Table Of ContentMATHEMATICAL
REASONING
Analogies, Metaphors,
and Images
STUDIES IN MATHEMATICAL THINKING AND LEARNING
A series of volumes edited by
Alan Schoenfeld
Carpenter/Fennema/Romberg. Rational Numbers: An Integration of
Research
Cobb/Bauersfeld. The Emergence of Mathematical Meaning: Interaction
in Classroom Cultures
English • Mathematical Reasoning: Analogies, Metaphors, and Images
Fennema/Nelson. Mathematics Teachers in Transition
Lajoie. Reflections on Statistics: Learning, Teaching, and Assessment in
Grades K-12
Romberg/Fennema/Carpenter. Integrating Research on the Graphical
Representation of Functions
Schoenfeld • Mathematical Thinking and Problem Solving
Sternberg/Bey Zeev • The Nature of Mathematical Thinking
MATHEMATICAL
REASONING
Analogies, Metaphors,
and Images
Edited by
Lyn D. English
Queensland University of Technology
I~ ~~o~~~~n~~~up
NEW YORK AND LONDON
First Published by
Lawrence Erlbaum Associates, Inc., Publishers
10 Industrial Avenue
Mahwah, New Jersey 07430
Transferred to Digital Printing 2009 by Routledge
270 Madison Ave, New York NY 10016
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
Copyright © 1997 by Lawrence Erlbaum Associates, Inc.
All rights reserved. No part of the book may be reproduced in
any form, by photostat, microform, retrieval system, or any other
means, without the prior written permission of the publisher.
Library of Congress Cataloging-in-Publication Data
Mathematical reasoning: analogies, metaphors, and images / edited by
Lyn D. English.
p. cm.
Includes bibliographical references (p. ) and index.
ISBN 0-8058-1978-9 (cloth: alk. paper). - ISBN 0-8058-1979-7
(pbk. : alk. paper).
1. Logic, Symbolic and mathematical. 2. Reasoning. I. English,
Lyn D.
QA9.M347 1997
51O'.1'9-<lc21 96-48234
CIP
Publisher's Note
The publisher has gone to great lengths to ensure the quality of this reprint
but points out that some imperfections in the original may be apparent.
Contents
Preface Vll
PART I: INTRODUCTION
1 Analogies, Metaphors, and Images:
Vehicles for Mathematical Reasoning 3
Lyn D. English
PART IT: COGNITIVE FOUNDATIONS FOR A MIND-BASED
MATHEMATICS
2 The Metaphorical Structure of Mathematics:
Sketching Out Cognitive Foundations for a Mind-Based
Mathematics 21
Nunez
George Lakoff and Rafael E.
PART ill: MATHEMATICAL REASONING: ANALOGIES
3 How Students Think: The Role of Representations 93
Robert B. Davis and Carolyn A. Maher
4 Analogical Reasoning and Early Mathematics Learning 117
Patricia A. Alexander, C. Stephen White,
and Martha Daugherty
v
vi CONTENTS
5 Children's Development of Analogical Problem-Solving
llill 1~
Barry Gholson, Dereece Smither, Audrey Buhrman,
Melissa K. Duncan, and Karen A. Pierce
6 Children's Reasoning Processes in Classifying
and Solving Computational Word Problems 191
Lyn D. English
7 Two Types of Reliance on Correlations Between
Content and Structure in Reasoning About Word
Problems 221
Miriam Bassok
8 Commentary: Mathematical Reasoning and Analogy 247
Mary Jo Rattermann
PART IV: MATHEMATICAL REASONING: METAPHORS,
METONYMIES, AND IMAGES
9 Reasoning With Metaphors and Metonymies
in Mathematics Learning 267
Norma C. Presmeg
10 Reasoning With Images in Mathematical Activity 281
Grayson H. Wheatley
11 Generalization Using Imagery in Mathematics 299
Norma C. Presmeg
12 Children's Mathematical Reasoning With the Turtle
Programming Metaphor 313
Douglas H. Clements and Julie Sarama
13 Commentary: On Metaphorical Roots of Conceptual
Growth 339
Anna Sfard
Author Index 373
Subject Index 381
Preface
How we reason with mathematical ideas continues to be a fascinating and
challenging topic of research. It has become even more so in recent years
with the rapid and diverse developments in the field of cognitive science.
Because cognitive science draws upon several disciplines, including psy
chology, philosophy, computer science, linguistics, and anthropology, it
provides rich scope for addressing issues that are at the core of mathe
maticallearning. One of these fundamental issues is how individuals men
tally structure their mathematical experiences and how they reason with
these structures in learning and problem solving. There has naturally been
considerable debate on this point, and indeed, some researchers would
argue that we cannot unlock the individual mind and should focus our
attention on how humans construct public bodies of knowledge. However,
if we analyze the powerful reasoning mechanisms we use in our everyday
communications and interactions with others, we begin to realize that
these same mechanisms playa significant role in our reasoning with mathe
matical ideas. This assumes, of course, that we have broadened our views
on reasoning itself.
Drawing upon the interdisciplinary nature of cognitive science, this
volume presents a broadened perspective on mathematics and mathemati
cal reasoning. In line with the thinking of George Lakoff and Mark
Johnson, the book represents a move away from the traditional notion of
reasoning as "abstract and disembodied" to the contemporary view of rea
soning as "embodied" and "imaginative." From this perspective, mathe
matical reasoning entails reasoning with structures that emerge from our
bodily experiences as we interact with the environment; these structures
extend beyond finitary propositional representations. Mathematical rea
soning is imaginative in the sense that it utilizes a number of powerful,
vii
viii PREFACE
illuminating devices that structure these concrete experiences and trans
form them into models for abstract thought. These "thinking tools" include
analogy, metaphor, metonymy, and imagery. They play an important role
in mathematical reasoning, as the chapters in this volume demonstrate,
yet their potential for enhancing learning in the domain has not been
acknowledged adequately. Given that "Mathematics as Reasoning" is one
of the curriculum and evaluation standards of the National Council of
Teachers of Mathematics (USA), it behooves us to give greater attention
to how these vehicles for thinking can foster students' mathematical power.
The contributing authors provide much food for thought here. Drawing
from backgrounds in mathematics education, educational psychology, lin
guistics, and cognitive science, the authors present a comprehensive and
diverse analysis of mathematical reasoning from the preschool years
through adulthood. The authors have all made significant contributions
to our understanding of human reasoning and continue to do so here
with their latest research on how reasoning with analogies, metaphors,
metonymies, and images can facilitate mathematical understanding.
One of the most rewarding aspects of editing a book is acknowledging
the colleagues who have made its production possible. First, of course, are
the contributing authors themselves. All have been most supportive and
understanding throughout the book's development, cheerfully responding
to my frequent, occasionally frantic, e-mails and faxes. The authors have
done a wonderful job of completing their chapters while trying to juggle
the increasing demands of academic life. I am indebted to them for their
contributions. In particular, Patricia Alexander, Norma Presmeg, and Mary
Jo Rattermann deserve special mention for their invaluable advice and
assistance with the book's production.
There are numerous other personnel who have contributed to the
book's development. In the early stages when I was desperately trying to
refine the book proposal, I turned for advice to Professor John Bain of
Griffith University, Queensland. As usual, John willingly gave of his time
and helped me rethink some of my original ideas. His efforts here are
greatly appreciated. Likewise, the editorial staff of Lawrence Erlbaum As
sociates, in particular, Naomi Silverman and Linda Henigin, have been
most supportive in my efforts to get the book into press. The research
assistants of the Centre for Mathematics and Science Education, Queens
land University of Technology, also deserve special thanks for providing
support when needed, in particular, Leone Harris, Lorraine English, Mar
tin Lambert, and Lynn Burnett. Last, but by no means least, is my mother,
Denise English, who not only supported and encouraged me, as always,
but also relieved me of some of the more tedious editorial tasks.
Lyn D. English
I
Part
INTRODUCTION
