The idea that aptitude for mathematics is rarer than aptitude for other subjects is merely an illusion which is caused by belated or neglected beginners.—J. F. Hebart
“I have a damned kink in my brain,” said the American logician and philosopher C. S. Peirce, and it is a commonplace of our culture that to be a mathematician you need such a kink, damned or blessed.
Our approach to math is based on the idea that anyone can learn to think like a mathematician and will, in the process, come to find pleasure in learning and creating math. Belief in talent is a holdover from simpler times, when what race you belonged to determined your character and the bumps on your head revealed it. It is much more romantic to picture geniuses, larger than life, suddenly bursting into view with something magical, almost diabolic, about them (Liszt, Ramanujan, Byron), than to take away the cape and the penetrating gaze and ask what the true story is. It also absolves the rest of us from responsibility: if a talent for math is inborn, no need to waste our time trying to develop one. Teachers are exonerated too: it is the Nature of Things that produces a stream of mathphobic failures.
What is the true story? The masters of any craft can off-handedly do what seems miraculous to outsiders: watch a magician shuffle a pack of cards and divide it into eight ordered sequences. Part of a long apprenticeship is served in learning techniques and trying them out in varied circumstances, until what were at first awkward gestures articulate smoothly into a whole. To the extent that math is a craft, its skills (such as unknotting tangled expressions and transforming the strange into the familiar) need to be practiced. Without your knowing or willing it, a threshold comes to be overstepped, and then you can do easily what before you couldn’t do at all (again, think of learning to ride a bicycle). This depends not on inborn genius but on doggedness and attention.
But beyond craft, mathematics is an art, and as such calls up all our mythology about artists. The craftsman may learn fluency, but those strokes of insight that mark significant works come out of the blue. Really? Ask any artist modest enough to put mystery and the glamour of genius aside, and he will tell you they come out of the gray. For while you are at work on a problem of painting, writing, music, or math, all sorts of hooks and eyes tumble by, and connections are made only to fall apart. What was clear becomes clouded; you begin to doubt the truth you are trying to establish, and then your own powers to establish it. Good leads peter out; fresh ones discordantly clamor for your attention. You lose track of bits of the composition and forget how others worked, until it all sinks chaotically out of sight. And then the solution dawns—when your attention was on something else.
When we read poets or chemists or painters or mathematicians on how they came by their ideas, we’re drawn toward agreeing that insight comes after playing intensely with the ingredients—and then leaving them to ferment in the mind’s cellars, out of attention’s light. Why and how this works isn’t yet clear, unless it is that inattention loosens the grip of firm misconceptions. In Inca mythology the head and body of Túpac Amaru, severed by the Spaniards, are slowly working their way toward each other through the earth—and when they rejoin, his people will rise again. Something subterranean like this may be at work in the subconscious, where the fragments of thought and their reflex shapes inch closer together.2
You will likely agree that skills can be taught—but doesn’t it take genius to move from good practitioner to artist? Can people be taught how to have insights? Suffice it for now to say that insights can be prepared for, by encouraging an imagination playful to the point of recklessness, along with a sort of experimental fervor that follows hypothesis out as if it were truth—but lets it go once revealed as error. It also helps to cultivate a healthy distrust of authority and a restless, ranging curiosity—not so much an anarchic spirit as the flexible feel for law that tricksters (in the tradition of Odysseus) have.
The phrenological theory of bumps marking the locations of mental faculties fell away a century ago, and the notion that one is ruled by such inborn faculties is falling away now too. We are beginning to sense how adaptable the mind-brain—our organ of adaptability—really is. Given the greater sophistication of the way we understand how the mind works, why has the myth of talent persisted? When we play Mozart to newborns in order to develop their musical sense, why do we still think a math sense is genetically determined? Certainly you can cite the Bernoulli family of mathematicians (like the musical family of Bachs), but far more often mathematicians come from parents startled by their progeny. Even among the Bernoullis, are we to deny the role that an atmosphere saturated in mathematics could have had on the rising generations?
Without any statistical evidence to back it, many still look for the mathematical rabbit to have jumped out of a genetic hat. Here are five out of probably many reasons for hugging this old husbands’ and wives’ tale to our bosoms.
First, we try to make sense of the extraordinary, and heeding the authority of the past, invoke the time-honored notion of a “calling,” usually in a sufficiently endowed family. We shy away from asking who calls, or how many of those who have heard the call answered it well. A sacred sort of aura—almost a superstition—puts inquiry off. It may well be that when you come to feel a calling (to math or anything else), you are interpreting passively a mixture of enthusiasm and confidence, along with a barely conscious sense of the line of fall your inclinations are gliding along. A hillock climbed early, or at a significant time, may make distant mountains in that direction seem inviting rather than daunting. If you don’t find something comes easily to you, you may believe (or society may encourage you to believe) that you have not been called to it—you are just not “meant” to be a musician or a mathematician or a poet or a parent. But you won’t get any calls if you’ve left your phone off the hook.
Second, there is also the problem of misleading analogies. Because you do need a certain flexibility for ballet and bulk for weight lifting, fast-twitch muscles to be a successful sprinter and slow-twitch for the longer distances, we come to speak of the “muscles of the mind” and imagine that here, too, genetic determinants rule. You need only look at the variety of backgrounds from which creative mathematicians have come to let go of this hypothesis. Look also at the variety of “traits” (as they were once called) that keep mathematical creativity company to see how short this kind of argument sells the intricate reality. Some marvelous mathematicians are cultivated, some boorish; some adept in the ways of the world, others extremely innocent. You will find brilliant mathematicians reticent and aggressive, humorous and stodgy, vain and humble, lawless and law-abiding, of every political, ethnic, and social stripe—and some brilliant mathematicians who are dense outside their field. It is inertia, really, that keeps our imagination on a monorail when it approaches the farther reaches of our thinking.
Third, the romanticism we spoke of before has much to answer for. We want signs that the gods love at least some of us. We want our lives lit up by comets of genius flashing through them: we want heroes to worship, the lucky to envy, the great to glance at us where we stand by the roadside, waving. Some mathematicians connive at this romantic image too—from self-love and pride on the part of a few, and others from a fashionably detached admiration of their talents, as if they were separate beings (like the tennis players who report that “the forehand was really on today”). Most of us know few mathematicians and very seldom see them at work; a perfectly understandable aesthetic motive hides the labor preceding insight, so that only the finished theorem and its proof get published, with the scaffolding and all the fallen workmen cleared away.
Fourth, mathematics has a reputation for being an uncompromising taskmaster: the road is narrow and steep; to be wrong by a little is to be wrong altogether, and its high-altitude truths care nothing for our weaknesses. Giants they, of another species, who stand on its Everests. It does indeed take a kind of blithe fortitude to put up with years-long frustration, and such a thorough sinking of self into the subject that every smallest gain in understanding enriches the harmony of a growing whole for which we have no pronoun: neither “you” nor “it.” But problems attractive enough to entice your putting in the effort develop your fortitude—who past the age of eight wants to go on beating five-year-olds at tic-tac-toe? The enticing morsels we dangle in The Math Circle are just beyond arm’s reach, so that grasp lengthens along with the span of attention, and we learn to delight in contriving new ways around old obstacles. Working together prying open the lid of a treasure chest (How can you add different sorts of fractions? Are a triangle’s altitudes concurrent? What is ii? How could you classify all knots?) blends individual egos into the common effort and unites them at last in the insights gained.
Finally, to round out this list on a cynical note, there is certainly a living to be made out of perpetuating the myth of talent and its levels of aristocracy: “gifted” to “deeply gifted” and “profoundly gifted” (is “abysmally gifted” next?). This is a horse worth looking in the mouth. It wouldn’t be hard to imagine a person setting himself up with a battery of multiple-choice tests and rewarding the parents who paid him by reporting that their child had an IQ of 183—but to keep them coming back, was also dyslexic, dyspractic, deficient in attention, or marginally autistic. Scores on intelligence tests do cover a wide range of qualities, though they are not unique determiners even of academic success: 175 is not necessarily greater than 100. Future generations will chuckle over the Binets we have in our bonnets, as we do over manuals of phrenology.
After a list such as this you may argue that we’re just avoiding the blatant truth that there are talented young: that prodigies of calculation star the benighted past; that experiments (as with MRI) demonstrate that parts of the brain, when mechanically stimulated, produce bursts of calculating brilliance; that articles appear yearly about eleven-year-olds leaving their college classmates in the calculus dust; that there is a remarkable correlation in some children between autism and a fascination and fluency with numbers.
Rapid calculation, however, has little to do with the structural insights that make up mathematics. You need only watch a tableful of mathematicians trying to calculate the tip, to see how many are inept at numerical tasks. Some, like Gauss, are exceptionally good at them, but Gauss himself employed an even more rapid calculator named Zacharias Dase, who had no idea what mathematics was about. The great Kummer, in nineteenth-century Germany, was notoriously unable to multiply even single-digit numbers together. “Seven times nine is … um … er …” he said from the lectern. A mischievous student spoke up: “Sixty-one,” and Kummer began to write it down. “Sixty-nine,” said another. Kummer stopped. “Come come, gentlemen,” he said, “it cannot be both. It must be one or the other.”
How clear is it that the symptoms we read as autistic in many mathematicians aren’t effects rather than causes of their devotion to the unknown god? “Abstract thought,” said Jacob Bronowski, “is the neoteny of the intellect.” But you might put autism aside and ask us what we have to say about someone like the three-year-old who announced at his first Math Circle class: “I’m very numbery”—and was. Children will tell you as quickly which of their classmates is good at math as they will who is good at baseball or drawing or spelling. We don’t question the presence of people in our society who are better, and in some cases very much better, than others at math. What we do ask is how they got that way, so that we can develop their skill in everybody.
So much of what seems innate depends on the luck of first encounters. The smiles and frowns that the world unwittingly gives to this or that little probe of ours can determine the landscape of our thought as thoroughly as a pebble diverting a rivulet will, in the end, make for a valley here and a mountain there. But behind the chance placement of the pebble lies the enormous force of gravity, just as behind our experience lies the deep structure of language: the capacity we all share for catching how the universal rules of grammar are applied in the particular language of our surroundings. That fluency with mathematics we see in some children is fluency with its language: another application of the same deep structure. It is unusual, of course, for this particular language to take precedence over, or even to flourish beside, our native tongue—and that’s where the luck of little encounters comes in. Something most likely unnoticed in the tumble of experience catches our inner ear and makes this conversation about number, pattern, and shape as appealing as that about people and things. Most of us learn math as if it were a second language, with all the ills of translation that implies, but after years on another track, it may be that one day a shunt opens, by chance, to structure itself: an architectural instinct, which delights in form and whose natural expression is mathematics.3
However you decide about talent on the basis of your own experience and inclinations, you will likely agree that it is best to act as if it were a myth—rather in the spirit, though with the opposite polarity, of what’s called Pascal’s Wager (act as if God exists; for if he does you may be saved, and if he doesn’t, nothing is lost). If there is no such thing as a talent for mathematics, then you do well to proceed as if everyone could be led to its delights; while if such a talent exists, your actions won’t harm those who have it and may give a spectator’s informed appreciation to those who haven’t.
Letting the unexamined assumption of talent run inevitably leads to building tracks for it to run on, perhaps disguised by jokey names, but no one is fooled about which is for the fast and which for the slow (one school we know of labeled their tracks “Peregrines” and “Yaks”—surely not without a touch of Dickensian malice in making sure people knew their place). Prophecies by labeling, such as these, can’t help but be self-fulfilling, as an experiment in a New York City public school a few years back showed: students were arbitrarily placed in two sections, but their teachers were told this was the slow group, that the fast … and by semester’s end, so they were.
Our own experience with students certainly reinforces our conviction that “talent” is not only a myth but a pernicious one. We’ve had any number of students in our years of teaching who sat inert as a noble gas through class after class, and then one day blurted out a stunning insight. Where had they been until then? They may tell you (like the young Macaulay) that there just hadn’t been an occasion before to speak, or they had been distracted or hadn’t caught on; those are flags variously signaling that an organic process hadn’t yet passed a crucial threshold (“Why couldn’t you balance on a bike before this moment?”—what would you expect a child to answer?).
We each have engraved on our hearts the names of students we had unjustly given up on. In one class of ten- to twelve-year-olds, we were coming to grips with different sizes of infinity (they were too young to know this was hard), and flying through the central ideas of sets whose elements matched up perfectly with those of other sets, and sets where this was impossible. They made up names when they needed them for these sets and for the operations on them. Eugenia talked eagerly all the while, but always about the names and their appropriateness: this was what engaged her imagination, and the ideas that the names stood for slid harmlessly by. In the next to last class—now onto very sophisticated topics about ever larger sizes of infinity—the boy next to her was having trouble seeing why there had to be more subsets of a set than there were elements in the set itself, and Eugenia turned and explained the proof neatly to him. We all fell silent in astonishment.
“What?” said Eugenia.
“But that’s perfect!” said another student.
“Well,” said Eugenia, “there it is.”
“But you didn’t understand anything last week!” said the boy.
“You mean we’ve been doing this all along and I didn’t know it?”
If you look beyond our possibly prejudiced evidence, you will find one example after another of famous mathematicians who showed no early signs of talent. The famous French mathematician Jacques Hadamard, whose accomplishments included proving the notoriously resistant Prime Number Theorem, held down the last place in his math class through seventh grade. Hermann Grassmann—one of the first to explore n-dimensional geometry and whose hypernumbers were a daring generalization of the complex numbers—was so backward as a child that his father hoped he might at least make it as a gardener. An interest in theology took him eventually to the University of Berlin, where he never attended a single lecture in mathematics. His many important papers in mathematics and physics (not to mention his mammoth translation of the Rig Veda, his collection of German folk songs, publications on botany, philology, theology, and music—while raising eleven children) were posted from the provincial German town where he spent his life as a schoolteacher.
And what of that eponymous hero of superhuman genius, Albert Einstein? “I know perfectly well,” he wrote, “that I myself have no special talents. It was curiosity, obsession, and sheer perseverance that brought me to my ideas. But as for any especially powerful thinking power (‘cerebral muscles’)—nothing like that is present, or only on a modest scale.” His biographer says that Einstein was slow to talk, formulating sentences before he spoke; that he was kept home with a tutor as unready for school but flew into rages with his tutor; and that his slow wondering was the basis of his always questioning what seemed obvious to more facile classmates.
It might shake loose the hold that this myth has on our point of view were we to put talent in the broader context of brain design and function. For connections in the brain aren’t linearly but spatially ordered, like a map (and a three-or-higher dimensional map at that) rather than a highway. Such an organization both calls for and begets plasticity, the forming and reforming of patterns that gel into routines at our peril.
Nothing so simple, then, as glimpsing a pattern (in the problem, in how to go about thinking, in math at large) and then snapping it into a point of view, but applying the new structure to old ones, or modifying it and the world to suit new situations—tinkering, remembering, forgetting, transforming in an organic rather than mechanical way, and watching with fascination rather than frustration as shape inchingly becomes form. “Talent” rings of rigidity, the world and everything in it as one thing or another, never both.
Compare for yourself the ways intuition emerges from practice in other parts of life: cooking, getting along with new companions, playing a sport, chess (and see Philip E. Ross’s suggestive article on the latter in the July 24, 2006, Scientific American: “The Expert Mind”).
Because we discount “talent” in our classes, students confident and enthusiastic about math sit mixed in with those who have joined because, although they are unsure about math, they had heard that these classes were fun. There will always be some people in a new group who assume that the point is to be best in the class, and are quick to put themselves up and others down. We are as quick to say that the math, not our little egos, is what matters. It is remarkable how well even feisty adolescents respond to this (almost with relief), and plunge together into problems whose answer is just around the corner—though the corner seems mysteriously to keep moving on just ahead of them. They excitedly pile suggestion on suggestion and glow with delight at one another’s insights. They even develop a thoughtful style of criticism: “That’s a good idea—but look, does it work in this case?”
We have even heard one rather precious six-year-old say to another: “I like your conjecture, Jeffrey, but I think I have a counterexample.”
You need a level field for play to be at its best, and egos are bumpy. Self-confidence grows, we’ve found, when people are focused on the work at hand: pride in belonging to a group, to a species, that can crack tough problems open—pride in a world that has such secrets in it—is much more satisfying than pride in your passing triumph. This is why our response to an insight is a workmanlike “Good. Now where does that take us?” A familiar mantra is “Teach the child, not the subject,” but the child is best taught while there is something external for him to think about; you don’t get many new insights when contemplating your ego.
We have another reason for treating all suggestions impartially, the least likely along with the most: by letting any conjecture run—rather than stopping it with a “No, that’s wrong”—we let the line of thought carry itself to a falsehood, a dead end, a more thoroughly explored uncertainty, or an onward-pointing truth; false conjectures often tell us more than true ones, narrowing down the landscape of likelihood.
Children aren’t as foolish, nor adults as gullible, as the media would have us believe. Learning in order to get ahead sounds very grown up, but all the trappings of competition look pretty shoddy in the watches of the night when we ask ourselves what the point of it all is. Since as a species we survive through curiosity, a hunger to understand is at least as great a driving force in us as self-aggrandizement. Nobel Prize winners, by and large, aren’t in it for the money; they are drawn on by the problem and rewarded by the admiration of their colleagues. It isn’t surprising, then, that a sense of pride in the cohort develops in a Math Circle class, which leads to more pleasure in what the idea is, than in whose idea it was. This is a subtle matter, for as George Polya pointed out long ago, the secret of solving a problem in mathematics is to sink your whole personality into it—but sinking in is different from slathering over. We need our idiosyncratic approaches and points of view, but not the distracting doting on them. The myth of talent disperses the energy needed for ideas to condense.