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Many years ago, I had the good fortune to board an air-plane ight across the USA and nd both Jim Greeno andDavid Clarke on the plane. The three of us quickly retiredto the back of the airplane, where we had a wide-rangingconversation for more than 5 h. We talked about our dif-ferent theoretical orientations to looking at classrooms.Clarke was involved in the study of student understanding;I was trying to model teaching; Greeno was focusing on theclassroom community. As we discussed each other’s work,we pursued points of commonality and difference, and thestrengths and limitations of the approaches we had taken.(None was adequate to explain everything; each explainedsomething that the others did not.) As always, a few peoplestood around the back of the plane, some stretching orotherwise occupied.At the end of the ight, the seat belt sign came on, andthe three of us shook hands and prepared to go on our way.At that point, one of the bystanders came up to us and said,‘‘Thank you so much. That was an absolutely wonderfulseminar. I learned a great deal from your exchanges.’’I had precisely that kind of feeling when I nishedreading Theories of Mathematics Education: Seeking NewFrontiers. This rather substantial volume is organized into19 parts, each of which, after the initial overview (‘‘Sur-veying theories and philosophies of mathematics educa-tion,’’ by the editors), typically contains a brief preface, asubstantial essay, and from one to three commentaries onthe essay. Thus, the book can be seen as a series of dialogsamong some of the top professionals in the eld, to whichreaders have the privilege of ‘‘listening in.’’ This is anexcellent format, one that is extremely well suited fordiscussions of theory. When issues are ‘‘settled,’’ a set ofdenitive, handbook-like chapters may be appropriate. Butwhen ideas are being sorted out, a set of dialogs helps toreveal more of the unsettled (as well as the settled) terrain.That is a good thing.To begin at the beginning, Jeremy Kilpatrick notes in hispreface to Part I (the survey by the editors mentionedabove) that Europeans have, in general, been more explicitand reective about issues of theory than have those in theUSA. Thus, the international authorship of the book bringsvalue added to the American context. Kilpatrick mentions,among other things, the need for greater clarity about therelationship between theories and models, a point to whichI shall return: as a reviewer I cannot resist the temptation toenter into the dialogs begun by the authors.The volume itself offers a tapestry of theory-relatedissues in mathematics education: some closely related,some parallel, and some orthogonal. Although there are noformal section breaks in the book, there is a tacit organi-zation in the order of the chapters, with the more broadlytheoretical and metatheoretical chapters coming in Parts Ithrough V, and a broad range of issues-oriented theoreticalchapters in the sequel.The volume begins with various perspectives on the‘‘big picture.’’ Sriraman and English, citing Lincoln andGuba (1994), reference some of the themes mathematicseducation should address:1. What is reality? Or what is the nature of the worldaround us?…2. How do we about knowing the world around us?…3. How can we be certain in the ‘‘truth’’ of what weknow? (p. 7)

Bharath Sriraman and Lyn English: Theories of mathematics education: seeking new frontiers. (Springer series: advances in mathematics education)