# Mathematical Reasoning: Analogies, Metaphors, and Images

# Mathematical Reasoning: Analogies, Metaphors, and Images

## Synopsis

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.

## Excerpt

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 psychology, philosophy, computer science, linguistics, and anthropology, it provides rich scope for addressing issues that are at the core of mathematical learning. One of these fundamental issues is how individuals mentally 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 play a significant role in our reasoning with mathematical 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 mathematical 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 reasoning as "embodied" and "imaginative." From this perspective, mathematical 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 reasoning is imaginative in the sense that it utilizes a number of powerful . . .