[International Perspectives on the Teaching and Learning of Mathematical Modelling] Modeling Students' Mathematical Modeling Competencies ||

International Perspectives on the Teaching andLearning of Mathematical Modelling

Richard LeshPeter L. GalbraithChristopher R. HainesAndrew Hurford Editors

Modeling Students' Mathematical Modeling CompetenciesICTMA 13

Modeling Students Mathematical ModelingCompetencies

International Perspectives on the Teaching and Learning of Mathematical Modelling

Editorial Board IPTL

EditorsGabriele Kaiser, University of Hamburg, GermanyGloria Stillman, Australian Catholic University, Australia

Editorial BoardMaria Salett Biembengut, Universidade Regional de Blumenau, BrazilWerner Blum, University of Kassel, GermanyHelen Doerr, Syracuse University, USAPeter Galbraith, University of Queensland, AustraliaToskikazu Ikeda, Yokohoma National University, JapanMogens Niss, Roskilde University, DenmarkJinxing Xie, Tsinghua University, China

For further volumes:http://www.springer.com/series/10093

Richard Lesh eter L. GalbraithChristopher R. Haines ndrew HurfordEditors

Modeling StudentsMathematical ModelingCompetencies

ICTMA 13

123

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EditorsRichard LeshSchool of EducationIndiana UniversityBloomington, IN, USA

Christopher R. HainesDepartment Continuing EducationCity UniversityLondon, UK

Peter L. GalbraithGraduate School of EducationUniversity of QueenslandBrisbane, QLD, Australia

Dr. Andrew HurfordSchool of EducationUniversity of UtahSalt Lake City, UT, USA

ISBN 978-94-007-6270-1 ISBN 978-94-007-6271-8 (eBook) DOI 10.1007/978-94-007-6271-8

S

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ordrecht Heidelberg New York LondonSpringer D

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IS 2211-4920 ISSN 2211-4939 (electronic) SN

2013933811

Contents

1 Introduction: ICTMA and the Teaching of Modelingand Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Gabriele Kaiser

Part I The Nature of Models & Modeling2 Introduction to Part I Modeling: What Is It? Why Do It? . . . . . 5

Richard Lesh and Thomas Fennewald

Section 1 What Are Models?3 Modeling Theory for Math and Science Education . . . . . . . . . 13

David Hestenes

4 Modeling a Crucial Aspect of Students MathematicalModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mogens Niss

5 Modeling Perspectives in Math Education Research . . . . . . . . 61Christine Larson, Guershon Harel, Michael Oehrtman,Michelle Zandieh, Chris Rasmussen, Robert Speiser,and Chuck Walter

Section 2 Where Are Models & Modelers Found?6 Modeling to Address Techno-Mathematical Literacies in Work . . 75

Richard Noss and Celia Hoyles

7 Mathematical Modeling in Engineering Design Projects . . . . . . 87Monica E. Cardella

8 The Mathematical Expertise of Mechanical Engineers The Case of Mechanism Design . . . . . . . . . . . . . . . . . . . 99Burkhard Alpers

v

vi Contents

Section 3 What Do Modeling Processes Look Like?9 Modeling and Quantitative Reasoning: The Summer

Jobs Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Christine Larson

10 Tracing Students Modeling Processes in School . . . . . . . . . . 119Nicholas Mousoulides, M. Pittalis, C. Christou, and Bharath Sriraman

Section 4 What Creates The Need For Modeling11 Turning Ideas into Modeling Problems . . . . . . . . . . . . . . . 133

Peter L. Galbraith, Gloria Stillman, and Jill Brown

12 Remarks on a Modeling Cycle and Interpreting Behaviours . . . 145Christopher R. Haines and Rosalind Crouch

13 Model Eliciting Environments as Nurseriesfor Modeling Probabilistic Situations . . . . . . . . . . . . . . . . 155Miriam Amit and Irma Jan

14 Models as Tools, Especially for Making Sense of Problems . . . . 167Bob Speiser and Chuck Walter

15 In-Depth Use of Modeling in Engineering Courseworkto Enhance Problem Solving . . . . . . . . . . . . . . . . . . . . . 173Renee M. Clark, Larry J. Shuman, and Mary Bestereld-Sacre

16 Generative Activities: Making Sense of 1098 Functions . . . . . . 189Sarah M. Davis

Section 5 How Do Models Develop?17 Modeling the Sensorial Perception in the Classroom . . . . . . . . 201

Adolf J.I. Riede

18 Assessing a Modeling Process of a Linear Pattern Task . . . . . . 213Miriam Amit and Dorit Neria

19 Single Solution, Multiple Perspectives . . . . . . . . . . . . . . . . 223Angeles Dominguez

Section 6 How is Modeling Different than Solving?20 Problem Solving Versus Modeling . . . . . . . . . . . . . . . . . . 237

Judith Zawojewski21 Investigating the Relationship Between the Problem

and the Solver: Who Decides What Math Gets Used? . . . . . . . 245Guadalupe Carmona and Steven Greenstein

Contents vii

22 Communication: The Essential Difference BetweenMathematical Modeling and Problem Solving . . . . . . . . . . . 255Tomas Hjgaard

23 Analysis of Modeling Problem Solutions with Methodsof Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . 265Gilbert Greefrath

Part II Modeling in School Classrooms24 Modeling in K-16 Mathematics Classrooms and Beyond . . . . 275

Richard Lesh, Randall Young, and Thomas Fennewald

Section 7 How Can Students Recognize the Need for Modeling?25 Modeling with Complex Data in the Primary School . . . . . . . . 287

Lyn D. English

26 Two Cases Studies of Fifth Grade Students ReasoningAbout Levers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Paula Guerra, Linda Hernndez, Ahyoung Kim, MuhsinMenekse, and James Middleton

27 Dont Disrespect Me: Affect in an Urban Math Class . . . . . . . 313Roberta Y. Schorr, Yakov M. Epstein, Lisa B. Warner,and Cecilia C. Arias

Section 8 How Do Classroom Modeling Communities Develop?28 Interdisciplinary Modeling Instruction: Helping Fifth

Graders Learn About Levers . . . . . . . . . . . . . . . . . . . . . 327Brandon Helding, Colleen Megowan-Romanowicz,Tirupalavanam Ganesh, and Shirley Fang

29 Modeling Discourse in Secondary Scienceand Mathematics Classrooms . . . . . . . . . . . . . . . . . . . . 341M. Colleen Megowan-Romanowicz

30 A Middle Grade Teachers Guide to Model Eliciting Activities . . 353Della R. Leavitt and Cynthia M. Ahn

31 The Students Discussions in the Modeling Environment . . . . . 365Joneia Cerqueira Barbosa

32 The Social Organization of a Middle School MathematicsGroup Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 373William Zahner and Judit Moschkovich

viii Contents

33 Identifying Challenges within Transition Phasesof Mathematical Modeling Activities at Year 9 . . . . . . . . . . . 385Gloria Stillman, Jill Brown, and Peter Galbraith

34 Realistic Mathematical Modeling and Problem Posing . . . . . . . 399Cinzia Bonotto

35 Modeling in Class and the Development of Beliefs aboutthe Usefulness of Mathematics . . . . . . . . . . . . . . . . . . . . 409Katja Maass

Section 9 How Do Teachers Develop Models of Modeling?36 Insights into Teachers Unconscious Behaviour

in Modeling Contexts . . . . . . . . . . . . . . . . . . . . . . . . . 423Rita Borromeo Ferri and Werner Blum

37 Future Teachers Professional Knowledge on Modeling . . . . . . 433Gabriele Kaiser, Bjrn Schwarz, and Silke Tiedemann

38 Theory Meets Practice: Working Pragmatically WithinDifferent Cultures and Traditions . . . . . . . . . . . . . . . . . . 445Fco. Javier Garca, Katja Maass, and Geoff Wake

39 Secondary Teachers Learn and Rene Their KnowledgeDuring Modeling Activities in a Learning CommunityEnvironment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459Csar Cristbal Escalante

40 An Investigation of Teachers Shared Interpretationsof Their Roles in Supporting and Enhancing GroupFunctioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471Betsy Berry

41 Mathematical Modeling: Implications for Teaching . . . . . . . . 481Maria Salett Biembengut and Nelson Hein

42 A Professional Development Course with an Introductionof Models and Modeling in Science . . . . . . . . . . . . . . . . . 491Genaro Zavala, Hugo Alarcon, and Julio Benegas

43 Modeling as Isomorphism: The Case of Teacher Education . . . . 501Sergei Abramovich

44 Mathematical Modeling and the Teachers Tensions . . . . . . . . 511Andria Maria Pereira de Oliveira and Jonei Cerqueira Barbosa

45 A Case Study of Two Teachers: Teacher Questionsand Student Explanations . . . . . . . . . . . . . . . . . . . . . . 519Lisa B. Warner, Roberta Y. Schorr, Cecilia C. Arias,and Lina Sanchez

Contents ix

46 Pre-service Teachers Perceptions of Model Eliciting Activities . . 531Kelli Thomas and Juliet Hart

Section 10 How Do New Technologies Inuence Modeling in School?47 Modeling Practices with The Geometers Sketchpad . . . . . . . . 541

Nathalie Sinclair and Nicholas Jackiw

48 A Principal Components Model of Simcalc Mathworlds . . . . . . 555Theodore Chao, Susan B. Empson, and Nicole Shechtman

49 Modeling Random Binomial Rabbit Hops . . . . . . . . . . . . . 561Sibel Kazak

50 Investigating Mathematical Search Behavior UsingNetwork Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 571Thomas Hills

51 Mathematical Modeling and Virtual Environments . . . . . . . . 583Stephen R. Campbell

Section 11 What is The History of Modeling in Schools?52 On the Use of Realistic Fermi Problems in Introducing

Mathematical Modelling in Upper Secondary Mathematics . . . . 597Jonas Bergman rlebck and Christer Bergsten

53 The Dutch Maths Curriculum: 25 Years of Modelling . . . . . . . 611Pauline Vos

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Contributors

Sergei Abramovich State University of New York at Potsdam, Potsdam, NY, USA

Cynthia M. Ahn Thorp Scholastic Academy, Chicago, IL, USA

Hugo Alarcon Tecnologico de Monterrey, Monterrey, Mxico

Burkhard Alpers University of Aalen, Aalen, Germany

Miriam Amit Ben-Gurion University of the Negev, Beer-Sheva, Israel

Cecilia C. Arias Rutgers University, Rutgers, NJ, USA

Jonei Cerqueira Barbosa State University of Feira De Santana, Feira DeSantana, Brazil

Julio Benegas Universidad de San Luis, San Luis, ArgentinaJonas Bergman rlebck Linkpings Universitet, Linkping, SwedenChrister Bergsten Linkpings Universitet, Linkping, Sweden

Betsy Berry Indiana University-Purdue University, Fort Wayne, IN, USA

Mary Bestereld-Sacre University of Pittsburgh, Pittsburgh, PA, USA

Maria Salett Biembengut University Regional of Blumenau (FURB), Blumenau,Brasil

Werner Blum University of Kassel, Kassel, Germany

Cinzia Bonotto Department of Pure and Applied Mathematics, University ofPadova, Padova, Italy

Jill Brown Australian Catholic University, Melbourne, VIC, AustraliaStephen R. Campbell Simon Fraser Simon Fraser University, British Columbia,Canada

Monica E. Cardella School of Engineering Education, Purdue University, WestLafayette, IN, USA,

Guadalupe Carmona The University of Texas, Austin, TX, USA

xi

xii Contributors

Theodore Chao The University of Texas at Austin, Austin, TX, USA

C. Christou University of Cyprus, Nicosia, Cyprus

Renee M. Clark University of Pittsburgh, Pittsburgh, PA, USA

Rosalind Crouch University of Hertfordshire, Hateld, UK

Sarah M. Davis Learning Sciences Lab, National Institute of Education,Singapore

Angeles Dominguez Tecnologico de Monterrey, Monterrey, Mexico

Susan B. Empson The University of Texas at Austin, Austin, TX, USA

Lyn D. English Queensland University of Technology, Brisbane, AustraliaYakov M. Epstein Rutgers University, New Brunswick, NJ, USA

Csar Cristbal Escalante Cinvestav-IPN, Mexico, MexicoShirley Fang Arizona State University, Tempe, AZ, USA

Thomas Fennewald Indiana University, Bloomington, IN, USA

Rita Borromeo University of Hamburg, Hamburg, Germany

Peter L. Galbraith University of Queensland, Brisbane, QLD, AustraliaTirupalavanam Ganesh Arizona State University, Tempe, AZ, USA

Fco. Javier Garca University of Jan, Jan, SpainGilbert Greefrath University of Cologne, Cologne, Germany

Steven Greenstein The University of Texas, Austin, TX, USA

Paula Guerra Arizona State University, Tucson, AZ, USA

Christopher R. Haines City University, London, UK

Guershon Harel University of California, San Diego, CA, USA

Juliet Hart Arizona State University, Phoenix, AZ, USANelson Hein University Regional of Blumenau (FURB), Blumenau, BrasilBrandon Helding Arizona State University, Tempe, AZ, USA

Linda Hernndez Arizona State University, Tucson, AZ, USADavid Hestenes Arizona State University, Tempe, AZ, USA

Thomas Hills University of Basel, Basel, Switzerland

Celia Hoyles London Knowledge Lab, Institute of Education, University ofLondon, London, UK

Nicholas Jackiw KCP Technologies, Emeryville, CA, USA

Contributors xiii

Irma Jan Ben-Gurion University of the Negev, Beer-Sheva, IsraelTomas Hjgaard The Danish School of Education, rhus University, DenmarkGabriele Kaiser University of Hamburg, Hamburg, Germany

Sibel Kazak University of Massachusetts Amherst, Amherst, MA, USA

Ahyoung Kim Arizona State University, Tucson, AZ, USA

Christine Larson Indiana University; Bloomington, IN, USA

Della R. Leavitt University of Illinois, Chicago, IL, USA

Richard Lesh Indiana University, Bloomington, IN, USA

Katja Maass University of Education Freiburg, Freiburg, GermanyColleen Megowan-Romanowicz Arizona State University, Tempe, AZ, USA

Muhsin Menekse Arizona State University, Tucson, AZ, USA

James Middleton Arizona State University, Tucson, AZ, USAJudit Moschkovich University of California, Santa Cruz, CA, USANicholas Mousoulides University of Cyprus, Nicosia, Cyprus

Dorit Neria Ben Gurion University of the Negev, Beer-Sheva, Israel

Mogens Niss IMFUFA, Department of Science, Roskilde University, Roskilde,Denmark

Richard Noss London Knowledge Lab, Institute of Education, University ofLondon, London, UK

Michael Oehrtman Arizona State University, Tucson, AZ, USA

Andria Maria Pereira de Oliveira State University of Feira de Santana, Feirade Santana, Brazil

M. Pittalis University of Cyprus, Nicosia, Cyprus

Chris Rasmussen San Diego State University, San Diego, CA, USA

Adolf J.I. Riede Ruprecht-Karls-University, Heidelberg, GermanyLina Sanchez Rutgers University, Newark, NJ, USA

Roberta Y. Schorr Rutgers University, Newark, NJ, USA

Bjrn Schwarz University of Hamburg, Hamburg, GermanyNicole Shechtman SRI International, Menlo Park, CA, USA

Larry J. Shuman University of Pittsburgh, Pittsburgh, PA, USANathalie Sinclair Simon Fraser University, Burnaby, Canada

Robert Speiser Brigham Young University, Provo, UT, USA

xiv Contributors

Bob Speiser Brigham Young University, Provo, UT, USA

Bharath Sriraman The University of Montana, Missoula, MT, USA

Gloria Stillman University of Melbourne, Melbourne, VIC, Australia

Kelli Thomas The University of Kansas, Lawrence, KS, USA

Silke Tiedemann University of Hamburg, Hamburg, Germany

Pauline Vos University of Amsterdam, Amsterdam, Netherlands

Geoff Wake University of Manchester, Manchester, UK

Chuck Walter Brigham Young University, Provo, UT, USA

Lisa B. Warner Rutgers University, Rutgers, NJ, USA

Randall Young Indiana University, Bloomington, IN, USA

William Zahner University of California, Santa Cruz, CA, USA

Michelle Zandieh Arizona State University, Tucson, AZ, USA

Genaro Zavala Tecnologico de Monterrey, Monterrey, Mxico

Judith Zawojewski Illinois Institute of Technology, Chicago, IL, USA

Chapter 1Introduction: ICTMA and the Teachingof Modeling and Applications

Gabriele Kaiser

Applications and modeling and their learning and teaching in school and universityhave become a prominent topic in the last decades in view of the growing world-wide importance of the usage of mathematics in science, technology and everydaylife. Given the worldwide impending shortage of youngsters who are interested inmathematics and science it is highly necessary to discuss possibilities to changemathematics education in school and tertiary education towards the inclusion ofreal world examples and the competencies to use mathematics to solve real worldproblems.

These concerns were the starting point for the establishment of the ICTMAgroup an international community of scholars and researchers supportingthe International Conferences on the Teaching of Mathematical Modeling andApplications. ICTMA has been concerned with research, teaching and practice ofmathematical modeling since 1983. From the beginning, the aim of the ICTMAgroup has been to foster both teaching and research about mathematical modelingand the ability to apply mathematics to genuine real world problems in primary,secondary, and tertiary educational environments as well as in teacher education,and education in professional and workplace environments.

From the outset ICTMA has maintained the integrity of its focus, which has botha mathematical and an educational component. This makes a distinction from amathematical focus on the one hand, and a mathematics education context in whichthe mathematics need not to have a connection with applications and modeling onthe other hand. Thus a distinctive aspect of ICTMA is the interface it provides forcollaboration between those whose main activity lies in applying mathematics, butwho have an informed interest in sharing these abilities and developing relevantskills in others (an educational focus), and those whose principal afliations arewithin education, but who have a commitment to supporting the effective applicationof mathematics to problems outside itself.

G. Kaiser (B)University of Hamburg, Hamburg, Germanye-mail: [email protected]

Gabriele Kaiser is Current President of ICTMA

R. Lesh et al. (eds.), Modeling Students Mathematical Modeling Competencies, 1

DOI 10.1007/978-94-007-6271-8_1, Springer Science+Business Media Dordrecht 2013 International Perspectives on the Teaching and Learning of Mathematical Modelling,

2 G. Kaiser

These goals have been promoted by biennial conferences the InternationalConferences on the Teaching of Mathematical Modeling and Applications takingplace since 1983 in various parts of the world, namely Great Britain (1983, 1985,1995, 2005), USA (1993, 2003, 2007), China (2001), Australia (1997), Germany(1987), Denmark (1989), Netherlands (1991), Portugal (1999).

At ICTMA conferences various themes and topics have been discussed, amongstothers:

How far are, at an international level applications and modeling examples,implemented in mathematics education at school and tertiary level?

How is instruction to structure which is adequate for the teaching and learning ofapplications and modeling and considering country-specic differences?

How can the development of modeling competencies be evaluated and promoted? How can authentic applications of mathematics in industry and technology be

introduced into mathematics instruction at various levels? How can modern technologies, necessary for authentic modeling examples, be

introduced in mathematics instruction at school and tertiary level?

At the last conferences and especially at the conference ICTMA13, of whichimportant chapters are presented in this book, various answers to these questionsand topics have been explored. However, it is obvious that real world and model-ing examples still do not have a high level of importance in mathematics educationat school and tertiary level, indispensable for a modern society. So, amongst othertopics, it seems necessary to investigate affective barriers of teachers and studentsagainst the inclusion of these kinds of examples at various levels. Furthermore cog-nitive barriers which prevent students to develop modeling competencies need tobe explored. First case-studies exist, but large-scale studies seem to be necessary inorder to broaden and deepen our knowledge about these aspects.

In order to broaden the audience ICTMA became, in 2003, an afliated studygroup of the International Commission on Mathematical Instruction, which givesthe opportunity to run special sessions during ICME and to carry out ICMI-study 14on that theme ICMI Study. With these strong relations to the mathematics educationdebate, and to the intense connection to the engineering and applied mathemat-ics discussion, ICTMA has a unique avour and will hopefully be able to bringin genuine real world and modeling examples into mathematics instruction aroundthe world. This book expresses this avour in a special way, it combines a designapproach to mathematics instruction at various levels with many empirical studiesfrom all over the world, not only from Western countries, but also in growing num-ber from Asian countries and countries of the southern hemisphere. This growinginternationality of the debate shows the joint worldwide concern about the lack ofreal world and modeling examples in mathematics education at school and tertiarylevel. But despite this growing internationality and the joint topics discussed all overthe world, approaches from various parts of the world are still inuenced by theirunderlying educational philosophies, their socio-cultural conditions and the studiesand results presented in the articles of this book displays the variety of this debate.Future trends concerning the teaching and learning of applications and modelingshould build on this variety and include it in its future debates.

Part IThe Nature of Models & Modeling

Chapter 2Introduction to Part I Modeling: What Is It?Why Do It?

Richard Lesh and Thomas Fennewald

At ICTMA-13, where the chapters in this book were rst presented, a variety ofviews were expressed about an appropriate denition of the term model andabout appropriate ways to think about the nature of modeling activities. So, it isnot surprising that some participants would consider this lack of consensus to be apriority problem that should be solved by a research community that claims to beinvestigating models and modeling.

We certainly agree that conceptual fuzziness is not a virtue in a research commu-nity especially if it impedes communication among members of the community.Furthermore, we agree that increasing clarity about key constructs is an impor-tant goal of research. Nonetheless, we also believe that, especially at early stagesof theory development, a certain amount of diversity in thinking is as healthy forresearch communities as it is for (for example) engineers who are at early brain-storming stages in the design of space shuttles, sky scrapers, or transportationsystems. Furthermore, we believe that the mathematics education research com-munity in particular has suffered from more than enough pressure for prematureideological orthodoxy.

The theme of ICTMA-13 was modeling students modeling competencies; and,in many of the research methodologies that are described throughout this book,students develop models to describe or design real life artifacts or tools; teach-ers develop models to describe students modeling competencies or to designproductive learning environments; and, researchers develop models of interactionsamong students, teachers, and learning environments. So, the goal of many ofour most productive studies focus on the development of powerful, sharable, andre-useable models which then, in turn, inuence theory development. But, model

R. Lesh (B)Indiana University, Bloomington, IN, USAe-mail: [email protected]

T. Fennewald (B)Indiana University, Bloomington, IN, USA


DOI 10.1007/978-94-007-6271-8_ , Springer Science+Business Media Dordrecht 2013 2International Perspectives on the Teaching and Learning of Mathematical Modelling,

6 R. Lesh and T. Fennewald

development research is not the same as theory development research. For exam-ple, in theory-driven research, the theory determines what questions are appropriateto ask, what kind of evidence needs to be collected, how the information should beinterpeted, analyzed, and assessed and when the question has been answered or theissue resolved. But, in model-development research (or design research), the prob-lem arises from a real world decision-making situation. Thus, this work is morelike engineering than a pure science. As engineers (and other design scientists)often emphasize (see Zawojewski et al., 2009):

Engineering is the science of solving real life problems where you dont haveenough time, or money, or other resources and when multiple stake holdersoften hold partly conicting views about the nature success (low costs versushigh quality, simplicity versus completeness, and so on).

Realistic solutions to realistically complex problems usually need to integrateideas and procedures drawn from more than a single discipline, or theory, ortextbook topic area.

When high stakes decision making issues arise in real life situations, it usuallyis important to design for success (power, sharability, re-usability) not simplyto test for it.

In design research, many of the things that we most need to understand andexplain are things that we ourselves are in the process of developing or design-ing. For example, in math/science education, these things range from studentsconceptual systems to curriculum materials which include systems of activitiesfor learning or assessment. And, in each of these cases, as soon as we come tobetter understand the thing being developed or designed, we tend to changethem so that another cycle of adaptation is needed. (Note: This is one reasonwhy scientists do not speak of developing a single theory of space shuttles.)

In the book, Beyond Constructivism: Models & Modeling Perspectives onMathematics Problem Solving, Learning & Teaching (Lesh and Doerr, 2003), wedescribe a variety of reasons why we consider MMP to be a blue color perspec-tive which is like engineering more than it is like pure sciences such as physics,chemistry, or mathematics. In fact, for many of the same reasons why Pragmatistssuch as James, Pierce, or Dewey considered Pragmatism to be more of a frameworkfor developing theories rather than being a theory in itself, we consider MMP to bea framework for developing models of students modeling.

According to MMP, students, teachers, and researchers all are in engaged inmodel development. Consequently, MMP research assumes that: (a) similar prin-ciples apply not only to the model development activities of students, but also tothe model-development activities of teachers and researchers, and (b) researchersearly-iteration models are expected to be characterized by conceptual inadequaciessimilar to those that characterize the early-iteration interpretation systems of stu-dents or teachers. . .. MMP research recognizes that we the researchers who studymodeling are humbly still in the early phases of our own model development.And, just as in the model development activities of the students and teachers that we

2 Introduction to Part I Modeling 7

study, we assume that we will only make progress if we go through multiple cyclesof expressing > testing > revising our current ways of thinking. So just as Darwinemphasized in the development of other kinds of complex adaptive systems (1859),evolution of ways of thinking about teaching and students learning is only likely tooccur if provisions are made to encourage diversity, selection, communication, andaccumulation (Sawyer, 2006).

In our own blue collar research on models and modeling, we take the followingto be a useful rst-iteration denition of a model. A model is a system for describing(or explaining, or designing) another system(s) for some clearly specied purpose.

The preceding denition seems simple and straightforward, but one thing that welike about it is that it has a clear history of pressing us toward important researchablequestions. Furthermore, best of all, and unlike terms such as cognitive structuresor schemes or other terms that are favored by cognitive scientists or constructivistphilosophers, the preceding denition uses the term model in the same way that theterm is used in mature science elds like physics or engineering. That is, a model issystem that is used to describe (or interpret) another system of interest in a purpose-ful way. However, whereas physicists and engineers tend to focus on only the writtensymbolic aspects of the models they develop, when math/science educators inves-tigate what it means to understand these models, it inevitably becomes apparentthat the written and spoken embodiments that scientists emphasize scientists usuallyrepresent only something like the tip of an iceberg. For example, in order for scien-tic models to be useful, understanding them usually involves a variety of diagrams,concrete models, experience-based metaphors, and other expressive media in addi-tion to technical spoken language and symbol systems, each of which emphasizesome aspects (but deemphasize others) for the thing that they are used to describe,or explain, or design. Furthermore, model development often involves dimensionsof development such as intuition-to-formalization, concrete-to-abstract, situated-to-decontextualized, specic-to-general, implicit-to-explicit, global-to-analytic,1 andso on. Furthermore, during early stages of development, models often function inways that are more like windows that we look through rather than as objectsthat we look at; and, they often function in ways that are rather unstable. That is,when the model developer focuses on forest-level or big picture interpretationsof the thing being described, they often neglect to keep in mind tree-level detailsabout the thing being described. In other words, even for experienced scientists,but especially for young students, and especially during early stages of model devel-opment, a model developers early interpretation of the thing being described isoften far more situated, piecemeal, and non-analytic than most traditional theoriesof learning consider it to be.

Should early interpretations (such as those that function intuitively, informally,or in piecemeal or unstable ways) be referred to as models? We believe that such

1 When we speak of these dimensions of development, we recognize that, in specic real lifesituations where model development is needed, the most useful model is not necessarily the onethat is most complex, most formal, most abstract, or most decontextualized.


questions should be resolved through research not through political consensusbuilding. This point brings out a point that we especially like about the term mod-els and modeling. That is the term allows room for debates among researcherswith opposing points of view. And, we believe that debates about the nature ofvarious models have exhibited a clear history of leading to researchable questions,testable hypotheses, and a variety of model validation activities. This is why we saythat Models and Modeling is not so much a view of how learning works as it is amethodological approach and framework for investigating learning.

Finally, we should not neglect to mention some other reasons why authorsthroughout this book view research on models and modeling to be especiallyimportant for mathematics education researchers. First, in nearly every eld whereresearchers have investigated similarities and differences between experts andnovices (or between successful learners or problem solvers versus those who areless successful), results have shown that expertise not only involves doing thingsdifferently (or better), it also involves seeing or interpreting things differently (orbetter). And, in mathematics and the natural sciences, interpretation developmentmeans model development. Furthermore, in students lives beyond school, the abil-ity to describe or explain things mathematically is one of the main factors that isneeded in order for mathematics to be useful.

Throughout this volume, even though most of the authors generally support theideas outlined above, conicting views emerged even during ICTMA-13. One rea-son why this happened is that the organizers of ICTMA-13 made special efforts toattract not only mathematics educators from more than 25 different countries, butequal efforts also were made to attract leading science educators, engineering edu-cators, and mathematics educators focusing on mathematical and scientic thinkingbeyond school. Yet, even with such diversity, we know of no cases where differencesin perspectives did not appear to be leading toward productive adaptations for thecommunity as a whole.

This volume beings with chapters that aim to answer the fundamental question:what are models? Hestenes outlines important relationships among various formsof modeling and elds that apply modeling theory. In his plenary article, he distin-guishes between uses of the term model to refer to a conceptual mental modeland uses of model to refer to a publically shared model. By so doing, he setsthe stage for a discussion that relates the act of modeling in professional and non-professional so-called everyday life to research in both science and mathematicseducation, thus giving a full view of modeling theory as describing the process bywhich a form of knowledge, modeling, is developed through lived acts that are bothmental and social in nature. Niss continues this examination of the question of whatmodels are and what modeling is with studies into how to model the learning ofmodeling itself. His study hints at perhaps the deepest and most philosophical of allquestions related to the origin of modeling competencies and knowledge: What isthe origin of understanding how to model? From his study of three different mod-eling problems, he suggests that the capability to model is a learned capacity. Theteaching of modeling, he notes, can be approached through activities that encouragecognitive dissonance of a type that drives the emergent modeling of Gravemeijer

2 Introduction to Part I Modeling 9

or the model eliciting activities described by Zawojewski or Carmona, or the kind ofscaffolded approaches described by Kaiser. Niss considers each as possible modesfor teaching modeling. Larson and colleagues also address the question of whatmodels are while also looking ahead to the future of modeling in the curriculum andexamining ways modeling can meet the challenge to provide students opportunitiesto create and rene ideas for themselves.

With a notion about what models are, discussion then proceeds to examinationsof where models and the modelers who make them are found. Noss and Hoylestake research about the pedagogical aspects of modeling beyond the classroominto the workplace, performing an investigation as to what Techno-mathematicalLiteracies are needed to understand manufacturing processes. Cardella examinesthe modeling of engineering undergraduates and graduates who might not other-wise see the connection of the math they are learning in the classroom to the realworld work they will perform. This study is followed by Alpers who examines themodeling of mechanical engineers outside of school contexts, looking at math andmodeling performed on the job. Together, these studies show both the applicationand real-world relevance of modeling in studies beyond the school setting while atthe same time demonsrating as well as to in-school modeling studies have to studiesin other disciplines, and in higher level mathematics course.

The modeling process is examined by Larson by employing one of Leshsoriginal modeling activities, Summer Jobs, in a study of how students developquantitative reasoning needed in real-world problems, supporting concluding thatchanges in the perception students have of relationships among quantities leadsto the development of better quantitative reasoning ability along a progression ofPiagetian-like stages. Using a related activity, University Cafeteria, Mousoulidesand Sriraman examine the progression in mathematical understandings made bymiddle school students over the course of their schooling.

The question of what is needed for modeling to occur is taken on by Galbraith,Stillman, and Brown, who provide insight into what has been recognized as oneof the most critical aspects of setting up a modeling experience for learners thatis, creating a meaningful context in which it is hoped the modeling learners willengage in signicant concepts. They explore this from a uniquely Australian per-spective, investigating the response of Australian students to Australian themedmodeling activities. Haines and Crouch further investigate this issue while explor-ing the intricacies of modeling cycles that distinguish modeling activities from otherassessments, as they expand their discussion to include the assessment of modelingcompetencies in general. Amit and Jan then present an extension of model-elicitingproblems into model-eliciting environments which are designed to optimize thechances that signicant modeling activities will occur. Speiser and Walter theninvestigate another aspect critical to modeling activities and models in general. Thatis, models are tools created and applied for a purpose. And, purposes are stronglyinuenced by communities and societies in which designers and modelers operate.Clark and colleagues then discuss lessons learned from a pilot course they designedto elicit systems thinking in which they employed and modied MEAs designed to


create a need for modeling. Davis then concludes the section by exploring the needfor tools that can help teachers make sense of data collected in generative activities.

The question of how models develop is examined by Riede who studies studentsas they explore and rediscover Webers law, progressing through modeling cycles,during their activities. Amit and Neria then examine the express > test > revisecycles of students engaged in generalizing pictorial linear pattern problems, ndingthat many students applied recursive strategies despite the appropriateness of globalstrategies. Work in the study of conceptual and model development is also importantin the work of Dominguez who uses modeling activities to reveal multiplicity instudents ways of thinking even when one nal answer is agreed upon.

Finally the question of what ways modeling is different from solving traditionaltextbook word problems is addressed starting with Zawojewski, who asks whatresearch implications and distinctions between the two can be drawn. She is fol-lowed by Carmona and Greenstein who nd that modeling is suggestive of a spiralcurriculum where powerful and underlying themes are revisited constantly notarrived at permanently. Jensen then shows and investigates distinctions betweenmodeling and problem solving competencies. Greefrath concludes with the exam-ination of students planning processes, noting the differences in strategies usedbetween modeling and problem solving questions.

These chapters reveal many facets of both modeling activities and the modelingresearch being conducted around student modeling activities. Although this collec-tion of contributions is not intended to provide a comprehensive overview of thework being done to study the nature of modeling, this collection certainly gives thereader a diverse introduction to some of the most important frontiers in research onmodeling and applications.

References

Lesh, R., and Doerr, H. (2003). Beyond Constructivist: A Models & Modeling Perspective onMathematics Teaching, Learning, and Problems Solving. In H. Doerr, and R. Lesh (Eds.),Hillsdale, NJ: Lawrence Erlbaum Associates.

Sawyer, R. K. (2006). Explaining Creativity: The Science of Human Innovation. New York: OxfordUniversity Press.

Zawojewski, J., Diefes-Dux, H., and Bowman, K. (Eds.) (2009). Models and Modelingin Engineering Education: Designing Experiences for All Students. Rotterdam: SensePublications.

Section 1What Are Models?

Chapter 3Modeling Theory for Math and ScienceEducation

David Hestenes

Abstract Mathematics has been described as the science of patterns. Natural sci-ence can be characterized as the investigation of patterns in nature. Central toboth domains is the notion of model as a unit of coherently structured knowledge.Modeling Theory is concerned with models as basic structures in cognition as wellas scientic knowledge. It maintains a sharp distinction between mental models thatpeople think with and conceptual models that are publicly shared. This supports aview that cognition in science, math, and everyday life is basically about makingand using mental models. We review and extend elements of Modeling Theory as afoundation for R&D in math and science education.

3.1 Introduction

Why should a theoretical physicist be concerned about mathematics education? Myanswer will be a long one, but let me begin by introducing you to some of myesteemed colleagues in Box 3.1. These fellows are such good physicists that mostif not all of them would be worthy candidates for a Nobel Prize if they were alivetoday. You may know that they are quite good at mathematics as well! Indeed, math-ematics textbooks often count them as mathematicians without mentioning that theyare physicists. I dare say, however, that they would be mightily offended to hearthat they are not counted as physicists. Likewise, I am more than miffed whenreviewers of my math-science education proposals discount my qualications asa mathematician because my doctorate is in physics. Like the fellows in the list,I regard my scientic research as equal parts mathematics and physics. The factthat the education establishment does not recognize that theoretical physicists areuniquely well-qualied to address education at the interface between mathematicsand science is traceable to a serious problem within the mathematics professionitself.

D. Hestenes (B)Arizona State University, Tempe, AZ, USAe-mail: [email protected]


DOI 10.1007/978-94-007-6271-8_ Springer Science+Business Media Dordrecht 2013 3,International Perspectives on the Teaching and Learning of Mathematical Modelling,

14 D. Hestenes

Box 3.1 Some distinguished theoretical physicists

NewtonEulerGaussLagrangeLaplace

CauchyPoincarHilbertWeylVon Neumann

Of course, the list of physicist/mathematicians in Box 3.1 is far from complete.Many of my favorite colleagues are omitted. But there are at least two good reasonswhy the list ends in the middle of the twentieth century. The distinguished Russianmathematician Arnold (1997) put his nger on both in a widely circulated diatribeOn Teaching Mathematics, wherein he asserts

Mathematics is a part of physics. Physics is an experimental science, a part of naturalscience. Mathematics is the part of physics where experiments are cheap.

In the middle of the 20th century it was attempted to divide physics and mathematics. Theconsequences turned out to be catastrophic. Whole generations of mathematicians grew upwithout knowing half of their science and, of course in total ignorance of other sciences.

Mathematician-cum-historian Morris Kline (1980) has thoroughly documentedthe disastrous divorce of the mathematics profession from physics, which began inthe latter part of the nineteenth century. He estimated that, by 1980, eighty percentof active mathematicians were ignorant of science and perfectly happy to remainthat way.

The divorce is thus an incontrovertible fact, but how disastrous can it be if onlya minority of mathematicians like Arnold and Kline are alarmed? Isnt it a naturalconsequence of necessary specialization in an increasingly complex society? Andisnt Arnolds claim of intimacy between math and physics merely a personal opin-ion? Surely the majority of mathematicians believe that mathematics is a completelyautonomous discipline.

I claim that the answer to all these questions is a resounding NO! Indeed, I submitthat the single most serious deciency in US math education is the divorce of math-ematics from physics in the education of mathematicians, in the training of mathteachers, and in the structure of the K-12 (-16-20) curriculum. Moreover, this is nota simple deciency in the breadth of education; it is a fundamental problem in con-ceptual learning and cognition. I claim that cognitive processes for understandingmath and physics are intimately linked and fundamentally the same! Indeed, I claimthat Physics is cognitively basic to quantitative science in all domains!!

Before delving into the deep cognitive issues, let us note some obvious academicconsequences of the math/physics divorce. Training in mathematics is essentialfor all physicists, amounting to the equivalent of a dual major in mathematicsfor theoretical physicists. But math courses have become increasingly irrelevant to

3 Modeling Theory 15

physics, so physics departments offer their own courses in Methods of mathemat-ical physics at both graduate and undergraduate levels, with additional courses inmore specialized topics like group theory. One consequence is a narrowing of thephysicists appreciation of mathematics. But a far more serious consequence is thereduction in opportunity for math majors to learn about vital connections to physics.This continues through graduate school, so the typical math PhD is ill-prepared forwork in applied mathematics. Some math departments have attempted to remedythis deciency with courses in mathematical modeling, but mathematical modelingwithout science is like the Cheshire cat: form without content! This is one of thedeep cognitive issues that we need to address.

Far and away the most serious consequence of the math/physics divide is thedecient preparation of K-12 math teachers! The neglect of geometry and excess offormalism that Arnold (1997) deplores in the university math curriculum has propa-gated to teacher preparation. There is abundant evidence that most teachers see theirjob as teaching formal rules and algorithms. Few have even a minimal understand-ing of Newtonian physics, so most are inept at applying algebra and calculus evento simple problems of motion. Consequently, high school physics courses are forcedto revisit the prerequisite math knowledge that students are supposed to bring fromyears of math instruction. As the math courses lack the intuitive base necessary forconceptual understanding, students are forced to rote learning, which has a shorthalf-life, so their recollection of math has decayed to nearly to zero by the time theyget to college.

I doubt that these crippling deciencies in math education can be fully resolvedwithout a sea change in the culture of mathematics. To drive such revolutionarychange we need a coherent theory of mathematical learning and cognition supportedby a substantial body of empirical evidence. My purpose here is to report on progressin that direction.

3.2 Origins of Modeling Theory

I have been investigating the epistemology of science and mathematics across thefull range of academic disciplines for half a century. As that may sound implausible,let me describe the unusual initial conditions that got me started.

My father was an accomplished mathematician who helped organize Americanmathematicians to support the war effort in WWII. Consequently, he got to knowmathematicians across the country on a rst name basis. That served him well when,shortly after the war, he was wooed from the University of Chicago to build a rstrank math department at UCLA. He was also appointed director of the Institutefor Numerical Analysis (INA), where the National Bureau of Standards installedthe rst electronic computer in western United States. With a solid backgroundin pure mathematics (Calculus of Variations), he blossomed then into a pioneerin the edgling elds of Control Theory and Numerical Analysis, for which hewas posthumously inducted into the Hall of Fame for Engineering, Science andTechnology (HOFEST). The well-funded, vigorous research activity at the INA and

16 D. Hestenes

the rapid emergence of the UCLA math department attracted a steady stream ofdistinguished mathematicians from around the world, for which my father was usu-ally the host. He was at the acme of his career when I entered graduate school in1956.

My undergraduate major in philosophy introduced me to the great conundrumsof epistemology, and I was inspired by Bertrand Russell to switch to physics insearch of answers. When I started graduate school, my father found me an ofce inthe INA where I was surrounded by a whirlwind of excited activity about the begin-nings of Computer Science and Articial Intelligence. That prepared me to followthe evolution of both those elds throughout my career, though my main efforts wereconcentrated on physics and mathematics. By my third year I hit upon GeometricAlgebra as the central theme of my scientic research. That induced me to rejectRussells logicist view of mathematics and sharpened my insight and interest in epis-temological and cognitive foundations. Throughout my graduate years in physics Ispent most of my time in the math department where I imbibed the culture of math-ematics. This strong association with mathematics continued throughout my careerin physics, anchored by relations with my father.

My diverse interests in cognitive science and theoretical physics converged inthe 1980s when I got embroiled in problems of improving introductory physicsinstruction. Like any conrmed theoretician, I framed the problems in the contextof developing a theory with testable empirical consequences. Largely from my ownexperience as a scientist I identied scientic models and modeling as the core ofscientic knowledge and practice, and I proceeded to incorporate it into the designof physics instruction with the help of brilliant graduate students Ibrahim Hallounand Malcolm Wells. Thus began a program of educational R&D guided by an evolv-ing research perspective that I called Modeling Theory in a 1987 chapter. Thatprogram has continued to evolve beyond all my expectations. An up-to-date reviewof Modeling Theory is available in a recent chapter (Hestenes, 2007). The presentchapter is a continuation, introducing new material with only enough duplication tomake it reasonably self-contained. Therefore, it contains many gaps, some of whichcan be lled by consulting the earlier chapter, and others that I hope will stimulateoriginal research. For Modeling Theory is an enormous enterprise that amounts toa thematic approach to the whole of cognitive science. The best we can do here issample the major themes.

As schematized in Fig. 3.1, research on Modeling Theory has developed alongtwo complementary strands. The strand on the right investigates scientic mod-els and modeling practices that are explicit and observable. It provides a windowto structure and process in scientic and mathematical thinking that we aim topeek through. That involves us with the strand on the left, which will be our mainconcern.

You may ask, Why should one adopt a model-centered epistemology ofscience? There are three good reasons:

1. Theoretical: Models are basic units of coherently structured knowledge, fromwhich one can make logical inferences, predictions, explanations, plans and


Fig. 3.1 Modeling theory a research program

designs. One cannot make inferences from isolated facts or theoretical principles.A model can serve as inferential tool for the kind of structure it embodies.

2. Empirical: Models can be directly compared with physical things and pro-cesses. A theoretical hypothesis or general principle cannot be tested empiricallyexcept through incorporation in a model. Empirical data is meaningless withoutinterpretation supplied by a model.

3. Cognitive: Model structure is concretely embodied in physical intuition, where itserves as an element of physical understanding.

The third reason is based on the Modeling Theory of cognition set forth in thischapter.

3.3 Models and Concepts

The term model is usually used informally (hence ambiguously), but to make crucialtheoretical distinctions we need precise denitions. Although I have discussed thisissue at length before, it is so important that I revisit it with a slightly different slant.I favor the following general denition:

A model is a representation of structure in a given system.A system is a set of related objects, which may be real or imaginary, physical or

mental, simple or composite. The structure of a system is a set of relations amongits objects. The system itself is called the referent of the model.

We often identify the model with its representation in a concrete inscription ofwords, symbols or gures (such as graphs, diagrams or sketches). But it must not be

18 D. Hestenes

forgotten that the inscription is supplemented by a system of (mostly tacit) rules andconventions for encoding model structure. As depicted in Fig. 3.2, I use the termsymbolic form for the triad of elements dening a model. I chose the term deliber-ately to suggest association with the great work on symbolic forms by philosopherErnst Cassirer (Cassirer, 1953).

MODEL

representation

referentstructure

Fig. 3.2 Symbolic form of a model

We are especially interested in scientic models, for which I have often used thedenition: A (scientic) model is a representation of structure in a physical systemor process.

This differs from the general denition only in emphasis and scope. Its scopeis limited by assuming that the objects in a physical system are physical things.Nevertheless, the denition applies to all the sciences (including biology and socialsciences). Models in the various sciences differ in the kinds of structure that theyattribute to systems. The term process is included in the denition only for empha-sis; it refers to a change in the structure of a system. Thus a process model is anabstraction of structural change from a more complete model including objects inthe system.

In most discussions of scientic models the crucial role of structure is overlookedor addressed only incidentally. In Modeling Theory structure is central to the con-cept of model. The structure of a system (hence structure in its model) is dened asa set of relations among the objects in the system (hence among parts in the model).

Universal structure types: From studying a wide variety of examples, I have con-cluded that ve types of structure sufce to characterize any scientic model. As thisseems to be an important empirical fact, a brief description of each type is in orderhere.

Systemic structure: Its representation species (a) composition of the system (b)links among the parts (individual objects), (c) links to external agents (objects inthe environment). A diagrammatic representation is usually best (with objectsrepresented by nodes and links represented by connecting lines) because itprovides a wholistic image of the entire structure. Examples: electric circuitdiagrams, organization charts, family trees.

Geometric structure: species (a) conguration (geometric relations among theparts), (b) location (position with respect to a reference frame)

Object structure: intrinsic properties of the parts. For example, mass and chargeif the objects are material things, or roles if the objects are agents with complexbehaviors. The objects may themselves be systems (such as atoms composed of


electrons and nuclei), but their internal structure is not represented in the model,though it may be reected in the attributed properties.

Interaction structure: properties of the links (typically causal interactions).Usually represented as binary relations on object pairs. Examples of interactions:forces (momentum exchange), transport of materials in any form, informationexchange.

Temporal (event) structure: temporal change in the state of the system. Change inposition (motion) is the most fundamental kind of change, as it provides the basicmeasure of time. Measurement theory species how to quantify the properties ofa system into property variables. The state of a system is a set of values for itsproperty variables (at a given time). Temporal change can be represented descrip-tively (as in graphs), or dynamically (by equations of motion or conservationlaws).

Optimal precision in denition and analysis of structure is supplied by mathe-matics, the science of structure. This agrees with the usual notion of a mathematicalmodel as a representation in terms of mathematical symbols.

Now, here is a perplexing question that bothered me for decades: If the meaningof a scientic model derives from its physical interpretation, from whence comes themeaning of a mathematical model? Mathematical models are abstract, which meansthey have no physical referent!

It dawned on me during the last decade that the emerging eld of cognitive lin-guistics provides a revolutionary answer. Cognitive linguistics has revolutionizedthe eld of semantics by maintaining that the actual referents of language are men-tal models in the mind rather than concrete objects in an external world. It followsthat, if mathematics is the language of science, then the referents of mathematicalmodels must be mental models. Likewise, the proper referents of scientic modelsmust be mental models of physical situations, which are only indirectly related toreal physical systems through data, observation and experiment.

This implies a common cognitive foundation for math, science and language: Justas science is about making and using objective models of real things and events, socognition (in mathematics and science as well as everyday life) is about making andmanipulating mental models of imaginary objects and events!Let me sum up this revolution in semantics with a modied denition:

A conceptual model is a representation of structure in a mental model.As before, the representation in a conceptual model is a concrete inscription

that encodes structure in the referent. However, we make no commitment as towhat the structure of a mental model may represent. Henceforth, scientic andmathematical models are to be regarded as conceptual models. But the referentof a conceptual model is always a mental model, so its structure in the mind isinaccessible to direct observation. How, then, can this be an advance in ModelingTheory?

The answer is: It enables transfer from a Modeling Theory of scientic knowl-edge to a Modeling Theory of cognition in science and mathematics. Much is known

20 D. Hestenes

about the structure of scientic models. We seek to solve the inverse problem ofinferring the structure of mental models from the objective structure of scienticrepresentations. If that seems like an impossible task, note that it is commonplaceto infer thoughts in other minds from social interaction. Can we not make strongerinferences with the full resources of science? Here we have a modeling approach tothe theory of cognition, so we can draw on the whole corpus of results in cognitivescience for support and critique. I will not duplicate my previous reference to thatenormous literature (Hestenes, 2007). However, I should emphasize the special rele-vance of cognitive linguistics and point out that two recent introductions to the eld(Evans and Green, 2006; Croft and Cruse, 2004) provide a comprehensive overviewthat was difcult to put together only a few years ago.

Let me now propose the First Principle for a Modeling Theory of Cognition:I. Cognition is basically about making and manipulating mental models.I call this the Primacy (of Models) Principle, noting I have already tacitly invoked

a variant of it for the Modeling Theory of scientic knowledge. Commitment to thisprinciple might seem extreme, for I must admit that it is not to be found in thecognitive science literature from which I draw most of the supporting evidence.However, I contend that for a guiding research principle the standard is not that it istrue but that it is productive, by which I mean that it leads to signicant predictionsthat are empirically testable. Even if proved wrong, that would be quite an interest-ing result! In the meantime, we shall see that the primacy principle can carry us along way.

For a start, the Primacy Principle helps sharpen the denition of a concept, asit implies that concepts must refer to mental models, at least indirectly. As donebefore, I dene a concept as a {form, meaning} pair represented by a symbol (orassembly of symbols). In analogy to Fig. 3.2, I dene the symbolic form of aconcept as the triad in Fig. 3.3. Much like a model, the form of a concept is itsconceptual structure, including relations among its parts and its place within a con-ceptual system. The meaning of a concept is its relation to mental models. All thisis close enough to the usual loose denition of concept to conform to commonparlance. It provides then a foundation for a more rigorous analysis of importantconcepts.

CONCEPT

symbol

meaningform

Fig. 3.3 Symbolic form of a concept

We are now prepared to propose the Second Principle for a Modeling Theory ofCognition:

II. Mental models possess ve basic types of structure: systemic, geometric,descriptive, interactive, temporal.


I call this the Principle of Universal Forms, where the forms are the ve types ofstructure. Obviously, this is direct transfer to mental models of the structural typesidentied above for scientic models. Thus, it provides us immediately with a richsystem of conjectures about mental models to investigate and amend if necessary.Moreover, it brings along a rich system of basic concepts involved in characterizingthe forms.

Scholars will note strong similarity of the Universal Forms to Immanuel KantsForms of Intuition and Pure Concepts of the Understanding (Categories) [1781,1787]. This should not be surprising, since Kant engaged in a similar analysis ofcognition with special attention to the mathematics and physics of his day. Kant pro-poses his Categories as a complete list of universal forms for logical inference. InModeling Theory this should translate into universal forms for the synthesis (to useKants term) of mental models. A brief account of Kants transcendental knowl-edge analysis is given below, but detailed comparison with Modeling Theory willnot be attempted here. Today we have so much more factual information about thestructure of science and cognition to guide and support our conjectures. Even so,the relevance of Kants thinking to current cognitive science has been examined byLakoff and Johnson (1999).

The Universal Forms are also similar to semantic structures identied in cog-nitive linguistics, especially in the work of Leonard Talmy (see Evans and Green,2006). This is another rich area for comparative research that cannot be pursuedhere, though we shall touch on more ideas to throw into the mix.

Modeling Theory must ultimately account for the origins of structure in mentalmodels and its elevation through the creation of symbolic forms into shareable con-cepts and conceptual systems. Let me comment on the second part of this ambitiousresearch agenda. Taking for granted the existence of structured mental models inperceptual experience, we posit the human ability to make distinctions with respectto similarities and differences in model structure as the basic mechanism for creatingcategory concepts.

Cognitive research has established that there are two general types of categoryconcepts, which I shall distinguish by the non-standard terms implicit and explicitto emphasize an important point. Implicit concepts are determined by their mentalreferents, that is, they derive meaning from a web of associations with one or moremental models. For example, the concept dog derives meaning from a stored mentalimage of a prototypical dog. Most category concepts in natural language are of thistype (Evans and Green, 2006), though my brief comments do not do justice to thesubject. Implicit concepts could well be called empirical concepts, because theirstructures are built from experience in the mind of each individual.

In contrast to implicit concepts, which are grounded in private mental images,explicit concepts are dened by public representations. For explicit concepts, cat-egory membership is dened by a set of necessary and sufcient conditions. Thisis, of course, the classical concept of category that we inherited from Aristotle.It was only recently realized that ordinary (i.e. implicit) concepts are not of thistype. Nevertheless, the crucial concepts of science and mathematics must be of theexplicit type to qualify as objective knowledge.

22 D. Hestenes

3.4 Imagination and Intuition

Modeling Theory provides a foundation for precise denitions of important con-cepts in cognitive psychology. Human imagination is one such concept, importantand familiar to everyone, but elusive in cognitive science. Let us reinvigorate it herewith a denition that embodies the First Principle of Modeling Theory:

Imagination is the faculty for making and manipulating mental models.This squares well with a well-established line of research on narrative and dis-

course comprehension, which supports the view that the linguistic function of wordsis to activate, elaborate and modify mental models of objects and events in an imag-inary unfolding scene (Goldman et al., 2001). We are most interested here in thethesis that the very same cognitive process is involved in thinking mathematics andphysics. To sustain that thesis we must account for the unique features of cognitionin the scientic domain.

Since the latter part of the nineteenth century, mathematicians and philosophershave vigorously debated the foundations of mathematics with no sign of consensus(Shapiro, 1997). But all agree on a crucial role for mathematical intuition. Even thesupreme formalist, David Hilbert, approvingly quoted Kants famous aphorism: Allhuman knowledge begins with intuitions, thence passes to concepts and ends withideas. Though mathematical intuition is never mentioned in formal publications, itoften comes up in informal discussion among mathematicians, and subtle hints of itspresence appear in choices of mathematical terms and symbols. Recently, however,Lakoff and Nez (2000) have dared to shine the light of cognitive science on therecesses of mathematical thought. My aim is to do the same from the perspective ofModeling Theory.

Physical intuition is privately held in the same high regard by physicists thatmathematicians attribute to mathematical intuition. I submit these two kinds intu-ition are merely two different ways to relate products of imagination to the externalworld, as indicated in Fig. 3.4.

MentalModels

Real thingsand events

Symbolicrepresentations

physical intuition

mathematicalintuition

cognitivestructure

physicalstructure

mathematicalstructure

Fig. 3.4 Intuition of structure

Physical intuition matches structure in mental models with structure in physicalsystems. Mathematical intuition matches mental structure with symbolic struc-ture. Thus, structure in the imagination is common ground for both physical andmathematical intuition.


I surmise that physical intuition is highly developed among experimentalphysicists, where they develop detailed mental images of experimental design,equipment, measurement procedures and data analysis. None of these abilities areinvolved in mathematical intuition, but theoretical physics requires integrating agood deal of both. Supporting evidence for this point will emerge as we move on.

Identication of intuition as the bridge between imagination and perception is asecure starting place for exploring the specics of mathematical intuition. It wouldbe helpful to have the testimony of procient mathematicians to guide the explo-ration. Much anecdotal testimony is scattered throughout the literature, but it wouldtake a major act of scholarship to bring it together. We shall have to be satised witha few telling examples.

Mathematician Jacques Hadamard (1945) surveyed 100 leading physicists andgives an introspective account of his own thinking, as well as that of others includingPoincar and Einstein. He documents two major facts about mathematical thinking:at the conscious level, much of it is imagistic without words; and, much of it isdone unconsciously, with clear insights or solutions emerging with sudden sponta-neousness into conscious thought. He does not discriminate between the thinkingof mathematicians and physicists. He quotes from a letter by Einstein:

The words or the language, as they are written or spoken, do not seem to play any rolein my mechanism of thought. . .. The physical entities which seem to serve as elements inthought are certain signs and more or less clear images which can be voluntarily reproducedand combined. . .. The above-mentioned elements are, in my case, visual and some of mus-cular type. Conventional words or other signs have to be sought for laboriously only in asecondary state. . ..

It is noteworthy that Einstein is famous for inventing thought experiments thatproposed new relations between theory and experiment. When this is presented asevidence for his singular genius, it remains unremarked that invention of a thoughtexperiment is an essential early step in the design of any experiment.

All this suggests that free play of the imagination (as Einstein put it) has afar more signicant role in math/science thinking (and human reasoning in gen-eral) than is commonly recognized in educational circles. Most intuitive structure isrepresented subliminally in the cognitive unconscious and is often manifested in pat-tern recognition and conceptual construction skills. Finally, it should be noted thatEinsteins description supports our view that intuition is grounded in the sensory-motor system; moreover, that ideas may be generated in the imagination before theyare elevated to concepts by encoding in symbols.

Hadamards report provides empirical support for general features of mathemati-cal imagination and intuition, but it lacks the detail we need to describe its structure.To remedy that, we can do no better than turn to Kants transcendental analysisin the Critique of Pure Reason (Kant, 1787). He was not a professional mathe-matician, but he did teach mathematics and physics for fteen years before writingthe Critique. Moreover, his analysis was greatly respected and highly inuentialamong mathematicians throughout the nineteenth century and beyond. My attemptto present the nub of Kants argument is indebted to clarications by philosopherQuassim Cassam (2007).

24 D. Hestenes

Kant conceded to his empiricist predecessors Locke, Hume and Berkeley thatall knowledge is derived from experience, but he rejected attempts to derive cer-tain knowledge from that. Rather, he turned the problem of knowledge on its headand accepted Euclids geometry and Newtons physics as objective facts. He thenasked the trenchant question How is such knowledge possible? This posing ofa how-possible question (as Cassam calls it) is the essential rst step in Kantstranscendental approach to epistemology. He completes his argument with amulti-level answer.

Kant applies his transcendental method to a number of epistemological problems.But the test case is Euclidean geometry, as that was universally acknowledged asknowledge of the most certain kind. Thus, he asked: How is geometrical knowledgepossible? This is just the kind of question we want to answer in detail. Kant beginshis answer by identifying construction in intuition as a means for acquiring suchknowledge:

Thus we think of a triangle as an object, in that we are conscious of the combination of thestraight lines according to a rule by which such an intuition can always be represented. . .This representation of a universal procedure of imagination in providing an image for aconcept, I entitle the schema of this concept.

Kant did not stop there. Like any good scientist, he anticipated objections to hishypothesis. Specically, he noted that his intuitive image of a triangle is alwaysa particular triangle. How, he asks, can construction of a concept by means of asingle gure express universal validity for all possible intuitions which fall underthe same concept? This is the general epistemological problem of universality forthe case of Kants theory of geometrical proof. Kants notion of geometrical proof isby construction of gures, and he argues that such proofs have universal validity aslong as the gures are determined by certain universal conditions of construction.In other words, construction in intuition is a rule-governed activity that makes itpossible for geometry to discern the universal in the particular.

Kant wants more. What still needs to be explained is the capacity of pure intu-ition to provide geometrical knowledge. Kants argument leads ultimately to theconclusion that space itself is an a priori intuition that has its seat in the subjectonly. He concludes famously that space and time are a priori forms of intuition,intrinsic features of mind that shape all experience.

We need not follow Kants argument to conclusions that have since proven to beuntenable, such as the claim that geometry of the physical world must be Euclideanbecause our minds cannot conceive otherwise. We now know that there are manykinds of non-Euclidean geometry, and the geometric structure of space-time is anempirical matter to be settled by interplay between theory and experiment. The bot-tom line is that Kants hypothesis of spatio-temporal constraints on cognition is stillviable today, but it must be recognized as an empirical issue to be settled by researchin cognitive science.

There is much more to be said in favor of Kants analysis. First, his charac-terization of geometric intuition has been universally approved (or, at least, neverchallenged) by mathematicians even to present day, as it is easily adapted to any


non-Euclidean geometry by simple changes in the rules. Second, his argument thatinference from the particular to the universal is governed by subsumption underrules is a profound insight that has not attracted the attention it deserves, even, itseems, from devoted Kantian scholars. Its import is evident in Modeling Theory, forit determines a mapping of structure in mental constructions (models) to structure indrawn gures, propositions or equations. That is, rules for parsing and manipulatingmental constructions correspond directly to rules for constructing and manipulatingmathematical representations. This is evidently a basic mechanism in mathematicalintuition as I dened it earlier. Moreover, it is a means for constructing and shar-ing objective knowledge, as the rules are publicly available to everyone, though it isnontrivial to learn how to employ them.

The power of rules was so evident to Kant that he posited a faculty of judgmentto administer it: If understanding as such is explicated as our power of rules, thenthe power of judgment is the ability to subsume under rules, i.e., to distinguishwhether something does or does not fall under a given rule. Judgment developedinto the central theme of Kants philosophy, but in the abundance of its applicationsto morals, religion and aesthetics, its fundamental role in mathematical intuition andobjective knowledge seems to have been lost.

We are now prepared for a more incisive comparison of mathematical and physi-cal intuition. To begin with, the intuitive structure of Euclidean geometry is commonknowledge for mathematicians and physicists. I submit, however, that their intu-itions of geometry gradually diverge as they employ geometry in different ways.The mathematician concentrates on construction and analysis of formal structures.The physicist uses geometry for modeling rigid bodies and measurement of length,which is the foundation for physical measurements of every kind. Such develop-ments in mathematics and physics do not have to go far before their common groundin Euclidean geometry is no longer obvious. With respect to the Kantian category ofcausality, intuitions of physicists and mathematicians diverge even more strongly,as we shall see.

It may be objected that our how-possible analysis of geometry is too limited forgeneral conclusions about mathematical intuition. As a remedy, I recommend a how-possible analysis of set theory, group theory, algebra and any other mathematicalsystem that the reader regards as fundamental. In fact, I submit that it would not bedifcult and perhaps enlightening to frame the math concept analysis of Lakoff andNez (2000) in how-possible terms.

3.5 Mathematical Versus Physical Intuition

Let me reinforce our conclusions about mathematical intuition with testimony byHilbert from an address delivered in 1927:

No more than any other science can mathematics be founded on logic alone; rather, asa condition for the use of logical inferences and the performance of logical operations,something must already be given to us in our faculty of representation, certain extralogicalconcrete objects that are intuitively present as immediate experience prior to all thought. If

26 D. Hestenes

logical inference is to be reliable, it must be possible to survey these objects completely inall their parts, and the fact that they occur, that they differ from on another, and that theyfollow each other, or are concatenated, is immediately given intuitively, together with theobjects, as something that can neither be reduced to anything else, nor requires reduction.This is the basic philosophical position that I regard as requisite for mathematics and, ingeneral, for all scientic thinking, understanding, and communication. And in mathematics,in particular, what we consider is the concrete signs themselves, whose shape, according tothe conception we have adopted, is immediately clear and recognizable. This is the veryleast that must be presupposed, no scientic thinker can dispense with it, and thereforeeveryone must maintain it consciously or not.

Note the coupling between concrete signs and intuitions, with logical inferencegrounded on the intuitive side.

For comparison, lets hear testimony about physical intuition from an eminentphysicist. Heinrich Hertz (1956) discovered the means to generate and detect elec-tromagnetic radiation, surely one of the greatest experimental achievements of alltime. He was equally accomplished as a theoretical physicist, though his tragic earlydeath deprived the world of his genius. His profound grasp of cognitive processesin science is exhibited in the following passage (Hertz, 1956):

The most direct, and in a sense the most important problem which our conscious knowledgeof nature should enable us to solve is the anticipation of future events, so that we mayarrange our present affairs in accordance with such anticipation.

. . .We form for ourselves images or symbols of external objects; and the form which wegive them is such that the necessary consequents of the images in thought are always thenecessary consequents in nature of the things pictured [Predictability]. In order that thisrequirement may be satised, there must be a certain conformity between nature and ourthought.

. . .The images we form of things are not determined without ambiguity by the requirementof [Predictability].

[I have inserted the term [Predictability] to compress the link between his last twoparagraphs.]

Hertz goes on to explain that images are constrained by certain ConformabilityConditions, including

Admissibility: Images must not contradict the laws of thought.Distinctiveness: Images should maximize essential relations of the thing.Simplicity: Images should minimize superuous or empty relations.

He adds that Empty relations cannot be altogether avoided.Hertz then explains that scientic representations (of our images) satisfy different

postulates.This passage (condensed for brevity) is studded with brilliant insights. First, note

that it is consistent with Kants account of geometric intuition (with which Hertzwas surely familiar), but it surpasses Kant in original detail. Next, note how sharplyHertz distinguishes between images (mental models) and their scientic represen-tations. He emphasizes that to have predictive value the images must satisfy certain


rules, which he sharply distinguishes from rules governing their public representa-tions. Finally, note the implication from Hertzs rst paragraph that the faculty ofintuition has evolved to guide effective action in the environment. This is currentlya major theme in the emerging eld of evolutionary psychology.

Differences between mathematical and physical intuitions emerge in meaningsattributed to mathematical expressions. We often speak of mathematical symbols asthough they have unique meanings that are the same for everyone. But we knowthat meanings are private constructions in the imagination of each individual, sotuning them to agree among individuals is a subtle social process. We have notedthat public access to geometrical gures provides common ground for geometricintuitions of both mathematicians and physicists. Now let us consider an importantconcept where intuitions strongly diverge, namely, the concept of force.

The general concept of interaction has been identied as one of the universalforms of knowledge in Modeling Theory. It corresponds closely to Kants causal-ity category. Though that category is construed broadly enough to include humanvolition, there is no doubt that the Newtonian concept of force was centermost inKants thinking. Force dynamics also appears as a major category in cognitive lin-guistics, especially in the work of Talmy and Langacker (see Evans and Green,2006). However, as I have explained before (Hestenes, 2007), linguistic research onthe force concept has yet to be reconciled with physics education research (PER).

Divergence of student intuitions about force from the Newtonian (i.e. scientic)force concept are reliably measured by the Force Concept Inventory (FCI). FCIassessment on large populations of students from middle school to graduate schoolshows conclusively that before physics instruction student concepts diverge fromNewtonian concepts in almost every dimension (Hestenes, 2007). Moreover, moststudents are far from Newtonian even after a year of university physics. I surmisefrom this that mathematics professors who have neglected physics in their educationwill likewise retain nave concepts of force. To check that out, it would be interestingto test a representative sample of such subjects with the FCI. But who dares bellthe cat?

Nave concepts of force have often been dismissed as misconceptions to bereplaced by the scientic Newtonian concept. But that is a serious mistake stem-ming from a nave view of cognition and learning. It should be recognized insteadthat student intuitions are essential cognitive resources developed through years ofreal world experience. We understand the world by mapping events into the mentalspaces of our imagination. The chief problem in learning physics is not to replaceintuitions but to tune the mapping to produce a veridical image of the world in theimagination.

To thoroughly understand what learning the Newtonian force concept entails, weneed an inventory of intuitive resources that students bring to the experience. AndreadiSessa (1993) has pioneered identication and classication of basic intuitions ofphysical mechanism that he calls phenomenological primitives, or p-prims.

Without going into detail that is readily accessible in the literature, I wish toexplain how diSessas theory of p-prims (or, at least, something very much like it)

28 D. Hestenes

ts naturally into Modeling Theory. It will be sufcient to comment on the p-primslisted in Fig. 3.5.

Fig. 3.5 Force p-prims (from Sherin, 2006)

Much like the image schemas in cognitive linguistics (Evans and Green, 2006),p-prims are stable units of mental structure employed in the construction of mentalmodels. Though diSessa identied the p-prims largely from interviews of scienti-cally nave students, I agree with Bruce Sherin (2001, 2006) that the same p-primsare involved in structuring the physical intuition of mature physicists. I regard thisconclusion as a major milestone in cognitive science, so I will return to it afterdiscussing the intuitive foundations.

diSessa found that, for nave students, each p-prim is a simple, separate anddistinct knowledge piece called forth for explanatory purposes by situational cues.Collectively, the p-prims compose a loose conceptual system that diSessa describedas knowledge in pieces. In contrast, Newtonian force is a complex, multidimen-sional concept (Hestenes, 2007). Lets consider how the p-prims can be integratedinto an intuitive base for the Newtonian concept.

Many p-prims in Fig. 3.5 have familiar names. This should not be surprising,because they derive from common human experiences. However, diSessa has giventhe peculiar name Ohms p-prim to the most important one in the lot. That doessuggest a historical role in the creation of Ohms law for electrical resistance. Butits most basic role is in the intuition of force. Ohms p-prim is schematized as anagent working against a resistance to produce a result. No doubt it originates inpersonal experience of pushing material objects, and it is projected metaphoricallyto other situations. diSessa notes that it serves as general intuitive schema for quali-tative proportional reasoning (hence its applicability to Ohms law by metaphoricalprojection). diSessa also suggests that it provides intuitive structure for the physi-cists understanding of F = ma, where the result of applying a force is acceleration(but not velocity, as is common in nave conceptions). Note the considerable adjust-ment in intuition required for a veridical match of mental model with physicalevents (in accordance with Hertzs conformability conditions). Indeed, ability

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