 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.
Robert Kaplan is a teaching assistant at Harvard University. His e-mail
address is kaplan@math.harvard.edu, and he has a web page at http://www.themathcircle.org/.
This article was taken from Notices of the American Mathematical Society,
Vol. 42, Num. 9. Sept. 1995.
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