When I was a boy, I loved numbers, and I still do. Only when I was a sophomore at Princeton did I shift my attention from mathematics to literature, in part because the same part of my mind that loves numbers also loves poetry. “I lisped in numbers, for the numbers came,” said Alexander Pope, describing his early childhood and his poetic fascination. I entered Princeton, wanting to become a mathematician and a poet, the former serving to support the latter. I did not know, when I entered, anything about the childhood of such mathematical geniuses as Pascal or Maxwell or Ramanujan. Nor had there been anyone in the schools I attended who could have made me aware of them.

I did not have teachers who enjoyed play with numbers. What used to be called Higher Arithmetic had long been eliminated from American curricula, as was mental math, which I was a whiz at. Sometimes, to help settle my brain to go to sleep, I square four-digit numbers in my head, or test four- or five-digit odd numbers not divisible by 3 to see if they are prime. Call it a pointless hobby. But there are a lot of things that people could have shown me, if they had known about them and if school had not been set up against their showing them. No advanced mathematics would have been necessary. These are doorways into the advanced.

Let me give an example. Suppose you have a prime number — say, 37. You are now going to divide by 37 a positive integer N that is not itself a multiple of 37; say, 43. The remainder may be anything from 1 to 36, with an equal chance for each remainder. But now multiply N by itself before you divide. The remainders suddenly are not all represented; 2, 5, 8, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 29, 32, and 35 are out. Raise N to the 6th power and then divide by 37. Now the only remainders possible are 1, 10, 11, 26, 27, and 36. Raise N to the 9th power, and the only remainders possible are 1, 6, 31, and 36. But for powers 5, 7, 11, 13, 17, and others, all the remainders are equally likely. For the 18th power, the only remainders are 1 and 36; for the 36th power, the only remainder is 1.

You don’t have to do all that cumbersome multiplication, though. You can show the child that for the sake of the problem, only the remainders count. If N/37 leaves a remainder of 2, for example, then N might as well be 2. Since 38/37 leaves a remainder of 1, that remainder is going to stay the same regardless of how often you multiply 38 by itself.

The results are quite pretty when you draw them out, especially if you label the higher half of the 36 remainders as negatives, so that, for example, the remainder of 36 in the problem above would be labeled as -1, because it falls 1 short of 37. The most fascinating patterns arise.

“But what is the point?” you ask. The professional point, assuming that we must have one, is that you are giving the child an important introduction to Number Theory, one of the most active branches of mathematics. The human point is that this is play, involving delight in pure knowledge and discovery. (RELATED: Invest in Education, Not the Department of Education)

Or you can do what the boy James Clerk Maxwell did, using twine and a ruler to draw a variety of interesting mathematical curves; this will be the child’s introduction to advanced geometry. Or you can prove the Pythagorean theorem without algebra, even without any numbers besides the idea of doubling, by drawing certain squares, rectangles, and triangles. You can open the door to topology by asking what sounds like a simple question but is not: “What shapes can you make from a flat piece of paper? What shapes can you not make?” Higher arithmetic has all kinds of applications. Why does a ballplayer’s batting average rise when he gets a hit? How is that like adding a drop of pure alcohol to a solution of alcohol and water? He is batting .300, but he goes 1 for 4, and his average drops a little. How is that like adding a little bit of a more dilute solution to one already prepared?

We can do similar things with physics, without advanced mathematics. Why does a billiard ball spinning clockwise strike the bank and veer to the left? Or how does the ball carom off another ball of the same weight, if the other ball is itself not resting on a bank? Or with chemistry: how can certain insects skate on water? What characterizes the water molecule that makes it possible?

And with poetry. Maxwell, taught by his mother till she died when he was eight, committed long passages from Paradise Lost to memory. Why not? The boys in Shakespeare’s troupe committed whole parts to memory — and played some of the greatest roles ever written for female characters: Cleopatra, Rosalind, Miranda, Juliet, Lady Macbeth, Viola, Beatrice, and many more. Teachers of English to children should have the grand vistas in mind, bringing before them whatever wonders they are ready for, and leading them to thirst for more. (RELATED: Leisure for Thought)

Nor were their schools meant to keep young people in holding tanks for twelve years, because nobody can figure out what else to do with them. I am not implying that there should be some ghastly national curriculum to integrate material taught in the earliest years with what real mathematicians, physicists, poets, artists, and engineers do. That would destroy the spirit of the enterprise, and it would universalize and perpetuate the inevitable mistakes. For 70 years, we have known what a disaster it was to dumb down children’s readers and to have children treat English words as if they were hieroglyphics. Yet we are still fighting against it. Besides, bureaucratization frustrates or repels the most imaginative among us. Set them free. (RELATED: Reclaiming Education for Boys)

Abolish all requirements that teachers have degrees or certificates in education. There is no such thing as “education” apart from the subject to be taught. If you are going to teach arithmetic, learn more and more about math, so you can direct your charges to higher things. Are you going to teach English? Learn more and more about the language and its literature. If you are not fascinated by language and grammar, you are in the wrong line of work. Be aware of what vistas grammatical knowledge opens up; know about other languages besides English. If you are teaching history, you may do well to collect old tools and learn how they were used, and why should you not have a stack of maps to pore over?

Maxwell was a very bright boy, no question, but his fellows in the Edinburgh Academy did not consider his knowledge of Milton to be all that prodigious — not by itself. We want those conditions again. We need teachers who know where the paths are that ascend the heights, and schools to encourage them or at least get out of the way.

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