“The Heart of Mathematics” and the Hearts of Students

HeartOfMathV2I 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.]

Baby’s First Infinite Series


Okay, okay, I admit this isn’t suitable material for babies, nor for any normal kid whose age can be expressed with one digit! The “baby” phrase just popped into my head and I found it too cute to resist. A more accurate title might be “Joanie & Johnny’s First Infinite Series”, or “Kids’ First Infinite Series”.

I’m always thinking about ways to get those apathetic middle- and high-school students interested in math again by showing them something wild and crazy, yet simple and concrete enough (for them, of course) that it shouldn’t be too difficult to understand. My latest idea is an infinite series with a surprising and very simple visual “proof”.  Can you guess what it is? Hint:  the “proof” is in each of the figures above (surprise, surprise). What it proves is…


I’m sure each of these informal proofs — really variations on a single proof  — has appeared in many places, but I ran across the one on the left first, in Roger Nelsen’s wonderful Proofs Without Words. (The one on the right is from the Wikipedia article “Geometric series”.) The two volumes of Proofs Without Words, published by MAA, contain dozens of marvelous and miraculous visual “proofs” on a wide variety of mathematical topics; I highly recommend browsing through them. Full disclosure, though: in my opinion, not many are as elegant or as easy to see as this one. Still, I believe that several of the proofs Nelsen has collected, certainly including this one, could be presented successfully on the middle-school level — though probably not with the above notation. Regardless of notation, of course, appreciating this proof requires some ability to do arithmetic with fractions.

I think most people with experience teaching high-school math would agree that (1) not many “apathetic” secondary-school students have much of an idea what a proof is (notwithstanding the emphasis on them in a typical geometry course), and (2) hardly any of them know why they or anyone should care! That may be a problem with my idea, but it strikes me more as an opportunity. I suspect that going through something like this proof would help with both problems; it would also expose students to the important idea of adding up infinitely many numbers and getting a finite sum.

Do you agree? Do you this could work (with a reasonable amount of scaffolding, naturally) as a discovery lesson for, say, 10th graders? If you think it’s worth trying, here’s an idea for “reasonable scaffolding”: Discuss the fact that 1/3 = 0.33333333…, with infinitely many 3’s, and ask the class what 0.33333333… with infinitely many 3’s actually means. Of course it’s a very simple infinite series, one with the same sum as the series I’ve been discussing:

InfiniteSumFormula2Since both series have the same sum, an obvious question might be whether it’s possible to draw a picture to illustrate the above equation, and if so, what it would look like. I leave those questions as an exercise for the reader.