A Puritan twist in our nature makes us think that anything good for us must be twice as good if it’s hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman—or more likely a robot—of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult.
What a perverse fate for one of our kind’s greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you’re an adult you’ll never have to listen to music again. And this is mathematics we’re talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down to our deepest intuitions and outward toward the nature of the universe—mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together—beyond, around, and within us. It doesn’t just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor from faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our architectural instinct—as deep in us as any of our urges.
Look around you: trees grow in countless shapes, birds in their infinite variety dart among them, we stroll in their shade, intent on divergent ends—yet we and the birds and the trees can only live and move because our collection of parts, and the forces acting on them, are in an ever-changing equilibrium, like a Calder mobile. The parts and the forces alike anatomize down to triangles, and these balance—singly and collectively—only because, no matter how different one triangle is from another, each has a center of gravity: the single point where its three medians meet. Nothing seems to demand that there always be this single point: why couldn’t two of the medians meet at one place and the third meet each of them elsewhere? Why should such a coincidence happen in any triangle, much less all? And yet we can prove, with a logical argument impervious to rhetoric, that this source of our animation must lie in every triangle that ever was, or is, or could be. The argument is of our devising, but it darts up all at once, away from our here and now and our strolling personae, like a bird, high above the trees, and even beyond the birds, to a timeless understanding of the hang of things.
This is the serene countenance of mathematics. How did so shining a beauty turn into such a wicked witch? Very young children are obsessed with numbers, counting everything in sight and playing counting-out games and fooling around with numbers and their names and their symbols the way they fool around with words in that exuberant gaiety that can blossom to meaning-bearing metaphor.
Then comes school.
By the time children are twelve the fun has all been leached out, and dread competes with boredom for its place. They begin to feel that math is in league with their enemies to make fools of them.
We teachers are largely to blame. Many of us are ourselves afraid of math. So many elementary school teachers love reading or writing or history or civics or ecology—math comes last on the list. And why shouldn’t it? The traumatized child is father to the traumatized man. Few teachers have a secure sense of how what they teach dovetails into the whole (or even that there is a whole). They aren’t sure what justifies the stupid rules you have to memorize (“‘My Dear Aunt Sally’: Multiply first, then Divide, then Add, last, Subtract”).1 They become the hated tax collectors on their students’ fund of good will, sent out by a remote authority that publishes Edicts of Right Answers. Of course they end up doing their best teaching not of math but of their fear of it. How could they not conspire with their victims to get through the hours and days and weeks and months as quickly and unquestioningly as possible?
We’ve described the worst of these teachers. What about the better ones? They want their students to understand how to add 1/3 and 1/5, and so show them that 1/2 and 1/3 can be rewritten as 3/6 and 2/6, which add up to 5/6. And 1/4 + 1/5 is the same as 5/20 + 4/20, making 9/20 … so look, 1/3 + 1/5 turns into what? 5/15 + 3/15 … and that’s … yes … 8/15.
Good; but do two examples prove a truth? Do twenty, or even two hundred? Are teachers going to derive the notion of least common multiple instead from the axioms for arithmetic (or, shades of the 1970s, from set theory!), and send their ten-year-olds screaming into the street? Are they going to sidestep the whole issue by turning fractions into decimals and letting fingers solicit the answer from the oracle of calculators, so that we all become the slaves of our mechanical servants?
Even teachers with the best intentions in the world find themselves compressed under the moving hand of the clock, which leaves neither leisure to explore nor the vital pause for doubting, and certainly no room to follow conjectures wherever they may lead. Everyone (except mathematicians) knows there is only one right answer. Everyone (except mathematicians) knows your thinking has to start salivating when the bell rings, and run panting until the next bell, and then stop, so that what were games become contests, and delight is turned into anxiety, and getting that right answer becomes the goal, instead of understanding.
Behind these many ways of blunting thought stands a problem peculiar to mathematics. Look back on your own career: you may have had only one good teacher of literature or history or language, preceded and followed by worse—but that one bright spot lit up the whole field and preserved it for you, so that you could take pleasure in reading or writing on your own. But learning mathematics is linked and linear: once you trip up, all the good teaching in the world is not likely to set you again on its path. Fall from a ledge and the odds are slim that you’ll climb back up to and past it.
Here are places at each of which many students have lost their footing. Each of us remembers where it happened to us:
place value
negative numbers
long multiplication, and worse, long division
adding fractions
letters for numbers
x, the unknown
x, the variable
rigorous proof
imaginary numbers
limits and calculus
non-Euclidean geometry
topology
category theory
In everything else we do there are degrees and stages of mastery, so that we can not only see where we’re going and how far we’ve come but can in fact enjoy ourselves at each level, with that fine capacity of ours for imagining the sandlot to be Yankee Stadium. The novice guitarist relishes the music he makes, strumming one or two chords behind the vocal and losing all sense of self in the performance: the amateur by definition loves, and love knows no hierarchy. Even professional athletes will remember with more pleasure a sunlit game when they were ten than a triumph at the peak of their careers. Only in mathematics does mastery of one level seem to count for nothing when entering the next. How did learning place value and memorizing your times tables prepare you for figuring out how to add 1/3 + 1/5? Back to square one! Mathematics makes mountains out of such molehills, and a Sisyphus out of each of its climbers.
Why should this be? The reasons go down to the very roots of the art. People must have struggled for a long time with adding fractions before someone thought it through and realized that you can only add together things of the same sort (3 apples and 5 oranges are either just 3 apples and 5 oranges—or 8 pieces of fruit). You need to turn thirds and fifths into the same kind of fraction. Then comes the second, tactical insight that not eighths (3 + 5) but fifteenths (3 × 5) were this kind (and why is that? Has it something to do with dividing a pie into pieces?). To teach it now as if it were A Rule, or (even more intimidating) The Law, is to pretend that what took years of experiment and ingenuity is as obvious as your nose. And then, because you never really had a chance to understand what was going on (“A negative times a negative is positive, because that’s the way it is!”), whenever you need this rule again it will come as just that—an arbitrary fiat, enforced by Them. And so the whole integrity of mathematics is compromised. The only reasonable conclusion for a struggling student to draw from such pretense is that he is irremediably stupid, or that Mathematics works in mysterious ways, its wonders to perform.
Kant said that mathematics is synthetic a priori: synthetic because we invent it, and a priori—prior to any experience—because we then see our inventions as discoveries. It is very hard to believe that the way to add fractions wasn’t already “out there”—wherever “there” is: independent of us and our hit-or-miss contrivings. Out there with the fractions—but of course they too were invented by us. We certainly are still very far from understanding the relation between thought, mathematics, and the world; but our ignorance is no excuse for pretending to others that what took effort (and perhaps well-prepared luck) to grasp should now be obvious to all. A teacher we knew told a student to take his feet off the desk. “I’ve been telling people for twenty years to take their feet off the desk!” Yes, but it was this student’s first day in school.
We fall so easily into taking new mathematical insights as eternal knowledge for several reasons. Once you get the hang of a tool you want to use it and, intent on the application, tend to forget or ignore the time and thought it took to get that hang. If you’re in the Tour de France, having learned to balance on a bike belongs to a prior existence. It is an understandable failure of imagination, then, coupled with impatience, that leads a teacher into thinking that what is obvious to him must be obvious to his student, so let’s get on with it. It is likely, too, that many a teacher of mathematics hasn’t realized that math has a history (a fault of the way we train our teachers, as if standing on the shoulders of giants meant we had no need to look down), and so a teacher, who is supposed to develop our powers of inquiry, becomes instead a messenger of Received Truth.
Mathematical shorthand plays a role here too. It is so easy and quick to write 1/3 + 1/5 = 5/15 + 3/15 = 8/15 that we’re swept past the thinking by the notation. If you can say it that quickly, there must not be that much to it. Impatience, slick symbols, and lack of imagination are external forces etching the ledges of mathematics. But these ledges are also intrinsic to the singular material of mathematics, which is structure. What can you get your hands on, what can you see? You may see three apples and five apples, and by an act of imagination (for regrouping is nothing less), realize that you have eight apples—but you never see “3” and “5” bare. Not even Euclid did. To recognize that behind the apples and pears, 3 + 5 = 8, is an act of abstraction unique, perhaps, to our species. How can we see 1/3 and 1/5? Taking two pies and cutting one up into thirds and the other in fifths may help; abstracting the pies to circles on the board and doing your dissections there may help yet more; but it isn’t even the symbols “1/3” and “1/5” we’re talking about; it’s what they stand for, which you can’t picture. Even the mind-boggling abstraction:
points to the immaterial relation we have in mind, which as teachers we would like another to have in mind too, and so grasp in all its generality. We have to recognize that this is an undertaking like no other (save perhaps music). The handholds seem to grow fewer the higher you climb.
Mathematics is all ledges. You no sooner acclimate yourself to breathing the thin air at this new height than the way opens up to one still higher (because math is freely—you might almost think arbitrarily—invented). No sooner do you feel comfortable at turning your new insight into the condensed form of symbols that can be used mechanically (because math is a priori: it underlies our thought) than you have to start thinking from scratch again about what some combination of these symbols suggests. 1/3 + x = 1/5: what am I supposed to do with that? This x isn’t like the a, b, c, and d in a/b + c/d, which were placeholders for any numbers (or almost any numbers): it is an unknown let loose like a wolf in the sheepfold of fixed quantities. Still, it is an unknown, which, if I can just contrive a way, I can make known as a fixed quantity too. But what about
what is x, what are x and y, now? Not fixed but variable quantities (if such a term makes any sense). Ten-year-olds the world round beg their teachers: “Won’t you please tell us after class what x really is?” And everything we’ve mastered so far seems to count for nothing; it all slips around and the very nature of number is put in doubt. Those ledges we’ve already struggled up hardly needed stepping over, compared to these rising endlessly above—and they rise not continuously but by jumps: new ways of looking, new definitions, new axioms that make you brace yourself for pushing the rock once more up the mountain.
Those who love to climb mountains have a very different view of them, and it may be no accident that so many mathematicians are also mountain walkers and climbers. It isn’t just the exhilaration of solving the rock face, but the fresher air along the way and the long views from the top that draw them on. “O height!” exclaimed Petrarch, the first man, they say, to climb a mountain for pleasure—and so say we in The Math Circle. We aim to take acrophobia away by having our students do the climbing however they will, with us as their Sherpas. We bring up the supplies and peg down the base camp; we point out an attractive col or a dangerous crevasse; but they do the exploring on a terrain we’ve brought them to.
Those kids you glimpsed finding fractions between 0 and 1 will either come up with the problem of adding fractions themselves, or we will drop the question, when it seems most natural to do so, into the ongoing conversation. Since it is a conversation, our questions are no more intrusive than theirs: they come in sideways, not from above. So absorbed are they in their creation that they rarely ask if we know what the answer is (and if they do, we answer with a question or a suggestion that will face them around in a useful direction). Because math is a human enterprise and they are humans, their thoughts can’t help but tack around the direct line. Look at a prehistoric ovoid hand ax from Périgord and a triangular hand ax from Tanzania: different in detail, their structure is the same because they solve, as humans would, a human problem.
Surprising and delightful things will come up. A common multiple, other than the least, of two denominators may be the first to surface, and even should the least then appear, affection for the firstborn—or the delight that small children take in large numbers—may lead them to settle on the larger one.
“1/3 is 20/60 and 1/5 is 12/60, so 1/3 + 1/5 is 32/60!” exclaimed Tom. This puzzled Sonya, who had gotten 1/3 + 1/5 = 15/45 + 9/45 = 24/45, and was sure she was right. That there were two different answers to the same question—and that both turned out to be right—was a revelation whose metaphorical value stayed with them.
They do the exploring, but each of us who leads a Math Circle course has clearly in mind from the start what the goal is; wandering around aimlessly is as boring as being marched from point to point. I wanted my young students to develop the arithmetic of fractions, to relate fractions to decimals, and at the end to invent a number that wasn’t a fraction. But had they followed another line of thought, which seemed to me fruitful—from least common multiples to greatest common divisors, say, and to the Euclidean algorithm, or primes—I would have unpegged my tent and moved it to their line (and indeed I have, in other semesters when the opening question was the same, but the conversation took off entirely differently). These are decisions that have to be made more or less on the spot, and are one of the quick-reflex challenges that make Math Circle teaching exciting fun. You’re in there doing math with other minds; the goal is clear but the ways to it are up for grabs and at the mercy of temperament and insight. It is just about the most exhilarating thing you can do.
Because the arena is conversation rather than competition, the students—whether they are five or fifteen or fifty—are first startled by, then quickly come to relish, other points of view. They take a collective pride in one another’s insights: “How did you ever think of that!” Since the majority of suggestions that come up are faulty or imprecise or incomplete, fear and embarrassment disappear in a mutual mulling over and reshaping of questions and answers. They like the image we’ve passed on to them from the mathematician Barry Mazur: we are all very small mice gnawing at a very big piece of cheese; no shame, then, in having bitten into a hole, nor any need to hoard up precious crumbs.
Rhythms of different depths pulse through an hour’s class and the ten weeks of a course. With the very young, attention flags after twenty minutes or so, or a threshold of frustration is reached. Time for their favorite sport: function machines. One of them announces that she has a secret rule; we put in numbers, she tells us what comes out (the Sherpa is now a blackboard amanuensis, keeping up an input-output chart underneath a baroque drawing of a function machine, or later, letting the machine “draw itself”—off-handedly introducing them to graphing, that is—and we almost always manage to guess the rule eventually, to universal delight). Then back to where we were, with spirits refreshed.
A deeper rhythm is the alternation between intuition and proof. An insight will usually be tested by several examples and, if it looks sound, will be promoted to a conjecture. But too many likely conjectures turn out to be faulty after a while, and the question of trying to derive our conjectures from foundations comes up. What foundations? How derive? How tell a false proof from a true one? This plays out in an unpredictable variety of ways. The point to notice is that rigor has its place: namely, where you can’t do without it. The need has to arise naturally and has to be intrinsic to the enterprise, as if it were a biological necessity. The younger students are often buoyed by waves of enthusiasm for a conjecture that we’ll conclude looks good, or even very good—but the shadow of that Scottish verdict, “Not Proven,” is always on the wall.
Waves of enthusiasm: it’s the sheer fun of it all that keeps eight-year-olds from leaving at the end of a class on a late winter afternoon, or that gets adolescents up for classes early on Sunday mornings. But before we say anything more about what works in The Math Circle and what doesn’t, about how it has evolved and what directions it is likely to go in, we should really go back to those barriers between mind and mathematics: for more than the ledges within and the teaching without stand in the way.
There is the myth of talent, to begin with—and after debunking it, we’ll talk about what the ingredients really are that go into cooking up a mathematician. Then there is the matter of the off-putting language of mathematics: its symbols, equations, and often glacial style. We’ll talk about the way not teachers but mathematics itself can be intimidating, and then about the fact that you can’t just jump into it in the middle; about the nerdish character attributed to mathematicians, which would put anyone off from wanting to keep them company; and last, the question so many ask when frustrated by a problem: Who Cares? Why should I spend a minute, much less my life, on this glass-bead game?
We’ll look at some of the ways math has been taught, and what The Math Circle approach owes to them; then at how mathematicians actually do their thinking. At that point we’ll be ready to look in detail at The Math Circle.
