Radical Constructivism in Mathematics Education
Mathematics is the science of acts without things - and through this, of things one can define by acts. 1 Paul Valéry The essays collected in this volume form a mosaik of theory, research, and practice directed at the task of spreading mathematical knowle
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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
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
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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’lnfomia
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