Mathematical Reasoning: Analogies, Metaphors, and Images

by English; Lyn D.

• ISBN13: 9780805819793
• ISBN10: 0805819797
• Format: Nonspecific Binding
• Copyright: 1997-07-01
• Publisher: Routledge

Summary

How we reason with mathematical ideas continues to be a fascinating and challenging topic of research--particularly with the rapid and diverse developments in the field of cognitive science that have taken place in recent years. Because it draws on multiple disciplines, including psychology, philosophy, computer science, linguistics, and anthropology, cognitive science provides rich scope for addressing issues that are at the core of mathematical learning. Drawing upon the interdisciplinary nature of cognitive science, this book presents a broadened perspective on mathematics and mathematical reasoning. It represents a move away from the traditional notion of reasoning as "abstract" and "disembodied", to the contemporary view that it is "embodied" and "imaginative." From this perspective, mathematical reasoning involves reasoning with structures that emerge from our bodily experiences as we interact with the environment; these structures extend beyond finitary propositional representations. Mathematical reasoning is imaginative in the sense that it utilizes a number of powerful, illuminating devices that structure these concrete experiences and transform them into models for abstract thought. These "thinking tools"--analogy, metaphor, metonymy, and imagery--play an important role in mathematical reasoning, as the chapters in this book demonstrate, yet their potential for enhancing learning in the domain has received little recognition. This book is an attempt to fill this void. Drawing upon backgrounds in mathematics education, educational psychology, philosophy, linguistics, and cognitive science, the chapter authors provide a rich and comprehensive analysis of mathematical reasoning. New and exciting perspectives are presented on the nature of mathematics (e.g., "mind-based mathematics"), on the array of powerful cognitive tools for reasoning (e.g., "analogy and metaphor"), and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool child to that of the adult learner.

Table of Contents

Preface	p. vii
Analogies, Metaphors, and Images: Vehicles for Mathematical Reasoning	p. 3
The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations for a Mind-Based Mathematics	p. 21
How Students Think: the Role of Representations	p. 93
Analogical Reasoning and Early Mathematics Learning	p. 117
Children's Development of Analogical Problem-Solving Skill	p. 149
Children's Reasoning Processes in Classifying and Solving Computational Word Problems	p. 191
Two Types of Reliance on Correlations Between Content and Structure in Reasoning About Word Problems	p. 221
Commentary: Mathematical Reasoning and Analogy	p. 247
Reasoning with Metaphors and Metonymies in Mathematics Learning	p. 267
Reasoning with Images in Mathematical Activity	p. 281
Generalization Using Imagery in Mathematics	p. 299
Children's Mathematical Reasoning with the Turtle Programming Metaphor	p. 313
Commentary: on Metaphorical Roots of Conceptual Growth	p. 339
Author Index	p. 373
Subject Index	p. 381
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