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Jim Holt has written in the New Yorker about a cognitive neuroscientist named Stanislas Dehaene.
Stanislas Dehaene is a French researcher whose field is numerical cognition.
Dehaene plots the contours of our number sense, puzzling over which aspects of our mathematical ability are innate, and which are learned — and how the two systems overlap and affect each other.
A colleague, Susan Carey, from Harvard, says “If you want to make sure the math that children are learning is meaningful, you have to know something about how the brain represents number at the level Stan is trying to understand.”
Dehaene has approached the problem from every imaginable angle. He has carried out experiments that get to how numbers are coded in our minds. He has studied the numerical ability of animals, tribespeople, and top mathematics students. He has used brain-scanning technology to see where precisely in the folds of the cerebral cortex the numerical faculties are located.
He has also weighed the extent to which some languages make numbers more difficult than others.
His work raises crucial issues about how we teach mathematics.
Dehaene believes we are all born with an evolutionarily ancient mathematical instinct. To become numerate, children have to capitalize on this instinct — but they also have to unlearn certain tendencies that were helpful to our primate ancestors and that clash with the skills we need today.
Some societies are evidently better at getting kids to do this. France and the US face a math teaching crisis — in comparison with countries like Japan, South Korea and Singapore.
To fix this situation means having to grapple with the major question of Dehaene’s career: what is it about the brain that makes numbers sometimes so easy and sometimes so hard?
Early in his career, Dehaene asked: How do we know whether numbers are bigger or smaller than one another? When we are asked to choose which of a pair — 4 and 7, for example — is larger, we respond “seven” immediately. But Dehaene’s early experiments found that although subjects could answer quickly and accurately when numbers were far apart (2 and 9), they slowed down when numbers were closer together( 5 and 6). And, performance got worse when the digits grew larger: 2 and 3 were much easier than 7 and 8.
Dehaene’s conjecture was that when we see numerals or hear number words, we automatically map them onto a number line that grows increasingly hazy above 3 or 4.
He found that no amount of training can change this. “It is a basic structural property of how our braiins represent number, not just a lack of facility.”
After learning of Michael Posner’s pioneering work using brain scanning technology in the late 80’s, Dehaene decided to “bridge the gap” between psychology and neurobiology. He set out to find exactly how the functions of the mind — thought, perception, feeling, will — are realized in the pulsing meat of the brain.
Because now, suddenly, researchers could create crude pictures of the brain in the act of thinking.
It had long been hypothesized, based on research with animals and infants, that the brain has an evolved number ability. Dehaene wanted to know where it might be found.
“In one experiment I particularly liked,” he says, “we tried to map the whole parietal lobe in a half hour, by having the subject perform functions like moving the eyes and hands, pointing with fingers, grasping an object, engaging in various language tasks, and of course, making small calculations.
“We found there was a beautiful geopmetrical organization to the areas that were activated. The eye movements were at the back, the hand movements were in the middle, grasping was in the front, and so on. And right in the middle, we were able to confirm, was the area that cared about number.”
This number area lies deep within a fold in the parietal lobe called the intraparietal sulcus, and it’s just behind the crown of the head.
The brain, however, is a product of evolution. It is messy and random in its processing. So although the number sense may be lodged in a particular bit of the crebral cortex, its circuity seems to be mingled with the wiring for a lot of other mental functions.
For example, in one experiment on number comparisons, Dehaene noticed that subjects performed better if they held the response key in the right hand for larger numbers, but they did better with the left hand when working with the smaller numbers. And strangely, if the subjects were asked to cross their hands, the effect was reversed. The actual hand making the response was irrelevant; it was space itself that the subjects unconsciously associated with larger or smaller numbers.
Dehaene hypothesizes from this that the neural circuitry for number and the circuitry for location overlap. He suggests that the reason travellers become dioriented upon entering Terminal 2 of Paris’s Charles de Gaulle Airport is because the small-numbered gates are on the right, and the large-numbered gates on the left.
“It’s become a whole industry now to see how we associate number to space and space to number,” Dehaene says.
Dehaene explains that new techniques of neuroimaging promise to reveal how a thought process like calculation unfolds in the brain. Since the brain’s architecture determines the sort of abilities that are natural to us, having a detailed understanding of that architecture should lead to better ways of teaching children mathematics.
The fundamental problem 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 be absorbed by areas of the brain that evolved for other purposes.
When what we are learning harmonizes with built-in circuitry, learning is easier. If we can’t change our brains we can at least adapt our teaching methods to the constraints imposed by it.
Researchers have seen that six-month-old babies, exposed simultaneously to images of common objects and sequences of drumbeats, will consistently gaze longer at the collection of objects that matches the number of drumbeats. It is now generally agreed that infants come equipped with a rudimentary ability to perceive and represent number. (The same also appears to be true for salamanders, pigeions, raccoons, dolphins, parrots and monkeys.)
But just as evolution has equipped us with a primitive number sense, culture has furnished two more methods for dealing with number: numerals and number words. This is both the good news and the bad news.
These three modes of thinking about number correspond to distinct areas of the brain, Dehaene believes. 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 the number words are processed by the language areas.
However, there is no clever computer chip in all this elaborate circuitry to help them communicate with each other instantaneously.
At first, children don’t experience problems. They have a number sense that gives them a crude feel for addition; even before they go to school, children can find ways to add things.
But multiplication is an “unnatural practice,” says Dehaene, because 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.
While the list of arithmetical facts to be memorized is short, it is fiendishly tricky — the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes.
The human memory is not like a computer; it is associative. That makes it badly suited to handle arithmetic, where bits of knowledge must be kept from interfering with one another. 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.
Adults, it has been found, make mistakes 10 to 15 percent of the time when multiplying double digit numbers; for the hardest problems (say 7×8) they make mistakes 25 percent of the time.
Dehaene questions why, given our inbuilt ineptness, we insist on drilling procedures like long division into our children. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them.”
Calculators, he says, will remove the need for children to spend hundreds of hours memorizing procedures and free them to concentrate on the meaning of those procedures.
Dehaene admires the mathematics curricula of Asian countries like China and Japan. These 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.
Working with a colleague, Anna Wilson, Dehaene has developed a computer game called “The Number Race” to help dyscalculic children. The software is adaptive; it detects the number tasks where the child is shaky and adjusts the level of difficulty to maintain an encouraging success rate of 75 percent.
Today, Arabic numerals are in use nearly everywhere in the world. The words with which we name numbers, however, differ from language to language — and those differences are not 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 say “twenty-ten, twenty-eleven…” French is bad, too, with vestigial base twenty monstrosities like quatre-vingt-dix-neuf (“four twenty ten nine”) for the number 99.
Chinese, however, is simplicity itself: its number syntax perfectly mirrors the base-ten form of Arabic numerals with a minimum of terms. The average Chinese four-year-old can count to forty; American four-year-olds struggle to get to 15.
And because Chinese number words are so brief — they take less than a quarter of a second to say — the average Chinese speaker has a memory span of nine digits, versus seven for English speakers.
sole source for everything in this piece is Jim Holt’s article in the New Yorker, in the March 3, 2008 edition. www.newyorker.com
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