The Math Circle
A Talk Given at the AMS Convention,
San Diego, January 8,1997,
by Bob & Ellen Kaplan
(BOB): Ellen and I began The Math Circle two and a half years ago, to bring people from six to sixty into the great conversation: that on-going dialogue between mind and the world, whose natural language is number and shape - because we were convinced by our long experience that the deep joys of inventing and discovering math belong to everyone, and that anyone is capable of savoring these pleasures.
Who are these anyones who come to The Math Circle? Since we haven't advertised, it is all word of mouth, which has brought us those good at math in school and hungry for more; those for whom it is a little scary, but who have heard that our Saturday mornings are fun; and parents who begin by sitting in (over-protectively) and end up hooked into joining - as long as they are well-behaved. The thirty or so in each of our first five semesters has grown to 55 this semester.
How can we manage such an age range? But dividing our three hour Saturday mornings into three groups, more or less by age: the youngest, or those without algebra; the middle, or those without calculus; and the seniors. Each group is led in the first hour by one of us - Ellen, me, and our third, new colleague, Mira Bernstein (whom we've taught since she was 14, and who is now a graduate student in math at Harvard. Then, after juice, cookies and socializing, we switch places and teach a second course; then all join to hear a guest speaker in the third hour.
We've had marvellous talks from a host of people - almost 50 so far, including Bary Mazur, Persi Diaconis, Raoul Bott, Andrei Zelevinsky, Joan Richards, Dick Gross, Kevin Oden and Tom Lehrer - talks that ranged from knot theory and perfect shuffles to p-adics and elliptic curves, and which generally involve a to and from with our eager students. This all happens on ten Saturdays in the fall semester, and ten in the spring (we've added a Thursday afternoon group for the very young, meeting at Harvard, and now a similar group meeting on Friday afternoons in nearby Arlington).
Why is it that most people don't know that math is their native language? Six reasons that strike me as central:
First, most students, whether good at it or not, are bored by their math classes - they are misled into thinking that the shining city of mathematics is just a huddle of old memory palaces.
Second, most - whether good at it or not - are at some level afraid of math, since the fear most teachers feel of it is the one thing they teach really well - so that the typical student is
"Like one who walks a lonesome road
And dares but look ahead,
Because he fears a loathsome fiend
Doth close behind him tread"
This fiend is named Embarrassing Error! "What! You didn't know that? Everyone knows that!" Because mathematical truths so quickly become impersonal, they have a devastating authority.
They haven't yet the momentum of engrossment behind them - and this is the third reason. They haven't yet - or haven't yet critically often - worked through dinner or forgotten other homework because they couldn't stop probing at an intractable problem - and haven't yet realised that the world is well lost for the solution. They haven't yet found themselves playing ninepins up in the mountains with the little men, while the decades rolled by. Fourth, they are put off by the ungainly symbolism of math - a symbolism both awkward and uncompromising - a secret language that shuts you out from the club of its users - yet is as half- formed as musical notation (think on the one hand of the common fear of subscripts - all that distracting decoration around our old trustworthy letters; and on the other, the deep issue of letting the wolf of a variable loose in the sheep-fold of constants).
Fifth, the passion of math is so intense that it can almost seem colorless to a spectator - and we know how shy we are about describing our work to "laymen' - there is still something of the Magus to the mathematician. Few students have any idea of what mathematicians do ("Learn the squares of all numbers?" "Solve simultaneous equations with 17 unknowns?") and therefore have no idea of what mathematics is.
Sixth, in particular, we're living at a time when schools and colleges, having lost their nerve - their self-confidence - try to cater to their students' enthusiasms rather than challenge them: as desperate to win the support of their future alumni as parents are to keep the affection of their children.
But rather than so try to buy into a consumers' world of fad and fashion - a world that can only leave its devotees trained to become obsolete - why not try to make one's student owners of their world, by letting them invent it - and then, with the shock of recognition, find that they have discovered together the world as it deeply is?
(ELLEN): So there are our aims - and you'll probably think, quite rightly, that what Bob has described are the aims we all have - that all good math teachers have had since Plato. The question is: how do you do it? How do you wake, tease, entice a mind into this new realm?
By the classic technique of intellectual seduction: trailing accessible mysteries in front of your students. In our classes that means posing problems that often take the form of scattered data crying out for a pattern. Some examples:
Bob's number theory course this fall for the middle group began with the question:what do you make of the sequence:
5, 13, 17, 25, 29, 37, 41, 53...?
This led, after ten intense weeks, through Pythagorean triples, Fermat's little theorem, Wilson' theorem and much more to the beautiful truth that k is a primitive hypotenuse if and only if each of its prime factors is congruent to 1 mod 4.
My very first course for the middle group began with students just making and then playing around with polyhedra. The actual scissors and glue work took far longer than seemed worthwhile; what I'd seen as a course on the Euler Characteristic looked as if it were turning into remedial pre-school. But then, during a conversation about how long it takes rubber cement to dry, someone hesitantly said: "I don't know that this is always true, but is the number of vertices plus faces two more than the number of edges?" And we were on our way, via Schlegel diagrams and Lakatos' great *Proofs and Refutations*, to a serious analysis of what "Eulerian" means. Had we spent too long on the concrete? No - as it turned out: that cutting and counting, gluing and holding had given them all an unshakeable sense that they *knew* polyhedra - that no fast-talking formalist could tip them off their foundation of intuition.
The tricky part for us is to recognize the moment when thought begins to act on matter, and at that stage to let them all go at it, wondering, arguing, conjecturing, refuting - sounding for all the world like a convention of mathematicians. Here's the way our colleague Mira put it:
"To kids, the only thing that matters is pattern, numerical or geometric - they look for order with relentless obstinacy and complete indifference to its possible meanings and interpretations, implications or causes. Of course 90% of the beautiful patterns they uncover aren't really there, but that doesn't seem to matter at all: once found, a pattern has already lost half its charm anyway, so they are only too happy to set out immediately in search of newer, shinier specimens. It seems sometimes that if I didn't prod them periodically toward my silly grown-up notions of verification and causality, they'd happily sit there all day just throwing out patterns, one after another. But then, isn't this the same insatiable appetite that drives us all?"
A third course was created on the spur of the moment, when a twelve- year-old said to me: "I'm sorry, but I just *have* to know: what is i^i?" By great good luck I held my tongue, and used his question and the energy behind it to generate a whole course, which I taught last spring: a ten- week safari through the complex plain, derivatives and Taylor Series, with our astonishing goal reached in the last fifteen minutes of the last class.
The students go from the fun of exploring to the deep pleasure of seeing this shiny bit in a larger context that adds to its sense and glory. They also discover the pleasures of encountering skills and ways of seeing unlike their own - for some have, when working on a particular problem, more of a capacity to invent, others to calculate, others to make elegant.
So, if William Lilly's story is true, when that astonishing English maker of log tables, Henry Briggs, who would extract 47 square roots by hand to produce approximations good to 15 decimal places, met the extraordinary Scot, Napier of Merchiston, who had invented the concept of logarithms, "Almost one quarter of an hour was spent, each beholding the other with admiration before one word was spoke." (Si non e vero, e ben trovato: if it isn't true it should be).
For we need, don't we,in the peculiar undertaking which is mathematics, both precision and imagination, both legality and insight. A successful solution to a mathematical problem depends on the proportion: free play is to rigor as discovery is to proof - opposites which (like the Apollonian and Dionysian) must both be as large as possible for the greatest work to be done. So in The Math Circle we coax our students away from either Math as messing around or math as mathematical logic toward a continual alternation of conjecture and validation. How do we do this?
The temptation always to be resisted is showing them how *we* do math - flying them to the top of Everest. Instead we make them climb, and we are their sherpas, pointing out good hand-holds, gesturing toward what seem likely routes, ready to lower a rope when they fall into crevasses, carrying what equipment they may need - and most importantly, moving the base camp up and up.
(BOB): We are helping them to discover - or recall - the grammar of the universal language, so that they will be able to write those poems in it where proof rhymes with insight.
How do you do this, so it isn't hit or miss? By luring them to step aside now and again from the on-going work to think about patterns - again - in their working.
So in the course for seniors I taught a year ago on the Pythagorean theorem, I began by asking if anybody knew *a* proof of it, expecting and getting the answer "yes" from each - and often, as I'd hoped, with the proof offered as *the* proof. I then handed out a variety of proofs - or hints of proofs - to each student, asking them to report on them next week. As I'd hoped, their discussion led to the question of how and why there could or should be so many proofs of the same theorem: "Will the right proof please stand up?" And as we went on, categorizing and relating the proofs, the background questions of "what *is* a proof?" and "do different proofs in fact really prove different statements?" began to emerge - as did issues of why some proofs seemed more natural, or more elegant, or more profound, or more general, than others - and to whom - and so to questions about mathematical taste.
As we went on with our work, generalizing the Pythagorean theorem in different directions, enjoying Fermat's proof that no integers satisfy x^4 + y^4 = z^4, and mulling over the open question of the Perfect Box, the two distinct strands of the course - the work on the Pythagorean Theorem and work on what proving entails - kept interweaving, as they do in a good novel.
Our approaches vary from course to course, influenced always by the particular material, and the emerging character of the students. Last summer, for example, we were invited by Urs Kirchgraber of the ETH to demonstrate The Math Circle approach at five high schools around Zürich, and the topic we chose was that straight road to glory, transfinite arithmetic. What happened was different in each group. In one of Ellen's classes a girl was in the process of coming to grips with one-one correspondences, torn between seeing that the evens were equinumerous with the naturals, but as troubled as Galileo had been 300 years before by all those orphaned odds. Her teacher, impatient, interrupted from the back of the room: "But all mathematicians agree that these sets are equinumerous, so that's that!" The girl turned to Ellen and said, "Well, I'm not a mathematician, so I don't have to agree."
Our overall aim is to give them the wherewithal to build the spiral from keen observation to deft experiment to extracting form from matter - and then to observe again - where these abstractions, now become familiar and comfortable, are the new data.
What are the difficulties we encounter in our courses? Not competitive egos, for we've found that by pitting us all against a hard problem, a wonderful conviviality and team spirit takes over - any insight is welcomed by all for the help it gives - and we offer no prizes, and never make odious comparisons. We are all mice nibbling at a very large piece of cheese.
No, the major difficulties come, it seems to me, from the baggage that people bring with them - truths their teachers told them that may not be so true; misleading analogies; false associations; vague language. I was trying, this last summer, to entice my cricket-club teammates in Scotland into transfinite arithmetic (I'll evangelize anywhere). "But infinity's only a concept, isn't it?" said one. I figured out eventually that what he meant was that, since it wasn't 'real', you could say what you liked about it. I tried to have them hi on a way of counting the integers (this was during a long car ride to a match, five of us trapped together in a nutshell) and pointed out that zero was in there, between the negatives and the positives. "Zero!" one of them said, "Not Zero! Don't talk about Zero! Zero is the Void, Nothingness, Death! Zero is he Bell-Jar!" The conversation moved on to beer.
But the chief difficulty, I think,is imagination constrained by unwarranted assumptions. You all know, I'm sure, the old nine-dot problem: connect a square array of nine points by a continuous line with four segments. Many a student will assume that the lines must remain within the square, and it is only on breaking out of those self-imposed limit that the solution comes.
The essence of mathematics is freedom, as Cantor once said; but that freedom is won by letting the problem at hand shape your language. Our role in a class at this point is to call our students' attention to the ways *their language* may be reshaping the problem.
The trick, as the great wide receiver Ray Berry once said, is for preparation to meet opportunity; and the hardest part for me in the classes is to watch for when the right openings happen in the discussions, and face the students toward them.
And we want to leave them, at the end of a course, triumphant, self- confident, with what was a mystery now accessible - but also with a sense of the greater mysteries ahead. In my class on sequences and series this fall with the 7- to 11-year-olds, we'd had a running battle throughout about
1 + 1/2 + 1/4 + 1/8 + ...
What did it all add up to, if anything? Had it a meaning, and if so, could we find it out? Not a class went by without someone jumping halfway to the blackboard, halfway again, and again - to prove that you would or (same proof) that you wouldn't get there... Claims and counter-claims rocketed back and forth. The class solved a lot of problems, but we'd gotten nowhere with this one, so in the last session I asked: "Do we agree that we have no idea what the answer is?" Yes, they agreed. "So what this series adds up to - " I wrote it yet again on the board - "is - we don't know what?" A chorus of yeses.
"Is it all right with you if i write 'is' as 'equals'?" Again yes. "And 'we don't know what' as 'x'?" Everyone nodded. I had our series equal to x, factored out 1/2 from all save the 1 on the left - a girl with a real flair for symbolic thought ran to the blackboard, said: "So 1 + 1/2x = x; so 1 = 1/2 x; so x = 2!" Stunned silence. Then her younger brother came up to the board, took the chalk from my hand, and pointed with it to my initial equation. "I should never have let you do that!" he said.
(ELLEN): I think you see the atmosphere of intensity and high humor in which The Math Circle operates. Is it exhausting for them or for us? No, it is exhilarating - even on a Saturday morning. Though I must admit we don't get much work done on Saturday afternoon.
It is exhilarating because it is important. Although not every day produces a completely unexpected insight, there are always the tiny triumphs - a long calculation that comes out (the spirit of Henry Briggs blesses you) - and every success gives a student's self-confidence a tremendous boost - in the way only an accomplishment you yourself recognize as significant can produce legitimate and secure self-respect. Better for someone to struggle, discovering how to bisect an angle, than to hear at length from one of us that in general trisection is impossible. Freud put his finger on something when he spoke about 'the conflict-free ego-sphere' - an ego almost not your own ("Listen not to me but to the logos", said Heraclitus). Kids - and not only kids - delight in moving into a realm where mind and imagination are set free from personalities and appearances.
It is important too because you come to know of what realm, of what Republic, you are a citizen, and your current address in it - and you learn that the politics of this Republic are neither conservative nor radical - or rather, are *both*, since math is both continuously created, and therefore radical, and exists outside of history - and is therefore conservative.In Kant's terminology, it is synthetic *and* a priori. This is demonstrated every day in math classes around the world. One student comes in and says:"That problem last night! I just couldn't get it!" - bemoaning the fact that math is synthetic. "You couldn't get that? It's obvious!" says the Obnoxious Twerp, pointing out rather crudely that math is a priori.
Recognizing that math is obvious only *after*, not *because of* your inventing is what gives it its unmatchable exhilaration, and at the same time abstract concreteness. once one is *in* math - once one is a citizen of this great Republic - there is no happier lot in life. Entrance should be the birthright of all our children: it is not only the glory of doing mathematics, but its universality and its accessibility that, above all, we aim to get across in our Math Circle classes.
copyright 1997 Robert & Ellen Kaplan