I recently ran across the book The Heart of Mathematics: An Invitation to Effective Thinking, by Edward Burger and Michael Starbird. Burger and Starbird are both math professors at well-respected colleges, and they’ve both won multiple awards for their teaching. But The Heart of Mathematics is hardly a conventional textbook for college math classes! The publisher’s website (http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000304.html) says: “Infused throughout with the authors’ humor and enthusiasm, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.” I’d hesitate to say anything contains THE “most important and interesting ideas” in any subject; but that’s a quibble, and otherwise I agree completely.
I’ve always felt the best hope for engaging most students in math at almost any level is to expose them to what I call “wild and crazy ideas” — i.e., to go for the gold medal of intrinsic motivation, not to try for the consolation prize of external motivation by attempting to convince students (without much justification, and and usually without much success) that knowing math will eventually be useful to them. (If you’ve read Nicholson Baker’s piece in a recent Harper’s Magazine, “Wrong Answer: The Case Against Algebra II”, you won’t be surprised to hear that I agree with 90% of what he says.) After hearing me rant for a few months about such things, Frank Lester loaned me his copy of The Heart of Mathematics, saying he thought it was very much my kind of book. He was right. His only real reservation, Frank said, was that it makes things too easy by letting students see the answers to the many challenges they pose — but that’s hard to avoid with paper. I think he’s right about that, too.
I’ve been working for years on a list of wild and crazy ideas for teaching math, and a lot of the topics The Heart of Mathematics covers (different sizes of infinity, the Monty Hall problem, Simpson’s paradox, Möbius bands, etc.) are on my list — and, it’s clear to me, a lot of the others should be! Frank, thanks so much for exposing me to this book.
But this book seems to be almost unknown to secondary-school math people. If it’s so good, why is it that? Probably because it’s explicitly intended for college-level courses for non-science majors, and for that audience, it’s been a huge success: according to the publisher, it’s “the most widely-adopted textbook in liberal arts and liberal studies mathematics and teacher preparation in over ten years”. But there’s plenty of material here for a book — or, perhaps better, an online course — for high-school students. More important, by the time students reach college hating math and having a hard time learning any, it’s too late!
I’d love to see Burger and Starbird come out with a high-school level version of the book, and Starbird tells me they’re interested. The problem, of course, is that this isn’t a textbook for any of the standard high-school courses, so it would be hard to be confident of its adoption by many school districts. Still, it’s worth a try; we as a nation need desperately to do something about our lame-brained approach to teaching mathematics — something other than pushing the same bad ideas even harder.
[revised January 2014: improved the illustration; updated the last paragraph.]