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.