How we reason with mathematical ideas continues to be a fascinating and challenging research topic. It has become even more so in recent years due to the rapid and diverse developments in the field of cognitive science. Because cognitive science draws upon several disciplines - psychology, philosophy, computer science, linguistics and anthropology - it provides a 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 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, 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, yet their potential for enchancing 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 authors have been able to provide a rich and comprehensive analysis of mathematical reasoning. New perspectives are presented on the nature of mathematics, on the array of powerful cognitive tools for reasoning, and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the pre-school child to that of the adult learner.