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  • Mathematics Education Library

    VOLUME 7

    Managing Editor

    A.J. Bishop, Cambridge, U.K.

    Editorial Board

    H. Bauersfeld, Bielefeld, Germany J. Kilpatrick, Athens, U.S.A.

    G. Leder, Melbourne, Australia S. Turnau, Krukow, Poland G. Vergnaud, Paris, France

    The titles published in this series are listed at the end of this volume



    University of Massachusetts


  • eBook ISBN: Print ISBN: 0-792-31257-0

    ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow

    All rights reserved

    No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

    Created in the United States of America

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    PHILIP H. STEEDMAN / There Is No More Safety in Numbers: A New

    JOHN RICHARDS / Mathematical Discussions

    JAMES J. KAPUT / Notations and Representations as Mediators of Construc-

    Conception of Mathematics Teaching 1


    tive Processes 53

    75 JACK LOCHHEAD / Making Math Mean

    NICOLAS BALACHEFF/ Treatment of Refutations: Aspects of the Com-

    plexity of a Constructivist Approach to Mathematics Learning

    JERE CONFREY / Learning to Listen: A Student’s Understanding of Powers

    of Ten 111

    CLIFFORD KONOLD / Understanding Students’ Beliefs about Probability 139

    PAUL COBB, TERRY WOOD, and ERNA YACKEL / A Constructivist Approach

    LESLIE P. STEFFE / The Constructivist Teaching Experiment: Illustrations

    JAN VAN DEN BRINK / Didactic Constructivism

    ROBERT G. UNDERHILL / Two Layers of Constructivist Curricular Interac-


    to Second Grade Mathematics 157

    and Implications 177


    tion 229



    While this volume was being prepared, the editor requested from the contribut- ing authors a brief statement concerning their past intellectual itinerary and present ideas. These statements, adjusted in length, are here prefaced by indications of the authors’ present academic positions.

    Nicolas Balachef (Currently research director at the French National Center for Scientific Research (CNRS), president of the International Group for the Psychology of Mathematics Education (PME), and editor of the international

    on principles for the design of Intelligent Tutorial Systems). I have been involved in mathematics education since 1977. For the past ten

    years the main focus of my research was on how students learn mathematical proofs, with specific attention to how they deal with counter examples and refutations. I did most of this work as a member of the Equipe de Didactique des Mathematiques et de l’lnfomiatique at the Joseph Fourier University in Grenoble. There I taught mathematics and mathematics education from 1972 to 1988. All along I have been concerned with the application of research results to practice and in particular to the practice of teacher education. The “intelligent tutorial systems” I am at present helping to design are dependent on the elaboration of a model of the active learner. Hence my profound interest in what and how students think.

    Paul Cobb (Ph.D. in Mathematics Education, University of Georgia, 1983. Currently directing research on mathematics instruction in the first three

    Having initially had several years of intensive interaction with Les Steffe, Ernst von Glasersfeld, and John Richards on a project on young children’s construction of arithmetical knowledge, I was much influenced by radical constructivism as a theory of knowing and as an orienting framework for educational issues. However, since 1983, I have been troubled by sociological and cultural phenomena which seemed to fall outside my admittedly immature grasp of constructivism. This feeling was further exacerbated by time spent in a classroom during a year-long teaching experiment. I have come to the view that

    E. von Glasersfeld (ed.). Radical Constructivism in Mathematics Education vii-xi. © 1991 Kluwer. Academic Publishers. Printed in the Netherlands.

    journal “Recherches en Didactique des Mathematiques” . He has recently joined the Laboratoire IRPEACS (CNRS), where he directs a research project

    grades at Purdue University ).

  • it is impossible to explain children’s mathematical learning unless psychologi- cal analyses are coordinated with anthropological analyses of classroom life. In this regard, I have been much influenced by a variety of theorists whose basic epistemology is compatible with constructivism (e.g. Goodman, Bloomer, Schutz, Mehan, Bauersfeld, Voigt, Barnes, Geertz, and Rorty). I now view classrooms as sites for action research in which to explore salient questions by having conversation-like interactions with the providers of one’s “data”.

    Jere Confrey (Ph.D. in Mathematics Education, Cornell University. Founder of Summer Math Program, Mount Holyoke College, Massachusetts. Currently directing research projects on exponential functions and multi-representational software at Cornell University.)

    I have come to constructivism through my interest in philosophical issues concerning knowledge and philosophy of science. As feminist, mother, and professor I had little difficulty in accepting the experience of multiple realities and I discovered my commitment to work against the silencing of student voices that is prevalent in our traditional ways of schooling. I met Ernst von Glasersfeld some ten years ago and he has continued to encourage me to examine my own understanding of cognition and mathematics. He, Les Steffe, and Paul Cobb have contributed to the development of my ideas about construc- tivism. My current work leads me towards the design of technological tools that promote students’ creativity rather than stifling it.

    James Kaput (Professor of Mathematics, Southeastern Massachusetts University; Coordinator of Mathematics Projects, Educational Technology Center, Harvard Graduate School of Education; Research Associate, National Center for Research in Mathematical Sciences Education.)

    My intellectual odyssey began when, having done my doctoral work as a mathematician in category theory, Bob Davis encouraged me in the early 1970’s to turn my attention to mathematics education. I became interested in finding better ways for undergraduates to learn mathematics and in preparing myself to help other teachers. Along the way it became clear to me that psychology, philosophy, mathematics, linguistics, anthropology, information science, and almost everything else I study and appropriate, are all part of the same enterprise. From my individual perspective, they all are branches of the science of re-presentation. Now I am especially intrigued by the fact that the act of building computer-based learning environments - and trying to make them work - forces me simultaneously to apply everything I have learned and to make my beliefs explicit and concrete. That all this is not an idle academic exercise, is regularly made plain by my three kids who are struggling through traditional school math, traditionally re-presented.

    Clifford Konold (Research Associate at the Scientific Reasoning Research Institute, University of Massachusetts. Currently directing the NSF-funded



    Probability and Statistics” .) As a graduate student in psychology I was interested in studying the effects

    of instruction on (as I would have said then) “the structure of concepts in long- term memory”. Probability and statistics offered a convenient research domain, both because I was teaching the topic and because it had a more definite, or agreed upon, structure than domains such as “Educational Psychology”. I soon became dissatisfied with the techniques that were then being used in psychol- ogy to characterize memory structure and found myself struggling to adapt Piaget’s clinical approach to my interests. I fell in with a group of like-minded individuals, got divorced, parted company with the religious group in which I had been raised, and finally convinced myself that it was OK to pay someone to do auto repairs that I was theoretically capable of doing myself. As a result of these changes, I now spend much less time on my back and my knees - all this because I wanted a different way to think about structure.

    Jack Lochhead (Graduate work in physics and Doctorate in Educational Research and Statistics, University of Massachusetts, 1973. Currently Director of Scientific Reasoning Research Institute, Director Basic Math Program, and Assistant Dean for Natural Science and Mathematics, University of Mas- sachusetts.)

    My interest in scientific and mathematical thinking stems from my own struggle to understand some of the apparently simple concepts in those fields. I believe that once we develop a clearer picture of scientific thinking and understanding, we may be able to design mor