**Ellen**: We’re burdened in the math world by two myths: the myth that mathematics is dreary (you all know how false this is, yet how hard to dispel), and the myth of talent. But this is a myth most of us live by. It affects how we think about math and about our students, and it affects how we teach.

Since math is our lost native language, there’s no more a talent for it than there is for reading – but romantic hero-worship by some and childish self-love by others keep the myth alive. Let’s look at it squarely.

“Experts are made, not born,” says Philip Ross in his August 2006 Scientific American article, *The Expert Mind.* “It takes approximately a decade of heavy labor to master any field,” he writes. Expert-mind experimenters use chess in order to quantify their results, and some of you may think that the combinatorial nature of chess confirms these results for math. Others think math much broader than combinatorics – but the conclusions still apply, for what he reports makes very good sense in any field.

If, as seems likely, we can only manipulate and hold in our minds seven or so chunks of information at a time, mastery comes from packaging greater and greater hierarchies into these few compact bundles – and ten years of enormous effort in any field will do it. ‘Effortful Study’ is how they put it: “Continually tackling challenges that lie just beyond one’s competence.” Experts-in-training, he concludes, “keep the lid on their mind’s box open all the time, so that they can inspect, criticize and augment its contents and thereby approach the standard set by leaders in their field.”

What does this mean for our unusual approach to teaching math in The Math Circle – whose spirit, we hope, captures some of the wonderful human warmth and generosity of Andrey Kolmogorov? For a start, it means we take whoever applies – no entrance exam, no skimming off the supposed cream of any crop. Our only criterion is that the student has to want to be there – either because she already loves math – or perhaps he’s scared of it, but has heard that our classes are fun. Curiosity is passport enough to enter the Republic of Mathematics. Remember what Cantor said a hundred years ago: “Mathematics is freedom!” What *we* want is what all of us want: to extend the franchise to this republic, to give citizenship in it to as many people as possible: for in it minds meet minds on equal terms and freely invent their way toward understanding the beautiful structure of things.

Extending the franchise: can this only be done outside of schools, in math circles? Why can’t it happen in schools, where teachers can have with a whole class the delightful experiences they remember having had with one or two? Once the myth of talent is exploded, teachers will no longer have to hide behind the excuse that failures weren’t their fault (“You never saw such a hopeless lot!”) – instead, they can make math as attractive as all those subjects that popularly outrank it – science, history, literature – by humanizing it.

This doesn’t only mean showing the historical and biographical contexts that the math they’re working on together lie in, but actually *working on it together:* letting their students feel the exhilaration of mini-discoveries, at least. Not telling them from on high some meaningless rigmarole that a x b = b x a, but having them discover for themselves the excellent labor-saving dodge of folding their times tables down the main diagonal.

Will this take too long, in a world geared to the MCAS? What starts slowly accelerates faster than mad minutes ever do, and leave a path in the mind that it can travel again. For our species lives by curiosity, and thrives on ingenuity. We sell kids short if we think they’re for MTV or a tale of bawdry; we sell math short if we think any prize is worth more than its own revelations.

So in our Math Circle, we don’t ask tricky questions and set students to racing against each other to get one puffed-up winner and a roomful of losers. We aim to let their curiosity loose on an **accessible** **mystery** – one of those questions that lie just beyond their competence, so that they will plunge into it together, conversationally, trying different approaches, re-shaping the question, testing a conjecture, refining their terms, giving up one insight and seizing on another, struggling to prove the likeliest and then stepping back to see what it was they had created – in short, doing mathematics.

**Bob**: And where has the dreariness gone to? Evaporated in enthusiasm; vanished, because they’re not being told what’s so or isn’t so, but are creating math themselves – new paths to universal truths. And you can just watch their confidence and their competence increasing together. This is why we treat all of their conjectures as we treat *them:* equally. If they’ve gone off on a wrong track, they’ll discover this for themselves soon enough – or with the least bit of nudging from us.

I was working with four-year-olds this fall on our semester-long mystery: are there numbers between numbers? They had invented halves, and later thirds, and figured out how to add halves together, and then thirds (even to the extent of adding up four of these thirds, which we all agreed was silly, because how could you ever have more than a whole?) But inevitably they found themselves asking: how much was a half and a third? This is one of those places where the boat carrying your self-esteem is likely to leave the ship of mathematics forever. They tugged and tore at it, and quite sensibly agreed that fifths were the key, since 2 + 3 = 5; and that the answer was probably 2/5.

But at last one brave soul stopped pushing symbols around long enough to notice that 2/5 of a pie looked like less than half a pie, and it would be a pretty funny sort of adding that ended up with less than we’d started with. I suggested that this might be a problem beyond human power, but frustrated as they were, they weren’t buying *that* (four-year-olds have no sense of tragedy). “Push them together,” somebody said – the half and the third. “Split them up,” said the class contrarian. So they did both, and the phoenix of sixths rose from the ashes, no one quite knew how, and of course it was 5/6, didn’t we know that all along? We didn’t – and did: the peculiar experience we’ve all had that what we couldn’t understand at all a minute ago is now just – obvious.

That myth of math being dreary comes from having it crammed down your throat. How could it possibly be dreary when you’re struggling together to make Rumpelstiltskin say his name – and at last he does?

This is where the most unusual aspect of our approach comes in: not telling anyone anything. At the outset we pose a question that looks harmless and bound to be answered in the next five minutes or so, but oddly enough leads on and on, until, in the course of a semester, we find ourselves on the frontier. Are there numbers between numbers? What’s a proof of the Pythagorean Theorem – what’s another – how can there be more than one proof? Are there fewer even numbers than natural numbers? Can you cut up a pea with a finite number of cuts and put it together again to be larger than the sun? Which polygons can you construct with straightedge and compass?

We step back and let them at it. At times we play the role of a skeptic who needs to be convinced; at times, an appreciative audience; and at times, a fellow explorer, who may point out a clearing in that direction and quicksand in this – but they rightly soon sense that the inventing and discovering are in their hands.

Here’s an example of how that works, in the course I mentioned on the Pythagorean Theorem. We’ll hand out a different proof to each of the students at the first class, and have them compare notes – a comparison that can continue by e-mail, if they wish, before the next meeting (no homework – the involvement is up to them; but curiosity inevitably tugs homeplay onward). In the next class, we’re ready to take up whatever issues one or more of these proofs have brought up – and may ask, what does it mean to have more than one proof of the same theorem – is the theorem really still the same if its context has changed from geometry to algebra?

We’re ready to open doors should they not, by asking, for example: do you think the theorem is true in three (or more) dimensions? Or for shapes other than squares on the three sides? Or (if we sense an inclination among them toward number theory): do you think there might be other triples of integers than (3, 4, 5) for which the theorem holds – can we find (and then prove!) some general pattern? Is there an analogue, or generalization, to non-right triangles? The openings are endless, because anything leads to everything.

**Ellen**: The path is never smooth in a Math Circle course, because it’s all improvisation – there’s the risk and the thrill. You have to cajole them out of shyness and fear of making mistakes, and wanting to agree with the group and trying to read you and to please their anxious parents.

But then there are the problems belonging to the craft itself – not the least, what happens when they look at published proofs or established techniques: all that off-putting symbolism. Following a train of someone else’s thought by reading straight through it is hard enough, now you have to scoop up subscripts and haul down exponents and try to scramble over those alchemical-looking integral signs, and make your way through a forest of symbols, tripping on the brackets, with never a word in sight. But as we say in our book about our Math Circle, ** Out of the Labyrinth: Setting Mathematics Free**, equations and formulas are the punch-lines that come at the end of the story; you’re not meant to understand them at the beginning.

But if you’re making up the symbols yourself, to stand for what you’ve already figured out and would just like to allude to now, while driving thought forward, they no longer bar the way but aid your progress. That’s why we encourage our students to devise their own signs and terms – it’s easy enough to show them afterward what other people use.

The symbols, though, are only the outward sign of what makes math, like a diamond, so hard and so beautiful: its impulse toward the ever more abstract. You thought you were right there, at the rockface of reality, trying to figure out why the three pairs of lines cross-connecting two triples of points on a circle’s circumference meet in three points that lie, miraculously, on a *straight* line – and it turns out this was only an instance of what’s true for the larger collection of conics; and *that* truth’s proper home isn’t Euclidean Geometry at all, but plane Projective Geometry, where it is equivalent to its fundamental theorem; and this truth is in fact none other than the commutative property of multiplication! – And…

No matter where you stand in math, there seems always to be a balcony above you, from whose higher standpoint your most general insights become particulars. Is this daunting? Or is this another version of what Hilbert called the Paradise whose doors Cantor opened for us? Some people turn away and look instead for chutes and ladders on which to dart from one floor to another, making math into a bag of magic tricks – as if this were the middle ages, and you needed a brittle formulary of simples (adding zero – or was it wolf’s bane? – to cure problems with your numerator, and mandrake root, or square root, to undo the Quadratic). But it isn’t homely nostrums and imaginary medicine; it’s imagination we want to cultivate.

So what we encourage, in our Math Circle, is that kind of utter immersion in the problem at hand that makes it all there is. This is the time for the timeless. Now you can take risks – yet also raise the ante of precision: for you can never be daring enough, nor ever careful enough. Those with a leaning toward the legal sit next to wild imaginers, and each, ideally, learns from the other and lives from then on with a mathematician’s split personality.

Now this sort of focused immersion vitally needs freedom from the worldly pressures of tests and competition. The more intense the thought, the more leisure it needs to support and surround it. Who wants to battle with those from whom you might learn something? Who wants a trivial triumph over some other person who happens to be in the room, when there’s the math itself to come to grips with: the deep connection hidden below so many surface relations; the tangles that succumb to technique, and the ideas that it takes imagination to reveal.

A collegial spirit not only lets the *math* grow, but the mathematicians too, keeping what is childlike while putting away the childish. What does this mean? Having an innocent eye, and unbounded expectation; letting a playful spirit loose and being game for the most audacious inventing (a specialty of rebellious adolescents); being ready as well to give up on a lead of yours that turns out not to work, since the math, not your little ego, matters. That’s the great gift of childhood: flexibility.

But the childish things we want our students to put by are the spirit of rivalry and self-trumpeting, which tarnish exchanges with colleagues and narrow your own pleasures down to the bitter and the abrasive. This stiffening is often mistaken for adult realism, in a world read as a fight to the finish.

**Bob**: Inevitably students will feel frustrated from time to time: mathematics, like nature, loves to hide itself. They’ll feel put down by the nature of things, and on a dark day wonder what the point of it is, after all. Rather than switch to short-term problems, like those in contests, that give a quick fix, or leap in with a soothing solution (which would indeed take the point and the pleasure away, and their freedom as well: here’s the answer, stupid – couldn’t you see it?), we’ll nudge as unobtrusively as possible, and count on the variety of outlooks in the room to hit on a new combination. This keeps the lid on their mind’s boxes open, so that they can inspect, and criticize, and rearrange its contents – until a shape all at once emerges.

The advantage of working collegially isn’t only psychological support when the going gets tough. The points of view combine as well, and in the intensity of all that focused attention, insight matures – since math is made by humans, for humans, and discoveries in it troll in the depths of our native, universal language.

This is the great conversation that has been going on since before Euclid, and which continues unbroken – a conversation we’re each so lucky to enjoy and contribute to. The starters of such conversations are ill-posed problems, because much of what a mathematician does is to rework a question until the least aperture of an answer begins to open in it – and we want our students to work as mathematicians do. Ten years of immersion to achieve mastery – perhaps that time can be shortened by approaches such as this. Or at least by starting it with the very young, 6 + 10 equals 16, rather that 16 + 10 being 26.

In the spirit of this great conversation, we’d like to leave you with a problem which has twice been the semester-long subject of a Math Circle course for 11- to 13-year olds. Can you tile a rectangle with non-congruent squares?

What do you mean, ‘tile’ – physical tiling? With grout? Can we put the tiles in crookedly – or vertically – or overlapping?

The meaning isn’t ours, it’s yours – what would you like?

And what do you mean, ‘** a** rectangle’? Which rectangle? How many rectangles? Any rectangle? Some? What about squares?

That’s up to you – choose, and see where the choice takes the conversation.

But what about some of the tiles having negative area, i by i? And you didn’t say how many tiles – an infinite number?

Your choice. Mathematics is freedom – the very freedom which will lift it, and us, out of the labyrinth.