Radical Constructivism In Mathematics Education (mathematics Education Library)

RADICAL CONSTRUCTIVISM IN MATHEMATICS EDUCATION 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 RADICAL CONSTRUCTIVISM IN MATHEMATICS EDUCATION Edited by : ERNST VON GLASERSFELD University of Massachusetts KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW eBook ISBN: Print ISBN: 0-306-47201-5 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 Visit Kluwer Online at: and Kluwer's eBookstore at: http://www.kluweronline.com http://www.ebooks.kluweronline.com This page intentionally left blank. TABLE OF CONTENTS ABOUT THE AUTHORS vii INTRODUCTION xiii PHILIP H. STEEDMAN / There Is No More Safety in Numbers: A New Conception of Mathematics Teaching JOHN RICHARDS / Mathematical JAMES J. KAPUT / Notations tive 1 13 Discussions and Representations as Mediators of Construc- Processes 53 JACK LOCHHEAD / Making Math Mean 75 NICOLAS BALACHEFF/ Treatment of Refutations: Aspects of the Complexity of a Constructivist Approach to Mathematics Learning 89 JERE CONFREY / Learning to of Ten CLIFFORD KONOLD / Listen: Understanding A Student’s Understanding of Powers 111 Students’ Beliefs about Probability 139 PAUL COBB, TERRY WOOD, and ERNA YACKEL / A Constructivist Approach to Second LESLIE P. and Grade STEFFE Mathematics / The Constructivist 157 Teaching Experiment: Illustrations Implications 177 JAN VAN DEN BRINK / Didactic Constructivism 195 ROBERT G. UNDERHILL / Two Layers tion 229 of Constructivist Curricular Interac- ABOUT THE AUTHORS While this volume was being prepared, the editor requested from the contributing 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 journal “Recherches en Didactique des Mathematiques” . He has recently joined the Laboratoire IRPEACS (CNRS), where he directs a research project 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 res