The young have a tendency to equate math with calculation – little boys thinking that even geometry is not abstract enough – and if you happen to mention that people came up with these ideas they really balk: that’s history – messy and unreliable, while math is pure and abstract – right or wrong, no opinions. Math as one encounters it in elementary school brings with it no sense of the mathematical mind.

I could give a whole impassioned talk on the metaphor of teaching music by practicing of scales, and of literature by perfecting spelling – and when you’d finished 12 years of boredom you’d be told you need never read a book or listen to music ever again.

Starting a class takes immense energy to overcome learned passivity: Blasé six year old: “you’re the teacher – your job is to tell me what to learn”. Don’t fall into that trap, or its dual, the “isn’t this neat!” play (making little men with coolie hats out of tangrams). You can use that to lure them in – but there has to be a real challenge to engage the mental muscles.

It is crucial to establish early the context of high humor, intensity – here’s what we’re doing – it is worthwhile, and though difficult, we can do it if we pool our brains. Establish the cohort: taking attendance at the start of class both serves to draw the line between this and the rest of the day, and is a way for the less socially adept to learn the names of other kids.

The first challenge is to get the group working collegially – neither passively nor competitively. Having them fixed on you is flattering, and does serve to get the class moving in the same direction, but fixed on the problem is what we are aiming for. The students shouldn’t be sitting enjoying your excitement; they should be exciting you.

And when they do, restrain the impulse to say “That’s brilliant!” to an individual, replacing it by “Great – what can we do with that . . .”, so that the mood of optimism and confidence is distributed to the whole group.

The big job of teaching is creating a seamless connection between Ends and Means: keeping your eye on the principle behind the local puzzle you’re all involved in. It’s not enough that your kids think math is fun – avoid the implication of ‘fun math’: yes, we want to lure them in, but into the real thing, not the Disney version. We need to let them know there is a high line and a deep structure in math, that they are learning to play the noblest game.

And also the widest ranging – anything can be grist for the mill of a questioning mind: when the eye glazes an intentional interruption of what you are working on joggles the brain awake: that’s the time for function machines, or a discussion of whether the columns of numbers are Doric, Ionian, or Corinthian, where the name zero comes from – using the same mental techniques on another topic.

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Why teach in the Math Circle Style?

The most disheartening question a math teacher is ever asked is “Why are we learning this?” and its variant “What’ll I need this for?”

In 1969 Congress was running a hearing on why the government should fund the building of a new collider (the superconducting supercollider, a tiny version of CERN’s large Hadron collider) and Senator Pastore from Rhode Island was questioning Robert Wilson, the head of Fermilab.

Pastore: Has the collider anything to do with promoting the security of the country?

Wilson: No sir, I don’t believe so.

Pastore: Nothing at all?

Wilson: Nothing at all.

Pastore: It has no value in that respect?

Wilson: It only has to do with the respect with which we regard one another, the dignity of men, our love of culture… It has to do with are we good painters, good sculptors, good poets? I mean all the things we really venerate in our country and are patriotic about… It has nothing to do directly with defending our country except to make it worth defending.

And that is the answer Math Circle teaching gives to the deadly question of practical advantage: People never ask ”What’s this used for?” or “What’s the point of this?” about something they invent (or discover) themselves.