The Math Circle

by Robert Kaplan

Back to New Directions in Pedagogy in Other Courses

The Math Circle (Part II)

Ellen chose to work on polygon construction with the 8-to-11 year- olds. Polygon construction meant using actual straight-edges and compasses; and while the hands were busy, a casual conversation about constructability steadily moved from the context to the content of the course. One student to another: "What do you mean, you can't trisect all angles? If you can trisect a 90-degree angle by copying 30-degree angles, can't you copy some angle twice to get any angle?" The crux of the matter is to seize on such assertions so as to let the students find out for themselves what's at stake (coming to grips with, among other things, the mysteries of quantification).

A generation brought up with calculators has difficulty manipulating fractions. Instead of addressing this directly, Ellen led them to construct an equilateral triangle and a regular pentagon in the same circle, and to figure out how to construct a regular 1 5-gon - thus discovering that 1/3 minus 1/5 could be seen as the useful 2/15 rather than the unenlightening .13333.... The confidence that followed this self-won competence made them feel this world was their oyster: they could construct whole series of regular polygons. Then, why were the heptagon and the ennagon so resistant to their efforts? Fermat primes came up before the course ended; but as in all the best dramas, exeunt omnes in mysterium.

With the 11-to-14 year-olds Ellen worked on polyhedra. Conversation accompanying scissors-and-paste constructions led very quickly to the discovery of the Euler characteristic. They tested it with Schlegel diagrams, studied and were convinced by Cauchy's proof, then read Lakatos' dialogue Proofs and Refutations (about the Euler characteristic), taking parts, and stepping out of character again and again to argue with the protagonists. They were startled to recognize in themselves the traits of monster-barrer and monster- adjustor, skeptic and omni-ameliorator. As in the dialogue, the semester ended with everything up in the air. Should things be neatly tied up?

With the same group, I worked on number theory beginning with this peculiarity: why do the digits in half the period of the decimal expansion of 1/7 yield, when added to the other half, all 9s? (One 11- year-old immediately said: "You mean, one less than a power of 1O"). And look, it's true of 1/13 too, and 1/1O1; but not of 1/2 or 1/5, much less 1/3 or 1/8. This took us on two long excursions: into geometric series and, through the idea of congruence, to Fermat's Little Theorem (which again they came up with themselves, just by messin' around). The farshining goal of our initial puzzle got us through difficult stretches. There is a push-me-pull-you rhythm to the best of these classes: convictions put together the week before turn out to have been soldered, not welded, together, and come apart with flexing (how was it we got the sum of an infinite series?). We reconstruct them more solidly under the pressure of doubt.

The 14-to-18 year-olds worked on infinite sequences and series with Guillermo, then did projective geometry with me. These were the most hard-fought of all the classes: they wanted nothing told them, all was to be invented. They came up with convergence criteria of their own (named after their new inventors), approximating ever more closely to the curve of the topic's history. By judicious choice of examples and nudges at critical moments I moved them to where they could - and did - come up with Desargues' Theorem, followed by their vigorous, critical role as sous-chefs in cooking up its proof. Because they were very puzzled by the maneuver of having to pass out of the plane and back to it, some doubting the validity of the proof, others the universality of the theorem, we had to digress to the free projective plane on four points - which they found startling and disturbing. They took an inventor's pride in coming up with a proof of the uniqueness of the fourth harmonic point, and that left us, at the end (ten sessions are too few), able to conjecture the Fundamental Theorem and prove its existence part. A real advantage of projective geometry for students whose graphing calculators usually do their visualizing for them is that their spatial imagination is awaken and exercised.

Our Saturday format has been two one-hour classes (milk and cookies in between), followed by guest lecturers (for example, Mazur on the ABC conjecture, Diaconis on the card-shuffling that led him to become a mathematician). A good high school mathematics course brings a student up to the eighteenth century. Here they could see contemporary mathematicians working on the frontier in the same manner that they had been developing for the last two hours.

Because our clientele is growing for next year we'll be taking on another hand. We're thinking too of branching out to other cities. What may be hard to export is our style: we entertain all conjectures and questions with equal seriousness, letting them follow their conversational course and turning the current of that conversation into fruitful directions as unobtrusively as possible. If a line of inquiry hits a wall we tend to let it lie and strike off in another direction, rather than throwing our students a sophisticated assortment of scaling-ladders. What's left fallow one week tends to produce a flurry of ingenious growths by the next (and these dead ends are the material of the week's homework). We do our best to hold off introducing a symbol until its abbreviatory power is welcomed for packaging up what had become an unwieldy complex of relations. Best when the students come up with the symbol - and the need for it - themselves.

What have we learned from this? That the appetite for real math, done neither competitively nor scholastically but as the most exciting of the arts, is enormous. I see no limits to what children can learn, and am convinced that if you want to teach them A, and A implies B, work on B with them: A will be mastered en passant, painlessly, absorbed in the bones. I'm certain too that removing any question of time - or achievement - pressure lets understanding and technique blossom, as well as developing a delightfully collegial feeling in those involved and a sense of the enterprise as contained within larger frameworks of question and significance. The students come away certain that math is mysterious, equally certain that its mysteries are accessible; unsure whether we discover or invent it; confident in their growing competence, and with that heightened threshold of frustration, that odd combination of watchfulness and willfulness, that characterizes the practitioners of our craft.

Go back to the first half of this article.

Robert Kaplan is a teaching assistant at Harvard University. His e-mail address is, and he has a web page at

This article was taken from Notices of the American Mathematical Society, Vol. 42, Num. 9. Sept. 1995.


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