with mathematical expressions and units;principles with formalization (13) – analogy (14)

IV. Questions (related to the frequency of occurrence of each type of question): qualitativedescription → explanation or justification → complement of description (15) - qualitativedescription → complement of description→ explanation or justification (16) - qualitativedescription → complement of description (17) - quantitative description → complement ofdescription→ explanation or justification (18) - explanation or justification → qualitativedescription (19).

V. Arguments: D → C with J and Rs (20) - D →C with J and M (21) - D →C with Rs (22).

Activities Working on misconceptions (1) - experimental activities and related questions (2) - analysis ofgraphics and tables (3) - explanations to everyday situations (4) - selection of alternatives (5) -research (6) - debates (7) - calculus (8) - design of devices where transformations take place (9) -use of a dictionary to search for meanings (10) - open situations (11).

4. ConclusionsIt has been observed that textbooks for EGB3 students offer a general treatment that begins withthe analysis of energy in natural processes; followed by the identification of different forms of

Conceptualdevelopment

Structure LanguageText-bookCode I II III I II III I II III IV V

Activities

A 2, 5 9 16 2, 8 9, 10,11

15 1 4, 5, 9 10 15 22x

2, 3, 4, 11

B 3, 4 10 16 4, 6 10 17 1,2

4, 5 11 17 22 4, 8

C7 2, 3 14 16 3, 4, 5 10 15 1,3

4, 5, 6 11 15 20 1, 2,4, 6,8

C8 2, 3, 4,5, 7

12,14 17 2, 3, 6 9, 10,11

14, 15,19

2,3

4, 5, 9 1013

15 22 3, 4, 7, 11

C9 5 11, 13 16 3, 4, 5,7

9 14 1 4, 5 11 15 20 1, 2, 3, 4,5, 6, 7

D 6 14 16 8 11 16 1 4, 5, 9 14 15 2021

1, 4, 6, 7,8, 10

E 1 9 17 2 11 18 1 4, 5, 7,8

12 17 20 1, 2, 7

F7 3, 7 12, 14 16 6 11 16 1,2

4, 5 10 16 20 2, 5, 8, 9

F8 6, 7 13, 15 17 6, 8 10, 11 16 1 4, 5 10 15 20 2, 4, 9F9 4, 5 10, 12, 14 17 7 9 17 1 4, 8 11

1419 20 2, 4, 5, 7,

8G 4 9, 11, 15 18 1 12 15,

201,2

4 1314

19 20 5

H 2, 4 9, 14 16 1, 2, 4 11, 13 19, 20 1 4, 6, 8,9

1314

18 20 1, 3, 4, 6,9, 11

I 3 8, 14, 15 17 3 10, 11,12, 13

14, 15,16, 17,19, 20

1 4, 5, 8,9

14 19 20 2, 3, 4, 5,6, 7, 8, 10

Table II: Main characteristics observed in the analysed textbooks

energy and a slight introduction to principles; while those developed for the 9th year deal with formsof energies that are important for human beings’ activities. They all have a colloquial style, writtenin everyday language to present examples as scaffolding mental activities, and they usephotographs, graphs, vignettes, and any other graphic design to accompany the explanations. Fewof them use historical references, experiments and questions as complements to the construction ofconcepts. Only one text, as can be seen in Table II, centres the treatment on problems derived fromenergy uses nowadays. As a general perspective, the organisation of these textbooks resembles aschema associated to the intention to construct answers to: why (is it important to know aboutenergy) → how and where (can energy be recognized) → how (do human beings obtain energy toperform their activities). The main purpose is to introduce scientific language by a closecorrespondence between words and facts, but not to organize formal thinking.Polimodal textbooks are basically centred on developing a deep and specific conceptualbackground. Examples are discussed with precision, presented by models and accompanied bydemonstrations, diagrams and mathematical expressions. It may be interpreted as the author’sintention to produce a first level of semantic activity within a scientific language, by a closerelationship between specific concepts and examples. Analogies are used in this level in order tosupport statements or as a scaffolding guide to reasoning. These textbooks abandon the narrativeand descriptive style seen in EGB3 textbooks to become explicative. Nevertheless, very littleargumentative structures, using counterexamples as refutation conditions, have been detected in allthe textbooks. Only some of them have been stated as proposed activities to be done by studentsto complement their study. A question arises in this sense: Do teachers use these activities toorganize and to produce an open perspective to science theory?

ReferencesÁlvarez Pérez, V., Los libros de texto, Alambique. 11, (1997).D’Amico H., Massa M., Argumentative Style In Teachers’ And Textbooks’ Discourses, PHYTEB, Barcelona, (2000).De Vega M., Carreiras M., Gutiérrez-Calvo M., Alonso-Quecuty, M., Lectura y comprensión. Una perspectiva

cognitiva, Alianza Editorial, Madrid, (1990).Halliday M., Martin J., Writting science: Literacy and Discursive Power, University of Pittsburgh, Press Pittsburgh,

(1993).Johnson - Laird P. N., Mental Models, Cambridge University Press, Cambridge, (1983).Kuhn D., The skills of argument, Cambridge University Press, Cambridge, (1991).Lemke J. L, Aprender a hablar ciencia. Lenguaje, aprendizaje y valores, Paidós, Madrid, (1997).Musheno B. V., Lawson A.. E.,. Effects of Learning Cycle and Traditional Text on Comprehension of Science

Concepts by Students at Differing Reasoning Levels, Journal of Research in Science Teaching, 36, (1), (1999), 23-37.

e-mail: [email protected]



3.10 Teacher Training

FORMALIZING THERMAL PHENOMENA FOR 3-6 YEAR-OLDS: ACTION-RESEARCH IN A TEACHER-TRAINING ACTIVITY

Luca Benciolini, Dipartimento di Georisorse e Territorio, University of Udine, ItalyMarisa Michelini, Dipartiment of Physics, University of Udine, ItalyAdriana Odorico, Facoltà della Formazione, University of Udine, Italy

1. IntroductionPrevious research has led us to identify teachers’ main needs for open and innovative activitiesregarding the scientific education of children from 6 to 14 [1]. Research on didactic innovationbased on new technologies for the teaching of physics have shown that it is necessary to integrateMeta-Cultural, Experiential and Situated models [2, 3, 4] into teachers’ in-service training. Arecent pilot project organized by the Italian Ministry of Education has made us aware that we needto completely change the ways in which we carry out our in-service teacher-training so as toinclude the dimension of educational and didactic research [5]. Action-Research, based on areflection on work done in the field, contributes in a unique way to the teacher’s professionalcompetence and gives him those elements of flexibility which make him able to follow and managedynamic mental models connected to the context [6]. For this reason the action of reflection andresearch which teachers can make is one of the factors which, with increasing attention, are locatedat the center of processes for improving the learning/teaching system [5]. There is now a generalagreement on the need to introduce science teaching right from play school. Science teachingperforms an educational function, by accustoming students to group work, to an open comparison,to explaining ideas, to recognizing the validity of results based on the sharing of common elements.It also performs a training function, stimulating representation, symbolization and modelling,which are essential in the development of cognitive skills. In the Italian school system, which is stillstrongly influenced by the reforms made by Gentile in 1923, the introduction of scientificeducation in the basic school is still an open problem [7].This work reports on an experience of teacher training in the Play school: it integrates research andinnovation while it introduces scientific education into a specific field of experience: thermalphenomena. The work focuses attention both on the teacher-training process and also on theactivities of the children (from 3 to 6 years of age): the indivisible nature of the two processes putsparticular attention on the children’s formalization processes on the experiences of constructingand structuring scientific thought.

2. Scientific education in the play schoolChildren are looking for stimulating activities in order to lessen their need of competence [8]. Thechild has to recognize sensorial information and learn to use it to collect information, includingquantitative information, on phenomena [9]. The child’s curiosity is the starting point for hisdevelopment of scientific knowledge. Through curiosity scientific education is constructed byasking questions, discovering problems, working on them with one’s thought, starting to observe orto make experiments [10]. In this way he develops a flexible attitude, a readiness to change opinionwhen this is reasonable, and an ability to analyze and synthesize [11]. Recent research has shownhow children from 3 to 6 do not limit themselves to asking questions and waiting for the answersfrom us: they look for personal ways to give answers [12]. Between 3 and 5 children developsystematic strategies and a generalized rule-governedness; there is an impressive variety ofsituations where children construct and use different rules, and this is particularly evident in theway they deal with new situations of problem solving [13].Scientific education in the play school must therefore be conducted in such a way as to stimulate

in play, in stories, in moments of didactic activity, that curiosity which activates processes toorganize knowledge, able to evolve in a continuous re- processing of the results of the child’sexploration of the world.The professional ability of the teacher who must produce this dynamism implies a combination oftechnical, social and organizational skills [2, 3, 14] and therefore the teacher training for this levelof school is particularly delicate.The degree course in the Science of Primary Education, which wasonly started in Italian Universities in 1999, has to deal with these problems. However, the problemremains of teachers already in service, especially with regard to the following needs:A) possessing a scientific competence which can be integrated with their pedagogical competence,

gained by their experience in the field [15]; understanding the cognitive skills of the children inthe scientific field at this age;

B) understanding the cognitive skills of the children in the scientific field at this age;C) being able to manage informal education, with play, to give the children the necessary

familiarity with the world around them;D) having experience in constructing contextualized formalization processes.Physics is one of the subjects which best lends itself to perform this task, because it usesmathematics in describing and interpreting phenomena and offers opportunities for symbolization,seriation and tabulation.We shall now illustrate the contents of a teacher training activity for play school teachers; it wascreated as a training course and transformed into a research gymnasium for the trainers andtrained, through analysing the cognitive processes of the children who were in the classes.

3. Training play school teachersThe activity of training as research, which we shall be reporting on here, had its origin in anexcellent experiment in the play school of Terenzano (Udine). It was called “Fisicando” andproposed physics as the subject matter and play as the didactic method for children of 5-6 years ofage.Both the planning part and the practical part of the project were followed, analysed and evaluatedin the context of the Degree course in Primary Education, with consequent influence on thepractical experience in the participating schools. This activity has stimulated great interest and 32teachers from the provinces of Udine and Pordenone have requested a specific training course atthe Interdepartimental Centre for Research in Education (Udine University), which has developedin the following five stages:I) Seminar on operativity in the construction of formal thought in the scientific fieldII) Visit to the GEI exhibition [16] with the children in the context of the “Science Days 2001”,

with their actions monitoredIII) Paths on thermal phenomena were discussed, simple experiments, including on-line

experiments, were performed and analyzedIV) Didactic activities were planned and feasibility studies were made in the field of thermal

phenomenaV) The experiments were carried out in class and there were periodic discussions of the paths

followed and the materials produced by the children.These five stages developed the following points in the training process of the teachers:

Role and method of scientific education for children:A) Contribution of science to general education. Science has to be used for rethinking andunderstanding experiences already made. Describing, interpreting, predicting and checkingpredictions is a strategy proper to scientific investigation.B) Role and duties of the teacher:- to help the children to pay attention to and reflect on the phenomenon;- to use common sense experiences and spontaneous interpretative patterns;



- to look for answers by exploring with hands and mind: the teacher must not impose answers whichhave not been identified/recognized;- to stimulate questions rather than give answers to questions which have not been asked. Inparticular, to educate the children to recognize that the decisive element for accepting a correctinterpretation of the phenomena is sharing:- to accastom the children to always insert the evidence of their prediction with respect to the resultof the experimental investigation.C) Processes of formalization, the role of operativity, ways and means of scientific communication(graphs, etc). Examining some examples taken from research with GEI [17]. Some examples ofexperiments with GEI were provided, some disciplinary contents in the field of thermology wereillustrated and some possible didactic activities were suggested.

Attitudes and cognitive processes activated in informal contexts by the childrenA group of 15 children of 4 and 5 years old, in the play school, was observed with parameters asused in other studies [18] during a visit to the GEI exhibition, to acquire experience of their waysof exploring in a scientific field. The children’s attention span and interest was surprisingly long:about 70 minutes, during which they asked a lot of questions, often redundant, showing in this sometypical features of their age, such as the need for a personal answer and for repetition [19].The cognitive path of the children on specific concepts of the subject dictated the criteria on whichthe activities proposed were set up in the subsequent teacher training step.

Planning didactic activitiesIt was mainly the task of the teachers to plan the activities to be carried out with the children andthe way to carry them out. The task was performednot in isolation but with a variety of humanresources and instruments. The human resources were not only the researchers, but also theteachers, in an integration of skills and experiences combined in a learning community. Theconceptual knots we pointed out were:1) Heat as an exchange quantity and temperature as state property;2) measuring the temperature of objects and the human body;3) thermal interaction and exchange processes between metal cubes at different initial

temperature;4) thermal interaction and exchange processes between water masses at different initial

temperature;5) specific heat and thermal conductivity of materials;6) analysis in energy terms of transformations.

Didactic activities with play school childrenEven though the proposed activities were elaborated, discussed and shared in the learningcommunity, the teachers actuated a wide variety of didactic strategies and ways of conducting theexperiments. Let us consider the following three points, dealt with by at least 5 teachers:I – thermal sensation (all teachers);II – thermal state (15 teachers);III – conductivity (5 teachers).

I – Thermal sensationThe didactic experiments on thermal sensation were for example, a thermal mapping of a space andclassification of a defined series of objects (cubes of polystyrene, wood, plastic, a rubber, a piece ofiron) according to thermal sensation.The children touched the objects with their hands and then had to formalize the thermal sensationthey experienced. Different formalization strategies were tried, as listed hereafter.• Seriations. The children classified the sensation produced, and seriate the objects in a diagram

along a diagonal. The objects which produced the sensation is symbolized using the objects,

photographs, sketches or symbols. They frequently used symbols or symbolic drawings, forexample: red balls for heat and blue balls for cold. From the cognitive point of view, the drawingwas equivalent to the symbol, since the children’s limited drawing ability always puts the factinto a symbolic context. In fact, when the teacher asks the child to draw objects, the similarity ofthe drawing to the real object is not very relevant, because either the sign is shared and thereforerecognized or it remains extraneous.

• Ven representations: they are used both to define groups and also to make correspondences.• Shared rules of spatial collocation or symbolic representation.• Double entry tables with seriations identical to those cited above. The correspondence with the

place or participating child is explained.• Spontaneous quantitative representations: diagrams and/or graphs. The intensity of the thermal

sensation is represented with squares of an area proportionate to the thermal sensationexperienced, or is translated into histogram bars or even graphs where the elements are orderedin increasing or decreasing fashion.

The initiative of one teacher (Emma) to include a step of intermediate binary classification proveduseful.

II - Thermal stateThe didactic experiments on thermal state consisted in putting some objects in the freezer and inhot water in order to take them to different thermal states. The children examined the sensationproduced by different objects which had been for a long time in the same environment (freezer,tank of hot water, environment) and by the same object in different environments. The childrenmade the deduction that the thermal state is determined by the environment and that the materialinfluences the rapidity with which every object reaches the thermal state without particular help.Formalization remained on the same level as the previous experiment: no formalization was notedwhich took the temporal variable into account. In order to overcome this limit, one teacher (Rita)translated this experiment into a game: three environments at three different temperatures (thefridge, the tank of hot water, the class room) were symbolized by different objects (foam rubbercubes / freezer, basin / tank of hot water, area of the gymnasium limited by a rope / environment)at different points of the gym. The children chose an action to carry out at random. If, for example,a child drew “it gets hot”, he went to the “hot water” and removed some clothes; if he drew “it getscold”, he had to go into the “fridge” and put his clothes on.

III – ConductivityThe didactic experiments concerning conductivity were conducted to evaluate the ways childrenrecognize process with respect to state. They led the child to observe the fact that ice placed on topof a metal melts more quickly than ice placed on top of polystyrene. In this case the question whichsome children asked was: “why does the object which I feel as colder make the ice melt quicker?”In this way there was a spontaneous separation and recognition of states with respect to processes.In fact, the children explained the process with a motivated animation (metal is a friend of ice andtakes it away) on the process variables,

4. ConclusionsWhen children report their experiences they often use drawings and this corresponds tosymbolizing, sharing the meaning of the symbol. The similarity of the symbol to reality is not veryimportant. At this age socialization begins and the comments of their peers are more importantthan those of adults (“cooperative learning”). Therefore, the object symbolized will be recognizedirrespective of its similarity to reality.However, the symbols are often too distant from reality to be understood outside the group. In thegame where the children “become hot or cold objects”, the child uses himself as a symbol (heidentifies) and this facilitates comprehension [19].In seriating “hot” and “cold” objects, the child has a problem: he is representing quantities of a

394 3. Topical Aspects 3.10 Teacher Training


single property, still not well identified, of a system outside himself. Seriating the objects along adiagonal line unifies these features and helps the child to recognize the first quantitative elements.When the experience is discussed and repeated, the children are faced by the problem ofreproducing the experiment and by the significance of the data. Representing thermal sensationwith an area-intensity translation, by means of squares, offers the child the opportunity to considertwo-dimensional spaces, which he can connect with geometry later, and immediately with therecognition of the space in which he is. In this case, when the thermal sensation is tabled for eachchild and object touched, the children can observe the “total thermal sensation” of a single objectfor an entire class of pupils, making integrations for classes. The concept of average also emerges:they are able to recognize an “average thermal sensation” and use it to obtain secondaryinformation and also a shared table of data.With the experiment of the ice cubes the predictions are compatible with the proof of thededuction.Our data show how very young pupils can manage very complex concepts. They discuss laboratoryexperiments and the meaning of data, they reflect on the experience finding different methods offormalization, also producing symbols, tables and types of seriations of the data.In order to obtain a shared seriation of the objects touched, different strategies were used: first ofall, the children discuss the different opinions [20], then they make a seriation of pairs of objects,they establish a series over the whole group, repeat the experiment after some days and finally thedata not shared are excluded from the representation.These results, of considerable interest, have emerged thanks to the learning environment which wasdeveloped: neither the teachers with autonomous activities, nor the researchers observing thechildren in unplanned activities, would have been able to recognize the obvious potential forformalization which the children show in the core of a subject. The dimension of the Action-Research associated with that of comparison has enabled the development of skills in the teacherson various levels (not least, the subject level). Although it was not expected, important skills wereacquired in managing the conceptual points which the children bring out in practical activities,creating a cognitive crisis in the teacher too; to overcome this crisis entails looking at the sameknowledge on many different levels. Reflecting on practical didactics is thus revealed to be theessential moment of in-service training for teachers.

AcknowledgmentsWe are greatly indebted with the 32 play-school teachers participating in teachers’ educationactivity and particularly Giuseppina Tirelli, Rita Maurizio, Emma Ciannavei, Ornella Milad andCarla Pecoraro. Their feeling and experience in teaching improved the work.

References[1] S. Bosio, V. Capocchiani, M. Michelini, F. Vogric and F. Corni, Problem solving activities with hands on

experiments for orienting in science in Girep Book on Hands on experiments in physics education, eds G Born,H Harries, H Litschke and N Treitz (for ICPE_GIREP Duisburg), (1998); Michelini M, Mossenta A andBenciolini L, Teachers answer to new integrated proposals in physics education: a case study in NE Italy in:Information and Communication Technology in Education, (Roznov pod Radhostem, Czech Republic, 18 - 21September 2000), Proceedings, ed E Mechlova (University of Ostrava, Faculty of Science), (2001), 149-154.

[2] M. Michelini and C. Sartori , Esperienze di laboratorio didattico in una struttura di raccordo scuola-università,Università e Scuola, III 1/R, (1988), 18-29.

[3] R. Martongelli, M. Michelini, L. Santi and A. Stefanel, Educational proposals using new technologies andtelematic net for physics in: Girep book on Physics Teacher Education Beyond 2000 (Barcellona), (2000).

[4] G. Marucci, M. Michelini and L. Santi 2000 The Italian pilot project LabTec of the Ministry of Education in: Girepbook on Physics Teacher Education Beyond 2000 (Barcellona).

[5] M.G. Dutto, M. Michelini and S. Schiavi, Reinventing in-service teacher training: research grants for teachers in:Practitioner Research - International trends and perspectives 2001, eds V Trafford and F Kroath (Treccani inprinting); M Michelini 2000 The contribution of institution to the improvement of physics teaching in: Girepbook on Physics Teacher Education Beyond 2000, Round Table (Barcellona).

[6] S. Vosnodiou, Capturing and modelling the process of conceptual change, Learning and Instruction 1994b, 22,(1994), 45-69.

[7] Italian earlier school levels have been poorly reformed after the organizationon on the lines laid down by Croceand Gentile. Giovanni Gentile’s reform was enacted in 1923, after that of Benedetto Croce in 1920.

[8] E.L. Deci and R.M. Ryan, The initiation and regulation of intrinsically motivated learning and achievement, in:Achievement and motivation eds AK Boggiano and TS (Pittman Cambridge, Cambridge University Press),(1992), 9-36.

[9] D.R. Olson and R. Campbell, Constructing representations in: Sistems of representation eds C Pratt and AFGarton (Chichester, Wiley), (1993), 11-26.

[10] Berlyne DE 1960 Conflict, arousal and curiosity (New York Mc Graw – Hill); Bijou SW Exploratory behavior ininfancy and early childhood Revista-Mexicana-de-Analisis-de-la-Conducta 24, (1998), 215-223

[11] S. Carey and R. Gelman (eds), The epigenesis of mind. Essay on biology and cognition, Hillsdale (N.J.), (Erlbaum,the Jean Piaget Symposium series), (1991).

[12] C. Pontecorvo, Narration and discursive thought in infancy in: Psychoanalysis and development: Representationsand narratives. Psychoanalytic crosscurrents, eds M Ammaniti and DN Stern, (New York University Press),(1994), 131-146.

[13] R.S. Siegler and D. KlahrWhen do children learn? The relationships between existing knowledge and theacquisition of new knowledge in: Advances in Instructional Psychology ed R Glaser, (Hillsdale, New Jersey, LEAvol 2), (1982).

[14] M. Michelini and A. Mossenta, 2000 The EPC project – Exploring Planning Communicating, Girep book onPhysics Teacher Education Beyond 2000, Round Table (Barcellona)

[15] J. Lave, The practice of learning in: Understanding practice. Perspectives on activity and contex eds S Chaiklin andJLave (Cambridge, Cambridge University Press), (1993).

[16] GEI (Giochi Esperimenti e Idee, www.uniud.it/cird/GEI) is a collection of 120 simple experiments in manydifferent fields of physics, well described in literature [10])

[17] S. Bosio, V. Capocchiani, M. Michelini, S. Pugliese Jona C. Sartori, M.L. Scillia and A. Stefanel, Playing,experimenting, thinking: exploring informal learning within an exhibit of simple experiments, in Girep book onnew way for teaching,, (Ljubljana), (1997); Bosio S, Ceccolin D, Michelini M, Sartori C and Stefanel A, Gamesexperiments Ideas from low cost materials to the computer on-line: 120 simple experiments to do and not onlyto see, in Girep Book on Hands on experiments in physics education, eds G Born, H Harries, H Litschke and NTreitz (for ICPE_GIREP Duisburg), (1998); Bosio S, Michelini M, Santi L, Sartori C and Stefanel A, A researchon conceptual change processes in the context of an informal educational exhibit, Wirescript, novembre(www.wirescript.com) (1999). Other articles can be down-loaded from the sitewww.fisica.uniud.it/GEI/GEIweb/ricerche/ricerche.htm.

[18] L. Cibin, A. Del Bianco, M. Michelini, G. Michelutti and A. Odorico Laboratorio di fisica per la didatticaUniversità e scuola in press

[19] T.B. Rogers, Emotion, imagery and verbal codes: a closer look at an increasing by complex interaction in: J YuilleImagery, memory and cognition (Erlbaum, Hillsdale, New York)

[20] One child said: “I took the object after my class mate and his hands had heated it up”

THE DIDACTIC LABORATORY AS A PLACE TO EXPERIMENT MODELS FOR THEINTERDISCIPLINARY RESEARCH

Margherita Fasano, University of Basilicata, ItalyFrancesco Casella, IRRE Basilicata, Italy

1. IntroductionBeyond what adults can imagine, the exchange of messages and information using a specialspontaneous and impromptu code, is nowadays more frequent between children and adolescents.In most of the cases, this code is organized at the moment and it is aimed at conveying instructionsin order to satisfy a practical need concerning an instrumental knowledge (e.g.: how you canadvance level in a computer game, how you can set in position the character movements, how aparticular game configuration can be determined...). The strong motivation and the need, in somecases, of a written communication favour the utilization of a rapid symbolic and very effectivelanguage from the point of view of the meanings to be conveyed.

The expression shown above is related to a communicative exchange between an 8-year-old child


Dx Sx x2 x9 x x5


and a 12-year-old adolescent about the use of a CD-ROM for the Playstation videogame platform.This expression may be variously interpreted by a mathematician or a computer scientist but to thetwo children it acquires a precise meaning, which permits to solve the problem.It can be pointed out that this language reflects most of the features of a more formal language: theuse of symbols, the relation among symbols, the assignments of meanings (in the expressionreported above the last but one x has been circled on purpose in order to give it a meaning whichis different from those of the other xs ), the conciseness and the uniqueness.In Italy, researches in Mathematics Education (see Arzarello 1994, Malara 1996) have oftenemphasized the difficulty in the teaching/learning process focused on the development of analgebraic thinking. The passage from arithmetic to algebra presents several pitfalls, due mostly tothe need of reading an expression not only from a strictly algorithmic point of view (technicalexecution of a calculation) but also as a mathematical structured object, which can be assigned ameaning. Being able to gather relations and analogies between mathematical objects and beingable to recognize the structures and meanings are fundamental competence required to developthe algebraic thinking, without which the application of rules and models to interpret the realitywould be extremely difficult.The habit of devoting the most of teaching time to the procedural aspects, which are typical ofarithmetic, prevents from gathering and properly facing the relational aspects, which involve somecompetence closer to the algebraic thinking.We believe that, from a didactic point of view, the recover of the approach which takes into accountthe motivation factors and the learning environment is of fundamental importance in mathematicsteaching. The motivation factors include not only the most pleasant and delightful situations butalso those elements relating to curiosity, game, individual and group experiences which maybecome a source of discussion, of working hypothesis, of discovery, of reflection among studentsand between the teacher and his/her students. The conditions of the learning environment derivemainly from a planning carried out by the teacher considering, in a dimension of a dialectical andin progress process, the work of both the student and the teacher and the didactic relationshipwhich is estabilshed each time.

2. The didactic laboratory as a research and learning environmentGenerally, the concept of didactic laboratory is associated to a place where various kinds ofexpriments (chemical, biological, physical...) are conducted. However, in the last few years, the ideaof a didactic practice based on a laboratory approach has been asserted. It does not necessarilycoincide with the common conception of a laboratory but it takes the form of a research andlearning environment whose focal point is making the most of the methodological aspects.On the one hand, this characteristic, particularly important for the innovation process in thetraining field, favours the development and the acquisition of the competence required to teachersfor the implementation, management and control of various teaching situations and on the otherhand it improves the motivation of the students in learning within a co-operative and researchenvironment.This approach, mainly focused on the methodological aspects, permits to realize more easily adidactic programme, indipendently of the contents chosen during the planning phase.The laboratory approach takes the shape of a work organized into projects, from the formulationof the project idea (on what we intend to work) to the definition of the objectives to achieve; to theanalysis of the available resources and those to be gathered; to the planning of the work stages andof the deadlines; to the organization of the activities to research, collect and ultimately process thedata; to the final evaluation of what has been obtained.The experimentation, being mainly cognitive because the student organizes and reorganizes his/herknowledge in a continuous “doing and being able to do”, is also supported by the emotionalinvolvement, by the enjoyment and the curiosity of the individual and of the group about theprocess of discovery which brings to the achievement of a shared objective.

3. The process of organization and of reorganization of knowledgeIn the process of organization and reorganization of his/her knowledge and particularly in sight ofan objective to achieve, the student has to identify, select and use the information proceeding fromthe situation to study. Moreover, the student has to connect and compare them with his/herscientific knowledge and then look for a new arrangement in a structure that represents andorganizes both the concepts and the relations among these concepts.During this activity, the oncoming difficulties are mainly due to the habit of behaving in a linearsequence of actions related to the daily routine activities (reading, writing, talking, solving variouskinds of problems...). This procedural aspect is so strong that it often limits or, sometimes, evenforbids the recourse to the relational thinking that, even if present, does not find either thepossibility or the conditions to be carried on.Consequently, from a didactic point of view, it is convenient to arrange in advance the learningenvironments, aiming at the development of both procedural and relational processes, whoseosmosis permits the construction of knowledge.The importance of the need of facing this problem is largely emphasized in the national andinternational literature, which underlines the relapse, according to the learning sphere, of theinsufficient development of the relational aspect compared to the procedural one. This can beexpressed in the difficulty in acquiring disciplinary competence and in developing logical capacitiesuseful in identifying crossing competence also between different knowledge fields.Among the most significant researches regarding this aspect, there are, for instance, those referringto the development of the algebraic thinking (Sfard, Arzarello, Malara), which is fundamental formodelling mathematically the reality and for estabilshing and representing the relations withmathematical objects through symbols ( the recognition of laws and structures). In particular, asMalara (1997) states: “ In teaching, algebra is introduced as an arithmetical generalization, using itssigns (the arithmetic operation signs and the equal sign) and its properties. However, while inarithmetic the calculation algorithms are privileged and the algebraic expressions are conceived assequences of operations to execute in order to obtain a certain result, in algebra, on the contrary,the study of the symbolic representations (expressions, equations, functions), conceived as amathematical object, prevails. This process-object passage, typical of the development ofmathematics (Sfard, 1991, 1994), is not enough emphasized in mathematics teaching”.Consequently, the teacher, aware of the necessity of not neglecting the interaction between theprocedural process and the relational one, must create ad hoc situations where the instrumentchoice and the usage modality are extremely coherent and functional with the development and/orthe strengthening of cognitive structures facilitating the knowledge construction process, in strictlydisciplinary and crossing contexts.The planning of a teaching activity in a multimedia environment may represent an excellentreference in order to realize the above-mentioned situation.For “multimedia learning environment” not only we do mean all the technological resources(hardware and software) necessary for producing a hypermedia but also all that precedes theimplementation phase, usually conceived as the most important and meaningful in the wholeprocess.This situation creates the conditions for realizing a laboratory activity where a learning andteaching model is experimented. This model is able to emphasize, at the same time, both thecognitive situation of a subject in a certain moment and in a specific reality and the way in whichhis/her information are organized and represented. The main interest in this psycho-pedagogicalstatement is to stimulate a training process, starting from the student real (and not presumed)experience that represents an unavoidable reference to obtain an effective didactic action.How is it possible to “ leaven” the subject cognitive and not cognitive experience? Whichinstruments should be used?

4. The conceptual map as an interaction instrument between the subject and the objectIt results from the researches and experiences carried on in the last few years that the conceptual



map is the most suitable instrument to represent the logical project of a multimedia productinformation network. Moreover, the researches that we have carried on (Fasano 1989) have shownthe importance of the construction of a conceptual map in order to promote and facilitate theinteraction between the cognitive processes of the subject and the learning object.By the way, the teacher who wishes to take advantage of this instrument has to be able to manageand control the methodology of the pertinent teaching activity.Indeed, the construction process of a conceptual map involves specific phases:a) choice of the object to study;b) recall, identification and selection of the concepts considered relevant in order to explore the

object to study;c) network organization and representation of the concepts by each student;d) argumentation of the choices made in terms of link-words, connections between the nodes of

the map, comparison and discussion about the various points of view;e) individual revision of the map, worked out on the basis of the stimulation induced by

comparison and discussion.The choice of the object to study (point a) is conditioned by several factors, implicit and explicit asregards the curriculum (course of studies), such as the disciplinary context, the interests related toa topic shared by the whole class, a common project involving several classes, even belonging toschools far from each other.It is important that the students do not experience the choice of the object to study as a classworkgiven by the teacher but, on the contrary, as a good opportunity to satisfy a curiosity or a need,shared by all the students or rather by the students and their teachers.The recall, identification and selection of the concepts (point b) has to appeal to both the schoolinternal and external resources (texts, films, interviews, network…) and the subject internalresources (his/her own information, experiences and competence…).The network organization and representation of the concepts (point c) can be also facilitated by agame involving the use of small cardboard circles (as many as the concepts identified before), eachone provided with the transcription of all the concepts. The handling of the small circles and thedifferent attempts aimed at organizing them on a sheet of paper (an A3 format is preferred)correspond to the activity of the cognitive organization and reorganization of the learner. It ispossible to have a visual idea of the activity thanks to the graphic representation.The argumentation of the options which have been chosen (point d) bears two positiveconsequences: the development of logical-linguistic capacities (pertinence, concordance,correctness, conciseness…) and the consideration of various points of view that can be found in theclasswork of the other classmates.The comparison and the discussion on classwork help the students to find out mistakes and tointroduce some corrections. This fact assumes a pedagogical value apart from a didactic one.The individual revision of the map, worked out on the basis of the stimulation induced bycomparison and discussion (point e), permits the self-regulation of one’s own learning process. Ifthe production of a hypermedia has been planned, it would be advisable, because of the managerialrole that the teacher has to play, to aim for a conceptual map shared by the whole class.Consequently, the maps which have been just produced will be revised again in order to make upthe various proposals into a new one.

5. A metacompetence for the interdisciplinary researchIn the last few years, the school and the training industry have undertaken a project-oriented wayof working. The main objective is usually the achievement of competence which are often certified.The methodologies are mainly heuristic aiming to involve all the students and one of the mostdesired and required training objective is being able to move in a flexible way among variousdisciplinary areas.A very favourable place where to realize this new pedagogical and didactic trend is the multimedia

laboratory in which a metacompetence can be gradually developed. It is built up on the habit ofworking on concepts whose meanings can be discovered and on the habit of making a proper useof the cognitive, procedural and relational processes, defining the sequences of a research itineraryand estabilishing connections between different kinds of objects to study.This kind of metacompetence fulsils effectively several needs sensed by modern professionals whooften require certain skills such as beeing able to work in team, being able to periodically reconvertone’s own competence, following the continuous innovations, and being able to interact with othercontexts greatly influenced by technologies.As far as the school field is concerned, a significant example of application of this metacompetenceis provided by the interaction between mathematics and physics: mathematics offers usefultheoretical models for physics in order to develop researches, inquires, checks on hypotheses thatfavour a greater knowledge of world and universe phenomena (for instance, Riemann’s geometryof the sphere and Einstein’s theory of relativity), while physics provides a context to favourproblem solving and to facilitate the construction of the meaning for the mathematical contents(for instance, the motion and the laws that bind velocity, space and time).Finally, we can deduce that the main problem is not the interaction, always possible, betweendisciplines but the capacity of planning learning environments that make possible theexperimentation of the interactions at different levels: research of both the information and theirmeanings, selection and conceptual organization and , in the end, representation and formalization.

ReferencesArzarello F., Bazzini L., Chiappini G., The process of naming in algebraic problem solving, atti PME XVIII, col. 2,

(1994), 40-47.Cafaro F., Casella F., Coronato A., Fasano M., Pandolfi C., Multimedialità e progetto didattico. Un’applicazione sul

problema dei rifiuti, Zanichelli, Bologna, (2000).Fasano M., Informatica come e perché, in M. Pellerey (edited by), SEI, Torino, (1989), 44-59.Fasano M., (edited by), Concetti in rete. Dalla costruzione della mappa concettuale alla produzione di un ipermedia,

Masson - Zanichelli, Bologna, (2000).Garito Amato, Antinucci F., Tecnologie e processi cognitivi. Insegnare ad apprendere con la multimedialità, Franco

Angeli, Milano, (1997).Malara N.A., Il pensiero algebrico: come promuoverlo sin dalla scuola dell’obbligo limitandone le difficoltà?, in

“L’Educazione matematica”, anno XVII, serie V, 1, (1996), 80-99.Malara N.A., Problemi di insegnamento-apprendimento nel passaggio dall’aritmetica all’algebra, in “La matematica e

la sua didattica”, N.2, (1997), 176-186.Pellerey M., L’agire educativo. La pratica pedagogica tra modernità e postmodernità, Ed. LAS , Roma, (1998).Polo M., Il ruolo dell’insegnante nella gestione delle attività in classe: risonanza degli interventi degli alunni sul progetto

delle insegnanti, Atti Internuclei, Scuola dell’obbligo, aprile 2001 (in course of publication by Pitagora Editrice).Sfard A., On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Object as Different Sides of

the Same Coin in “Educational Studies in Mathematics”, 22, (1991), 1-36.Sfard A., The Gains and Pitfalls of retification: The case of Algebra in “Educational Studies in Mathematics”, 26, 191-

228.

A MODERN TEACHING FOR MODERN PHYSICS IN PRE-SERVICE TEACHERSTRAINING

Marco, Giliberti, Dipartimento di Fisica Università degli Studi di Milano, INFN sezione diMilano, Italy

1. IntroductionSince many years the Research Unit in Physics Education of the Università degli Studi di Milano hasbeen working in the field of Modern Physics Education. In the last two years the members of thisresearch group have been involvend in the SeCiF (Explaining and Understanding in Physics)national Project together with other five Research groups, from the Universities of Napoli, Palermo,Pavia, Torino and Udine with the aim of studying a coordinated approach to Mechanics,



1 Also other two research unities involved in he SeCiF Project, those of Udine and Torino, are working on ModernPhysics Education[19]-[21] but with an approach slightly different from ours.

Thermodynamics, Optics and Quantum Physics Education, bringing their contribution especially onthe last topic1.The starting points of our work are some conclusions of our (and other’s) previous studies aboutModern Physics Education [1]-[9] and may be roughly summarised as follows: High-School Studentshave great difficulties in understanding some of the principal conceptual points of Modern Physics, i.e. the meaning of quanta, the necessity of a statistical description of Nature, the structure of the Worldaround us (beginning from the structure of the atom) etc. mostly because of the inadequate“traditional” teaching approach. In this situation is not surprising that even High-School teachershave shown problems in constructing a coherent picture of Modern Physics and, especially, inbuilding a didactical framework that is appealing and has a real cultural meaning for Students and,consequently, for all people.To overcome these difficulties, the educational materials now available (text-books, CD roms and soon) are, in general, not immediately useful because lacking of a sufficiently coherent framework andbecause leading to didactical routs that, as it has been generally highlighted by many studies[3], [5],[9], are clearly not effective.For these reasons, in the context of the SeCiF project, we studied, and started to test, the newapproach to Modern Physics Education that is sketched in Appendix 1. It’s too difficult for me toexplain in few words this scheme, nonetheless I’ll try to give the principal starting points of ourapproach and some very rough examples of its peculiarities, in the hope of stimulating interest.

2. Starting points of our Quantum Teaching ProjectOur project has 5 starting points:1. Physics is one and we should not create a gap constructing Modern Physics against Classical

Physics.2. History is not our reference guide and thus we must construct a general coherent framework of

Quantum Physics independently of how it is grown.3. Quantum Physics is not equivalent to Quantum Mechanics and its paradigm; the Quantum

Theory we have now is Quantum Field Theory and it we’ll be our principal guide.4. Physics does not end in the ‘30s and thus we should try (the best that we can) to have in mind the

whole last century and not only its first 30 years.5. Popularisation is not our field of interest.In fact our aim is to teach Quantum Physics and our teaching should be as effective as possible. Webelieve that this goal may be achieved using the ideas of Quantum Field Theory. Whereas QuantumMechanics is the adaptation of the Newtonian world of point-like particles to quantum phenomena,Quantum Field Theory is the evolution of Classical Field Theory to give reason of quantumphenomena. In this framework: the primary object of the theory is the field, not the particles; whilethe statistical and particle aspects of Quantum Physics are all embraced in the description ofinteractions.Our project is ambitious: starting from the preceding points and, merging some modern experimentsof the last decade with some “classical” experiments, as well as with some results of Quantum FieldTheory, it has the purpose to lead to a new vision of the Quantum World Teaching.In my opinion it goes straight on to the main goal and gives a “natural” and unitary referenceframework.

3. Some peculiar AspectsI give here, very shortly, just some rough examples of the differences among our and other more orless “standard” strategies, while a scheme of our approach can be found in the Appendix 1.The first peculiarity refers to the meaning of the word “classic”. We call “classic” every non quantised

field description of both force and matter. In this line of reasoning the well known Davisson-Germerdiffraction pattern is a classical effect, in which a matter pencil undergoes “classical” (that isexplainable without quanta) diffraction. With this sense of “classic”, a lot of beautiful example of“classical” (or, in more standard words, high intensity) experiments on the wave behaviour of mattercan be seen in very recent works of Tonomura, Rauch, Zeilinger[10]-[12] and others. From this pointof view there is a strong similarity of behaviour between matter and radiation if we describe matter,as we do, through a classic matter wave field.Another peculiar aspect may be seen in the way in which we introduce particle-like interactions. Weknow that Chemistry interactions may be well understood in terms of local and universal exchangeof Quanta (atoms). It’s not surprising, therefore, that other “similar” interactions, for instance ofelectromagnetic radiation impinging on a metal, may be well interpreted in terms of other Quanta(photons)… In such a way Photoelectric as well as Compton effect receive an explanation that is notconstructed “against” Classical Physics in the same sense in which Chemistry is not. The discovery ofPhotons may be considered an important improvement to the already known Physics and there’s noneed of a long and difficult discussion about the inadequacy of Classical Physics for explaining sucheffects…This is not a way of hiding the differences between the Classical and Quantum World but,instead, it’s a way of putting it in the right place, that is in the (statistical) description of interactions.Another typical example of the difference between our and “traditional” Modern PhysicsCurriculum concerns the logical route towards the structure of the atom. We don’t follow the usualway of presenting the Thomson, Rutherford, Bohr, Sommerfeld, De Broglie and, last, the orbitalmodels (six models!). We just present first some fundamental experiments (the most important arethose of Geiger and Marsden) [13] and then, starting from the knowledge of the behaviour of aquantised field enclosed in a box, we arrive immediately at the energy levels and the structure of theatom.

4. Testing the Approach with Teachers (some numbers)This approach has been tested for two years in the course “Quantum Theories” of the S.I.L.S.I.S.-MI(Inter-University Lombard School of Specialisation for Secondary Teaching, section of Milan), that isone of the Italian, regional, post-graduate, 2-Years, Schools of pre-service Teachers training. Bothyears the duration of the course was of nearly 30 hour, 20 of which of disciplinary education and 10(held with the help of Prof. L. Cazzaniga) of Educational Laboratory.In the following some numbers about the course are given.The first year we had 13 students of the overall 25 students attending Physics courses of the SILSIS-MI.The Mathematicians were 4, the Physicists 8 and there was also 1 Engineer.Every student of the first year, but one, passed the examination.The marks were between 26 and 30 cumlaude (In Italy sufficiency is 18 and the maximum is 30 cum laude) while the average mark was 29.The second year we had 19 students of the overall 21 students attending Physics courses. TheMathematicians were 8, the Physicists 10 and there was also 1 Engineer.The examinations for the second year’s students have not yet started.To the Students of the second year we proposed the anonymous questionnaire about the course thatcan be seen in Appendix 2.

5. Few Final Considerations on the obtained ResultsThe answers given to the questionnaire and the colloquia we had during the examinations (as well asthe exchange of opinions during laboratory work) greatly encouraged us in carrying on our work onQuantum Physics Teaching. In fact most of the students declared themselves very satisfied by ourapproach and asked for a similar approach even in the University degree courses. Everyone wasfavourably impressed by the educational content of the course and 3 of them (the only that werealready teaching, as supply teachers, in a final class of a Scientific High–School) spontaneously triedour approach in their classes and discussed, during laboratory lessons, the results obtained with theirHigh-School students. They declared themselves very satisfied.Nonetheless some problems appeared. The only Engineer of the first year course did not make the



examination while the Engineer of this second year course explicitly declared himself uncomfortablewith the ideas of the course.4 of the 8 Mathematicians had some problems because they felt uneasy with electromagnetism whilethe others declared themselves greatly encouraged by our strategies.During laboratory work, 40% of the students had problems in preparing lesson unities about theatom without the usual route through Thomson, Bohr, De Broglie, Sommerfeld-Models and someefforts had to be made to clarify our view.Most of the students pointed out a difficulty in preparing the examination because there were (and,that I know, there are yet) no textbooks with our line of reasoning. To this respective we hope thatthe materials we are preparing for the SeCiF Project can be useful2.In conclusion we believe we have reached some general important results and we have a clear wayto go through, but we have still a lot of work to do. First of all we have to improve our testingstrategies, experimenting with a larger number of Teachers and monitoring the High-School Studentsresponses. Some preliminary attempts, made in High-School classes in the last 6 years and alreadypresented in past Girep Conferences[14]-[17], give us confidence in good results in real classes but wehave never really monitored what our Teachers will teach when left “alone”. We do hope that thiswe’ll be done in the next years with the help of every Teacher and Researcher interested in our work.

Aknowledgments I’m very grateful to Professors L. Lanz and G. M. Prosperi for the helpful and deep discussions aboutFoundations of Quantum Theories and Teaching. I address a lovely thought to memory of late DoctorC. Marioni that with competence and helpfulness initiated me to Physics Education. And last, but notleast, I’m very indebted Professor G. Vegni, under whose precious guide I’m doing my research, andto all the friends of the SeCiF Project. A special thank to all them.

References[1] G., Rinaudo, “Introduction” , L.F.N.S. quaderno 7, 1, (1997).[2] A., Hobson “Teaching Quantum Theory in the Introductory Course” The Phys. Teach. 34, (1996), 202-210 [3] D., Johnston K., Crawford P. R. Fletcher “Student difficulties in learning quantum mechanics” Int. J. Sci. Educ.,

vol. 20, no 4, (1998), 427-446.[4] F. Hermann et al. “Der Karlsruher Physikkurs” par. 3. E. Taylor, S. Vokos [5] H. Fischler, M. Lichtfeldt “Modern Physics and Students’ Conceptions” Int. J. Sci. Educ. 14 (2) 181-90.[6] I. Lawrence, “Quantum Physics in school” Phys. Educ. 31, 278-87.[7] J. O’Meara “Teaching Feynman’s sum-over-paths quantum theory” Computer Physics, vol. 12, no. 2,. (1998).[8] M. Giliberti, C. Marioni “The Introduction of Modern Physics at High-School Level. A new Approach based on

the Analysis of Student’s Conceptions”; IFUM 529/FT, (1996).[9] D. Gil, J. Solbes “The introduction of modern Physics: overcoming a deformed vision of science” Int. J. Sci. Educ,.

15, (3), 255-60.[10] A. Tonomura “The Quantum World Unveiled by electron Waves”; World Scientific Publishing (1998).[11] H. Rauch, Proc. 3rd Int. Symp. On Found. Of Q. M., (1990).[12] A. Zerilnger Letter to Nature 401, (1999), 680.[13] N. Bergomi, M. Giliberti, “Modelling Rutherford Scattering: A Hypertext for High-School Teachers Net-

training”; Proc. GIREP ICPE “Physics Teacher Education Beyond 2000”, Editions Scientifiques et MedicalElsevier, (2001).

[14] M. Giliberti, C. Marioni, “Revisiting the H-Atom and Electron Configurations in Atoms: a Case Study at High-School Level”; Proceedings of the GIREP ICPE International Conference: Teaching the Science of CondensedMatter and new Materials, Udine 24-30 Agosto, (1995), 168-171.

[15] M. Giliberti, C. Marioni, “The Introduction of Modern Physics at High-School Level. A new Approach basedon the Analysis of Student’s Conceptions”; IFUM 529/FT, (1996).

[16] M. Giliberti “Teaching about Heisenberg’s Relations”; Proceedings of the GIREP ICPE InternationalConference: New Ways of Teaching Physics, Ljubljana 21-27 Agosto, (1996), 533-534.

[17] M. Giliberti. “Popularisation or Teaching ? How much Math in Physics Courses?”; Proceedings of the GIREPICPE International Conference: New Ways of Teaching Physics, Ljubljana 21-27 Agosto, (1996), 408-411.

[18] G. C. Ghirardi, R Grassi, M Michelini, “A Fundamental Concept in Quantum Theory: The SuperpositionPrinciple”, in Thinking Physics for Teaching, Aster, Plenum Pub Corp, (1996).

2 They should be ready by the end of November 2001.

[19] Ghirardi G. C., Grassi R., Michelini M., “The linear superposition principle and non classical features ofmicrophenomena”, in GIREP-ICPE Book, Forum, (1996).

[20] A. Cuppari, G. Rinaudo, Robutti O., Violino p., “Gradual introduction of some aspects of quantum mechanicsin a High School curriculum” , Phys. Ed., 32 , 302-308 , ISSN 0031-9120, (1997).

[21] D. Allasia, C. Bottino, G. Rinaudo, A. Cuppari, I. Giraudo, “A simple hands-on experiment to appreciate thelimits between Classical and Quantum Physics”, Proceedings of GIREP Int. conference, Duisburg, (1998).

Appendix 1


GROUND STATE OF THE HYDROGEN

ATOM

E. M. INT.: E. M. QUANTA

MATTEROPTICS

MATTERWAVE EQ.

E. M.WAVE EQ.

E. M. OPTICS

MATTER AS A CONTINUUM CONTINUA OF FORCE FIELDS(E. M. FIELD)

FROM CLASSICAL PHYSICS TO MODERN PHYSICS

CHEMISTRY:MATTER QUANTA

INTERFERENCE, DIFFRACTION, ETC. (DOUBLE SLIT)

STATISTICAL INTERPRETATION

FIELD OF WHICH THE PARTICLE IS THE QUANTUM

WAVEPACKETS

STATIONARYSTATES

RADIOACTIVITY

RUTHERFORDMODEL

HEISENBERG’S RELATIONS SPECTRA FRANCK-HERTZ

EXPERIMENT

ENERGYLEVELS

HYDROGENATOM

ATOMIC

STRUCTURE

KEY

CONCEPT


Appendix 2

19 (all) Students answered.

Evaluation Questionnaire of the Course “Quantum Theories”

Please answer the following Questions giving a mark between 1 (worst) and 5 (best); 3 is sufficiency.(In brackets and in Italics the average result for each question)

• I’m satisfied by the course (3.8)• The contents of the course have been

Pertinent (4.8)Examined carefully (3.8)

• The Teachers have beenUnderstandable (3.7)Well-integrated (4.2)

• The teaching methodology has been effective (3.6)• The organisation has been appropriate (3.5)

(In brackets and in Italics the most given answers)* Please point out, among all the lessons and the laboratory works, the most interesting aspects(Educational Aspects, New Experiments, Conceptual Revision)

* Please give indication about the aspects that should be improved(Lecture notes and textbooks on the subject)* Please list three topics you found particularly interesting for your disciplinary preparation(Matter Waves, Analogy Matter Fields-Force Fields, Introduction of Radiation Quanta)* Please list three topics you found particularly interesting for your class work(Preparation of lesson unities, Analysis of classroom experimentations, Preparation of initial and finaltests for High-School students)* Please list three topics you found particularly difficult(Klein-Gordon Equation, Electromagnetic Field as a gauge Field)* It should be useful that the teachers of the course….(Gave reference Textbooks, Gave a better Bibliography, Used Multimedia) The teaching rhythm has been. (In brackets and in Italics the number of answers received) Too slow (0) Adequate (14) Too fast (5)The participants to the course have been involved Too much (0) Adequately (19) Too little (0)The group work have been Useless (0) Useful (4) Very useful (19)

Please give an overall judgment on the course (4)

TEACHING ENERGY IN HIGH SCHOOL: CRITICAL ANALYSIS AND PROPOSALS

J. Ll. Domènech, D. Gil-Pérez, Universitat de València, SpainA. Gras-Martí, J. Martínez-Torregrosa, Universitat d’Alacant, SpainG. Guisasola, Euskal Herriko Unibertsitatea, SpainJ. Salinas, Universidad Nacional de Tucumán, Argentina

1. IntroductionThe importance given to the study of energy both at high-school and University level has beenaccompanied by a progressive understanding of the many difficulties encountered in teaching andlearning this domain. Published studies range from specific research items to proposals for in-classdealing with the subject. These works include: Duit (1981 and 1986), Black and Solomon (1983),Solomon (1983 and 1985), Watts (1983), Driver and Warrington (1985), Trumper (1990), Nichollsand Ogborn (1993), Koliopoulos and Ravanis (1998), and many others.All the studies about the teaching and learning difficulties of energy, like those referred to above,have addressed, in general, various specific aspects. However, we believe that the difficultiespointed out in the literature are interrelated and demand a more global approach, which we haveundertaken.

2. A proper comprehension of energyAs a result of the analysis of the abundant research literature on the subject, plus a number ofinterviews with teachers (Doménech, 2000), we have made an attempt to spell out in 26propositions what we consider a proper understanding of energy and related issues by high-schoolstudents. These propositions are divided into three groups: conceptual (18 propositions),procedural and axiological (8 propositions). The propositions are mutually dependent andinterconnected, and should be discussed accordingly in the teaching process. We have only spacehere to mention the general guidelines and a few propositions in each group.The conceptual propositions are structured in five groups:I) About the meaning of concepts. (“Energy is not some kind of fluid, nor a kind of fuel which is

needed to produce transformations (…) The transformations that a system undergoes are dueto interactions with other systems or to interactions among its parts”).

II) About the systemic and relative nature of energy. ( “Speaking about the energy of an isolatedobject lacks all meaning”).

III) About the relations between energy, work and heat. (“From the kinetic-molecular theory, heatarises as a magnitude that encompasses mechanical work performed at a submicroscopic level.Therefore heat, like macroscopic work, is not a form of energy but an exchange of energy”).

IV) About transformation and conservation of energy. (“The total energy of an isolated systemremains constant, but whenever such a system experiences changes there must necessarily beenergy transfers and/or transformations of energy in its interior”).

V) About energy degradation. (“When we talk about ‘energy consumption’ or ‘energy crisis’, wedo not mean that energy disappears, but that it has become ‘homogenized’, i.e., that theconfiguration of the system does not allow for macroscopic changes to occur”).

A fair scientific knowledge cannot be limited to conceptual aspects; a correct conceptualunderstanding by the students must also incorporate procedural and axiological aspects, whichare structured in three groups:

VI) About the origin and relevance of the concepts. (“It is necessary to be aware of the problemsthat led to the introduction of the concepts, if one wants to stress the rational character ofscientific knowledge”).

VII) About the scientific approach. (“Students must have the opportunity to use criteria andstrategies of elaboration and validation which are characteristic of scientific work -namely, tomake assumptions, to conceive experimental designs, etc.- in such a manner that they canconfront their tentative constructions with those of the scientific community. In our particular



instance, students must know the scientific criteria that laid the foundations for the acceptanceof the theory of caloric, and its ulterior rejection, and they should understand that theintegration of mechanics and heat was a real revolution”).

VIII)About a full understanding of this conceptual domain. (“It is necessary to overcome thetendency of students to disregard the use of the energetic approach in solving problems ofmotion and to use instead, routinely, only the kinematics-dynamics approach”).

With this set of propositions we have attempted to offer a global view of what we consider to be anadequate comprehension of the energy concept and its implications. The set of conceptual,procedural and axiological aspects has to be contemplated in its totality, in order to make theunderstanding of this field of knowledge possible.It is our assumption that the students’ difficulties pointed out by numerous researchers (Duit, 1981;Pintó 1991) may be due to the fact that usual teaching practice leaves out a good number of theaspects that have been mentioned above. In order to check this assumption we have analyzed theteaching of energy in high school and have also tested a modified teaching approach thatincorporates explicitly, and in various contexts, all the propositions enumerated above.

3. Textbook analysisIn order to test the assumption that the usual pattern followed to introduce the concepts of energy,work and heat, is not in agreement with the propositions we have just outlined, therebycontributing to students’ lack of comprehension, we have analysed 33 Spanish high-schooltextbooks corresponding to the following levels: 13 textbooks for compulsory secondary education(students aged 14 to 16), and 20 textbooks for the equivalent of A-levels (students aged 16 to 18).We show in Table I some of the results obtained from this analysis.Let us briefly comment upon these results, which have been reinforced with interviews addressedto high-school teachers (Doménech, 2000). One observes the strikingly small percentage (alwaysbelow 20 %) of textbooks that incorporate any of the aspects considered in our analysis. So, in none

% (Sd) 1. Is the development of the topic presented as a strategy in order to solve problems

of interest? 0 (-) 2. Is the knowledge introduced in a meaningful way? In particular, do the operative definitions proposed for the concepts respond to qualitative ideas?

2a. Work 2b. Kinetic energy 2c. Gravitational potential energy

18 (7) 15 (6) 12 (6)

3. Is the dynamic character of the process of construction of knowledge pointed out? In particular, are the limitations or changes in the proposed definition of energy stated?

0 (-) 4. Are situations analyzed where a connection with students’ alternative conceptions

is possible? 4a. Is it stated explicitly that the absolute energy of a system cannot be defined? 4b. Is it pointed out that speaking about the energy of an isolated object is meaningless? 4c. Are the differences between heat and internal energy spelled out? 4d. Is the apparent contradiction between energy conservation and depletion of its resources dealt with?

12 (6)

0 (-) 3 (3)

18 (7) 5. When studying heat phenomena, is the integrating character of thermodynamics

emphasized? 0 (-) 6. Are the advantages and limitations of the energetic treatment pointed out, in

comparison with the dynamic-kinematic treatment, when studying the motion of

objects?

0 (-)

Table I: Analysis of how the concepts of energy, work and heat are introduced in textbooks

of the 33 books is the development of the subject matter associated with the treatment of problemsof interest that need to be solved by scientific means. All these textbooks proceed, with no link tothe problems stated at the onset of the chapter, to introduce the concepts of work and energy. Thisway of introducing the concepts must appear quite arbitrary to students since they cannot evenguess the reason for this procedure, at this initial stage of the discussion.In most cases, the operative definitions of the concepts are introduced ad hoc, as a starting point.This conveys some arbitrariness: why do we define work the way we do, and not, for instance, as theproduct of force and the time during which it acts?If students should acquire a dynamic vision of science, in which the meaning of concepts isconstantly evolving, it is necessary that they perceive this characteristic in the manner in which thescientific concepts are introduced in the classroom, and they must become acquainted with thescientific criteria for acceptability or rejection (Gil, 1996). The energy concept is especiallyadequate in this respect, because it has undergone substantial changes before reaching its presentstatus (Harman, 1982). None of this happens in any of the textbooks examined.Within the constructivistic approach of learning, which we share, question 4 in Table I is essential.Even though constructivistic models of knowledge have oriented different teaching strategies, allof them coincide in the necessity, as a requisite to achieve a meaningful learning, that studentsrelate the new knowledge with their initial conceptions. The scarcity of arguments that textbookssuggest with the purpose of clarifying the intricacies of energy, heat and work, do not facilitate thatstudents achieve an adequate comprehension of these concepts.None of the textbooks point out that, historically, mechanics and the study of heat evolvedseparately, and that it was the integration of both fields which allowed an adequate concept of heatto develop, and to advance in the construction of a more general and powerful body of knowledge.We have found also that none of the textbooks suggest any consideration of the advantages anddisadvantages of using the concepts of energy and work to study the problems that are usuallyattacked with a kinematic-dynamic treatment. Neither do the textbooks discuss the use of theexpression W + Q = ∆E to tackle the study of motion.

4. A new teaching approachWe have developed a new teaching approach that takes into account research in science educationand, in particular, the proposals of science teaching and learning as a guided research. We haveinvestigated the effects of reorganizing the teaching of energy by taking into account theorientations synthesized in the 26 propositions mentioned above.In this perspective, we have prepared programs of activities oriented towards students that learn,as “novel researchers”, under the guidance of the teacher (Gil, 1996). Our aim is to facilitate theattempts by students to tackle problematic situation of interest, related to transformations thatoccur in Nature. We expect students to be able to (re)construct in a significant manner theknowledge about energy, heat and work, that is usually delivered to them as a finished product.We have prepared three working guides (programs of activities) with the following titles:a) A deeper study of changes: introduction of the concepts of work and energy.b) Heat as an agent of change.c) Energy, work and heat: integration of two fields of knowledge.37 class hours were spent in developing this program. The detailed “research programs” can befound in (Doménech 2000). They have been tested both with high-school students and with in-practice teachers.We have compared the learning achieved by an “experimental” group of students with thatobtained by a “control” group. We have performed “surprise” tests a few weeks after completion ofthis program, and we have also checked the long-term recall of both groups of students.The results obtained are quite promising (Doménech 2000), in that students make noticeableprogress both in the quality and duration of their learning and in the abilities developed to addressnew situations. In particular, those students that have worked through the program of activities



take into account in their analyses the relevant aspects of the concepts of energy, work and heat,and its mutual relationships. Also, the arguments they use are more elaborated than those used bythe students in the control group.

5. Conclusions and perspectivesThe overall conclusions of our study of textbooks and student performance with the currentteaching materials and practices (Doménech 2000) are these:1) Generally speaking, teachers do not introduce the concepts of energy, work and heat as

hypothesis put forward with the aim of solving problems.2) A meaningful introduction to the concepts is seldom made. In particular, it is not shown that

the operative definitions of the concepts arise from certain qualitative ideas. Nor the limitationsencountered by the proposed definitions of the concepts are ordinarily exposed.

3) Situations that favor a connection of students’ alternative conceptions to questions beingstudied are not frequently introduced.

4) The integrating and universal character of the energy concept is not stressed.5) Students are not motivated to appreciate the advantages that the energetic treatment has, over

using exclusively the dynamics and kinematics treatment.All these facts back up our initial assumption that the difficulties shown by students in thecomprehension of energy and related concepts may be due, at least partly, to deficiencies in theorientation of their teaching, like the ones we have detected.The development of new teaching materials, presented and worked out as guidelines for a guidedresearch towards development of the concepts needed for solving certain scientific problems, helpsto remedy this situation.

ReferencesBlack P. and Solomon J., Life-World And Science World- Pupils’ Ideas About Energy, Entropy In The School, 1,

Roland Eotvos Physical Society. Budapest, (1983).Doménech J.L., L’ensenyament De L’energia, L’educació Secundària. Anàlisi De Les Dificultats I Una Proposta de

millora, PhD. Thesis. Universitat de València, Spain, (2000).Driver R. and Warrington L., Students’ Use Of The Principle Of Energy Conservation In Problem situations. Physics

Education, 20, (1985), 171-176.Duit R., Understanding Energy As A Conserved Quantity. European Journal Of Science Education,. 3, (3), (1981),

291-301.Duit R., In Search Of An Energy Concept. En Driver, R.And Millar, R. (Eds), Energy Matters. University Of Leeds,

(1986).Gil D., New Trends In Science Education. International Journal Of Science Education, 18, (8), 1996, 889-901Harman P.M., Energy, Force And Matter. The Conceptual Development Of Nineteenth-Century Physics, (Cambridge:

Cambridge University Press), (1982).Koliopoulos D. and Ravanis K., L’ensegnement De L’energie Au Collège Vu Par Les Enseignants. Grille d’analyse

de leurs conceptions. Aster 26, (1998), 165-182.Nicholls G. and Ogborn J.,. Dimensions Of Children’s Conceptions Of Energy, International Journal Of Science

Education, 15, (1993), 73-81.Pintó R., Algunos Conceptos Implícitos En La Primera Y Segunda Leyes De La Termodinámica: Una Aportación Al

estudio de las dificultades de su aprendizaje, PhD. Thesis. Universitat Autònoma de Barcelona. Spain, (1991).Solomon J., Learning About Energy: How Pupils Think In Two Domains, European Journal Of Science Education, 5,

(1983), 49-59.Solomon J., Teaching The Conservation Of Energy, Physics Education, 20, (1985), 165-170.Trumper R., Being Constructive: An Alternative Approach To The Teaching Of The Energy Concept, Parts 1&2,

International Journal of Science Education, 12, (1990) 343-354, 13, (1990), 1-10.Watts D.M., Some Alternative Views Of Energy, Physics Education, 18, (1983), 213-217.

OBSTACLES TO THE DEVELOPMENT OF CONCEPTUAL UNDERSTANDING INOBSERVATIONAL ASTRONOMY: THE CASE OF SPATIAL REASONING DIFFICULTIESENCOUNTERED BY PRE-SERVICE TEACHERS

Ch. Nicolaou, C. P. Constantinou, Learning in Physics Group, University of Cyprus, Cyprus

1. IntroductionLearning in Physics is a complex and multidimensional enterprise which can be analyzed into anumber of constituent components: the acquisition of experiences with natural phenomenaprovides the basis for the subsequent development of concepts; the mental representation of thestructure of organization of scientific knowledge that is needed to avoid knowledge fragmentationand meaningless use of jargon comes with the development of epistemological awareness; scientificand reasoning skills provide the strategies and procedures for making operational use of one’sconceptual understanding in order to analyze and understand everyday phenomena but also toundertake critical evaluation of evidence in decision making situations. Finally, positive attitudestowards inquiry feed student motivation and safeguard sustainable engagement with the learningprocess.Traditionally, our educational systems fail to connect these components into a coherent learningparadigm for physical science. The Learning in Physics Group at the University of Cyprus conductsa coordinated program of research, curriculum development and teaching, which fundamentallyrelies on the premise that real learning can only emerge when all these components are promotedin unison.This article is part of an on-going research program, through which we aim to investigate theconceptual and reasoning difficulties that pre-service teachers encounter when guided to constructa model for the relative motion of the sun and the earth, which is capable of offering detailedexplanation of the phenomenon of day and night. In this part of the project, we recorded theprevalent initial ideas of undergraduate students on the day-night cycle and then exposed them tothe Astronomy by Sight module in Physics by Inquiry [1] We subsequently investigated weather thestudents had developed an appropriate model of the sun-earth relative motion that could accountfor the day-night cycle.

2. The day-night cycleOn a daily basis, we all observe and feel the consequences of the day-night cycle [2] However, onlya small percentage of us can explain in an appropriate manner how this happens, probably becauseof the complexity of the phenomenon itself or of the inherent difficulty in the process of modelingand understanding this phenomenon.• The apparent motion of the sun in the sky.In this approach, we assume that an observation record of the apparent daily motion of the sun in

the sky is a pre-requisite to the construction of a model for the day-night cycle. Rise and settimes, the altitude and the direction of the sun in the sky are important data which need to beincluded in the observation record. Such data can serve as a basis for eliciting students’ initialideas and supporting the process of model construction through negotiated reformulation andevolution of those ideas. The same observational records can also serve as instruments forevaluating the validity of competing models.

• An observation record of the apparent motion of the sun in the sky.Shadow plots can be used as convenient records of the apparent motion of the sun in the sky.From this we can infer the direction of the sun in the sky at any one instant. In order to constructa shadow plot we need a flat planar board with a nail to act as gnomon placed perpendicular tothe plain of the board. This is the shadow plotting board. We place a sheet of paper on the boardso that the nail sticks up roughly through the middle of the paper. We tape the paper onto theboard so that it cannot move and we note the height of the nail and the date on the paper. It isalso important to state on the paper the locations of several nearby landmarks so that we can



subsequently identify the orientation of the paper. During the day we record the shadow of thetip of the nail at intervals of approximately half an hour. The locus of the shadows is directlyrelated to the apparent motion of the sun in the sky. Repeated daily observations of thephenomenon provide important additional information as to the time variation of the apparentmotion of the sun.

• Appropriate accounts of the relative motion of the sun-earth system (two alternative, equivalentmodels)The information that arises from analysis of the shadow plots and the direct observations of thelearner form the basis for the construction of two alternative, equivalent models, each of whichis acceptable as an explanatory account of the mechanism of formation of the day-night cycle.In each model the system consists of two bodies: the sun and the earth.Model 1According to this model, the day-night cycle results from a possible spin of the earth around itsown axis at a rate of one complete revolution every 24 hours. In this model, the sun is stationary.Any one point on the earth’s surface faces the sun for 12 hours, the time period needed for halfa revolution. During this time interval, this point has daylight. For the twelve hours remaining tocomplete one revolution, this point on the earth’s surface does no see the sun and experiencesnighttime. At every instant in time, half the earth’s surface experiences day and the otherhemisphere experiences night.Also at any instant, all points on the earth’s surface that fall alonga great circle through the poles experience transition from light to dark or vise versa as they arerotating into the dark or the light, respectively.

Model 2 According to this model, the day-night cycle is a result of the rotation of the sun around theearth once every 24 hours. In this model the earth is stationary at the center of the suns circularpath. At any one instant in time, half of the earth’s spherical surface is oriented towards the sunand experiences day. The rest of the surface, which does not face the sun, experiences night. Apoint on the earth’s surface has 12 hours day and 12 hours night, the time periods thatcorrespond to one half of the sun’s rotation.

It is important to note that both models can be modified in order to introduce a slant to thespinning axis in order to account for the deviation from equal 12 hour intervals for day and night.It also important to note that based only on observations of the sun-earth system, both models 1and 2 can explain all of the observations (such as those recorded on a shadow plot) and thereforeneither of can be rejected. The two models can also adequately explain all other the personalobservations related to the day-night cycle. The equivalence of the two models only breaks downwhen a third (astronomical) object is entered into the system.The equivalence of the two models (in the context of the two-body system) and as a consequence,their equal validity in explaining shadow plot observations are major epistemological revelationsfor students in typical astronomy classes.This happens for a number of reasons: students often tendto judge the validity of a model with respect to its correspondence to what they or an expert knowsand not in relation to whether it can account for their observations or not. Students also fail todifferentiate between a model and a phenomenon; hence they find it difficult to accept that twomodels can be valid at the same time.Finally, students also encounter conceptual and reasoning difficulties (with respect to model

equivalence and reversibility in relative motion, respectively) in their effort to understand thedifferent mechanism underlying the two models.

3. Methodology • Population

The research was conducted at the University of Cyprus during the spring semester 2000. Thedata was collected in the context of a course on Physical Science in the Elementary Grades

attended by 82 Students enrolled in the Primary Education Program at the University of Cyprus.The course used the Greek version of Physics by Inquiry with special emphasis on the moduleObservational Astronomy: The sun, the earth and the stars. Data was collected through a series ofpre-tests administrated throughout the semester at the beginning of every section in thecurriculum.•The pre-test In one of the pre-tests, given roughly half way through the semester, the students were asked torespond to the following question:“The sun rises roughly in the east and sets roughly in the west. Therefore, someone located insidea spaceship hovering over the North Pole will observe the earth spinning counter-clockwise. Doyou agree or disagree with this statement? State clearly whether you think this statement iscorrect or not and explain your reasoning. You will find it helpful to include a diagram in youranswer.”Only a small percentage of the students were able to identify that the statement is true andexplain their reasoning. A typical correct response might include the following:

Suppose that someone is positioned at location K on the earth’s surface. Consider also that s/he islooking towards the North Pole. If s/he points her/his right arm towards her/his right parallel to theground, s/he will be pointing eastwards. On the other hand, if s/he points her/his left hand towardsher/his left parallel to the ground, s/he will be pointing westwards. If the earth is spinning aroundits axis counterclockwise, when observed by another person over the North Pole, the person atlocation K will see the sun rise from an easterly direction in the morning and set in a westerlydirection in the evening.


: The face of a person as observed from the front

N.P.: North Pole

The drawing is not to scale

SUN .

earth

D

N.P.

Top view diagram

An alternative way to explain this phenomenon is the following. According to figure 1, country Ais located to the east of country D. Country D is located to the west of country K. So, if earth isspinning around it’s axis counterclockwise, when observed over the North Pole, the sun will firstappear in the horizon of country A, then country K and then country D. Thus the sun is rising firstin the easternmost of the three countries, as it is observed in real life.Since the pre-test was given prior to the intervention both of these responses were deemedappropriate and where evaluated as correct.

4. Data analysisPhenomenographic analysis was used to categorize the student responses. Subsequent qualitativeanalysis of the responses in each category revealed the models which students used in theirattempts to respond to the question and the difficulties they encountered in the process. Thefollowing 4 difficulties [3] were identified through phenomenographic analysis of the student’sresponses:1. Many students interpret the geographical directions as absolute locations or points in space.2. Many students fail to distinguish the concepts clockwise and anti-clockwise1.

Figure 1: The rotation of the earth around its axis as a mechanism for formation of the day-night cycle

1 In Greek language clockwise rotation is called a right sense rotation and anti-clockwise rotation a left sense rotation.


3. Many students orient the four directions erroneously with respect to each other.4. Many students fail to appreciate the manual reversibility of the relative motion in the two

equivalent models.Evidence for difficulty 3 usually appears in diagrammatic representations. One example is shown

in figure 2We have evidence from other data that this difficulty is relatedto spatial reasoning. However, for the sake of conciseness wewill not discuss it further.According to the fourth category, pre-service teachers do nothave “reversibility” in their thinking when they present twoalternative models for the day-night cycle. They often mentione.g. that it is the earth, and not the sun, that moves, and then, inthe same response, proceed to include a diagram indicatingmotion of the sun around the earth. Other answers refer tosimultaneous motion of the sun and the earth. Other studentsdescribe the day-night cycle as a result of the rotation of eartharound the sun. The ability to reverse two alternative models is

indirectly connected to spatial reasoning ability, mainly as far as spatial rotations in space areconcerned.Category 1 and 2 will be analyzed in greater detail below.• Absolute interpretation of the four directions in spaceStudents tent to interpret the four directions (north, south, east, west) as absolute locations in space.They believe that the directions north, south, east, west are permanent points, which are firmly locatedon the earth or in space.They do not conceive them as directions that change in respect to the positionof the observer.According to student’s answers, 59.5% (44/74) of pre-service teachers encounter this difficulty.Figure 3 shows the diagram sketched by student 4. Student 4 clearly presents East and West aspoints to the left and right side of North Pole respectively.

E

N

S

W

Figure 2: Erroneous representationof the four geographical directions.20% of the students in our sampleencountered this difficulty.

E

W

N

S

N WE

E: East W: West N: North Pole S: South Pole

earth

Figure 3: Diagram sketched by student 4 to explain the rotation of earth around it’s axis.

Student 4 obviously does not consider the position of the observer (points 1 or 2 in figure 4) asrelevant to the model. So, if the observer is at point 1 or at point 2 and look towards the north pole,east will be for observer 1 at her/his right hand side and for observer 2 at her/his left hand side.Consequently, country X (figure 4), which is visible by both observers 1 and 2, is located to the east.And, to take this thinking further, if for observer1 the sun rises at 6 a.m. from the east and movesto the West, by noon, the sun will rise for observer 2 from the west!

According to student 15:“The earth is turning around itself during the day. The sun is eastward in the morning and westwardin the evening. So, someone in a spaceship, which hovers over the North Pole, will observe the earthspinning counterclockwise.”


E

W

N

S

1

2

BD A

earth

Figure 4: Predictions based on the response of student 4. The letters N, S indicate the North and South Polerespectively. The letters E and W indicate easterly and westerly directions respectively.

Earth

East-morning West-night

W

S

E

Figure 5: The diagram drawn by student 15 for the earth’s spin around it’s axis.

East-morning

C

West-night

W

N

S

E

Figure 6: Predictions based on the answer of student 15.

The response of student 15 leads us to the following thinking: in figure 6 it is morning for Cyprus(C). For another country X, or a point on the surface of the earth, which is positioned diametricallyopposite to Cyprus, the sun is in the East. Simultaneously, according to figure 6, it is local noon forCyprus and midnight for the other country (X). The diagram of the four directions (figure 5)reinforces the idea that student 15 interprets the four directions as absolute locations or as pointsin or out of the earth.


The fact that students tent to interpret the four directions as absolute locations or points on or outof earth may occur because in Greek Language they are labeled as the four points of the horizon,a devious term as it implies four points on the earth or in space. This notion is probably reinforcedby the educational system through the lack of distinction between the north and south directionsand the North and South Pole as points on earth!The first conclusion of the study is that an understand that the four directions of the horizon arenot points on or out of the earth, but directions in space, is an important pre-requisite to developingunderstanding of the day-night cycle.The identification of the four points of the horizon as points inside the earth or in space is offundamental importance and its resolution should be considered as a prerequisite to the teachingof simple astronomical phenomena, such as the cycle of the day and night.• Differentiation of clockwise and anticlockwise rotationStudents believe that the concepts clockwise and anticlockwise are absolute and are defined in respectto right and left. Specifically, they cannot distinguish between rotational relations and relations due tothe perspective from the locations of two objects.According to the answers of the students in the pre-test, 39.2% (29/74) of preservice teachersencounter this difficulty.In the Greek language, the clockwise rotation has the meaning of a right sense rotation and anti-clockwise rotation the meaning of a left sense rotation.According to student 15 (figure 5):“The earth is turning around itself during the day. The sun is eastward in the morning and westwardin the evening. So someone in a spaceship, which hovers over the North Pole, will observe the earthspinning counterclockwise.”Student 15 draws an arrow, which describes a clockwise rotation. However, s/he mentions that therotation is anticlockwise.Student 31 wrote:«The sun is stationary. I am standing on a point on the surface of the earth and I am moving to the

left (not me but earth), the sun will be on my right hand side, that is, westward from the place I amstanding. So, the sun rises in the east and sets in the west because of the earth’s anticlockwise rotation.» Student 31 believes that an anticlockwise rotation is one in which an object (in this case the earth)moves towards left. According to this opinion and based on figure 7, moving towards the left iscaused both by a clockwise and an anticlockwise spinning of the earth.

Left Right (when I look at the sheet) Earth

Figure 7: Predictions based on the response of student 31.

Student 48 states the following:«I agree with the above statement. The earth spins counterclockwise. This is clear from the followingdiagram. The earth spins counterclockwise so that the sun sets due west. » Student 48 also drew the following sketch (figure 8).The analysis of the explanation of student 48 showed that s/he encounters the two difficulties,which are analyzed in the present article.

According to figure 8, the student believes that the position of the sun in space is east. So, s/hethinks that east is a point in space. S/he does not state the position of the observer, for whom at thisspecific moment, the sun is due east. Moreover, while in figure 8 the arrows indicate a clockwiserotation, at her/his explanation, s/he refers to an anticlockwise rotation.According to the data of pre-test 3 many students cannot distinguish between clockwise andanticlockwise rotation. This happens probably because of the distinctiveness of Greek language.Clockwise in Greek is and is a compound word. The word means “right”and the word «______» means “rotation”. Similarly, anticlockwise in Greek is «______________»and is a compound word. The word «________» means “left” and the word «______» means“rotation”. So in Greek the concepts clockwise and anticlockwise are conventional. Often, studentsdo not understand those concepts and even though they refer to a clockwise rotation they draw ananticlockwise rotation and vise versa. There are also students who interpret clockwise rotation of abody in an absolute way. They believe that a rotation is always only counterclockwise or onlyclockwise despite of the position of the observer.The difficulty has to do with the fact that studentsthink that a body, which turns left is spinning counterclockwise and a second one, which turns rightis spinning clockwise. Moreover, students tent to define the two concepts in connection to front andin confrontation to backwards. Someone spins clockwise or counterclockwise if he/she turns fromhis/her nose to his/her right or left shoulder, respectively. They do not trace differences indescription of a motion for different positions of the observers and cannot accept the fact that aclockwise rotation as it is seen when an observer is over a spinning body is counterclockwise asviewed by an observer under the body.

5. Developmental and structural analysis of the curriculum materialThe Learning in Physics Group at the University of Cyprus is revising the program Physics byInquiry so that it responds to the conditions and restrains of the Greek Educational System. Theunderlying aim is to safeguard the development of conceptual understanding, thinking and otherabilities relative to the process of learning in physical science. Physics by Inquiry contains narrative,experiments and exercises, and supplementary problems at the end of each module. Through in-depth study of simple physical systems and their interactions, students gain direct experience withthe process of science. Starting from their own observations, they develop basic physical concepts,use and interpret different forms of scientific representations, and construct explanatory modelswith predictive capability. Physics by Inquiry is explicitly designed to develop scientific reasoningskills and to provide practice in relating scientific concepts, representations, and models to realworld phenomena.The aim of the module Astronomy by Sight, according to McDermott (1996) [4] is for students tomake observations of the motion of the sun, the moon, and the stars in the sky, to identify patternsin the changes that occur during the course of a day and a month and to develop models that enablethem to determine the present, past and future appearance of the sky.Physics by Inquiry: Astronomy by Sight is based on a sequence of activities which aim to guide


Sun is due east

Earth

E

N

S

W

Figure 8: Diagram of student 48 of the spinning of earth around itself.

« » « »« »

« » « » « »


students to overcome the difficulties identified through research, so that real conceptualunderstanding is achieved for the majority of students. Below we will present a brief structuralanalysis of the module Observational Astronomy (sections 1-5).

Section 1Section 1 starts with the record of student’s observations for the apparent motion of the sun in thesky during the day using newly constructed shadow plots [5]. Prior to their first shadow plot,students make predictions of how it will look before they construct it. Then, they make and studytheir plot in comparison with their predictions. Based on the shadow plot, they act out andrepresent the apparent motion of the sun in the sky during the day. At the end students develop anoperational definition[6] of the concept “local noon”

p

Shadow Plot

Apparent Motion of the Sun

Measurements of the sun’s altitude

Consideration: the sun rays reach earth parallel

The Shape of Earth

Estimation of the Circumference of the Earth-Eratosthenis’ Method

Measurement of the sun’s azimuth

Scientific Model of the Motion of the Earth and the Sun

Section 1 «Sun Shadows»

Section 2 «Observing Changes in the Sky»

Section 3 «The Size and Shape of the Earth»

Section 4 «Daily Motion of the Sun»

Record of Moon Observations

Patterns of Moon Shapes

Relative positions of Sun-Moon according to the Phase of the Moon.

Scientific Model of Motion of the motion in the System of Earth-Moon-Sun.

Section 5 «Phases of the Moon»

Diagram of the epistemological structure of the curriculum material

Figure 9: Brief structural analysis of Physics by Inquiry: Observational Astronomy, which was designed to guidestudents to overcome specific difficulties.

Section 2Section 2 begins with directions for students to measure the altitude of an object with the “fist-over-fist” process [7], they develop operational definitions of the concepts “horizontal”, “vertical” and“altitude of an object” and they take measurements of the altitude for different objects. Thenstudents estimate the altitude of the sun for different moments in time during a day using theirshadow plots and they come to the conclusion that sunrays reach earth parallel. Finally, discusswhat they know about the moon distinguish between their observations, for which they have directevidence and those facts they know because they were told them or read them. This is done so thatthey recognize the degree of confidence that our own observations give us and the critical attitudewe must have towards anything that is offered as given knowledge.

Section 3The third section has to do with the shape of earth. Firstly, students discuss the disagreementbetween the measurement of the altitude of a mountain peak by a person who assumes a flat earthand the estimation of its altitude using data from a map.Then they try to explain why some celestialbodies are visible from north and not from south regions of the earth. Based on these observationsand exercises, they come to the conclusion that earth’s surface is not flat, but curved. Then theyestimate the circumference of the earth with Eratosthenis’ Method and assuming a spherical earth.

Section 4Section 4 starts with the comparison of two or more shadow plots (spanning a period of 3-4 weeks)as far as the general characteristics are concerned with special emphasis on the shortest shadow.Then students define the “azimuth” [8] and estimate the azimuth of several objects. The sameprocess is used for the estimation of the azimuth of the sun through a shadow plot for differenttimes. Then students construct a graph of the azimuth versus time of day, and they discuss the formof the graph and how it represents the observations. Consequently, students construct the scientificmodel of the motion of sun and the earth, based on which they explain their observations asrecorded by the first shadow plot. Finally, they use a compass [9] to find the magnetic declination[10]. Students discover through an exercise two ways of geographical orientation (use of compassand through a shadow plot).

Section 5Before section 5, students observe the Moon for a month and they record their observations in arelevant sheet. At the beginning section 5 students group and order chronologically theirobservations on a “Moon Observation Summary Chart (MOSC) [11]. Based on MOSC, studentslook for patterns in the behavior of the moon, which repeat in a period of a day, week or month,and record the phase/form of the moon for this pattern during the synodic period [12]. Then, basedon their observations, they trace the position of the moon in relation to the position of the sun, andthe direction of the moon for different moments. Consequently, they construct a scientific modelfor the relevant motion of the bodies in the system sun-earth-moon, based on which they canexplain the cycle of moon phases. Students study the sun-moon angle during a synodic period ofthe moon and estimate the time at which the moon rise and set for its different phases. Finally, theydiscuss the relative distances between sun, moon and earth and the possible consequences whichderive when we alter them in the scientific model we constructed. At the end of the section,students try to construct alternative model that explain their observations.

6. DiscussionA basic conclusion deriving from this research is the existence of reasoning difficulties, whichcomplicate student’s efforts to construct understanding in Astronomy and Physics. There arecertain common difficulties that many students encounter that should be made explicit and whereappropriate confronted in a learning environment so that conceptual understanding is achieved.Very often reasoning difficulties are not identified either by the students or by the teachers andtherefore remain in the conceptual ecology of the learners and affect or even determine the



learning process. The design of curriculum should include the development of strategies andactivities that encourage students to express their opinion. Through the expression of studentsopinion difficulties come up and enable learners and teachers to discuss them openly. Pre-tests likethe one presented in this article can contribute much as instruments of revealing thinking and otherdifficulties.On the other hand, the results of this research indicate the importance of integrated developmentof conceptual understanding and scientific reasoning as well as other abilities closely connected tolearning in physical science. Moreover, in physical science there is a need to emphasize more thedevelopment of connections between the formal information, the epistemological structure of thesubject, the means of representation of the information and the real phenomena. Physics by Inquiryis a program in which students themselves make observations and use them in combination withspecific reasoning patterns and concepts that they formulate to construct conceptual models withpredictive capability. The models are gradually refined through application and furtherobservations of related but more complex systems. As a result of this process students emerge withexperientially developed intuitions of inquiry as the process of science, which is constructedaccording to clear epistemological criteria. In contrast to traditional instruction, the method ofinquiry does not emphasize on the transmission of information, which if preoccupied, stayfunctionally unutilized for the explanation of everyday phenomena and their application indecision making.

References[1] L. C. MacDermott and the Physics Education Group, Physics by Inquiry, J. Willey, New York, (1996).[2] M. Summers, J. Mant, A survey of British primary school teachers’ understanding of the Earth’s place in the

universe, Educational Research, 37, (1), (1995), 3 – 19. A. Lightman, Ph. Sadler, The earth is round? Who areyou kidding?, Science and Children, (1988), 24 – 26. St. Vosniadou, Designing curricula for conceptualrestructuring: Lessons from the study of knowledge acquisition in astronomy, J. Curriculum Studies, 23, (3),(1991), 219 – 237. St. Vosniadou, Mental models of the earth: a study of conceptual change in childhood,Cognitive Psychology, 24, (1992), 535 – 585. J. G. Sharp, R. Bowker and J. Merrick, Primary astronomy:conceptual change and learning in three 10 – 11 year olds, Research in Education, 57, (1997), 67 – 83.

[3] L. C. McDermott, Millikan Lecture 1990: What we teach and what is learned – Closing the gap. AmericanJournal of Physics, 59 (4), (1991), 301-315.

[4] C. McDermott and the Physics Education Group, Physics by Inquiry, J. Wiley, New York, (1996).[5] Shadow plot is the diagram, which is created by the changing of the track of the top of the shadow of an object

caused by the sun during the day.[6] Operational definition is a set of instructions which allow anyone to measure the specific dimension, identify an

example of the concept or alter it for others. An operational definition cannot be misunderstood.[7] Fist over fist process is a method of measuring angles in space. The method is based on counting of our fist in

degrees, having our arm stretched in front of us.[8] Azimuth of an object is the angle measured clockwise from north, as viewed for above) to the object.[9] Compass is the scientific instrument with which we orient, as it points always toward magnetic North.[10] Magnetic declination is the angle between magnetic north and the true north.[11] Moon Observation Summary Chart (MOSC) is a chart on which we record daily, for a specific period, moon

observations.[12] Synodic period of the moon is the cycle from the new moon, through the full moon and back to the new moon

in about 30 days. Local noon is the time of the day for which the sun is located at its highest position in the sky.e-mail c.p.constantinou @ucy.ac.cychristiana78 @hotmail.com

A STUDY ON THE COMPETENCE AND PROFESSIONAL DEVELOPMENT OFSCIENCE/PHYSICS TEACHERS IN ANTALYA PROVINCE OF TURKEY

Isik, S., Ustuner, Akdeniz University, Antalya, TurkeyYasar, Ersoy, Middle East Tech. University, Ankara, Turkey

1. IntroductionThe improvement of both science/physics education in schools, and of the professionaldevelopment of teachers has been points of interest and important issues in almost all developedand developing countries [1-3]. In recent years, the findings in this particular area of studies havegained impetus, enlightened several issues, and guided both decision-makers and educators [4-6].Thus, for all countries, independent of the fact that they are developed or not, there have beenvarious contemporary problems waiting to be resolved and handled by the researchers andscience/physics educators. In the present study, we overview a group of science/physics teachers’views on their competence and needs for the professional development. Before introducing thedetails of the study, we like to briefly give general information about the Turkish education systemand the initial training (pre-service education) teacher program.The Turkish education system is based on a three–tiered principle (primary, secondary andtertiary/higher education). The primary level encompasses primary schools (lower primary stage)and junior high schools (upper primary stage) and provides basic education for children from 6 to13 years of age (Grades 1–8). School attendance at primary level is free of charge and compulsoryfor all children. Secondary education is available for pupils’ aged 14 to 17 and is provided invocational, technical and general/academic high schools. These secondary schools qualify studentsfor entry into higher education[7,8]. Therefore, there are some differences within the curricula ofthe initial or pre-service of science and physics teacher education and training (INSET) and withinthe higher education institutions. In the present study, we consider a group of science/physicsteachers’ views and competence in the upper primary/middle school, i.e. the junior high schools ina province of Turkey, namely Antalya.The pre-service (initial training) education and INSET should be complementary, andcompromised by the designed programs. In Turkey, the organization, rules and the syllabi of theinitial teacher-training program have been revised for many times in the last fifty years. Accordingto the new rule, the secondary school physics teacher has four year study in the Science Faculty(SF) at university.After graduation from the SF (Physics, Chemistry or Biology) they continue theirtraining in a post graduate program which is considered to be equivalent to master program orfollow a set of didactic courses (35 credit-hours) in three semesters. The science/physics teacher ofprimary school has 4 years’ of studies in the Education Faculty (EF), where he/she getsknowledge and skills in both discipline related and professional courses. Before 1982, the ruleswere, however, different, and to become a science/physics teacher one had to get a diplomafrom Egitim Enstitusu (EE), which was a three years higher education school for trainingteachers. The EE is affiliated with the Ministry of National Education (MEB) and, following achange of the law of higher education, each of them became EF in 1982. Now there are morethan 40 EFs in Turkey, and train prospective teachers and have the same syllabi in pre-serviceeducation except the elective courses.

2. MethodologyMain and Sub-Problems: In Turkey, we have faced many issues and several problems related toscience/ physics education in general, and teacher education and training in particular. Therefore,we search for science/physics teachers’ needs for the professional development, and attempt todesign more efficient and effective INSET for them. Therefore, we decided to find out the answersto three main questions below and test the interrelated hypothesis.• P1: What is the professional experience of the science/physics teacher in the upper primary

school in Turkey and their needs for the professional development?



• P2:What is the professional (in)competence of the science/physics teacher in the upper primaryschool in Turkey and their needs for the professional development?

• P3: Do teachers’ personal characteristics affect professional experiences and (in)competence?Hypothesis: The present study aims at to discuss the science/physics teachers’ views on pre–serviceeducation, and to present and exchange our experiences to improve the current situation withrespect to age, seniority, gender and type of diploma. In this respect, following two hypothesesH0(1), H0(2) are considered and, will be tested.• H0(1): There is no significant difference between various personal characteristics (i.e. age,

seniority, gender, type of school graduated, etc.) and professional experience of teachers.• H0(2): There is no significant difference between various personal characteristics (i.e. age,

seniority, gender, type of school graduated, etc.) and physics competence of teachers.Instrument: To reflect the practicing teachers’ needs, views and suggestions the data have beengathered by using the Likert-type scale. The scale, i.e. the instrument for the study, was developedin EUPEN Project [1], but translated and adapted to Turkey by the researchers. The questionnairewas delivered to 392 science/ physics teachers who work in the upper primary/secondary schools inAntalya, i.e. a province in the southern part of Turkey. The number of replies was 265, i.e 68% ofthe teachers filled in the questionnaire and returned. The researchers received a great help fromthe Directorate of the MEB in Antalya to collect the data and make the interviews. The gathereddata were then analyzed in PC-SPSS program, and some results and findings are presented below.The reliability of the questionnaire, i.e. RK is relatively high, i.e α = 0.96.

3. Analysis of data and resultsSome information about the background of the sample, i.e. the group of science/physics teachers,and the analysis of the gathered data is presented below.

3.1. Demographic InformationGraduation: Among the respondents to the administrated questionnaire, we found that 157 (59 %) ofthe science teachers were male (M), 105 (40%) were female (F), and 4 of the respondents’ gender isnot stated. About half of the teachers 136 (51%) teach in the city of Antalya, and 129 (49%) teach inthe suburbs, i.e. the various towns of Antalya.About 60% graduated from EE, while 40% from EF/SF.After the graduation or during their study in EF/SF the teachers took various didactic courses and hadthe qualification in teaching science in the upper primary school or junior high school.Age and Seniority: The average of the seniority/experience of the science teachers is about 17years, most of the teachers’ age is between 40 and 49, and have less then 20 years experience. Forexample, only 2% of the science/physics teachers are older than 50 years. This means that almostall teachers more than 50 years old retire and do other jobs instead of working in public schools.First Appointment and Status: Most teachers (70%) taught science/physics courses when theystarted their first appointment. Remarkably 95% have permanent position, which is not obviousfrom the seniority distribution. This means that almost all teachers work in public schools and gettheir salary from the government, not from the owner of private schools. They have permanentposition of being teacher, and do not have an unemployment problem even though there is not avacant position.

3.2 Professional Experiences of Teachers and Use of Various StrategiesThe summary of the gathered data is displayed in tables. Table 1 shows how the respondents use“Methodological strategies”, “Plan their teaching”, “Assess their students” and “Survivaltechniques in case of difficulties”. According to the analyzed data, two thirds of the teacher makesuse of experiments in teaching science, but the frequency is not mentioned here. We think that theexperiment is done by demonstration purpose and not done frequently. More teachers (%82) statesthat they use “A problem-base teaching approach” and (80%) “Lecture to class”. However, themeaning of ‘a problem-based teaching approach’ here was not the same as the teacher perceivedit. Instead of a problem-based teaching they usually do drill and practice, and exercise basic skills

in solving closed-ended problems in the textbooks. “The use of concept maps”, “Visual material”and “Use of computer” are mentioned less. This means that the teacher is not using moderntechnologies, in particular information and communication technology and new methodologicalstrategies in teaching science. Moreover, about 43% of the respondents frequently plan their“Teaching strategy alone”. About half of the respondents ask for help in the “Class council andtheir colleagues”. “Diagnostic assessment” (23%) and “Summative assessment” (8%) of thestudents’ performances is used less than “Formative evaluation”.

On an individual base there are, of course, some differences in certain items of Table 1 for thegroups of the teachers in Turkey and between teachers in Turkey and those in the countriesparticipated in the EUPEN Project. In Turkey, this difference appears when a teacher hasdifficulty, he/she refers first of all to a book or journal (86%), and the interior resources (55%).Only 7% refer to the headmaster and 11% to the experts (e.g. consultants at the Ministry ofEducation). Unfortunately, there are lacks of resources for the professional development ofscience/physics teachers in Turkey.Therefore, there is an urgent need for the periodicals and severalbooks for the professional developments of the science/physics teachers.

3.3 Subject-Knowledge Competence of Science/Physics TeachersTable 2 lists the usefulness of disciplinary knowledge for teaching competence. The usefulness isscored in an interval between 1 (less) and 5 (more). If the mean of scores (X) is more than 4.0, thedisciplinary knowledge is considered to be absolutely necessary and useful. In the second columnfrom the right of Table 2 one can find the average value of the point of view of the science/physicsteachers on the usefulness of the subject-knowledge.


Scoring: ( Frequently: 3 point; Seldom: 2 point; Never: 1 point)

Statements

Nev

er (

%)

Sel

dom

(%

)

Fre

quen

tly

(%)

Om

itted

(%

)

Mea

n, X

Std

Dev

.,S

1. Didactic strategies used by the teachers

1.1 • Experiments in the school laboratory 3 28 66 3 2.7 0.50

1.2 • Problem-based teaching 1 10 82 7 2.9 0.33

1.3. • Concept maps (use and/ or construction) 28 33 11 29 1.9 0.70

1.4 • Computers (PCs) 57 9 3.4 28 1.2 0.5

1.5 • Demonstration 3 26 62 9 2.6 0.54

1.6 • Lecturing to class 1 12 80 7 2.9 0.36

1.7 • Visual material 6 40 37 17 2.4 0.61

2. Teaching strategy planed by the teachers

2.1 • Alone 5 26 43 26 2.5 0.6

2.2 • In a class council 2 33 59 6 2.6 0.5

2.3 • With colleagues of your discipline 5 25 51 19 2.6 0.6

3. Assessments the teachers make use

3.1 • Diagnostic 17 40 23 20 2.1 0.7

3.2 • Formative 5 29 55 11 2.6 0.6

3.3 • Summative 49 10 8 33 1.4 0.7

4. In difficult moments (pedagogic, disciplinary, etc) teachers address themselves to

4.1 • Colleagues 2 43 43 12 2.5 0.5

4.2 • Books and / or journals 0 9 86 5 2.9 0.3

4.3 • Your own interior resources and reasoning 2 20 55 23 2.7 0.5

4.4 • Experts 27 33 11 29 1.8 0.7

4.5 • Headmaster 35 22 7 36 1.6 0.7

Table 1. Percentage (%) of Scores, Mean (X) and Standard Deviation (S) of Science/Physics Teachers’ Use ofDidactic and Teaching Strategies, Assessment, etc


The means of the scores assigned by the group of upper primary school science/physics teachers to thedifferent items are the following. They are “Usefulness of basic physics (4.3)”; “Modern andcontemporary physics (4.0)”; “Relations between physics, technology and society (4.0)”, “Problemsolving strategies (4.2)” and “Aspects of experimental research (4.5)”;“Other science at the basic level(4.0)”. It is noticeable here that the teachers gave more emphasis on several aspects of experimentalresearch (namely, Item 9.1-9.4). The same group of teachers, however, gave less emphasis andconsidered uselessness and omitted answers to some items in the questionnaire. More specifically, thepercentages are the following: “4: Knowledge of the history of physics (14%)”; “5: Epistemology ofphysics (25%)”; “3: Critical analysis of contents and conceptual structure (36%)”. The means of thescores for these items are 2.9, 3.2 and 3. 9, respectively.The reason of the relatively lower usefulness ofsubject-knowledge competence on these items might be the teachers’ unawareness of the aims andcontents of such courses. Indeed, there is no course related to these items in the pre-service educationof science/physics teachers in Turkey.Therefore, the in-service education program should be revised byincluding new courses stated in Item 4 and giving more emphasis on Item 9. Moreover, as seen in items7 (7.1-7.3), the teachers considered the usefulness of “Good knowledge of mathematics” at the ratewhich is less than 4.0. The figures about the items 3, 4, 5, 7 possibly indicate that the teachers areuncertain about what they mean and how much they are useful for teaching science to pupils.

4. Testing the hypothesis and resultsIn Table 3, and Tables 4a, 4b the results of some analysis of gathered data are displayed. The figuresin the tables show the significant differences in the professional experience and competence of thevarious groups of the teachers, in particular the higher education institutes (HEI) from which theygraduated. Furthermore, hypotheses H0(1) and H0(2) were tested by using ANOVA, PC-SPSS inthe analysis. The analysis, i.e. mean (X), standard deviation (S), F-value, t-value, degree of freedom(df) and level of significant (Sig.) are displayed in Table 3, the results are explained and interpretedbelow very briefly.

Statements 1 2 3 4 5 Omitted

X S

1 Good knowledge of basic physics 5 5 3 27 54 6 4.3 1.09

2 Good comprehension of modern and contemporary physics 8 7 6 30 41 8 4.0 1.26

3 Critical analysis of the contents and the conceptual structure of physics 8 5 9 30 36 12 3.9 1.27

4 Knowledge of the history of physics 24 10 10 25 14 17 2.9 1.51

5 Knowledge of the epistemology of physics 22 5 7 25 25 16 3.3 1.57

6. Knowledge of the relations between physics, technology and society 12 3 5 21 45 14 4.0 1.42

7 Good knowledge of

7.1 • Algebra and Euclidean geometry 15 5 5 26 40 9 3.8 1.47

7.2 • Differential and integral calculus 16 5 8 30 29 13 3.6 1.47

7.3 • Probability and statistics 16 7 3 26 37 11 3.7 1.50

8 Knowledge of strategies for selecting routes and solving

8.1 • Qualitative physical problems 7 3 3 30 47 10 4.2 1.18

8.2 • Quantitative physical problems 6 4 4 28 48 10 4.2 1.17

9 Understanding the following aspects of experimental research

9.1 • Experiment designing 4 1 3 24 62 6 4.5 0.97

9.2 • Measurement techniques and data gathering 3 1 3 30 53 10 4.5 0.84

9.3 • Analysis and interpretation of data 5 3 5 24 51 12 4.3 1.09

9.4 • Use of information technologies in the laboratory 5 1 1 18 66 9 4.5 1.00

10 Knowledge of the applications of physics in other sciences 7 2 5 26 48 12 4.2 1.19

11. Possess the additional knowledge of

1.1 • Another science at the basic level 10 2 4 35 37 12 4.0 1.26

1.2 • A foreign language 14 3 6 20 48 9 3.9 1.45

Table 2. Discipline Related Competencies of the Science/Physics Teachers

As seen in Table 3, the means of the scores of the group of science/physics teachers with respect to theHEI they graduated, i.e. the higher education institutes, namely FS/E or EE are relatively close. Thisseems very reasonable since they have more or less the same facilities in the schools in which they workas well as similar background when they graduated from the higher education institute in which theygraduated. Indeed, although the name of EE changed to FE and new rules were in applications after1983 the instructors were more or less the same as well as the resources, e.g. classrooms, laboratories,library, etc. The mean and standard deviation of the scores of “Summative assessments of students’performance”and of “In difficult moments,help from experts”of both groups are more or less the same,i.e. they are (1.3 +/-0.57) or (1.4 +/-0.75) and (1.7 +/- 0.75) or (1.8 +/-0.67), respectively.


HEI N X S F t df Sig.FS/E 92 2.5 0.62 -3.33 242 (1.1) Using experiments in the school laboratory EE 152 2.8 0.43

37.17 -3.05 145

.00

FS/E 91 2.5 0.56 -2.12 228 (1.5) Using demonstration didactic strategies EE 139 2.7 0.53

6.10 -2.10 185

.01

FS/E 93 2.9 0.28 2.03 232 (1.6) Using lecture didactic strategies to class EE 141 2.8 0.41

18.23 2.18 231

.00

FS/E 81 2.3 0.69 -0.46 206 (1.7) Using visual material EE 127 2.4 0.56

6.46 -0.44 146

.01

FS/E 90 2.5 0.54 -1.64 233 (2.2) Planning teaching strategy in a class council EE 145 2.6 0.51

4.61 -1.61 179

.03

FS/E 72 1.3 0.57 -1.34 167 (3.3) Summative assessments of students’ performance EE 97 1.4 0.75

9.06 -1.40 167

.00

FS/E 82 2.8 0.45 2.20 191 (4.4) Teacher own interior reasoning in difficulty EE 111 2.6 0.57

17.23 2.29 191

.00

FS/E 78 1.7 0.75 -0.91 180 (4.5) In difficult moments, help from experts EE 104 1.8 0.67

4.05 -0.89 156

.05

FS/E 84 3.8 1.30 2.11 217 (7.1) Good knowledge of algebra and Euclidean geometry EE 135 3.4 1.57

15.30 2.20 200

.00

FS/E 86 3.9 1.27 1.40 222 (7.2) Good knowledge of differential and integral calculus EE 138 3.6 1.63

19.48 1.49 211

.00

Age N X S F t df Sig. 30-39 58 4.3 1.07 2.78 209 (7.1) Good knowledge of algebra and Euclidean

geometry for science/physics competence 40-49 153 3.7 1.52 17.22

3.24 146 .00

30-39 57 4.1 1.09 3.39 197 (7.2) Good knowledge of analytical geometry for science/physics competence 40-49 142 3.4 1.54

29.21 3.91 145

.00

30-39 60 4.2 1.06 2.91 204 (7.3) Good knowledge of the probability and statistics for science/physics competence 40-49 146 3.6 1.61

30.22 3.44 164

.00

Gender N X S F t df Sig. M 144 2.8 0.39 -2.54 241 (1/1.1) Using problem based teaching strategies for

professional experience F 99 2.9 0.22 29.53

-2.80 234 .00

M 139 2.5 0.63 -1.38 230 (1/3.2) Using formative assessment of students’ performances F 93 2.6 0.53

6.93 -1.43 217

.01

M 141 3.9 1.35 -0.57 229 (2/3) Good knowledge of the conceptual structure of physics for physics competence F 90 4.0 1.15

5.89 -0.59 211

.02

M 145 3.7 1.54 -1.38 237 (2/7.1) Good knowledge of algebra and Euclidean geometry for science/physics competence F 94 3.9 1.34

5.57 -1.42 217

.02

M 138 3.5 1.54 -1.33 226 (2/7.2) Good knowledge of analytical geometry for science/physics competence F 90 3.7 1.35

6.97 -1.37 208

.01

Table 3. The Analysis of Data wrt Higher Education Intuitions (HEI) Teachers Graduated and Various Items

Table 4a. Analysis of Data with respect to Age and Some Items

Table 4b. Analysis of Data with respect to Gender and Some Items in Table 1/ and Table 2/


The results of testing the hypothesis with respect to some items and either age or gender (M: male,F: female) are displayed in Table 4a and 4b respectively. We discuss briefly if there are significantdifferences between personal characteristics (i.e. gender, age, seniority, type of school graduated,etc.,) and comment on some items they are stated explicitly and explained very briefly.The analysis of Hypothesis H0(1) shows that:• There is a significant difference between ‘gender’ and the following two items of the professional

experience category; (a) Using problem based teaching strategies; and (b) using formativeassessment of students’ performances

• There is a significant difference between ‘age’ and the following three items of the professionalexperience;(a) Good knowledge of algebra and Euclidean geometry, (b) Good knowledge ofanalytical geometry, (c) Good knowledge of the probability and statistics.

• There is a significant difference between ‘type of school graduated’ and the following eight itemsof the professional experience: (a ) Using experiments in the school laboratory, (b Usingdemonstration didactic strategies, (c) Using lecture didactic strategies to class, (d) Using visualmaterial, (e ) Planning teaching strategy in a class council, (f) Summative assessments of students’performance, (g) Teacher own interior reasoning in difficulty, (h) Help reached from experts indifficult moments.

• There is no significant difference between ‘gender’, ‘age’, ‘type of school graduated’, ‘seniority ‘and the rest of the items of the professional experience listed in Table 2.

The analysis of H0(2) shows that:• There is a significant difference between ‘gender’ and the means of the scores of three physics

competence: (a) Good knowledge of the conceptual structure of physics; (b) Good knowledge ofalgebra and Euclidean geometry; (c) Good knowledge of analytical geometry.

• There is a significant difference between the ‘higher education institutions graduated’’ and themeans of the scores of the physics competence: (a) Good knowledge of algebra and Euclideangeometry, (b) Good knowledge of analytical geometry.

• There is no significant difference between ‘gender, higher education institutions graduated, age,seniority ‘ and the rest of the items of the physics competence listed in Table 2.

5. Concluding remarksAlthough it is generally accepted that there is no good school if there are no competent andqualified teachers. Therefore the pre-service and INSET is a matter of primary concern in acontemporary society in both developed and developing countries. In this respect, the quality ofscience/physics teacher education is a current issue in most countries and there is a call forcurriculum reform to meet the needs of the information society. In the present study, we attemptedto reflect a group of science/physics teachers’ views on the usefulness of various strategies inteaching science/physics in the upper elementary school in Turkey and perceptions of subject-knowledge competence. The data were gathered by means of the questionnaire designed in theEUPEN Project, but adapted to Turkey. The data were gathered from 265 upper primary schoolscience/physics teachers working in Antalya, a province in the southern part of Turkey, in 2000-01school year.The usefulness of various subject-knowledge for teaching science/physics in schools varies.According to the teachers’ responses some have higher priority with respect to others. The analysisof data showed that there are some significant differences in the professional experience andsubject-knowledge competence with respect to the groups related to gender, age, and the THIteachers graduation, but no significant difference between seniority and all these items. Finally,further research would be useful to answer many questions and find out why the teachers havedifferent views and subject-knowledge competence. Therefore, we continue doing research to findout the reasons of incompetence and developing some instructional materials, in particularunderstanding experimental research, use of technology in teaching/learning physics etc. We do

hope that our effort, namely research, seminar and workshop help the teachers to meet their needsto become more competent science/physics teachers.

References[1] H. Ferdinande, S.P. Jona, Latal, The Training Needs of Physics Teachers in Five European Countries: An Inquiry,

Proceedings of the third EUPEN General Forum 99 (EGF99) part II, 4, Universities Gent, London, (1999).[2] Y. Ersoy, M. Ve Sancar, Okullarda Fen/ Fizik Eîrtimi : Boyutlar ve Ö^retmen Deî¨keni, Fizik Dergisi 12, Aralık,

Özel Sayı, (1988), 4-7 (in Turkish).[3] Y. Ersoy, A study on the Education of School Mathematics and Science Teachers for Information Society, METU

Education Research Report, I ,ODTU Eîtim Fak. Yay, Ankara, (1992), 39-54.[4] Y. Ersoy, & I.S Üstüner, A Project for the Professional Development of Physics Teachers in the Antalya Region”,

Int. Conference on Physics Teacher Education Beyond, (Aug 27-Sep 1, 2000, Barcelona-Spain).[5] I. S.Üstüner, M. Sancar, Ö^retmenlerin Fizik Dersinin Amaçları ve Ö^rencilerin Bilimsel Dü¨üncelerini

Geli¨tirme Konusundaki Görü¨leri” Türk Fizik Derneî 18.Fizik Kongresi (TFDK-18), Çukurova Üniversitesi,Adana, (1999), 102-106 (in Turkish).

[6] A. Erdem, I.S. Üstüner, M. Sancar, “Edirne ve Kırklareli Illerinde Ö^retmenlerin Fen-Fizik Eîtimi KonusundakiGörü¨leri”, IV. Fen Bilimleri Eîtimi Kongresi 2000, Hacettepe Üniversitesi Eîtim Fakültesi,Ankara 6-8, (2000),175 (in Turkish).

[7] http://www.deu.edu.tr/prospectus98/general_.htm[8] http://www.turkembassy.dk/principles.htm


DEVELOPING FORMAL THINKING IN PHYSICSUniversity of Udine, 2-6 September 2001

M. Cobal, L. Santi, Physics Department of the University of Udine (Italy)

SeminarIn 1997 the GIREP Committee started to express the intention to organize intermediate Seminarsbetween the GIREP Conferences. They were thought of as minor Congresses between colleagues ofneighbouring countries. For several reasons, for long time it was not possible to realize this project.The First International Seminar of the Group International de Recherche sur l’Einsegnement dela Physique (GIREP) was held in Udine in the period 2-6 September 2001, on the subject:“DEVELOPING FORMAL THINKING IN PHYSICS”.This first Seminar therefore, represents a dream come true.The Seminar was organized by the University of Udine (through its Centro Interdipartimentaledella Didattica –CIRD-), with the collaboration of the Conferenza Nazionale dei CentriUniversitari di Ricerca Educativa e Didattica (Concured).The scientific responsibility was assigned to Ian Lawrence, of the Birmingham University and toMarisa Michelini, of the Udine University. More details on the organizational aspects of theSeminar can be found in appendix.

Subject of the Seminar: the development of the formal thinkingThe subject of the first GIREP Seminar has been the development of the formal thinking.Why thistopic?Physics offers us that knowledge of the world around, which allows us to describe and interpret it,to appreciate the beauty of its organization and to foresee its evolution. It offers us views fromdifferent perspectives: mechanics, thermodynamics, electromagnetic, optics, macroscopic, microscopic,classical, quantum; ways that pre-suppose different kinds of formalization and use of mathematics. Thevery language of physics uses mathematicalconcepts and instruments so that we can describe,interpret and predict.The game of understanding in physics throughexperiments and theorizing, which builds linksbetween equation and data, makes physics morefascinating for us, but not many young people areso enthusiastic about it anymore.We can find many reasons for this: thesubordinate role assigned to physics in manyfields in which it has a great importance, thescarce recognition in society its researchers andworkers receive, the lack of importance givento physics in education and the lack ofunderstanding of its educational value, thelarge amount of hard work required to study it.The new means of communication, withmeanings constructed through intuitiveassociation, languages through images and

4. The Seminar

4.1 First International Girep Seminar

Figure 1: The GIREP Committee: I. Lawrence, S.Oblack, M. Michelini, M. Euler, C. Ucke.

verbal languages, seem to alienate students even more from the process of formalization of themathematical language of physics. The development of formal thinking in physics represents theacquisition of a network of contacts which assign meaning to figurative elements and which allowstudents to navigate around the land of physics.If we don’t want to loose this knowledge and if we still believe it to be important, we must makean effort so that physics will become more familiar to all. We must work to have it introduced earlyto young children in the primary school, we must work to make it more easily understood, in orderto connect the language of physics to everyday language.The necessary prelude to a new way of approaching the teaching of science and especially ofphysics, might well be an analysis of formalization processes at all educational levels and adiscussion of the connection between mathematics and physics.

Structure of the SeminarThe opening Ceremony took place in Udine, in the same room -Salone del Parlamento- of theUdine castle, where, in 1995, the International GIREP-ICPE Conference on the teaching of thecondensed matter and new materials science was opened.The Ceremony was held on September the 3rd, and saw the participation of:Marisa Michelini, GIREP VicepresidentSergio Ceccotti, Udine MajorFurio Honsell, Rector of the Udine UniversityManfred Euler, GIREP PresidentMarzio Strassoldo, President of Udine ProviceCarlo Del Papa, Director of the Physics Department at the Udine UniversitySergio Focardi, SIF CommitteeGiuliana Cavaggioni, AIF CommitteeCarlo Rizzutto, Sincrotrone TriesteDaniele Amati, SISSA, TriesteAlessandro Borgnolo, LIS, Trieste

The structure of this Seminar was discussed at length: the initial idea – to do a workshop to select paperswhich would be useful for teachers on a certain topic, in order to contribute to the work which ICPE isdoing – has been transformed by the way the participants interpreted our proposal.A preliminary work of examining the various papers on a certain topic was supposed to be the basis ofthe Seminar which, as one of its purposes, was supposed to collect papers to inform teachers about, andalso,possibly, to identify a standard protocol for drawing up articles which teachers could easily use.This choice stimulated the interest of numerous colleagues all over the world, and they submittedto the Seminar numerous specific contributions on the development o formal thinking in physics.Thus a new and interesting working scenario came into being: an environment for discussion onresearch and practice in didactics.The task of selection and scientific organization has been hard, but the whole Advisory Committeehas carried it out in order to group together the contributions into thematic areas and to preservethe two important dimensions of the Seminar: to have through scientific discussions in thematicworkshops and to produce shared results about the problem posed.The Seminar has been organized in 8 General Talks (for invitation), 11 Sessions for thepresentation of the various contributions and 7 Workshops.During the Seminar it was also possible to visit, in a room of Palazzo Antonini, a postersexhibition, for those works related to the topics of interest.The General Talks had the important task to offer an overview of the subjects that were thendiscussed in detail in the various Workshops.In the following, the titles of the presentations are given, with the respective authors– Imagery and formal thinking (M. Euler Germany)– Physics curriculum reform (R.G. Fuller, USA)

428 The Seminar

– Jumping toys: a topic for interplay between theoryand experiment (C. Ucke, Germany)

– Real-time approaches in the development of formalthinking in physics (E. Sassi, Italy)

– Differences between the use of mathematical entitiesin mathematics and physics and the consequencesfor an integrated math and science learning (T.Ellermeeijer, Netherlands)

– An epistemological framework for laboratory (M.Vicentini, Italy)

– Is formal thinking helpful in everyday situations (S.Oblak, Slovenia)

– The formal reasoning of quantum mechanics: can wemake it concrete? Should we? (D. Zollmann, USA)

In the Sessions dedicated to the presentation of thevarious contributions, all the papers accepted for theSeminar have been presented: more than a hundred.Allthese Sessions were given in parallel during the first day.This has been done with the purpose to allow a scientificdiscussion of the papers presented in the next days.The leaders of each Session have been acting as bothcoordinators and chair-person reporteurs with respect to the responsible of each Workshop. Thisdouble role appeared to be a functional arrangement, since it allowed plenty of time for theworkshops, it precluded presentations from the discussions, it offered outcomes which could beseen from at least two points of view: that of the reporteurs and that of the Workshop leader.The Workshops –divided in 4 sessions, each one two hours long and divided in two consecutiveafternoons - have been the core of the Seminar, and have determined its outcomes. They havebenefited of the fundamental contribution of the General Talks and of the contributions of theparticipants. Many goals have been fulfilled in the four sessions of each workshop:• To give a general view of the problems involved in the subject of the Workshop• To develop a discussion on these problems• To take a further and more detailed look at some specific aspects• To give a general summary of the conclusions reached

The subjects of the various Workshops, with the respective Workshop leaders, are given in thefollowing.

- Interplay of theory and experiment (A. De Ambrosis, G. Rinaudo)The discussion was guided by the followinglines: a) What is the meaning of “theory”,“experiment” and “interaction” betweenthe two b) definition of the goals to beachieved c) kinds of experiment, and theirefficiency in helping the formalizationprocess. This Workshop was characterizedby the presentation of severalexperiments, and by the attempt tounderline the aspects which are relevantfor the formalization process.- Learning physics via model construction(R.M. Sperandeo-Mineo)Presentations and discussions have beenmainly connected to two themes of the


Figure 3: The poster exhibit.

Figure 2: Opening Ceremony - Marisa Michelini,chair; Furio Honsell, Rector of the University ofUdine; Marzio Strassoldo, President of the UdineProvince, past Rector of the University of Udine.

Seminar: a) modelling the world and b) mathematics.The discussion ended up with the analysis of the roleof developing models inside a curriculum.- Modelling for younger learners (I. Lawrence)Modelling has taken root in some undergraduatecourses. The models built here are often numerical,with explicit use of algebra and /or arithmetic. ThisWorkshop concentrated on the possibility of using thecomputer for engaging with the modelling processlower down the age ranges.- Toys for learning physics (C. Ucke)The starting question was: which role has the formalthinking when physics is learned through the use oftoys? Many toys allow the first step toward the goal ofdeveloping the formal thinking. And their role doesnot end even at the level of University studies.However, teachers need books, publications, catalogsand pointers to the available toys (this information isnot always easily available).The idea to organize a seminar/workshop specializedon toys, and to create a database of the existing toyshave been discussed as well.- Early start in physics understanding (P. Guidoni)

Today’s voiced problem in the transmission of the physics culture is however not one of early startin schools’ teaching/learning sequences: rather one of systematic, coherent, carefully mediated rootingof explaining/understanding dynamics into the extremely rich virtual field of children’s potentialities Several approaches to this fascinating research theme have been developed in the last two decades.Aim of the Workshop is to compare these approaches and to use them as a base to develop newideas.The works of this Workshop have been integrated with those of the Workshop devoted to themodeling for younger learners (I. Lawrence)- New technology and computers in physics learning (L. Rogers)This Workshop has been finalized to compare differentapproaches to the uses of Informatics Technologies (IT)for teaching physics in different countries.After a first discussion, the following questions havebeen identified:a) Why the IT use in the school is so limited, even if theyhave been introduced already several years ago? b)How can be teachers motivated in using IT? c) Howthey can find time to learn the use of IT? d) Whichproblems can arise in using IT? e) Which role shouldhave the teacher when pupils are using IT? f) What arethe components of training which teachers need to useIT effectively? - Textbooks as an image of philosophy of teaching (Z.Golab-Meyer)The main points of discussion have been the following:a) How do textbooks reflect new trends in physics

education?b) Who is the real target of the textbooks? Teachers or

students?c) Do we know students preferences?

430 The Seminar

Figure 4: Professor C. Ucke during hispresentation.

Figure 5: Professor E. Sassi during herpresentation.


Figure 6: A moment of relax…

d) Who is examining and evaluating the new textbooks?e) Where are the possible mistakes which have to be absolutely avoided?

Social programmeThe social programme included a visit to the “Casa del Vino” of Ersa, , situated in a 16th centurybuilding with the possibility to taste typical wines of the Friuli Venezia Giulia (on 3rd September), aconcert of the Graz Capella Musicae in the Church of San Antonio (on 4th September), an informaldinner at a typical restaurant of the Friuli land, offered by the Organizing Commette (on Septemberthe 5th) , a guided trip to Cividale del Friuli and Gorizia, including some of the most beautiful historicalsights, and – finally – a scientific excursion to the Sincrotrone-Elettra in Trieste (on the 6th ).

Participants and results The Seminar put together scientists, researchers in the didactic of physics, school teachers anduniversity professors, into the common effort of discussing the formalization of the scientificlearning, considering the physics phenomena and their interpretation, the role of the technologicalapplications, the results in the didactical research, the educational strategies, the new curricula, theteaching resources.This first GIREP Seminar has provided an unique opportunity for working on defined themes forexperts on science teaching from all over the world, for starting research together and as apreparatory activity for thematic sessions of subsequent GIREP Conferences. In spite of the factthat the International Scientific Committee previously decided a selection of the participantsallowing a maximum numbers of 50-60 of them, the number of contributions that arrived broughtto the decision of doubling this quota.About 125 experts of didactics took part to the Seminar (98 from Europe, 14 from America, 7 fromAsia and 1 from Africa). The papers that appear in this book have been selected after a carefulrefereeing work. The other papers –belonging to the hundred which have been selected by theInternational Scientific Committee- will be published on the web.The positive evaluation received on both the scientific and organizational grounds, has in additionbrought to the decision of having the next (2003) GIREP Seminar on the eaching formation, again inUdine.On the www.uniud.it/cird/ web page, one can find all the material related to the Seminar. s

4.2 Welcome of the Rector

Furio Honsell, University of Udine, Italy

Over the last decade, probably due to the acceleration of scientific and technologica innovation,there has been growing interest on the part of the scientific community and higher educationinstitutions throughout the world on the problems concerning with the teaching of sciences, thetraining of teachers and the development and spread of the scientific and technological culture. Inthis respect the University of Udine has been very alert and receptive, and has considered it to be animportant part of its cultural mission and social responsibility to contribute meaningfully to theeducation and training of teachers and the growth and dissemination of scientific and technologyawareness in the society at large. Udine University has therefore developed a number of importantresearch initiatives and projects in these areas, especially within the activities of theInterdepartmental Center for Educational Research (CIRD), founded and directed by ProfessorMarisa Michelini.The University of Udine is therefore very pleased to host this important international conference,which will gather in this beautiful and hospitable city, with is young and innovative University,scholars, researchers, and teachers, working in physics, pedagogy, education, phychology and variousother cognitive sciences, all with the common purpose of addressing the crucial question in theteaching of science: how does formal thinking arise and develop?The cognitive process of formalization is indeed the cornerstone of modern western science and thekey to its success.Looking at the conference programme I am very impressed and pleased to notice the extremely widerange of different teaching levels at which this problematic issue is addressed. I am confident that thisSeminar will provide very stimulating pedagogical suggestions both to school and university teachers,and it will contribute to our understanding of how scientific thought develops and hence how it canbe enhanced.Finally, on behalf of the University of Udine I wish you all a very fruitful and pleasant week!


First GIREP Seminar, Udine 2001It is a great honour for me to open a GIREP iniziative in Udine again. In 1995, in this sameParliament Hall in the castle of Udine, we opened the GIREP-ICPE International Conference onthe teaching of the science of condensed matter and new materials.At that time, Girep had not been in Italy for 20 years, and it was an important commitment for us.Organizing the first Girep Seminar is perhaps even more important and demanding, because a newinitiative is starting for which there are great expectations: let us hope that it will be fruitfull asexpected fruit and become a regular initiative, starting from this first year of the new millennium.

A Seminar, not a ConferenceWe can sai without a shadow of a doubt that the GIREP Conferences, traditionally organized withthe International Commission on Physics Education (ICPE) of the International Union on Pure andApplied Physics (IUPAP), are one of the main reference points for the international communitywhich deals with the teaching of physics. They offer a rich environment for the comparison andexchange of research results and proposals based on practice. In fact, GIREP has specifically chosento bring together experts in didactic research and those who have professional experience indidactics, leading to an explicit collaboration between research and teaching at all levels wheredidactics are practised, from play school to university.The GIREP Conference take place every two years, and are getting bigger all the time. In the fewdays when they take place, it is hardly possible to make contact with our numerous colleagues, andto keep up with their ever broader spectrum of papers.We need to make time for discussion and new ways of comparing progress, in order to study specificproblems more deeply. This has led to a proposal for intermediate Seminars between twoConferences for a limited number of researchers and teachers.The GIREP Seminars have been conceived of to provide opportunities for working on definedthemes for experts of science teaching from all over the world, for starting research together and asa preparatory activity for thematic sessions of subsequent Girep Conferences. They are also ameeting place for the GIREP Thematic Discussion Groups: this is another recent initiative, still in thestart-up hase, intended to lay foundations for distance research collaboration, with the support of aelectrom forum.The Seminars and the Thematic Discussion Groups were presented in the GIREP Newsletter n° 43in November 2000. They are a proposal for a new, deeper and more frequent method of interactingbetween scholars of science teaching.

The subject of this Seminar: Developing formal thinkingPhysics offers us that knowledge of the world around us which allows us to describe it and interpretit, to appreciate the beauty of its organisation and to foresee its evolution. It offers us views fromdifferent perspectives: mechanics, thermodynamics, electromagnetics, optics, macroscopic,microscopic, classical, quantum, ways that pre-suppose different kinds of formalisation and use ofmathematics. The very language of physics uses mathematical concepts and instruments so that wecan describe, interpret and predict.The game of understanding in phjysics through experiments and theorising, which builds linksbetween equations and data, makes plysics more fascinating for us, but not many young people areso enthusiastic about in any more.We can find many reasons for this: the subordinate role assigned to physics in many fields in whichit has a pivotal importance, the scarce recognition in society its researchers and workers receive, thelack of importance given to physics in education and the lack of understanding of its educationalvalue, the large amount of hard work required to study it. The new means of communication, withmeanings constructed through intuitive association, languages through images and verbal languages,

4.3 Introduction to the Seminar

Marisa Michelini, Vicepresident of Girep, Director of CIRD, University of Udine (Italy)

434 The Seminar

seem to alinate students even more from the processes of formalization of the mathematical languageof physics.The development of formal thinking in physics represents the acquisition of a network of contactswhich assign meaning to figurative elements and which allow students to navigate around the land ofphysics.If we do not want to lose this knowledge and if we still believe it to be important, we must make aneffort so that physics will become more familiar to all. We must work to have it introduced early toyoung children in the primary school, we must work to make it more easily understood, in order toconnect the language of physics to everyday language.The necessaryp relude to a new way of approaching the teaching of science and especially of physicsmight well be an analysis of formalisation processes at all educational levels and a discussions of theconnection between mathematics and physics.The Seminar will bring together scientists, researchers in education, teachers from schools anduniversities in a common effort to discuss formalisation in science learning, considering physicalphenomena and their interpretations, the role of technological applications, the results of didacticresearch, educational strategies, new curricula, teaching resources.The subjects which the discussion will focus on are those of the Workshops, around which theproblems of the connection between theory and experiment, modelling, matematics and informallearning are articulated. Let us consider some aspects.I would like to say just a few words about them.1) Interplay of theory and experiment

The role of practical work in physics education has been the subject of many debates. The impactof ITC on the process and understanding of measurement has renewed many discussions under anew guise, especially for new learning opportunities offered by real time measurements.

2) Modelling the worldIn physics models are the way for interpreting the world around us. They are built organizing bya formal representation of the phenomena in the reference frame of a theory.Their capabilities indescribing a whole class of phenomena and to give previsions on the evolution of the processes,express the most fascinating aspects of physics and explicit the role of maths in physics.

3) MathematicsMathematics, in fact, has long been a useful language for developing physics in a way which isunderstandable for students and young people.

4) Informal learningToday’s challenge is to build the links between common knowledge and science, betweenspontaneous interpretative models and formal scientific models and finally to lead children andyoungsters to learn stimulated by curiosity through the informal and playful exploration ofphenomena. For the development of formal thinking it is also important to promote scientificculture through the spreading of scientific knowledge. This implies diffusing the awareness that itis actually possible to describe the world, that some physical quantities can enable us to forseewhat will happen, that different phenomena can be looked at in similar ways and that the samemathematical concepts can be used to describe them.

Issues such as how to achieve these goals and how to activate the learning process will be introducedby general talks and explored in parallel workshops.

A little history will clarify the nature and program of the SeminarIn 1997 the Girep Committee started to express the intention of organizing intermediate Seminarsbetween the Conferences. They were thought of as minor Congresses between colleagues ofneighbouring countries. For various reasons it was not possible to put these hypotheses into practice.This first Seminar therefore represents a dream come true, and we hope it will meet the requirementsof the participants and will continue in the future.I would like to thank Ian Lawrence for sharing the responsibility and commitment of this firstoccasion. We have debated it a lot in the last few months, because the initial idea has beentransformed by the way the participants interpreted our proposals. The current organization is theresult of a process of interpretation and proposals, which has been carefully followed throughout itsevolution. The initial idea was to do a workshop to select papers which would be useful for teachers

on a certain topic, in order to contribute to the work which ICPE is doing, starting from the Girepproocedings. A preliminary work of examining the various papers on a certain topic was supposed tobe the basis of the Seminar which, as one of its purposes, was supposed to collect papers to informteachers about, and also, possibly, to identify a standard protocol for drawing up articles whichteachers could easily use.This choice stimulated the interest of numerous colleagues all over the world, and they submitted tothe Seminar numerous specific contributions on the development of formal thinking in physics. Thusa new and interesting working scenario came into being: an environment for discussion on researchand on practice in didactics. The task of selection and scientific organization has been hard, but thewhole Advisory Committee has carried it out in order to group together the contributions intothematic areas and to preserve the two important dimensions of the Seminar: to have thoroughscientific discussions in thematic workshops and to produce shared results about the problems posed.Linking the Panel Talks (which present all the works in parallel on the first day) with the Chair-persons who perform the function of reporteurs to the workshop leaders, appears to be a functionalarrangement, since it allows plenty of time for the workshops, it precludes presentations from thediscussion, it offers outcomes that can be seen from at least two points of view: that of the reporteurand that of the workshop leader.The latter collect, for discussion in the workshops, both contributionsconnected with the problems which they are intended to focus on, and also the reports from thereporteurs.The workshops are the heart of the Seminar, because they shape the outcomes.They therefore make use of the contributions of the General Talks, which offere them an overview ofthe problems being discussed.

The Seminar will last only a few days, the topic it deals with is broad and important: we hope it willserve to lay the basis for a later work which will be effective in this sense, perhaps in the form of aGirep Thematic Discussion Group. We hope, in any case, that every participant will be able to gohome having enriched his skills in this field, having established opportunities for collaboration andhaving perfected proposals for the future.

Ian Lawrence and I have worked a great deal: many people have helped us and without them wewould never have succeeded in the task.The following people’s help has been fundamental: ManfredEuler, Christian Ucke, Seta Oblack and Lorenzo Santi, members of the Advisory Committee and theother committees published in this booklet. We would like to express to them our deepest gratitude.Particular thanks go to our numerous Italian and Slovenian colleagues, who believed in and becameinvolved in the initiative.I would personally like to take this opportunity to officially thank the many people from theUniversity of Udine who made the Seminar possible. There are so many, it is not possible to mentionthem all, we have done that in the booklet. But allow me to mention 7 people only: the two RectorsMarzio Strassoldo and Furio Honsell, who have followed and supported the initiative, DonatellaCeccolin and Mauro Sabbadini of CIRD, Plioni De Zorzi of CESA and two little starlets: the studentsElisa Ius and Elizabeth Chamberlain, between whose competent hands the majority of work haspassed.Certainly, some things could have been done better: we dide what we were able to do with the greatestcommitment.We hope that the Seminar will be useful for all.Welcome to Udine, I look forward to being able to learn toghether.


4.4 The organization of the Seminar

Marisa Michelini, Vicepresident of Girep, Director of CIRD, University of Udine (Italy)

Groupe International de Recherche sur l’Enseignement de le Physique (GIREP)University of Udine, ItalyInterdepartmental Centre for Research in Education (CIRD)Department of Physics with the National Italian Conference of the University Centre for Research in Education (CONCURED)

People in chargeMarisa Michelini (Udine University) and Ian Lawrence (University of Birmingham),Vicepresidents of GIREP

International Advisory BoardManfred Euler, Department of Physics Education, IPN (Institute ofr Science Education), Kiel, GermanyMarisa Michelini, Dipartimento di Fisica, Udine UniversityIan Lawrence, Department of Education, University of Birmingham, UKSeta Oblak, Board of Education, Ljubljana, SloveniaChristian Ucke, Physikdepartment E20, Tech, Universität München, GermanyGiunio Luzzatto, President of CONCURED, Genoa UniversitySilvia Pugliese Jona, Vicepresident of AIF

Local Organizing CommitteeFurio Honsell, Rector of Udine UniversityMarzio Strassoldo, President of Udine ProvincePier Luigi Rigo, Committee for Promotion of Scientific and Technological Studies, Udine UniversityMarisa Michelini, Director of CIRD, Udine UniversityCarlo del papa, DIrector of Physics Department of Udine UniversityLorenzo G. Santi, Physics Department, Udine University

Collaborating Scientific InstitutionsEUPEN, European Physics Education NetworkCISM, International Centre For Mechanical SciencesSIF, Italian Physical SocietyAIF, Association for Physics TeachingCoordinating Commission for Physics in Friuli Venezia GiuliaSincrotrone-Elettra, TriesteSISSA, High School for Advanced StudiesCLDF, Centre-Laboratory for Physics Education, University of Udine

Supporting CommitteeM. Sestito, F. Giorgetti, E. Montiglio, School for Interpreters and Translators of Udine UniversityM. Cobal, A. Stefanel, Physics Department and CIRD, Udine UniversityM. Jarc, teacher at the Slovenian Language School, GoriziaD. Ceccolin, M. Sabbadini, in charge of Services at CIRD, Udine UniversityG. Cabras, D. Cobai,A. Di Marzio, S. Zuccaro in charge of Services at the Physics Department, UdineUniversityF. Caufin, M. De Anna, P. De Zorzi, S. Di Zanutto, S. Fabbris, A. Lucatello, A. Missana, D. Sillani, E.Vecchio, in charge of Services at the Udine UniversityMaria Julia Passalenti, Peter Fersila, students of Diploma Course for Translators and Interpreter,Udine University

Secretary and Technical Support of the SeminarG. Cavasino, D. Ceccolin, E. Chamberlain, E. Ius, M. Sabbadini

The Seminar is under the patronage of the collaborating Scientific Institutions SIF - Italian Physical SocietyAIF - Association for Physics Teaching

Friuli-Venezia Giulia RegionRegional Direction of the Ministerium of Education for the Friuli-Venezia Giulia RegionIRRE - Regional Institute for Educational Research of the Ministerium of Education for the Friuli-Venezia GiuliaUdine ProvinceGorizia ProvinceComune di Udine

Further support byMURST - Ministero dell’Università e della Ricerca ScientificaUNESCO - Representative office in the Islamic Republic of IranCISM - International Centre for Mechanical Sciences, UdineSincrotrone-Elettra, TriesteRegione Friuli-Venezia Giulia - Presidenza della Giunta - Assessorato alla Cultura e all’IstruzioneIRRE del Friuli-Venezia GiuliaIstituto per l’educazione della Repubblica di SloveniaEureka, CIRD of the University of TriesteLIS - Laboratorio dell’Immaginario Scientifico, TriesteScienza Viva, Associazione per la diffusione della cultura scientifica e tecnologica, Calitri (AV)Comune di UdineConsiglio di Amministrazione dell’Università di UdineComitato per la Promozione degli Studi Tecnico Scientifici, UdineFacoltà di Agraria dell’Università di UdineFacoltà di Scienze della Formazione dell’Università di UdineFacoltà di Lettere e Filosofia dell’Università di UdineFacoltà di Lingue e Letterature Straniere dell’Università di UdineSchool for Interpreters and Translators of Udine UniversityCIRF - Centro Interdipartimentale di Ricerca sulla cultura e la lingua del Friuli, University of UdineDipartimento di Scienze della Produzione Animale dell’Università di UdineDipartimento di Economia Società e Territorio dell’Università di UdineCLAV - Centro Linguistico e Audiovisivi dell’Università di UdineCORT - Centro Orientamento e Tutorato dell’Università di UdineCECA - Centro di Calcolo dell’Università di UdineERDISU - Ente Regionale per il Diritto allo Studio Universitario di UdineConvitto “C. Mander”, UdineFondazione CRUP, UdineCRUP - Cassa di Risparmio di Udine e PordenoneSezione AIF, PordenoneSezione AIF, UdineSezione AIF, VeneziaCIDI - Centro di Iniziativa Democratica degli Insegnanti, Gemona del Friuli, UdineGEI - Games Experiments Idea Exhibit, Udine UniversityItineraria, Associazione Guide Turistiche Autorizzate per la Regione Friuli-Venezia GiuliaEstate in Città 2001, del Comune di UdineELITALIA PASCO Scientific, MilanoMedia Direct, Bassano del Grappa, VicenzaERSA, Casa del Vino, UdineConsorzio Tutela Formaggio Montasio, Rivolto di Codroipo, UdineConsorzio per la Tutela del Marchio Gubana di Cividale del FriuliNonino Distillatori spa, Percoto, UdineFriul Service s.r.l., Fraz. Colugna, UdineOsteria Al Vecchio Stallo dei F.lli Mancini, UdineAzienda Regionale per la Promozione Turistica, UdineParco Naturale Prealpi Giulie, Prato di Resia, UdineNicastro Emilio snc, Fogliano Redipuglia, GoriziaSALVIUS, Lavorazione artistica del vetro e della ceramica, Villa Vicentina, UdineHallo Sport, ModenaLitho Stampa srl, Pasian di Prato, UdineThe English Language Centre


438 The Seminar

4.5 Structure of the seminar


Rapporteur

Each Workshop consists of four sessions, each one 2 hours long and divided in two successiveafternoons. These four sessions are meant to:1) give a general view of the problems involved in the subject2) develop a discussion on these problems3) take a further and more detailed look at some specific aspects4) make a general synthesis of the result








