from The New Yorker, March 3, 2008
According to Stanislas Dehaene, humans have an inbuilt "number sense" capable of some basic calculations and estimates. The problems start when we learn mathematics and have to perform procedures that are anything but instinctive. But here is a quarrel with Holt:
Dear New Yorker Editor:
Jim Holt, writing about Stanislaus Dehaene's research on neurobiology and arithmetic, tells us about findings which are new and tantalizing
("Numbers Guy," March 3rd.) But one might expect a writer on research to have done some research himself. As a longtime curriculum developer in school math and researcher in the learning of
mathematics I can say without hesitation that Holt's claim that the "new math" was grounded in the theories of Jean Piaget is utterly false. "New math" or "modern math" as known in the USA amounted to the imposition on children of mathematics as known by university mathematicians, most of whom would not have heard of Piaget Ã¢€” most certainly not in the nineteen-fifties, a time in which Holt claims Piaget's views were "standard." At that time, Piaget's research was virtually unknown in the United States.
Second, the notion that "reform math" calls for children to discover things their own way gives the impression that they wander aimlessly and randomly through math. A substantial body of research shows that this is fundamentally wrong. The one thing that we humans do is to make sense of things; what "reform math" curricula do is to challenge pupils to make sense of math, usually in the face of thoughtfully constructed problem situations. This is not chaotic or chance discovery. Rather it provokes adaptation involving the evolution of children's networks of ideas in terms of complexity, stability, economy and generalizability (and, not the least, the quest to go further in their investigations).
Holt's slippery characterization of reform math could support the authoritarian approach to the teaching of mathematics so common these days, a policy virtually identical to "modern math" with the widespread message to children, "You're incompetent. You are capable only of following orders."
Third, one should keep in mind that there is an enormous gap between research showing that the notion of subtraction, for example (not to say anything about a wide range of more complex mathematical thinking), resides in these or those neural folds and evidence for the success of educational methods/school curricula which enable children to subtract.
Thomas C. O'Brien
North Atlantic Treaty Organization Senior Research Fellow in Science
by Jim Holt
One morning in September, 1989, a former sales representative in his mid-forties entered an examination room with Stanislas Dehaene, a young neuroscientist based in Paris. Three years earlier, the man, whom researchers came to refer to as Mr. N, had sustained a brain hemorrhage that left him with an enormous lesion in the rear half of his left hemisphere. He suffered from severe handicaps: his right arm was in a sling; he couldnÃ¢€™t read; and his speech was painfully slow. He had once been married, with two daughters, but was now incapable of leading an independent life and lived with his elderly parents. Dehaene had been invited to see him because his impairments included severe acalculia, a general term for any one of several deficits in number processing. When asked to add 2 and 2, he answered "three." He could still count and recite a sequence like 2, 4, 6, 8, but he was incapable of counting downward from 9, differentiating odd and even numbers, or recognizing the numeral 5 when it was flashed in front of him.
To Dehaene, these impairments were less interesting than the fragmentary capabilities Mr. N had managed to retain. When he was shown the numeral 5 for a few seconds, he knew it was a numeral rather than a letter and, by counting up from 1 until he got to the right integer, he eventually identified it as a 5. He did the same thing when asked the age of his seven-year-old daughter. In the 1997 book "The Number Sense," Dehaene wrote, "He appears to know right from the start what quantities he wishes to express, but reciting the number series seems to be his only means of retrieving the corresponding word."
Dehaene also noticed that although Mr. N could no longer read, he sometimes had an approximate sense of words that were flashed in front of him; when he was shown the word "ham," he said, "It's some kind of meat." Dehaene decided to see if Mr. N still had a similar sense of number. He showed him the numerals 7 and 8. Mr. N was able to answer quickly that 8 was the larger numberÃ¢€”-far more quickly than if he had had to identify them by counting up to the right quantities. He could also judge whether various numbers were bigger or smaller than 55, slipping up only when they were very close to 55. Dehaene dubbed Mr. N "the Approximate Man." The Approximate Man lived in a world where a year comprised "about 350 days" and an hour "about fifty minutes," where there were five seasons, and where a dozen eggs amounted to "six or ten." Dehaene asked him to add 2 and 2 several times and received answers ranging from three to five. But, he noted, "he never offers a result as absurd as 9."
In cognitive science, incidents of brain damage are natureÃ¢€™s experiments. If a lesion knocks out one ability but leaves another intact, it is evidence that they are wired into different neural circuits. In this instance, Dehaene theorized that our ability to learn sophisticated mathematical procedures resided in an entirely different part of the brain from a rougher quantitative sense. Over the decades, evidence concerning cognitive deficits in brain-damaged patients has accumulated, and researchers have concluded that we have a sense of number that is independent of language, memory, and reasoning in general. Within neuroscience, numerical cognition has emerged as a vibrant field, and Dehaene, now in his early forties, has become one of its foremost researchers. His work is "completely pioneering," Susan Carey, a psychology professor at Harvard who has studied numerical cognition, told me. "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 kind of level that Stan is trying to understand." [emphasis added]
Dehaene has spent most of his career plotting the contours of our number sense and puzzling over which aspects of our mathematical ability are innate and which are learned, and how the two systems overlap and affect each other. He has approached the problem from every imaginable angle. Working with colleagues both in France and in the United States, he has carried out experiments that probe the way numbers are coded in our minds. He has studied the numerical abilities of animals, of Amazon tribespeople, of top French mathematics students. He has used brain-scanning technology to investigate precisely where in the folds and crevices of the cerebral cortex our numerical faculties are nestled. And he has weighed the extent to which some languages make numbers more difficult than others. His work raises crucial issues about the way mathematics is taught. In Dehaene's view, we are all born with an evolutionarily ancient mathematical instinct. To become numerate, children must capitalize on this instinct, but they must also unlearn certain tendencies that were helpful to our primate ancestors but that clash with skills needed today. And some societies are evidently better than others at getting kids to do this. In both France and the United States, mathematics education is often felt to be in a state of crisis. The math skills of American children fare poorly in comparison with those of their peers in countries like Singapore, South Korea, and Japan. Fixing this state of affairs means grappling with the question that has taken up much of Dehaene's career: What is it about the brain that makes numbers sometimes so easy and sometimes so hard?
Dehaene's own gifts as a mathematician are considerable. Born in 1965, he grew up in Roubaix, a medium-sized industrial city near France's border with Belgium. (His surname is Flemish.) His father, a pediatrician, was among the first to study fetal alcohol syndrome. As a teen-ager, Dehaene developed what he calls a "passion" for mathematics, and he attended the Ãƒ‰cole Normale SupÃƒÂ©rieure in Paris, the training ground for France's scholarly ÃƒÂ©lite. Dehaene's own interests tended toward computer modelling and artificial intelligence. He was drawn to brain science after reading, at the age of eighteen, the 1983 book "Neuronal Man," by Jean-Pierre Changeux, France's most distinguished neurobiologist. Changeux's approach to the brain held out the tantalizing possibility of reconciling psychology with neuroscience. Dehaene met Changeux and began to work with him on abstract models of thinking and memory. He also linked up with the cognitive scientist Jacques Mehler. It was in Mehler's lab that he met his future wife, Ghislaine Lambertz, a researcher in infant cognitive psychology.
By "pure luck," Dehaene recalls, Mehler happened to be doing research on how numbers are understood. This led to Dehaene's first encounter with what he came to characterize as "the number sense." Dehaene's work centered on an apparently simple question: How do we know whether numbers are bigger or smaller than one another? If you are asked to choose which of a pair of Arabic numeralsÃ¢€”-4 and 7, sayÃ¢€”-stands for the bigger number, you respond "seven" in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene's experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the Ãƒ‰cole Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number.
Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. "It is a basic structural property of how our brains represent number, not just a lack of facility," he told me.
In 1987, while Dehaene was still a student in Paris, the American cognitive psychologist Michael Posner and colleagues at Washington University in St. Louis published a pioneering paper in the journal Nature. Using a scanning technique that can track the flow of blood in the brain, PosnerÃ¢€™s team had detailed how different areas became active in language processing. Their research was a revelation for Dehaene. "I remember very well sitting and reading this paper, and then debating it with Jacques Mehler, my Ph.D. adviser," he told me. Mehler, whose focus was on determining the abstract organization of cognitive functions, didnÃ¢€™t see the point of trying to locate precisely where in the brain things happened, but Dehaene wanted to "bridge the gap," as he put it, between psychology and neurobiology, to find out exactly how the functions of the mindÃ¢€”thought, perception, feeling, willÃ¢€”are realized in the gelatinous three-pound lump of matter in our skulls. Now, thanks to new technologies, it was finally possible to create pictures, however crude, of the brain in the act of thinking. So, after receiving his doctorate, he spent two years studying brain scanning with Posner, who was by then at the University of Oregon, in Eugene. "It was very strange to find that some of the most exciting results of the budding cognitive-neuroscience field were coming out of this small placeÃ¢€”-the only place where I ever saw sixty-year-old hippies sitting around in tie-dyed shirts!" he said.
Dehaene is a compact, attractive, and genial man; he dresses casually, wears fashionable glasses, and has a glabrous dome of a head, which he protects from the elements with a chapeau de cowboy. When I visited him recently, he had just moved into a new laboratory, known as NeuroSpin, on the campus of a national center for nuclear-energy research, a dozen or so miles southwest of Paris. The building, which was completed a year ago, is a modernist composition in glass and metal filled with the ambient hums and whirs and whooshes of brain-scanning equipment, much of which was still being assembled. A series of arches ran along one wall in the form of a giant sine wave; behind each was a concrete vault built to house a liquid-helium-cooled superconducting electromagnet. (In brain imaging, the more powerful the magnetic field, the sharper the picture.) The new brain scanners are expected to show the human cerebral anatomy at a level of detail never before seen, and may reveal subtle anomalies in the brains of people with dyslexia and with dyscalculia, a crippling deficiency in dealing with numbers which, researchers suspect, may be as widespread as dyslexia. One of the scanners was already up and running. "You don't wear a pacemaker or anything, do you?" Dehaene asked me as we entered a room where two researchers were fiddling with controls. Although the scanner was built to accommodate humans, inside, I could see from the monitor, was a brown rat. Researchers were looking at how its brain reacted to various odors, which were puffed in every so often. Then Dehaene led me upstairs to a spacious gallery where the brain scientists working at NeuroSpin are expected to congregate and share ideas. At the moment, it was empty. "WeÃ¢€™re hoping for a coffee machine," he said.
Dehaene has become a scanning virtuoso. On returning to France after his time with Posner, he pressed on with the use of imaging technologies to study how the mind processes numbers. The existence of an evolved number ability had long been hypothesized, based on research with animals and infants, and evidence from brain-damaged patients gave clues to where in the brain it might be found. Dehaene set about localizing this facility more precisely and describing its architecture. "In one experiment I particularly liked," he recalled, "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, like thirteen minus four. We found there was a beautiful geometrical 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 an area that cared about number."
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