*The Puncturing of Space: a Developing Pedagogical Tool* by Dr. Robert Emmett Mason IV, does not fit an easy description. It is part authoritative teaching handbook, part textbook, and part philosophical discourse from a master pedagogue with thirty years teaching experience. It is a marvelous resource for all teachers of mathematics and the natural sciences.

From his very first teaching job in 1973 at a school for gifted children in New York, Dr. Robert Mason has been taking careful notes and developing a philosophy of teaching that he shares in this elegant and incisive volume. The intellectual framework for his teaching approach is modified from Jack Whitehead’s “Living Educational Theory of Teachers,” in which teaching becomes an iterative process of self-discovery on the way to unlocking the intellectual capacities of students. Robert Mason develops a model in which the teacher moves from instructionalist to constructivist educator, constantly modifying and improving didactic approaches in response to the dynamics of a classroom.

The first chapter of the text is a highly personal and riveting account of Dr. Mason’s earliest experiences and development as a mathematics teacher. It chronicles his transformation from a novice teacher clinging to the Teacher’s Edition of a standard textbook to a skilled craftsmen who challenges and is challenged by the dynamic interactions of the classroom.

This personal theme is continued in the second chapter in which Dr. Mason applies classical models of learning theory such as those by Piaget and Vygotsky to further define his pedagogical approach. The third chapter is a fascinating meditation on the most basic notions of consciousness that are a prerequisite to understanding the basis of the classical admonition of the teacher: “please give me your undivided attention.” The most abstract section of this volume, this chapter touches on a wide range of themes, including theories of neuronal group selection and brain physiology. The chapter thus attempts to define the “boundedness of the contents of mental activity.” Such biological and psychological foundations, Mason argues, are critical for the teacher to understand and improve strategies of communication of mathematical concepts.

Chapter Four provides more of the details of the central model of the text, the approach to teaching as “puncturing space with pedagogical objects” and defines such notions as “concept proxies” by use of specific examples and actual transcribed dialogue between teacher and student. Chapter Five begins by introducing the “geo-board” as an example of a concept proxy and cites certain advantages over the chalkboard. This chapter explores the use of concept proxies in illustrating basic algebraic and geometric identities.

Chapter Six is a detailed description of the theoretical debate between Vygotsky and links the notion of concept proxies with Vygotskian notions of cultural artifacts as pedagogical objects. Mason embraces the theory of “Genetic Epistemology” in which knowledge is viewed as a process of continuous self-construction. With this philosophical framework established, Chapter 7 and 8 offer detailed descriptions of curricula largely designed by the author and now put into practical application. These chapters include the introduction of Euclid Two, an advanced geometry curriculum developed by Dr. Mason that builds on his earlier seminal work in this area. His approach to the teaching of geometry is characterized by a syncretic interaction in which the students are encouraged to question and discover for themselves the theorems and definitions of transformational geometry. Utilizing his prior (classic) annotation of “Geometer’s Sketch Pad,” Dr. Mason provides a series of annotative lessons that can be easily followed by teachers.

These latter chapters also introduce a second curriculum entitled “An Introduction to Experimental Research Skills for the Scientist.” This curriculum is appropriate for 5^{th} through 8^{th} graders. Dr. Mason introduces the familiar matrices, classically taught as part of linear algebra courses at the college level, but here brought to life for younger students. Through concrete examples, matrices are applied to, for example, financial transactions as well as geometric transformations in a way that will engage students’ imaginations. The introduction to research methods for the young scientists consists of annotative experiments that draw on basic mathematical constructs. Dr. Mason essentially provides an introductory curriculum in biostatistics, classically reserved for graduate students. By introducing statistical concepts in the context of interesting and simple scientific experiments, Dr. Mason challenges traditional approaches to teaching of mathematics at the pre-graduate level.

*Puncturing of Space: A Developing Pedagogical Tool,* represents that rare synthesis of cognitive/learning theory and a practical annotated curriculum. Dr Mason is the creator of Euclid Two, as well as the prior annotation of Geometers Sketch Pad that has been widely utilized in classrooms for years. The two new curricula proposed in this volume could stand by themselves and provide exciting tools for teachers seeking to provide novel course outlines for mathematic students in either public or private educational settings. What is unique about the volume is the juxtaposition of a highly innovative, detailed mathematics curriculum with the philosophical construct from which it emerged. Having witnessed Dr. Mason as a teacher, I can also testify that he is an expert practitioner of his pedagogical theories. Dr. Mason’s sensitivities to cultural and gender differences among students have nothing to do with “political correctness,” but rather stem from his heartfelt philosophical commitment to the individualized didactic approach to education as articulated in the opening chapters of this volume. While there is perhaps no substitute for witnessing Dr. Mason’s teaching in the classroom, this volume provides an example of an eloquent and highly readable ”concept proxy” devised by a gifted educator.

Kenneth Offit, M.D., M.P.H.

Professor, Medicine and Public Health

Cornell University Medical College

Chief, Clinical Genetics Service.

Memorial Sloan-Kettering Cancer Center

- By Michael Sturm
Robert Mason, affectionately known as Doc by both faculty and students, alike, has taught middle school math at Dalton for the last 20 years. ...

- Frank A. Moretti, Ph.D
There are times when a rare person, for mysterious reasons, transcends this set of circumstances and feels the inner necessity to locate practice in the context of theory. Dr. Robert Emmett Mason IV, however, has taken on the challenge of integrating his range of experience in a way ...

- Kenneth Offit
The Puncturing of Space: a Developing Pedagogical Tool by Dr. Robert Emmett Mason IV, does not fit an easy description. It is part authoritative teaching handbook, part textbook, and part philosophical discourse from a master pedagogue with thirty years teaching experience ...

- Victoria Geduld
Dr Robert E. Mason's Teaching Mathematics might seem far removed from productive pedagogical reading that would be assigned to an incoming Ph.D. teaching assistant in a History department. Indeed, this book should be mandatory for teachers in all disciplines at both the beginning and more advanced levels. ...