Fermat’s Last Theorem on the Greenboard

FermatsLastThmOnTheHSGreenboardTRIn the first week of my very brief career as a regular full-time high-school math teacher, my students and I somehow started having a conversation in the lower-right corner of the greenboard at the rate of one statement by each party per day. Their first two comments were very complimentary, and I responded as shown above. Of course the math in the first sentence is Fermat’s Last Theorem. Not that I expected them to understand it (despite the fact that I had all honors classes), or even to be interested in understanding it! But they definitely were interested. So I had the delightful experience of talking to my Algebra II students about number theory for a few minutes.

This is one of several experiences that convince me that a significant fraction of high-school math students — at least of honor students, but probably of ordinary students as well — haven’t had all their natural curiosity about things mathematical squeezed out of them, despite the worst efforts of the American math-education system. It’s well worth a few minutes of classtime now and then for a teacher to try to take advantage of it.

“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.]

Zeno’s “Achilles and the Tortoise” Paradox and Geometric Series

When I decided to retrain as a secondary-school math teacher and applied for a Woodrow Wilson Indiana Teaching Fellowship a few years ago, I had to teach a 5-minute lesson. It could be on any subject I wanted, and I chose Zeno’s first paradox of motion, the “Achilles and the Tortoise” Paradox. As you may know, the Greek philosopher Zeno (5th century BCE) devised several “paradoxes of motion” that baffled all of his contemporaries. His first paradox involves a race between Achilles (a very fast runner, said to be “the fleetest of foot of all mortals”) and a turtle (a rather slow crawler), with the latter having a head start. Zeno gives a simple but surprisingly convincing argument that Achilles would never catch up: “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”


In my brief lesson, I explained that this argument assumes that the sum of an infinite geometric series cannot be finite, and showed why that’s not true, i.e., why such a series can converge to a finite value.

Why, in my interview lesson, did I teach something as advanced as convergence of infinite geometric series? In two words, intrinsic motivation. High-school Algebra II (at least in Indiana) covers geometric series and their sums, and infinite geometric series are very likely to be touched on. Still, students won’t be expected to understand why the sum of infinitely many numbers can be finite until they take calculus. But, while the details require methods of calculus, the concepts are simple; there’s no reason they can’t understand immediately. I’m fond of the infinite, that fruitful source of crazy ideas, and I’ve wanted from the beginning to expose students to the wildest and craziest ideas I could possibly get them to understand. And why did I want to do that? The main reason is that, based on my own experience, I believe that wild and crazy ideas can be a great source of intrinsic motivation for a lot of kids—certainly not the majority, but a lot. And intrinsic motivation is generally agreed to be far more reliable than external motivation. Along the same lines, Sawyer’s fine book Prelude to Mathematics comments “A non-mathematician learning mathematics…often has to plough through routine procedures, which can be extremely dull… An education should also contain elements that perform the functions of a cold bath—to provide a shock and keep one awake.”

I’ve given versions of this lesson three times now, including once to a high-school “math exploration” class, and I think it went over well each time. The latest version — available in narrative form — has two parts: “Resolving the Paradox” (roughly the original 5-minute form) and “Boring Details, A Related Example, and the General Situation” (about infinite geometric series in general and when they converge). The only technology needed is a yardstick and a greenboard drawing.