According to the editors, this handbook provides a “comprehensive overview of the most recent theoretical and practical developments in the field … spanning established and emerging topics, diverse workplace and school environments, and globally representative priorities.” Packed into this 726-page paperback third edition are 12 revised and 17 entirely new chapters spread across five sections: (1) priorities in international research in mathematics education — 4 chapters; (2) democratic access to mathematics learning — 7 chapters; (3) transformative learning contexts — 6 chapters; (4) advances in research methodologies — 5 chapters; and (5) influence of advanced technologies — 6 chapters, with one final commentary. There are new perspectives on such topics as embodied learning, educating future mathematics education professors, multi-modal technologies, and e-textbooks. The authors, of these sometimes multi-authored chapters, stem from a variety of countries including Australia, Cyprus, Canada, United States, Mexico, Greece, Israel, Portugal, Denmark, United Kingdom, Italy, Norway, Netherlands, and France.

This handbook is not to be confused with the similarly named *Third International Handbook of Mathematics Education*, edited by M. A. (Ken) Clements, Alan J. Bishop, Christine Keitel, Jeremy Kilpatrick, and Frederick K. S. Leung, which focused on the “social turn” in mathematics education, rather than on mathematical cognition. My somewhat critical comments on that handbook can be found in a separate *MAA Review*. Also, in the interests of full disclosure, I should say that I was a pre-publication reviewer of one chapter of this book, but the rest of the contents were unknown to me prior to publication.

There is much to choose from in this handbook — a given chapter might speak to one reader, but not to another, and vice versa. After the introductory first chapter written by the editors, Chapter 2 provides an overview of research over the past decade on mathematics teacher knowledge and practice and on student learning. Chapter 3 covers a range of topics on, or leading to, embodiment of learning. What follows is an eclectic sampling of various other chapters.

In Chapter 4, titled “Configuring Learning Theory to Support Teaching,” David Kirschner discusses the challenges of using various learning theories from psychology, each with its own independent view of learning. The problem, he asserts, is that psychology is preparadigmatic and that psychologists have various competing views of learning — behavioral, functional (Dewey), Gestalt, developmental, cognitive, and sociocultural — that create a “Babel Effect,” making it hard for mathematics educators to establish a coherent basis for teaching. He suggests a “genres approach” that involves one’s own views of learning, or aspects of learning, and then using an appropriate psychological theory to inform one’s teaching. He considers, in turn, learning as habituation, which can be informed by behavioral psychology; learning as construction, which can be informed by developmental psychology (Piaget); and learning as enculturation, which can be informed by sociocultural psychology. Personally, I found this chapter a hard slog, but it contains worthwhile insights.

In Chapter 6, titled “Powerful Ideas in Elementary School Mathematics,” David Carraher and Analúcia Schliemann assert that “functions, and more generally, relations have a critical role to play throughout elementary school mathematics education.” The authors’ argument is that “functions and relations are already implicit throughout the contents of the present-day elementary mathematics curricula,” that they are “accessible to young students,” and that bringing them out can “deepen and integrate” elementary students’ knowledge, “increase their ability to make mathematical generalizations, and better prepare them for a later, more formal introduction to algebra and fractions.” (p. 191). To make their case, the authors provide a number of mathematical problems and samples of third- fourth-, and fifth-grade students’ work on these problems. A sample third-grade problem is: *Tom is 4 inches taller than Maria. Maria is 6 inches shorter than Leslie *(*along with a sketch of Maria next to an unmarked vertical line with an arrow pointing up at the tip*)*. Draw Tom’s height, Maria’s height, and Leslie’s height. Show *[*on the vertical line*] *what the numbers 4 and 6 refer to.* (p. 204).

For those who teach proof-based mathematics courses, Chapter 8, written by Joanna Mamona-Downs and Martin Downs, on “Mathematical Structure, Proof, and Definition in Advanced Mathematical Thinking” may be of special interest. While the authors note the many “slants” have been taken on the character of structure, for them structure can take two forms: “structure that is identified in the environment” and “structure that is specified by a property.” The first form is “accessible to young students, but the second depends on [an] appreciation of given abstract definitions” and the axiomatic character of mathematics. (p. 241). While the former is based on perception and observation, the latter is based on imposing certain conditions and seeing what follows. In this chapter, the authors view structure as incorporating both of these forms and see this as important for “analyzing students’ behavior in proof making/reading, and in designing/handling definitions.” (p. 242). Their discussion of structure is followed by a discussion of recent research on proof and proving and on definitions and defining. By way of personal indulgence, I would like to mention that Martin Downs was a group theorist, with a keen interest in mathematics education, who frequently conducted mathematics education research with his wife, Joanna. His untimely death before this volume’s publication is noted in memoriam in the opening acknowledgements. Martin was a friend, and his wife, Joanna, also a friend, feels his loss very deeply.

Chapter 18, written by Patrick Thompson, is titled, “Researching Mathematical Meanings for Teaching,” and indeed the first ten pages are a discussion of the distinction he sees between knowledge and meaning. The rest of the chapter is a discussion of the construction of the assessment instrument that he, colleagues, and graduate students devised: *Mathematical Meanings for Teaching Secondary Mathematics*, a 43-item diagnostic instrument designed for use in high school mathematics teacher professional development. Four actual assessment items are given, along with samples of teachers’ work on the items, and the holistic scoring rubric devised for each of these items. To give a flavor of these items, here is one: *A college science textbook contains this statement about a function f that gives a bacterial culture’s mass at moments in time. The change in the culture’s mass over the time period *D*x is 4 grams. Part A. What does the word “over” mean in this statement? Part B. Express the textbook’s statement symbolically. *The point of this item is to get at teachers’ use of the word “over” as in “distance over time.” Are they only referring to a spatial arrangement of symbols? Or, are they thinking that two things happened concurrently? Developed under an NSF grant, it is interesting to note that each item of this assessment together with its scoring rubric, cost on average, over $35,000. (p. 458). While the thinking and work that went into the development of the instrument and the discussion of the example items are very informative, it would also have been useful to have included more about how the instrument is, or is intended to be, used to design professional development for high school teachers.

Of course, there are many other chapters that I could have mentioned. One cannot read this tome at one sitting, however. Indeed, some chapters may require a great deal of thought. One thing I like about this Handbook is that each chapter has its own set of references at the end, instead of homogenizing them as a whole, at the end of the book, making it easier to see who is being cited at any given point. Also, the citations are in *APA Style* (e.g., Thurston, 1995), which I personally like — if one knows the author(s), one can continue reading without having to consult the end of the chapter, as one would with numbered citations (e.g., [4]). There is also a reasonably thorough 14-page index at the end.

That said, the question is always: Who is this book for? Clearly, it is meant for mathematics education researchers and their graduate students, but others can gain insights and ideas, as well as a few interesting mathematical problems, together with examples of student work, from some of the chapters.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.