Mathematical and Analogical Reasoning of Young Learners

235 Pages·2004·9.52 MB·English

by  Lyn D. English

MATHEMATICAL AND
ANALOGICAL REASONING
OF YOUNG LEARNERS
STUDIES IN MATHEMATICAL THINKING AND LEARNING
Alan H. Schoenfeld, Series Editor
Artzt/Armour-Thomas • Becoming a Reflective Mathematics Teacher: A Guide for
Observation and Self-Assessment
Baroody/Dowker • The Development of Arithmetic Concepts and Skills: Constructing
Adaptive Expertise
Boaler • Experiencing School Mathematics: Traditional and Reform Approaches to
Teaching and Their Impact on Student Learning
Carpenter/Fennema/Romberg (Eds.) • Rational Numbers: An Integration of Research
Cobb/Bauersfeld (Eds.) • The Emergence of Mathematical Meaning: Interaction in
Classroom Cultures
Cohen • Teachers' Professional Development and the Elementary Mathematics Classroom:
Bringing Understandings to Light
Clements/Sarama/DiBiase (Eds.) • Engaging Young Children in Mathematics: Standards
for Early Childhood Mathematics Education
English (Ed.) • Mathematical and Analogical Reasoning of Young Learners
English (Ed.) • Mathematical Reasoning: Analogies, Metaphors, and Images
Fennema/Nelson (Eds.) • Mathematics Teachers in Transition
Fennema/Romberg (Eds.) • Mathematics Classrooms That Promote Understanding
Lajoie (Ed.) • Reflections on Statistics: Learning, Teaching, and Assessment in Grades
K-12
Lehrer/Chazan (Eds.) • Designing Learning Environments for Developing Understanding
of Geometry and Space
Ma • Knowing and Teaching Elementary Mathematics: Teachers' Understanding of
Fundamental Mathematics in China and the United States
Martin • Mathematics Success and Failure Among African-American Youth: The Roles
of Sociohistorical Context, Community Forces, School Influence, and Individual
Agency
Reed • Word Problems: Research and Curriculum Reform
Romberg/Fennema/Carpenter (Eds.) • Integrating Research on the Graphical
Representation of Functions
Schoenfeld (Ed.) • Mathematical Thinking and Problem Solving
Senk/Thompson (Eds.) • Standards-Based School Mathematics Curricula: What Are
They? What Do Students Learn?
Sternberg/Ben-Zeev (Eds.) • The Nature of Mathematical Thinking
Wilcox/Lanier (Eds.) • Using Assessment to Reshape Mathematics Teaching: A Casebook
for Teachers and Teacher Educators, Curriculum and Staff Development Specialists
Wood/Nelson/Warfield (Eds.) • Beyond Classical Pedagogy: Teaching Elementary School
Mathematics
Yoshida/Fernandez • Lesson Study: A Japanese Approach to Improving Mathematics
Teaching and Learning
MATHEMATICAL AND
ANALOGICAL REASONING
OF YOUNG LEARNERS
Edited by
Lyn D. English
Queensland University of Technology
LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS
2004 Mahwah, New Jersey London
Copyright © 2004 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.
Lawrence Erlbaum Associates, Inc., Publishers
10 Industrial Avenue
Mahwah, New Jersey 07430
Cover design by Kathryn Houghtaling Lacey
Library of Congress Cataloging-in-Publication Data
Mathematical and analogical reasoning of young learners / edited by Lyn D. English.
p. cm.
Includes bibliographical references and index.
ISBN 0-8058-4102-4 (acid-free paper) — ISBN 0-8058-4945-9 (pbk. : acid-free paper)
1. Mathetmatics—Study and teaching (Elementary). 2. Reasoning in children. I. English,
Lyn D.
QA135.6.M374 2004
510'.19—dc22 2003064155
CIP
Books published by Lawrence Erlbaum Associates are printed on acid-free paper,
and their bindings are chosen for strength and durability.
Printed in the United States of America
10 9 8 7 6 5 4 3 21
To Ben, Jasmine,
Lauren, and Paige
Contents
Preface ix
1 Mathematical and Analogical Reasoning
in Early Childhood 1
Lyn D. English
2 Seeing the Possibilities: Constructing and Validating
Measures of Mathematical and Analogical Reasoning
for Young Children 23
Patricia A. Alexander and Michelle M. Buehl
3 Longitudinal and Cross-Cultural Trends in Young
Children's Analogical and Mathematical
Reasoning Abilities 47
Michelle M. Buehl and Patricia A. Alexander
4 Classroom Discourse and the Development of
Mathematical and Analogical Reasoning 75
Shuhui Chiu and Myriam O. Tron
5 Portraying Mathematical and Analogical Reasoning
in the Young: Three Case Studies 97
Debby Deal and Shanon Hardy
vii
VIII CONTENTS
6 Teachers' Knowledge, Beliefs, and Practices and
Mathematical and Analogical Reasoning 127
C. Stephen White, Debby Deal, and Carla Baker Deniz
7 Case Studies of Teacher Beliefs, Knowledge, and Practices 153
C. Stephen White and Debby Deal
8 Commentary: Analogical Reasoning
and Mathematical Development 169
Usha Goswami
9 Commentary: Toward a Cognitive-Sociocultural
Psychology of Mathematical and Analogical
Development 187
Giyoo Hatano and Tomomi Sakakibara
10 Promoting the Development of Young Children's
Mathematical and Analogical Reasoning 201
Lyn D. English
Author Index 215
Subject Index 221
Preface
This book is intended for those interested in the mathematical development
of young children, in particular, the development of their reasoning proc-
esses. The book will appeal to mathematics educators, researchers in mathe-
matics education, educational psychologists, and teachers in early childhood
classrooms.
As we illustrate in the following chapters, young children spontaneously
pose and solve problems that involve a broad range of mathematical ideas
and processes. They enthusiastically embrace problems involving pattern,
shape, number, and measurement as they explore their physical and social en-
vironments. Of particular significance in young children's mathematical de-
velopment are the reasoning processes they use in learning about their world,
such as spatial and quantitative reasoning, deduction and induction, and an-
alogical reasoning. However, existing research has yet to capture the range
and extent of young children's reasoning processes and the ways in which
they use these in their informal and formal learning. In particular, longitudi-
nal studies exploring the development of children's reasoning in the early in-
formative years of schooling are lacking. This book is an attempt to redress
this situation.
In compiling the book, we drew on rich data sets from a longitudinal
crosscultural study of young children's reasoning development as they pro-
gressed from preschool through the end of Grade 2. The participants in-
cluded the children and their teachers from schools in Australia and the
United States. We pay special attention to the children's development of
IX
X PREFACE
mathematical and analogical reasoning in their informative years as our ear-
lier studies had highlighted the importance of analogical reasoning in young
children's mathematical development. Background information on analogi-
cal and mathematical reasoning is provided in chapter 1 and is elaborated on
in subsequent chapters.
One of the key issues explored in the volume is the relation between mathe-
matical and analogical reasoning. Until recently, this link has received little
interest from the research community, especially studies involving in-depth
analyses of young children's mathematical and analogical development over
time. In conducting our investigations, it was imperative that we design de-
velopmentally appropriate and theoretically sound ways to expose young
children's reasoning capabilities. Our approaches to doing so are addressed
in chapter 2 where the construction and validation of our measures of mathe-
matical and analogical reasoning are described.
The crosscultural nature of our investigations enabled us to examine
whether developmental trends in the young children's reasoning are the same
for both American and Australian children. Our findings on this aspect are
presented in chapter 3, where profiles of children's analogical and mathemat-
ical reasoning development are examined. Chapter 5 presents in-depth case
studies of individual children's development across the three years.
Also of interest was how changes in children's reasoning relate to the
implicit instruction they receive in their classrooms. Although children's
learning is influenced by the way in which teachers construct a context for
classroom discourse, the way individual children interpret that discourse is
equally important. Hence, another aspect of our research was on individual
learners, their learning environments, and the interaction between the two.
Chapter 4 examines the role of classroom discourse in supporting the devel-
opment of young children's mathematical and analogical reasoning.
During each of the three years of the study, we interviewed the teachers of
the participating children to gain insights into their knowledge, beliefs, and
practices with respect to mathematical and analogical reasoning. In chapter
6, we present interview data addressing these aspects and discuss how the
teachers viewed the teaching of early mathematics and the development of
young children's reasoning. Chapter 7 presents in-depth case studies of the
three teachers.
Commentaries on our investigations are presented in chapters 8 and 9;
chapter 10 addresses some ways we can promote the mathematical and ana-
logical reasoning development of young children.