Out of the Labyrinth: Setting Mathematics Free by Robert and Ellen Kaplan (Oxford University Press, 2007).
I have a deep appreciation for this book. Authors Robert and Ellen Kaplan speak of mathematics and math education from a perspective I've rarely encountered in writing, although I read a great deal on mathematics and math education. Namely, they combine a love of advanced math with an appreciation for students of mathematics coming to their own conclusions rather than taking anything on faith. I identified with this point of view deeply as I read.
For fourteen years, the Kaplans have been practicing a rare (but happily, decreasingly so) form of math education, working with students who come voluntarily, after school or on the weekends, to study mathematics purely for its own sake. They facilitate the students' exploration of deep mathematical content, for the sake of nurturing the students' latent powers of reasoning, perseverence, and creativity, as well as mathematical technique. The book begins and ends with descriptions of what happens inside their classrooms. For me, these were the most vital parts of it. The middle of the book is an extended meditation on the nature of mathematical inquiry, mathematical symbols, and the history of math teaching. Mathematicians, and philosophers of mathematics, will find this discussion very thought-provoking. Educators reading the book with a more utilitarian purpose may find this section unnecessary to digest in detail.
The book is fun to read, with a sort of analogical style both ornate and provocative. This provides for a good number of memorable one-liners (in an extended critique of math competitions - "Having long since stopped beating students to make them learn, why do we still encourage them to beat one another?") including the best one-sentence definition of math I've ever heard ("...what is mathematics but the study of structure?"). In the middle of the book, it also sometimes makes the main lines of argument harder to follow. From the point of view of a math educator, this isn't a problem, a propos of the last paragraph. The Kaplans also draw on a wealth of fabulous references, quotations, historical anecdotes, etc., although with a somewhat consternating lack of footnotes or bibliography.
All in all, a unique and remarkable book, articulating a point of view the world of math education deeply needs. Mathematicians and educators will be inspired alike. Certainly, this one was.©2009 Ben Blum-Smith, 2009