Why are most children bored by math - or worse, frightened
of it - when its beauties are so many and its pleasures so great?
Probably because it is usually taught by people who fear it, and who
therefore teach their fear.
To counter this (locally, but with hope of disseminating the ideas),
three of us - my wife Ellen, Tomas Guillermo and I - have begun a
Saturday morning Math Circle in space given us by Northeastern
University in Boston (thanks to Andrei Zelevinsky), for interested
8-to-18 year-olds. Word of mouth alone brought us twenty-nine students
for the first semester and forty for the second. Their giving up
Saturday mornings of sleep, sports or music lessons is a sign of how
much enthusiasm there is for math done with intensity and delight. We've
now had to add a Thursday afternoon session at Harvard (thanks to Danny
Goroff) for the overflow.
Our approach in The Math Circle is to pose questions and let
congenial conversation take over. Conjectures emerge from a
free-for-all, examples and counterexamples from the conjectures. Two
steps forward are followed by a step back. What really is at issue here?
How will we know when we've understood something? Is proving different
from seeing? Where and with what should proofs begin, and how validate
these beginnings? And if we get it, need we formalize it? Yes -
following my old fencing-master's adage about holding the foil like a
bird: tightly enough not to let it get away, not so tightly as to crush
it. We don't want a '6Os feel-good sense of math as expressive
hand-waving. We explain that rigor without mortis consists in fluency at
making a connected path back to foundations that will stand up to
scrutiny. In our exchanges the students are developing the knack of
pushing insight adventurously ahead while protecting the supply-lines
that fuel it.
Here's an example of what happens, from the Thursday afternoon
sessions with 8-to-11-year-olds. I began by asking them each to give me
very large numbers, and a rather conservative list of integers developed
on the board (Littlewood calculated that the gigantic stretches of time
in Indian mythology "only" amounted to years - for example, for kids,
25O is, as it should be, way out there). Was there a last integer? Of
course not, said 8-year-old Anna at once: if there were, just add one to
it. I was struck by this instantaneous freeing of imagination with the
passage to the general. They all took easily to one - one
correspondences, via the need to show Martians who count 1, 2, 3, many;
that a heap of thirteen pencils contains a larger "many" than a pile of
eight coins (some sorted by threes, some paired up, all relished the
Martian astonishment that there were two sizes of 'many'). By the end of
the first hour each was pairing up the naturals with the evens or the
odds or multiples of 7 ("Let's walk through the numbers in seven-league
boots!" said one) or the negatives and positives, or the integers (a
puzzling moment, followed by a cry of "Shift!"). The children left that
first session in two minds: they saw and could not but assent to the
results of their work, but they "knew" it couldn't be right - there were
clearly only half as many even numbers as naturals.
I was glad to see one boy return to the second session who had come
to the first the way you go to a horror movie: frozen between
fascination and flight. He had been brooding about the infinite, his
mother told me, and found it terrifying, and had signed up for this
class to overcome his fear. As you might expect, his were the most
daring conjectures in what followed.
Could the positive rationals be counted? Passionate arguments raged
about the lack of a starting place (the abyss of density opened, for the
first time, for several there). I put the first few rows and columns on
the board to claims ringing through the room that we could never get out
of the row we once committed ourselves to - until a girl, quiet before
this, asked if she could come up and began spinning a drunken spider's
web from the upper left-hand corner. Another girl suggested how we might
regularize the pattern and despair turning to glee, each now constructed
a bus-route through these streets and avenues (to the accompaniment of
popping insights and questions:."Look! Those diagonals are all repeats!"
"But how can we tell ahead of time that the sixteenth stop will be
2/5?") The diagonal proof belongs now to each of them, for as Locke
pointed out, "Property is that with which one has commingled one's
labor."
We headed toward Cantor's second diagonal proof through the largely
unfamiliar territory of decimals, browsing among repeating and
nonrepeating growths (much ingenuity here, on their part, in concocting
and explaining). Then a long digression on what it would mean not to
have a one - one correspondence between two sets, and at on proof by
contradiction (this pushes the limits of how much can equably be held in
mind), heralding the metaphorical entrance of a figure modeled on the
late and unlamented Senator McCarthy, patting his jacket-pocket: "I have
in my pocket a list of all the card-carrying decimals between O and 1",
each card bearing its ordinal. They followed the proof complaining all
the way, arguing with and explaining it to one another, ending up as
flabbergasted as I am every time I think it through.
By the time I arrived for the next class, a girl with an uncannily
precise mind was explaining Cantor's proof to a die-hard skeptic,
countering his attempts to put "our decimal made up of 5s and 6s 'at the
end of the list'". Now the pace accelerated as we came back to the
visual: they called the proof they devised of the one - one
correspondence between points on closed segments of different lengths
"the circus tent". Giving names, as Adam first knew, is a sign of
confidence. Since I'd told them about, they christened our new
cardinality "zig-zag null".
And the open interval and the real line? This led to the greatest
frustrations as well as ingenuity. At last, exhausted, with time running
out and skepticism about everything creeping out of every corner, the
boy who feared infinity had the saving insight. Tremendous relief,
appreciation, exhilaration. But the last note was struck by the girl
with the precise mind: "I see the proof and I accept it", she said, "and
at the same time I don't. It doesn't tell us, does it, just which real
number goes with the decimal we choose in (O, 1)? That makes me uneasy."
A good note to end - and begin afresh - on.
The topics we chose - but more particularly the ways we worked on
them - were designed to avoid two things which tend to go wrong with
accelerated or "enrichment" math courses. One is children being taught
the punch-lines without having worked their own way up to them ("I've
already had Pascal's Triangle", says the child who knows neither its
genesis nor implications). The second is teaching the magic that looks
like math: Pavlovian training of ten-year-olds to push the right symbols
to take derivatives of polynomials.
Why did we choose infinite sets for the youngest group? Because
little kids love large numbers, the way they like elephants and
dinosaurs: powerful friends in high places, but a liking tinged with
titillating fear. Children have no reputations to protect - because they
don't even know yet about reputation - so they are much better than
older students at making and getting far-fetched ideas - not a card is
held close to a chest. Unlike philosophy, math begins with awe and ends
with wonder.
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 9O angle by copying 3O 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 15-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.
This article was taken from Notices of the American Mathematical
Society, Vol. 42, Num. 9. Sept. 1995.