Ambition, Distraction, Uglification, and Derision

The Numbers Guy, Jim Holt, has a great piece in The New Yorker. In the article Are Our Brains Wired for Math? Holt discusses the work of the French neuroscientist, Stanislas Dehaene who argues that we are born with “number sense” that makes us capable of basic calculations and estimates. However, Dehaene clarifies, the procedures that we learn to carry out in mathematics are not instinctive at all.

The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools—symbols and algorithms—that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes.

For nearly two decades, American educators have pushed “reform math,” in which children are encouraged to explore their own ways of solving problems. Before reform math, there was the “new math,” now widely thought to have been an educational disaster. (In France, it was called les maths modernes, and is similarly despised.) The new math was grounded in the theories of the influential Swiss psychologist Jean Piaget, who believed that children are born without any sense of number and only gradually build up the concept in a series of developmental stages.

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Piaget’s view had become standard by the nineteen-fifties, but psychologists have since come to believe that he underrated the arithmetic competence of small children. Six-month-old babies, exposed simultaneously to images of common objects and sequences of drumbeats, consistently gaze longer at the collection of objects that matches the number of drumbeats. By now, it is generally agreed that infants come equipped with a rudimentary ability to perceive and represent number. (The same appears to be true for many kinds of animals, including salamanders, pigeons, raccoons, dolphins, parrots, and monkeys.) And if evolution has equipped us with one way of representing number, embodied in the primitive number sense, culture furnishes two more: numerals and number words. These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.

And if evolution has equipped us with one way of representing number, embodied in the primitive number sense, culture furnishes two more: numerals and number words. These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.

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 crude feel for addition, so that, even before schooling, children can find simple recipes or adding numbers. If asked to compute 2 + 4, for example, a child might start with he first number and then count upward by the second number: “two, three is one, our 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.

Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo.

This attitude might make Dehaene sound like a natural ally of educators who advocate reform math, and a natural foe of parents who want their children’s math teachers to go “back to basics.” But when I asked him about reform math he wasn’t especially sympathetic. “The idea that all children are different, and that they need to discover things their own way—I don’t buy it at all,” he said. “I believe there is one brain organization. We see it in babies, we see it in adults. Basically, with a few variations, we’re all travelling on the same road.” He admires the mathematics curricula of Asian countries like China and Japan, which provide children with a highly structured experience, anticipating the kind of responses they make at each stage and presenting them with challenges designed to minimize the number of errors. “That’s what we’re trying to get back to in France,” he said.

It makes for fascinating reading. I still don't agree with the give a calculator to a five-year-old stuff, though. Nope. I just don't buy that.
Are Our Brains Wired for Math?
Jim Holt
The New Yorker
March 3, 2008