David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick, UK. He received his Ph.D. in mathematics from Oxford University in 1967 under Professor Michael Atiyah, recipient of the Field’s Medal and Abel Prize. While completing his Ph.D. in mathematics, he became interested in mathematical thinking as a subject of study, but few theoretical frameworks existed. In 1986 Tall completed a second doctorate with Richard Skemp on *Building and testing a locally straight approach to the calculus*. Richard Skemp, a major pioneer in mathematics education, is the author of The Psychology of Learning Mathematics. At this time David Tall met Ed Dubinsky, whose APOS theory approach to learning mathematics is based on Piaget’s concept of encapsulation of process to object. The theory of encapsulation unlocked new avenues for Tall’s research. From this time on research on *Mathematical Thinking* became is mission and passion.

Tall’s book is well written for a general audience, accessible to students and educators from elementary school to university, and provides profound insights and theories for the professional educator and researcher. This book provides a comprehensive framework for understanding mathematical growth, from the basics of arithmetic, through symbolic formulation, to the more sophisticated axiomatic thinking. “This book takes you on a journey from the early conception of a newborn child to the frontiers of mathematical research.” Tall presents not only his own view but a also compares it to other theoretical frameworks. Specifically, chapter 3 provides a comparison to the SOLO Taxonomy by Biggs and Collis, the APOS Theory by Ed Dubinsky et al., and the Operational Structural approach developed by Anna Sfard.

The book is divided into 15 chapters. In chapter 1, titled *About This Book*, Tall outlines the content of the book and introduces the major ideas that form the foundation of how humans learn to think mathematically. He observes that children of the same age may have very different levels of understanding of mathematics. For example, Tall videotaped two children of the same age in the same school thinking very differently. One child, age 6, had only learned to count on his fingers while the other had an understanding of place value and number relationships. This example and the following lead to the idea of the *procept theory*. Tall worked with Eddie Gray while Eddie was completing his PhD on how children performed calculations in arithmetic. If children were asked to find \(13-9\) and they did not know the answer, how did they approach the problem? Did they use a method of counting or did they recognize that 9 is one less than 10, so the difference is 4? Tall and Gray recognized that these represented two rather different ways of thinking, one using counting processes, and the other manipulating number concepts. “The idea of ‘procept’ suddenly occurred. It linked dramatically earlier experiences of processes encapsulated as concepts through the ideas of Dienes, Dubinsky, and Sfard.”

In chapter 1 Tall also introduces the long-term development of mathematics and the *three forms of mathematics*: (1) Objects and their properties, such as geometric constructions, involving graphs and diagrams, (2) Operations and their properties, advancing to generalizing these concepts in algebra, (3) Axiomatic Formal Mathematics based on formal definitions of properties and deduction by mathematical proof. These three forms of mathematics build the foundation for the more sophisticated model of *three worlds of mathematics*.

Tall proposes three different ways in which mathematical thinking develops:

*Conceptual embodiment* builds on human perceptions and actions developing mental images that are verbalized in increasingly sophisticated ways and become perfect mental entities in our imagination.

*Operational symbolism* grows out of physical actions into mathematical procedures. Whereas some learners may remain at the procedural level, others may conceive the symbols flexibly as operations to perform and also to be operated on through calculation and manipulation.

*Axiomatic formalism *builds formal knowledge in axiomatic systems specified by set-theoretic definitions, whose properties are deduced by mathematical proof.

At the end of chapter 1 there is a complete outline for the rest of the book.

Chapters 2 to 8 focus on the development of mathematics in school and transition to formal mathematical thinking, beginning with the experiences of the young child involving shape and arithmetic. Chapter 3 develops the compression of knowledge into crystalline concepts. Chapters 4 and 5 study the foundational ideas of set-before and met-before and the related emotional aspects that affect long-term learning. Chapter 6 details the three worlds of mathematics and chapter 7 focuses on the relationship between the embodiment and symbolism in school mathematics. Chapter 8 studies problem solving at all levels and the long-term development of mathematical proof. These chapters offer an overview of the development of mathematical thinking, cognitively and affectively, for mathematics in school.

Chapter 9 is an interlude, using the three-world framework to analyse the historical evolution of mathematical ideas.

Chapters 10 to 14 follow the development of formal mathematical thinking and its continuing relationship with embodiment and symbolism to the frontiers mathematical research.

Chapter 10 considers the transition from the natural Mathematics embodiment and symbolism to formal mathematical thinking in which individuals progress in a range of ways.

Chapter 11 applies the three-world framework to the calculus as a blend of dynamic visual change and operational symbolism leading to the formal ideas of the limit concept in mathematical analysis.

Chapter 12 studies the development of formal knowledge into rich crystalline concepts and the proof of ‘structure theorems’ that expand formalism to a various blends of embodied thought experiment, symbolic manipulation and formal proof.

Chapter 13 applies the three-world framework to the infinitely large and infinitely small, proving a structure theorem that offers a way of seeing infinitesimal quantities with the physical human eye, vindicating the three-world framework that blends together embodiment, symbolism and formalism as a coherent basis of mathematical thinking.

Chapter 14 carries the blending of embodiment symbolism and formalism to the boundaries of mathematical research.

The final chapter reflects on the whole framework and its relationship with other theories.

The book closes with an appendix tracing the evolution of this theory, to reveal its origins in the insights of others to whom I am forever in debt.

This book is an invaluable reference for educators at all levels, from early childhood, graduate school, to the professional mathematics researcher. For the novice there are numerous examples in the book that provide important insights on how we learn mathematics from early years through advanced degrees while for the professional researcher, the comparison and development of theoretical frameworks can form a foundation for further research.

Gertrud L. Kraut is Professor of Mathematics at Southern Virginia University, a liberal arts college in Virginia’s Shenandoah Valley. She enjoys teaching undergraduate mathematics and statistics courses. Her training is in Numerical Analysis; her current interests are in cognitive psychology and the methods of learning and teaching mathematics.