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.

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.

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. Working with his colleague Anna Wilson, Dehaene has developed a computer game called “The Number Race” to help dyscalculic children. The software is adaptive, detecting the number tasks where the child is shaky and adjusting the level of difficulty to maintain an encouraging success rate of seventy-five per cent.

Despite our shared brain organization, cultural differences in how we handle numbers persist, and they are not confined to the classroom. Evolution may have endowed us with an approximate number line, but it takes a system of symbols to make numbers precise—to “crystallize” them, in Dehaene’s metaphor. The Mundurukú, an Amazon tribe that Dehaene and colleagues, notably the linguist Pierre Pica, have studied recently, have words for numbers only up to five. (Their word for five literally means “one hand.”) Even these words seem to be merely approximate labels for them: a Mundurukú who is shown three objects will sometimes say there are three, sometimes four. Nevertheless, the Mundurukú have a good numerical intuition. “They know, for example, that fifty plus thirty is going to be larger than sixty,” Dehaene said. “Of course, they do not know this verbally and have no way of talking about it. But when we showed them the relevant sets and transformations they immediately got it.”

The Mundurukú, it seems, have developed few cultural tools to augment the inborn number sense. Interestingly, the very symbols with which we write down the counting numbers bear the trace of a similar stage. The first three Roman numerals, I, II, and III, were formed by using the symbol for one as many times as necessary; the symbol for four, IV, is not so transparent. The same principle applies to Chinese numerals: the first three consist of one, two, and three horizontal bars, but the fourth takes a different form. Even Arabic numerals follow this logic: 1 is a single vertical bar; 2 and 3 began as two and three horizontal bars tied together for ease of writing. (“That’s a beautiful little fact, but I don’t think it’s coded in our brains any longer,” Dehaene observed.)

Today, Arabic numerals are in use pretty much around the world, while the words with which we name numbers naturally differ from language to language. And, as Dehaene and others have noted, these differences are far from trivial. English is cumbersome. There are special words for the numbers from 11 to 19, and for the decades from 20 to 90. This makes counting a challenge for English-speaking children, who are prone to such errors as “twenty-eight, twenty-nine, twenty-ten, twenty-eleven.” French is just as bad, with vestigial base-twenty monstrosities, like *quatre-vingt-dix-neuf *(“four twenty ten nine”) for 99. Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)

In 2005, Dehaene was elected to the chair in experimental cognitive psychology at the Collège de France, a highly prestigious institution founded by Francis I in 1530. The faculty consists of just fifty-two scholars, and Dehaene is the youngest member. In his inaugural lecture, Dehaene marvelled at the fact that mathematics is simultaneously a product of the human mind and a powerful instrument for discovering the laws by which the human mind operates. He spoke of the confrontation between new technologies like brain imaging and ancient philosophical questions concerning number, space, and time. And he pronounced himself lucky to be living in an era when advances in psychology and neuroimaging are combining to “render visible” the hitherto invisible realm of thought.

For Dehaene, numerical thought is only the beginning of this quest. Recently, he has been pondering how the philosophical problem of consciousness might be approached by the methods of empirical science. Experiments involving subliminal “number priming” show that much of what our mind does with numbers is unconscious, a finding that has led Dehaene to wonder why some mental activity crosses the threshold of awareness and some doesn’t. Collaborating with a couple of colleagues, Dehaene has explored the neural basis of what is known as the “global workspace” theory of consciousness, which has elicited keen interest among philosophers. In his version of the theory, information becomes conscious when certain “workspace” neurons broadcast it to many areas of the brain at once, making it simultaneously available for, say, language, memory, perceptual categorization, action-planning, and so on. In other words, consciousness is “cerebral celebrity,” as the philosopher Daniel Dennett has described it, or “fame in the brain.”

In his office at NeuroSpin, Dehaene described to me how certain extremely long workspace neurons might link far-flung areas of the human brain together into a single pulsating circuit of consciousness. To show me where these areas were, he reached into a closet and pulled out an irregularly shaped baby-blue plaster object, about the size of a softball. “This is my brain!” he announced with evident pleasure. The model that he was holding had been fabricated, he explained, by a rapid-prototyping machine (a sort of three-dimensional printer) from computer data obtained from one of the many MRI scans that he has undergone. He pointed to the little furrow where the number sense was supposed to be situated, and observed that his had a somewhat uncommon shape. Curiously, the computer software had identified Dehaene’s brain as an “outlier,” so dissimilar are its activation patterns from the human norm. Cradling the pastel-colored lump in his hands, a model of his mind devised by his own mental efforts, Dehaene paused for a moment. Then he smiled and said, “So, I kind of like my brain.” ♦

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