ERIC ED378043: Constructing Mathematical Knowledge: Epistemology and Mathematics Education. Studies in Mathematics Education Series: 4.

297 Pages·1994·4.7 MB·English

by  ERIC

DOCUMENT RESUME
ED 378 043
SE 055 588
AUTHOR
Ernest, Paul, Ed.
TITLE
Constructing Mathematical Knowledge: Epistemology and
Mathematics Education. Studies in Mathematics
Education Series: 4.
REPORT NO
ISBN-0-7507-0354-7
PUB DATE
94
NOTE
297p.; For the companion volume, "Mathematics,
Education, and Philosophy: An International
Perspective," see SE 055 587.
AVAILABLE FROM
Falmer Press, Taylor & Francis Inc., 1900 Frost Road,
Suite 101, Bristol, PA 19007.
PUB TYPE
Books (010)
Collected Works
General (020)
EDRS PRICE
MFO1 /PC12 Plus Postage.
DESCRIPTORS
*Constructivism (Learning); Elementary Secondary
Education; *Epistemology; *Hermeneutics; *History;
*Mathematics Education; *Psychology
ABSTRACT
This book illustrates the breadth of theoretical and
philosophical perspectives that can be brought to bear on mathematics
and education. Part 1, "Constructivism and the Learning of
Mathematics," contains the following chapters:
(1) "A Radical
Constructivist View of Basic Mathematical Concepts" (E. von
Glasersfeld);
(2) "Interaction and Children's Mathematics" (L. P.
Steffe & R. Tzur);
(3) "Radical Constructive Criticisms of von
Glasersfeld's Radical Constructivism" (R. S. D. Thomas);
(4)
"Articulating Theories of Mathematics Learning" (S. Lerman);
(5) "Is
Radical Constructivism Coherent?" (M. Otte);
(6) "Social
Constructivism and the Psychology of Mathematics Education" (P.
Ernest); (7) "Mathematics, Computers and People: Individual and
Social Perspectives" (E. Smith); and (8) "The Context of Cognition:
The Challenge of Technology" (K. Crawford). Part 2, "Psychology,
Epistemology and Hermeneutics," contains:
(9) "Another Psychology of
Mathematics Education" (D. Pimm);
(10) "On Interpretation" (D.
Tahta);
(11) "Potent'ial Space and Mathematical Reality" (P. Maher);
(12) "Towards a Hermeneutical Understanding of Mathematics and
Mathematical Learning" (T. Brown); and (13) "The Myth of Mathematics"
(F. Seeger & H. Steinbring). Part 3, "Enquiry in Mathematics
Education," contains: (14) "The Problem of the Problem and Curriculum
Fallacies" (S.
I. Brown); (15) "Enquiry in Mathematics and in
Mathematics Education" (J. Mason);
(16) "Demystifying Mathematics
Education through Inquiry" (M. Siegel & R. Borasi); and (17) "Reading
to Learn Mathematics in the Primary Age Range" (C. W. Desforges & S.
Bristow). The final section, Part 4, "History, Mathematics and
Education," contains: (18) "The Idea of 'Revolution' As an Instrument
for the Study of the Development of Mathematics and Its Application
to Education" (F. Speranza); and (19) "Mathematical Practices,
Anomalies and Classroom Communication Problems" (A. Sfard). Contains
references with each chapter and a subject index. (MKR)
nstru tin
Mati.ematical
Knowledge:
Epistemology and
Mathematical Education
U.S DtPARTMENT OF EDUCATION
Office of Educational Research and Improvement
DUCATIONAL RESOURCES INFORMATIOh
CENTER IERIC)
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Edited by
AM
Paul Ernest
BEST COPY
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The Fahner Press
I*
Constructing Mathematical Knowledge
1
Studies in Mathematics Education Series
Series Editor
Paul Ernest
School of Education
University of Exeter
Exeter
The Philosophy of Mathematics Education
1
Paul Ernest
Understanding in Mathematics
2
Anna Sierpinska
Mathematics Education and Philosophy: An International Perspective
3
Edited by Paul Ernest
Constructing Mathematical Knowledge: Epistemology and Mathematics
4
Education
Edited by Pau! Ernest
Investigating Mathematics Teaching: A Constructivist Enquiry
5
Barbara Jaworski
Radical Constructivism: A Way of Knowing and Learning
6
Ernst von Glasersfeld
4
Studies in Mathematics Education Series: 4
Constructing Mathematical
Knowledge:
Epistemology and Mathematics
Education
Edited by
Paul Ernest
The Faimer Press
(A member of the Taylor & Francis Group)
Washington, D.C.
London
r1;
UK
The Falmer Press, 4 John Street, London WC1N 2ET
USA
The Falmer Press, Taylor & Francis Inc., 1900 Frost Road, Suite 101,
Bristol, PA 19007
P. Ernest 1994
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without permission in writing
from the Publisher.
First published in 1994
A catalogue record for this book is available from the British
Library
Library of Congress Cataloging-in-Publication Data are
available on request
ISBN 0 7507 0354 7 cased.
Jacket design by Caroline Archer
Typeset in 9.5/11pt Bembo by
Graphicraft Typesetters Ltd., Hong Kong.
Printed in Great Britain by Burgess Science Press, Basingstoke on paper
which has a specified pH value on final paper manufacture of not less than
7.5 and is thertftre 'acid free'.
Contents
Series Editor's Preface
Introduction
Part 1
Constructivism and the Learning of Mathematics
1
Chapter 1
A Radical Constructivist View of Basic Mathematical
Concepts
5
Ernst von Glasersfeld
Chapter 2
Interaction and Children's Mathematics
8
Leslie P. Steffe and Ron Tzur
Chapter 3
Radical Constructive Criticisms of von Glasersfeld's
Radical Constructivism
33
Robert S.D. Thomas
Chapter 4
Articulating Theories of Mathematics Learning
41
Stephen Lerman
Chapter 5
Is Radical Constructivism Coherent?
50
Michael Otte
Chapter 6
Social Constructivism and the Psychology of
Mathematics Education
62
Paul Ernest
Chapter 7
Mathematics, Computers and People: Individual and
Social Perspectives
73
Erick Smith
Chapter 8
The Context of Cognition: The Challenge of
Technology
92
Kathryn Crawford
Part 2
Psychology, Epistemology and Hermeneutics
107
Chapter 9
Another Psychology of Mathematics Education
111
David Pimm
Chapter 10
On Interpretation
125
Dick Tahta
Chapter 11
Potential Space and Mathematical Reality
134
Philip Maher
Chapter 12
Towards a Hermeneutical Understanding of
Mathematics and Mathematical Learning
141
Tony Brown
V
sit
Constructing Mathematical Knowledge
The Myth of Mathematics
Chapter 13
151
Falk Seeger and Heinz Steinbring
Part 3
Enquiry in Mathematics Education
171
Chapter 14
The Problem of the Problem and Curriculum Fallacies
175
Stephen I. Brown
Chapter 15
Enquiry in Mathematics and in Mathematics Education
190
John Mason
Chapter 16
Demystifying Mathematics Education through Inquiry
201
Marjorie Siegel and Raffaella Borasi
Chapter 17
Reading to Learn Mathematics in the Primary Age
Range
215
Charles W. Desforges and Stephen Bristow
Part 4
History, Mathematics and Education
237
Chapter 18
The Idea of 'Revolution' As an Instrument for the
Study of the Development of Mathematics and Its
Application to Education
241
Francesco Speranza
Chapter 19
Mathematical Practices, Anomalies and Classroom
Communication Problems
248
Anna Sfard
Notes on Contributors
274
Index
276
List of Tables and Figures
Table 17.1
Incidences of Different Purposes Expressed as Pupils
Engaged with the Maths Packs
228
Table 17.2
Levels of Constructive Activity in Learning from Text
231
Table 17.3
Incidence of Levels of Constructive Activity in Response
to Text in Science and in Area
232
The Computer Screen of Sticks
Figure 2.1
14
Figure 2.2
The Screen after Arthur's Attempt
17
The Screen during Nathan's Attempt
Figure 2.3
18
Figure 2.4
The Srreen after Arthur Had Used 'Pull Parts'
22
Figure 2.5
Computer Screen after Nathan's and Arthur's Work
23
Figure 2.6
Social Interaction, Learning and Development
Figure 2.7
Types of Interaction
29
Figure 7.1
Individual Constructivist Perspective
78
Figure 7.2
Social Constructivist Perspective
79
Figure 7.3
Computer from Social Constructivist Perspective
86
Figure 7.4
Computer from an Individual Constructivist Perspective
87
Measuring the Height of a Tree
Figure 13.1
154
The Standard Problem-solving Paradigm
Figure 14.1
177
Figure 14.2 A Reversal of the Standard Problem-solving Paradigm
179
Figure 14.3
A Summary of Some Non-standard 'Problem' paradigms
182
A Sheet front the Averages Pack
Figure 17.1
218
Figure 17.2 A Sheet from the Fractions Pack
219
Full Learning Pack for Area
Figure 17.3
220
Figure 17.4
An Extract from the Fractions Learning Pack
224
An Extract from the Fractions Learning Pack
Figure 17.5
224
Figure 17.6
Examples of the Science Text Presented to the Pupils
229
Vii
t.)
Preface by Series Editor
Mathematics education is established worldwide as a major area of study, with
numerous dedicated journals and conferences serving national and international com-
munities of scholars. Research in mathematics education is becoming more theoret-
ically orientated. Vigorous new perspectives are pervading it from disciplines and
fields as diverse as psychology, philosophy, logic, sociology, anthropology, history,
feminism, cognitive science, semiotics, hermeneutics, post-structuralism and post-
modernism. The series Studies in Mathematics Education consists of research contri-
butions to the field based on disciplined perspectives that link theory with practice.
It is founded on the philosophy that theory is the practitioner's most powerful
tool in understanding and changing practice. Whether the practice is mathematics
teaching, teacher education, or educational research, the series intends to offer
new perspectives to assist in clarifying and posing problems and to stimulate
debate. The series Studies in Mathematics Education will encourage the development
and dissemination of theoretical perspectives in mathematics education as well as
their critical scrutiny. It aims to have a major impact on the development of
mathematics education as a field of study into the twenty-first century.
Unusually for the series this book (Volume 4) and Volume 3 are edited
collections. instead of the sharply focused concerns of a research monograph the
books offer a panorama of complementary and forward-looking perspectives. In
the spirit of the philosophy of the series Volumes 3 and 4 illustrate between them
the breadth of theoretical and philosophical perspectives that can fruitfully be
brought to bear on mathematics and education. The companion to the present
volume is Mathematics, Education and Philosophy: An International Perspective.
It
offers a reconceptualization of mathematics from a range of philosophical, educa-
tional and social perspectives, as well as philosophical reflections on mathematics
education itself. The present volume provides a complementary focus. Its empha-
sis is on epistemological issues, encompassing multiple perspectives on the learn-
ing of mathematics, as well as broader philosophical reflections on the genesis of
knowledge. The two books aim to set a research agenda for the philosophy of
mathematics education, a rapidly developing area of enquiry. Together they sur-
vey research, providing a report on advances made so far, as well as indicating
orientations for potentially fruitful work in the future.
Paul Ernest
School of Education
University of Exeter
March 1994
ix
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