March 05, 2008

Math Is Hard

In The New Yorker, Jim Holt channels French neuroscientist Stanislas Dehaene, who has a theory that our brains come with a built-in sense for numbers, but not for doing mathematical calculations like multiplication or long division. That stuff is unnatural:
Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—"Ambition, Distraction, Uglification, and Derision," as Lewis Carroll burlesqued them—a chore.

It's not so bad at first. Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: "two, three is one, four is two, five is three, six is four, six."

But multiplication is another matter. It is an "unnatural practice," Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.)

The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you're trying to retrieve the result of multiplying 7 X 6, the reflex activation of 7 + 6 and 7 X 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 X 8, the error rate can exceed twenty-five per cent.
A "double terror." I like that. From this, Dehaene goes on to suggest that maybe first-graders shouldn't be forced to memorize times-tables until they puke, but ought to be given calculators so that they can learn "the meaning of these procedures." This debate always gets heated, but I can't say I feel strongly. I've got my times tables under control, but it only really comes in handy when I'm resizing images for blog posts. By contrast, I have a very smart friend in private equity who still balks at 11 X 12 and 8 X 9 and the rest. But who cares? (I should say, however, that the finger system for multiplying by 9 is great fun regardless.)

Another coffee-table tidbit: In Chinese and Japanese, number systems are base-ten, rather than our slightly screwy system (for instance, we say "eleven" rather than "ten-one," as it is in Japanese). As such, the average Chinese speaker can hold nine digits in her head, rather than seven for English. French is particularly horrible on this front ("four-twenty-ten-five" is the way you say 95), and despite having been drilled repeatedly, I will never be able to do long division in French in my head. That doesn't bother me.
-- Brad Plumer 12:13 PM || ||