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Naze sugaku ga tokui na hito to nigate na hito ga irunoka?
(Why are some people good, but others bad at maths?)

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What Counts - How Every Brain Is Hardwired For Math
Brian Butterworth



So you think you're bad at maths? Meet Charles, he has a normal IQ and a university degree yet has problems telling whether 5 is bigger than 3. And what about Signora Gaddi, an Italian woman who hears and sees normally but, following a stroke, is deaf and blind to all numbers above 4?

Their stories and others are told by neuropsychologist Brian Butterworth in his book The Mathematical Brain. For Butterworth, they are living evidence that the brain contains a special device for making sense of numbers. It's just a little knot of cells over your left ear, but when it's working properly, this number module doesn't just allow us to see the world in terms of numbers - it compels us to. We can't stop enumerating, says Butterworth, any more than we can avoid seeing in colour. Even as a baby, it was making you notice discrepancies in, say, how many spoonfuls of food came your way compared with how many came out of the jar.

But if most people have this innate and unstoppable number sense, why do so many numerical skills seem so hard to acquire? And why aren't most of us in the Einstein league of maths brains? Or perhaps we are? Alison Motluk talks numbers, brains and genes with Butterworth at his office in University College London.

True grit

Look at a spring leaf and your brain instantly grasps the "greenness" of it. You don't have to think. The greenness just happens. Now imagine looking at 4 dots on a page. Doesn't your brain just as effortlessly grasp the "fourness" of it, even without any conscious counting? And if there were 4 people standing next to, say, 3 cars, would you have to count them to grasp there were more people than cars? No, you'd know - and laboratory studies confirm this - just by looking.

This superquick understanding of ours is one of the things Butterworth is so keen on. But why? It's a neat trick, and the survival benefits of being able to "subitise", as experts call it, are obvious: five of them, two of us ... run! But we're hardly talking fancy maths skills here. And, disappointingly, the ability seems to peter out when the numbers are greater than five. So what's left to be said about it?

Plenty if you believe Butterworth. He thinks the brain circuit that enables us to subitise underpins virtually all our numerical understanding of the world. Mastering long division, spreadsheets and tax forms are obviously all skills that involve many different brain circuits and which have to be developed the hard way. But, says Butterworth, without a number module, that learning wouldn't take place. If maths is the pearl, the module is the grit in the oyster - it's what tells the brain about the sizes of numbers and what those sizes mean.

Butterworth hasn't always been so obsessed with how the brain handles numbers. He spent his early career in the realm of words, studying dyslexia. He did, however, once take a masters degree in mathematical logic, and in 1984 two things happened to nudge him back to numbers. First, he says, he met the American psychologist Prentice Starkey, then on sabbatical in London. Starkey was one of the first to argue that even babies have a sense of number. And secondly, Butterworth's first child, Amy, was born, allowing him to see that sense in action. "I started to believe it:" he says.

Apes and babies

And what Butterworth clearly believes with a passion is that the number module - the grit - is there in the brain from day one. Take counting. Like times tables and calculus, we tend to think it's something kids have to be formally taught. Wrong, says Butterworth - it's an instinct. Sure, we have to learn the names and symbols of numbers to develop that instinct, but, because the number module is hardwired into the brain, basic counting comes naturally. Remote tribes can count even when they have no R words for numbers. And ingenious experiments have shown that even babies and apes can grasp what Butterworth calls "numerosities" - the threeness of three and fourness of four. In maths as in language he believes, "kids start off with little starter kits" And their maths starter kit is the number module.

All of which is more controversial than it sounds. Others say we have no special device for representing numbers in the brain and that far from being an independent ability or instinct, our number sense flows from general intelligence and reasoning, or spatial awareness, or linguistic abilities - or some combination of all three. So why is Butterworth convinced this is wrong?

Meeting and studying people who lacked the normal version of the sense was a big factor, he says - people such as Signora Gaddi and Charles. Following a stroke, Butterworth explains, Signora Gaddi has normal language and reasoning skills but has no idea whether 20 is bigger than 10. She cannot use the phone, recognise which bus to catch, or remember any facts at all involving numbers above 4. And up to 4, she has to count the numbers to herself to know how many of anything they represent. But what's really intriguing, says Butterworth, is she can't even subitise. Even this, the most basic number sense, is lacking.

Charles can't subitise either. If he saw 2 cars in a car park, he'd have to count them: 1, 2 . . .. If he then saw three in a neighbouring car park, he couldn't tell you which car park had more cars. He can't even work out which chocolate bar costs more or if he got the right amount of change. And, lacking the grit that Butterworth thinks is so important, what Charles has so far learnt about numbers in his life has been pretty ineffective. He has to count on his fingers and can barely do subtraction, division or any problem involving multiple digits. In effect, says Butterworth, he is blind to the underlying meaning of numbers. He can say 4 and 3 as words and recognise the numerals, but lacking a proper number instinct, he has little feeling for what they represent.

But is it a special number sense that these patients lack, or something more general to do with reasoning? The fact that Charles is impaired in maths and nothing else suggests it's a specific number problem, says Butterworth. His IQ is normal. He even has a university degree in psychology (Charles says he can handle statistics because the computer does all the calculations).

Or maybe subitising is really the work of some general purpose brain circuit for recognising the way objects are positioned in space. After all, four objects often adopt a quadrilateral pattern, three objects a triangle of some sort. Surprisingly, Butterworth thinks people's awareness of their fingers - rather than spatial patterns in general - plays the bigger role in the development of the number instinct.

The number module is genetically programmed to understand the sizes of numbers only up to 4 or 5. But we can grasp much bigger numbers, Butterworth suggests, because the module links up during development with the circuits that control finger movements. All children instinctively use their fingers to represent the sizes of numbers. These finger representations, Butterworth believes, are the stepping stones that enable the brain to generalise from our limited innate number sense.

In other words, your number skills are pre-programmed, but not predestined. "What you end up knowing about numbers is a function of your experience in your culture, " As an example, Butterworth points to Chinese-speaking children who are so much better at counting and maths than their English-speaking counterparts. It's nothing to do with the genes, says Butterworth, but rather reflects the fact that their language uses more logical number words - 11, 12 and 20, for instance, are represented verbally as 10 plus 1, 10 plus 2, and two 10s, whereas in English they have special names. At first nearly all kids get tripped up by "twelve" and "twenty"

So what about maths geniuses? Intriguingly, the Einstein study that made so much news last month found that his brain was enlarged in the very area where Butterworth thinks his number module can be found, the so-called inferior parietal lobule. Doesn't this suggest some maths starter kits are better than others? Surely if a bad bit of brain can account for number blindness, a supercharged number module could help to explain people like Einstein, or at least those savants who can calculate in a flash what day my birthday will fall on in 2033?

Butterworth is surprisingly doubtful. He believes the number module is either there and intact, or it's not (as in Charles). There may be people with extraordinary number abilities, he says, but these have less to do with the number module than with obsession and hard work. Really? "We don't know how trainable it is," Butterworth admits, but he says one intriguing study found career cashiers were just as quick as maths prodigies at multiplying four and five-digit numbers in their heads.

As for Einstein's extra brain cells, Butterworth admits he doesn't know what put them there. "Do geniuses have more parietal lobe brain cells than the rest of us at birth, have they recruited more to that region, or have they allowed fewer to die?" His suspicion is that what makes the Einsteins of this world good is what makes everybody good- hard work and practice.

The Mathematical Brain by Brian Butterworth, Macmillan, £20, ISBN 0333735277. Published in the US next month by Simon & Schuster as What Counts, 25 dollars.

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A cheerful English voice, crisp and elegant, asked her the question again. "How many coins do you have there, Signora?"

Signora Gaddi stared at the coins in her hand for a long time, and then looked up to smile apologetically at the doctor. It was a soft smile, warm, but tenuous and sad. The corners of her lips trembled delicately when she tried to explain the inexplicable: she knew that there were more than four, but she could not imagine how many. Were there eight? Or ten? Or some other strange number, whose name hung heavily on her tongue and could not be uttered?

"It's all right, Signora. There are six coins." The doctor's voice was kind; he understood. He knew of other people like Signora Gaddi, people who had little or no sense of numbers. These people were not simply bad at math, nor were they poorly educated. The clinical terms are acalculia, for people like Signora Gaddi who lost her sense of numbers after a stroke, and dyscalculia for people who were born without numbers. But clinical terms don't go very far towards describing the people who lead lives almost completely devoid of numbers.

S. Gaddi is a charming, middle-aged Italian woman. Before her stroke, she managed the books at her family's hotel, and led a life filled with numbers. Room assignments, charges, debits, profits, expenses - S. Gaddi was more than proficient at rapidly and accurately performing arithmetical calculations. But since the day that a small blood vessel in her left parietal lobe burst, S. Gaddi has been blind to numbers greater than four. She can readily perform addition and subtraction, she can list number names in sequence - so long as all the digits involved are less than or equal to four. Dr. Brian Butterworth, the University College London neuroscientist who worked with S. Gaddi, writes that,

"Since her stroke, [S. Gaddi's] life had been one of frustration and embarrassment. She was unable to do things that previously had been second nature to her. She could not give the right money in shops; she had no idea how much she was spending or how much change she was getting. She could not use the phone. There was no way to call her friends. She was unable to tell the time, or catch the right bus."

S. Gaddi's number blindness, a condition called acalculia, is evidence that the brain may be biologically "wired" for mathematics. Dr. Butterworth's powerful "mathematical brain hypothesis" has important implications for how teachers should approach math education. If the brain really is wired for mathematics, then it may be necessary to reconceptualize our views on what math is, and how it affects our daily lives.

BC (Brain Connection): How does the brain process math, and what are the stages of calculation?

BB (Brian Butterworth): Well, there are different aspects of mathematics, and there are different levels within these aspects. If we take numbers as one domain of mathematics, operations on numbers, arithmetical operations for example, but not only arithmetical operations, then we can actually trace a few brain areas that are involved. If you look at the most basic numerical processes, for example, just saying which of two numbers is larger, that seems to be a process which can be done in both parietal lobes of the brain. Then there's just seeing the number of objects in an array, seeing the twoness or threeness or fiveness of an array - seeing their numerosity. We're not really clear which parts of the brain do that. But it's a very important issue, because we know that infants and some animals are able to recognize the numerosity of arrays up to five or six.

BC: Which brings up the question that, experimentally, how do you distinguish a recognition of quantity from recognition of numerosity? What do you look for?

BB: People have looked into this for animals. If you set up a situation, let's say, where animals have to choose more bits of food versus fewer bits of food, are they picking on the basis of numerosity, or are they picking on the basis of just the quantity of food? To find out, you do an experiment in which you change the quantity of food without changing the number of bits, or change the number of bits without changing the total quantity. Then you can see in a practical way whether the animal is responding to number or quantity. And there are quite a number of experiments which show that animals can respond to the number of things rather than the total quantity of things. In fact, in apes, in monkeys, in rats, and in birds it has been shown that they can respond to numbers specifically.

BC: You wrote about a number of different cases in your book. Can you tell me about one or two of the most significant cases that you've studied?

BB: One of the things that I'm very concerned with this year, because this year as you know is the International Year of Mathematics, is the way in which some people are excluded, socially excluded, because they're really very bad at mathematics, because of some congenital deficit.

These are cases of dyscalculia. It's a bit like being born dyslexic, or being born colorblind - nothing you can do about it, and it's very handicapping. In fact, some research in England has shown that when it comes to getting and keeping a job, being bad at maths is worse than being bad at reading and spelling. It's a very serious problem, and we don't really know how many people are affected. Some estimates suggest it's as many as 5% of the general population, which is one or more in every classroom. So it's a problem that's as widespread as dyslexia in English speaking countries. Some of the cases that I've been particularly interested in are the first cases of these developmental dyscalculics that have been fully described.

BC: Developmental dyscalculia, as distinguished from another later-onset dyscalculia?

BB: There's late-onset, you know, if you have a stroke or get a bang on the head, then that would be an acquired acalculia. Developmental dyscalculias are something that we assume you're born with.

One case that was really very striking for us was the case of Charles, whom I described in the book. He is a highly intelligent young man with a degree who is very, very disabled when it comes to numbers. If he goes into a shop, he can't make sense of the prices on the products. He can't add up the prices, he doesn't know how much money to offer when he goes to the checkout, he doesn't know if he's getting the right change. It's very embarrassing, because he has to open his wallet and say, "Take as much as is necessary, and give me the change," without being able to check it. Even if people are entirely honest with him, which they probably are most of the time, it's still deeply embarrassing. We discovered that the most basic numerical processes are defective in Charles, that is he's not very good at telling how many objects there are in an array. He can't seen the twoness of two objects, he has to count them, he can't see the fiveness of five objects. Most of the rest of us can just look at this and say, oh it's two, or perhaps five, but he can't. He's extremely slow even when he's comparing two numbers, so if you ask him to compare five and nine, it takes him a while to come up which of those two is the larger. And he uses his fingers to do it. So he's an intelligent, well-educated, hard working man and yet he has the most extraordinary difficulty with numbers. Just applying hard work and intelligence has not been able to remediate this problem.

BC: Charles, as I recall from the book, approached you. What was it like for you to meet him, to meet people like him who have what seem to be profoundly strange ways of seeing the world?

BB: I think, in a way, it's stranger for them than it is for me, because I've now seen quite a lot of these cases.

BC: And people like Charles must feel very alone.

BB: That's right, he's in the same position that a dyslexic would have been in 25 years ago. They think there's nobody else like them, and they think that they're really stupid. And they're often treated as being really stupid, because they can't do what other people are able to do. Charles has a degree in psychology, so he knows that there's a specific deficit, that there's nothing he can do about it. Recently I've been seeing a bunch of kids who have problems similar to Charles. The children are worried about it, but the parents are desperate, because the schools don't have the diagnosis, and they don't have a system for treating it.

BC: So the schools in Britain have started recognizing this as a clinical issue?

BB: I'm afraid they haven't been recognizing it, no. Parents who've heard about my work get in touch, often desperate, and say, look, my child is doing really badly at school because of his mathematics. Everything else he or she does quite well, but his mathematics is terrible. And no one seems to know what the problem is, he's not dyslexic, he's had his IQ tested, he's not stupid. So what's the matter?

BC: What kinds of mistakes do dyscalculic children make, is there a tell-tale pattern or are they just generally deficient?

BB: They're deficient across the board. What some of them can do, and what Charles can do for example, is that they can learn an ordered sequence. They can learn to count, they know the number words in order. So they haven't got a problem with sequencing things. But I have recently seen some kids of eleven or twelve who can't count above 20, and who can't count backwards from 20. They have trouble really having a sense of number size, that seems to be the core of their difficulty. This has an effect on everything they do.

Let me give you a very simple example. When a child learns to do addition, a child of four or five or six, you can ask them "What's three plus five?" and they start off by going, "Three," and they count up with their fingers three, and they count up with their other fingers five, and they count all of their fingers, and they go "One two three four five six seven eight." Then they get to a stage when they don't count out the first number, they go "Three," and then they count with their fingers, "four five six seven eight." They can see that they've held up five fingers, so they say, "Well, that's eight."

Then they go to a stage called "counting on from the larger." This is very important, because this is the stage at which they start to remember number facts. Here they start, "Five," and then they count up with their fingers "six, seven, eight." Now, two things that dyscalculic kids, like Charles, can't do very easily is select the larger of two numbers. Children who are dyscalculic can't count on from the larger, because they can't pick the larger number. So that's one problem.

The other thing is that they seem to have a problem in seeing the number of things that there are. That means they can't see that they've got five fingers up, they have to count their fingers as well. And that makes the whole process of learning arithmetic using your fingers difficult, because when you count on from five, "six, seven, eight," you can see that there are three fingers there. But the child who has this profound dyscalculia, like Charles, can't see that there are three fingers there. He has to double count to know how many fingers he's raised, whether he's raised the number of fingers he needs to do the addition. That whole thing makes the very simple, nearly universal process of learning how to do addition extremely difficult. It turns out that these kids just don't ever have a good body of number facts at their disposal, because they find it very hard to do the things that we normally do when we're learning addition.

BC: How important is the ability to visualize numbers in a spatial sequence, that is, a number line, for learning number facts?

BB: People like Dehaene, in Paris, have been enthusiastic proponents of the idea that we have a mental number line onto which we map numerical expressions. However, most people, if you ask them, say that we don't have a mental number line, at least not one that we're aware of. Maybe 10 or 15% of people do. It's not clear how important this number line is for carrying out calculations. Even for people who have a number line... my partner, for example, has a number line, one in fact that's been written up. She says she uses it for calculations, but if you ask her how she does it, she, like everybody else in this area, is just a bit unclear about how they use it. So it's not clear, even for people who have conscious number lines, how they're used.

So that's one perspective. The other is that we did see somebody who had the wrong number line, and this caused her enormous problems in life. This was somebody called Cathy, who we described in the book. She had a very weird number line, it was one where she went 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 - that was the first bit of the number line - then the next bit was 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. And then it was 20, 21, 22, and so on. It meant that 10 was represented twice on the number lines, 20 was represented twice. And in fact within the tens, she had 1, 2, 3, 4, 5 for the first half of the tens, and then 5, 6, 7, 8, 9, 10 for the second half. So nothing worked, with that number line, because she had too many numbers in it. This caused her problems, but the question that I asked is, "How could she live for 25 years with a number line like that without realizing that it was wrong?"

She had a profound disorder, which meant that for some reason or other she'd got the wrong number line, and she couldn't get out of it. We taught her the right number line, and it may have helped, but she kind of disappeared before we had a chance to follow that up. So I think number lines are interesting, but they're not terribly important. The only piece of really serious evidence that they are important is that there was one or maybe two cases now where somebody who had a conscious number line had some brain damage, lost the number line, and also lost the ability to calculate. So they may be causally connected.

BC:Some studies in recent years have suggested a connection between music and math, in particular that instruments like the piano and the violin, which emphasize linearly arranged numerical relationships, help with math. Does that correlation relate to the concept of the number line?

BB: Ah, if only it were that simple! We did a study in which we compared students at University College, so some of them would be students of the science, with matched students at the Royal College of Music. What we found was that the students at the Royal College of Music were actually better at calculation than our students. So that's one mark for the music-maths theory. On the other hand, Einstein was meant to not be a very good violinist. Somebody came up to me after a lecture last week and said, "All my family are very good musicians, and we're all terrible at maths." I said, "Right, come into the lab please," my standard response. So I'm not sure that there's a very close connection between the two. It's not clear that the brain areas involved are closely related, either. I think that there's more work to be done on this. It may be the case that learning something which is quite mathematical, like Western music, does actually help build math skills. I wouldn't want to deny that.

BC: Evidence from patients with Gerstmann's syndrome suggests that there is a connection between awareness of individual fingers and math. Do you think that increasing finger awareness can help mathematical ability?

BB: I think that's a good hypothesis. The evidence is rather different, though. It may suggest that you need to reconceptualize this a bit. We know that if you use your fingers a lot for some particular task, you increase the brain representation of those fingers. But that seems to be rather specialized, that is, if you use your fingers for music, or for reading Braille, what you get is an extended representation of the fingers for music or for Braille. It's not clear that you get general improvement of finger representation in the brain.

It's not been tested, so your hypothesis is still in there with a fighting chance. Here are some things that would bear upon it. What about people unfortunate enough to have been born, whose mothers took thalidomide, and whose hands and fingers are very strangely arranged, possibly not on the ends of arms but sort of growing out of their shoulders. So they can't see them, and they would have a different sense of them. Do these people have problems with mathematics? We don't know. What about blind kids, who can't see their fingers? They have a different kind of neural representation of their fingers, one that is proprioceptive, not visual. How does that affect the acquisition of numerical abilities? No one knows.

BC: I wanted to ask you a bit more about your own experience. What was your mathematical education like, and how did you come to study the neuroscience of math?

BB: I had a very poor maths education. I did maths up to sixteen at school, as everybody does. I was not very well taught, and I gave it up after sixteen. I kind of picked it up again, in a strange way, as a philosophy student, when I became interested in the foundations of mathematics. I know quite a lot about the foundations of mathematics, but I don't know a great deal about mathematics! I spent a lot of time reading Gödel, and Russell, and Hilbert, and people like that. I often thought about how foundational issues in mathematics might relate to psychological issues. What really transformed idle speculation into a research program was seeing people in the clinic who seemed to have specialized numerical disabilities. Their language was ok, their memory was all right, but they could no longer do the kinds of calculations they used to be able to do. They found this very handicapping. I started to think that maybe I should do a bit of work in this area, because nobody at the time seemed to be doing very much.

BC: Did you have any mentors as you were coming into the field?

BB: The only one, I suppose, was Elizabeth Warrington at the National Hospital for Neurology here in London, who did what was really the first modern neuropsychological study of numerical abilities. Again, it was really just one study to begin with. She had a patient who had just lost the ability to remember number facts. She wrote that up, and everybody thought, hello, maybe numerical abilities really are rather specialized, and that they have a special neural representation. And maybe within that there is a structure, it's not all just thrown together. So there are facts here, and there are procedures like carrying and borrowing and using counting over here, and maybe this can go, but not that. That got me thinking, and then I started to see patients myself. I got a very good student that I was working with, and we continued to look at our patient. We actually ended up working with Warrington at the National, and we wrote up a few papers there. Then I started to see patients who had developmental problems, and I got interested in that. Now I'm seeing patients who have genetic problems, because I'm now interested in the genetics of numerical ability. Are there genes which code for the building of the specialized brain circuits in the parietal lobes? We're looking at people with genetic disorders which seem to affect that bit of brain and that function.

BC: Have you had any reaction to your work from teachers of mathematics?

BB: Yes, enormous reaction. They find that their eyes have been opened by this book. My kids teachers have all started to buy the book.

BC: Really! How old are your children?

BB: Well, Amy is sixteen tomorrow, and Anna is thirteen next month. And their teachers have begun to get interested in it. Also, educational psychologists have tests for diagnosing dyslexia and ADHD and all kinds of other problems, but not for numerical problems. They don't know what to do about it. I've been talking to a lot of educational psychology groups as well as the teachers' groups, because these people are worried about the children under their care, that they are not doing very well, and they want to understand why these kids are not doing well, and what they can do to help.

BC: Have teachers found that once they start viewing this as a biological issue they're able to make changes to the educational program?

BB: I think it's early to say. They can make changes, but it's not as simple as that. In England we have a national curriculum, which means that teachers aren't free to teach whatever they like however they like. They have to follow government prescriptions to some extent. It's a question of trying to persuade the government, really, and the education department, to take this seriously. I spend more time than I like to try and persuade the government that they ought to be taking this problem as seriously as they take the problem of reading. Last year, they started a new literacy program. It so happens that there is a well-known writer here called Ken Follett who writes best selling thrillers (quite good, responsibly written best-sellers), and he's also quite closely associated with the Labor Party, his wife's a member of Parliament. Well, he dropped a line to the minister saying that he ought to do something about dyslexia, because he's president of the dyslexia group as well as being a writer. They actually have changed the literacy program in response to this kind of representation. Now, I obviously don't have the clout of Mr. Follett. I've been trying to persuade the government to modify its numeracy program, which it started this year, to take into account that there are kids who are going to need special help. They haven't done very much yet, but I'm going to keep on phoning them up and writing them letters.

BC: I wanted to appeal to your philosophy background for a moment. If the brain has an innate capacity for mathematics, does this speak to the issue of whether math is invented or discovered in nature?

BB: It does a bit, yes. I don't think that it means that math is invented, at least not just invented. It does mean that the material world indeed has numerical properties. It does mean that there are numbers in nature, and that we've evolved a capacity to identify numbers in nature and respond appropriately to them. But it doesn't mean that the numbers are actually there. What we're looking at are not properties of objects, but properties of sets of objects. It's slightly more abstract than to say "the greenness of my apple," and "the solidity of my mug." It doesn't have that kind of physical invariance, because you can have three of anything. But it is a detectable property of sets.

BC: You've interacted with a number of remarkable patients in the course of your work, people whose sense of numbers is so profoundly different from what most of us experience that it's hard to imagine what their thoughts are like. In the case developmental dyscalculics like Charles, will his condition ever improve?

BB: No. No, this is one of the things that's really striking, what makes it different from dyslexia and more like colorblindness. Dyslexics can become good readers. They'll never stop being dyslexic, the genotype doesn't change, but they can become efficient readers. But people who've got dyscalculia never become good calculators. What they have to do, and what Charles and other people like him have done, is they've found ways around it. They've found strategies for coping in real life, like Charles always carries a calculator around with him. That's perfectly sensible, and is absolutely the right thing to do. But he's tried to improve, we've tried to help him. But it doesn't really... it's just like trying to help a colorblind person see color.

BC: One of the things that was most striking about your book was that a lot of people think that they're sort of bad at math, that math is difficult for them. Your patients illustrate that numbers are so much more a part of our lives than we think they are.

BB: Yeah, absolutely. Where would we be without them!


Interview by Ashish Ranpura



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Britannica & Australasian Science: October 1999

A large company of chimps travels through the forest, headed by their fearless leader Brutus. To increase their chance of finding food they break up into several bands, but keep in touch by "pant-hooting" and drumming on resonant trees.

When it is Brutus doing the drumming, however, the other chimps treat the number of drum beats as instructions. One beat means "change direction." Two beats means "rest"--always for between 55 and 65 minutes. A beat on one tree and two on another combines these, instructing the chimps to change direction and then rest. Most remarkably of all, four beats instructs the other bands to rest for two hours.

These observations were made on animals in the wild, not ones that had been patiently trained by humans. They appear to provide proof that some animals are able to count and measure time. The evidence is particularly remarkable when one considers that counting is not believed to be automatic in humans, but has been passed down the generations from some "ancient Einstein." This observation, recorded by the zoologist Christophe Boesch, is one of the many extraordinary anecdotes and facts used by cognitive neuropsychologist Brian Butterworth, of University College London, to support his theory of a "mathematical brain." According to the theory almost all humans, and many animals, have an in-built ability to do mathematics.

Butterworth, promoting his book The Mathematical Brain , cites evidence from diverse fields to support his theory, such as the body counting systems used in the New Guinea highlands. Here each number is associated with a body part, so that the word for "three" might be the same word used to denote the middle finger. The attachments vary across the thousands of different communities in the area.

Among the Yupno the little finger of the left hand is one, the right big toe is 20 and the left ear is 21. When only one man is present--women are forbidden to count in public--numbers stop at 33 for the penis. With more men present it is possible to go to higher numbers.

Isolated communities in New Guinea have similar solutions to keep track of possessions, but it seems that most, if not all, have invented this for themselves rather than copied from their neighbours. Butterworth concludes this on the basis of the diversity of counting patterns--some go left hand first and then right hand, while others cover the whole left side before crossing to the right. Some can go as high as 68 by including extra body parts. There does not seem to be a geographical pattern to this as one would expect if trade had enabled counting to diffuse outwards from the community that invented it.

Numerical skills appear to be concentrated in a particular section of the brain, the inferior lobule of the left parietal lobe. Within this it seems that different subsections control distinct mathematical abilities. Patients who have suffered strokes or other forms of brain damage can lose the ability to tell which number is larger, or subtract or multiply. Most remarkably, some patients lose one ability while others are largely unaffected, suggesting that different parts of the brain have been compromised.

Adding Cells

Butterworth believes that almost everyone is capable of being good at mathematics if they are taught well. He believes ability at math, like many other skills, fundamentally comes down to practice. For example, Braille readers develop far more brain cells in the areas associated with messages from their "reading" finger than those who do not read Braille.

In other words, using the sections of the brain that are good at math causes them to expand, and therefore get better at mathematical processing. Thus we should not be surprised that Albert Einstein's left parietal lobe hosted an unusually high concentration of brain cells.

Butterworth admits that practice doesn't always make perfect when it comes to math. For some, a defect in the relevant area of the brain means that no matter how much they practice they will never catch up with the rest of the community. However, Butterworth believes that this group accounts for only 4 percent of the population, far fewer than those who consider themselves to be inherently bad with numbers.

If 96 percent of the population is born with the same mathematical potential, why do some reach genius level while others struggle to calculate the change on a simple transaction? According to Butterworth something, usually early in life, stimulates some people to think about math more. They solve mathematical puzzles in their heads and the relevant brain areas expand. Once these people reach school age they find they are good at math, and consequently enjoy it. This leads them to do more, and they become even more gifted.

In an interview with Australasian Science Butterworth argued that what mattered was not the time spent studying for the exam but the total time spent on mathematical thoughts. Thus those who sail through their math education with little study might be spending much of their free time solving mathematical puzzles plucked from observations.

Good mathematicians, Butterworth argues, love math and do it at every opportunity. In interview Butterworth retreated slightly from his claim that mathematical ability is entirely a function of practice. He admitted it was possible that some people are born with brains better suited to this area. However, he stressed: "Practice is vital. There is currently no evidence that anything else matters." He remains uncertain about why some people gain a desire to practice math more, but describes this as "individuation, not ability." The implications of Butterworth's theory are profound. Even if we reject the argument that ability is entirely based on exercise he makes a powerful case that activity is more important than variations in natural talent. If he is right we need to radically rethink our school curricula.

If engaging in mathematical puzzles and games is an effective way to increase ability then it stands to reason that making math fun should be the priority for teachers, at least in the early years. Yet Butterworth documents repeated attempts by the previous conservative government in Britain to enforce more rote learning and less activity likely to stimulate children towards engaging in math out of hours.

What is more, Butterworth alleges that nowhere in the reports recommending more drill is there the slightest evidence that this works, and that a major report contained just one sentence on the importance of enabling students to understand what they are doing. He attributes this to a slavish subservience to the prejudices of politicians. In interview he noted that many of the English educators setting policy in this area were actually telling the politicians what they wanted to hear, but doing something different.

Butterworth also notes that an inquiry into math education was specifically banned from allowing the use of calculators in primary school. While he maintains an open mind on whether early exposure to calculators is beneficial he would welcome some research on this topic, rather than an order based on "common sense."

Butterworth contrasts this approach with the way Chinese students are taught multiplication. Instead of having to learn a complete table up to 9 x 9 it is explained that 3 x 5 is the same as 5 x 3. Consequently their five-times table starts at 5 x 5 = 25. This cuts the memory load from 81 facts to 36 (the one-times table is also deleted), and also creates the perception that multiplication is about logic rather than abstract facts. Butterworth believes this contributes to the fact that 12-year-old children in Shanghai perform better at math than 17-year-olds in America.

Just how ingrained the rote learning system is can be observed from Butterworth's point that he had to learn tables up to 12 x 12 because he went to school before decimal currency. Yet long after shillings and pence were abolished in Australia children in many schools are still forced to recite tables up to 12.

Getting Math Right

While Butterworth believes that some mathematical skills are ingrained within us, getting to more complex concepts is much harder. For example, our number system uses 10 as a base, with all larger numbers defined in relation to that base. Thus 128 is 1 x10 x10 + 2 x10 + 8. A base-5 system would write this as 1003: 1 x 5 x 5 x 5 + 3 (the last number represents the units, the second last represents the number of fives, the next represents the number of 25s, etc.). In theory any number other than 1 can be used as a base.

Westerners tend to think of a base-10 number system as "natural", but many other cultures have come up with a wide range of bases, most famously the Sumerian base 60, from which we get our hours. Furthermore, many of the cultures Butterworth deals with use a mixed base-5 and base-20 system, skipping over 10 as just another number. Then there is the Base-12 Society, a group dedicated to the profoundly fruitless task of converting the entire world to a number system that, having more factors, should be easier to learn.

Base-10 is not entirely unchallenged within Western culture. As Butterworth points out, we say "eleven", not "tenty-one", and the French and Danish are heavily influenced by base-20 counting. More profoundly the concept of zero, so obvious to those brought up with it, was totally alien to most ancient cultures, even those that developed quite advanced mathematics. The difficulties Archimedes must have gone through to calculate the number of grains of sand in the universe without the use of a zero for place counting are as staggering as the numbers he came up with. The power of numbers has been recognised by some authorities who have attempted to prevent further development. In 1299 the Guild of Florentine Bankers forbade the use Arabic numerals, demanding instead that numbers be written out using letters.

Yet despite all the obstacles in its path, the study of mathematics has flourished.

Stephen Luntz

©, Australasian Science, 1999.



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Our second book is by Brian Butterworth who's in Melbourne this week from Britain. I'll tell you his title in a minute. At the moment he's working on where numbers go in the brain and it really matters.

Brian Butterworth: Well, some people like, for example, Piaget, argued that really mathematics was no more than an extension of logic and the mathematician Keith Devlin in a recent book called the "Maths Gene", has argued that maths is nothing more than an extension of language. Now modern scientific approaches to how the brain deals with numbers and other aspects of mathematics show that really there are separate parts of the brain that deal with maths on the one hand, deal with reasoning on the other deal and deal with language on the third hand.

So there is evidence that there is independence in the brain. Of course it doesn't mean that there's functional independence. Clearly you learn most maths through language. But once you've learnt it, does it get stored with other things that you've learnt through language or does it get stored somewhere else? One of the things that we've been working on - and we work on these things entirely opportunistically, it depends who comes into the clinic, is about whether reading words, reading numbers and recently, reading music all use the same brain circuits or whether they use separate ones.

There's a very interesting novel by Ken Follett called the "Hammer of Eden" in which one of the central characters, I'd better not say whether he's a hero or a villain, is unable to read either letters or numbers but actually truth, as is so often the case, is stranger than fiction. We've found that there are people who can read numbers but can't read words, and my colleague Lisa Cipolotti here at the National Hospital for Neurology, has found that there are people who can read words much better than they can read numbers. I mean, they've had brain damage which has affected their number reading but left their word reading intact.

Now recently with some colleagues at St. Thomas's Hospital, we examined a patient called Mrs C. Now Mrs C was a professional musician and a professional musical composer, and she had some brain disease which affected not the language part of the brain and not what we believe to be the number part of the brain, but a part of the brain that we haven't really paid much attention to before, in the right hemisphere. As a consequence of this she was left with a very, very specific deficit. Although she could still play music, although she could still sing, she could no longer sight-read music; something that she was able to do fluently for over 30 years, nearly 40 years in fact. Now she can't sight-read music at all. In fact she can't read musical notes at all. Nevertheless she can still read numerals, she can still read words perfectly well, and this shows that very soon after the visual information enters the brain it gets split into separate channels. There's a channel for words, there's a channel for numerals and there's a channel for musical notation as well.

Robyn Williams: But is it just a matter of the numbers, recognising the numbers or is it this ability for computation? In other words, putting them together Is it all in one bag?

Brian Butterworth: Well we actually have found a disassociation here. In a subject I called in my book Charles. He was able perfectly well to recognise 17 and he could read aloud the number 17 but he had no idea what it meant; he couldn't do computation with it. So deeper into the brain than, if you like, the recognition centre, there's comprehension and calculation and that's clearly something that goes beyond the reading and comprehension.

Actually, we found another patient who was rather poor at reading numerals but was very good at calculating with them. So if you asked what is 17 plus 35 he would give you the correct answer, which is 52, I think. But if you asked him to read aloud the numerals 17 plus 35 he would often make mistakes in doing that. So there seems to be something rather separate about the calculation.

Robyn Williams: Yes. What are the implications of this independence of mathematical ability?

Brian Butterworth: Well, one of the things that's coming up now as an important social issue is people's numeracy. Certainly our government and the government of the United States have been worried that the numeracy of our citizens is really much lower than our economic competitors and we wonder what we should do about that. Now one of the questions that comes up is can we treat mathematics and arithmetic as being separate from other school subjects. If it's separate in the brain what are the education implications of this? Now that's not completely clear yet, but one thing is becoming clear - and this is something that I have to say neither government recognises - and that is that there are people who are born without a properly developed number centre in the brain and these people are always going to be very bad at calculation. They might be quite good at other aspects of maths but at calculation and understanding numbers they'll always be awful.

They are in a similar position to dyslexics twenty or thirty years ago. It's not a condition that's recognised, no one really knows how to treat it. We're beginning to assemble guide lines for best practice here and it's not absolutely clear how you diagnose it, but I get frantic parents ringing me up and saying, "Look, my kid does very well at school in geography, history, English. He's not dyslexic but he's just hopeless at maths. Is it bad teaching? Can I get him extra tuition, will that help?" So I say, "Well bring him in and we'll give him some tests."

And often we find that these kids are very poor at stuff to do with numbers that really doesn't depend upon education at all. Like, for example, counting dots or saying which of two numbers is larger; is 6 larger than 2? These kids have trouble with that in a way that's very striking. As I say, it's not absolutely clear how to help these children but it is becoming clear how to diagnose them and that's by using these very simple, non-education-dependant tests like dot counting or number comparison.

Robyn Williams: I don't suppose the same thing would happen with music comprehension because music is often considered to be an add-on. You won't have someone brought to you - "Look I'm afraid my child can't play the violin to concert performance or sight-read Bach?"

Brian Butterworth: That's absolutely true. It's not regarded as a great national problem that we're not very musical. But actually not being able to multiply 8 x 7, which our current First Secretary of the Treasury was unable to do in public, isn't regarded as a great social problem, whereas saying "we was" instead of saying "we were" still is regarded as a great social problem.

Robyn Williams: Going to the brain itself, I can actually work out how words might be represented in the brain but how on earth are the numbers?

Brian Butterworth: Well, there's a big debate going on at the moment. On the one hand there are people who think that the way in which we represent the meaning of numbers and what 5 stands for really isn't specifically numerical. Their idea is that we have a kind of quantity sense, so 5 is kind of approximately this much; it's a bit more than 4 and a bit less than 6 but it's all rather approximate. It's as though you have a line in your head and 5 is approximately here and 4 is approximately there on this line. Now I take a different view. I think that we're born with a sense of a collection of things and a number for that collection so that a collection with an extra member has a very definite precise extra representation.

At the moment we are going through some experiments trying to see whether I'm right or whether most of the rest of the scientists in this area are right. They've actually got some rather good evidence, but we've got some experiments that we're doing now which I think may well tip the balance in our favour, but I can't tell you about that at the moment because we haven't finished the analysis.

Robyn Williams: Well I won't ask you about it if you haven't finished the analysis, but what I will ask you about are those animals which, I think, dogs can count fairly comfortably up to 4, specially mine which are border collies which are the most intelligent and can go even further. But parrots can do calculation and not simply say, well there are two things over there but two entities we've brought together, you know two apples and one pear, they can say, the parrot or one or two parrots can say, This is 3, as if animals indeed have a lower level of cognition evolution have got this wired in as well.

Brian Butterworth: Well, that's certainly my belief which is that our specific number sense is something that we've inherited from animals. We know that animals can do quite a lot of number tasks, some of which they have to be taught and some of which seem to come rather naturally. They can be taught for example to pick the larger of two numerical arrays, so they can pick, if you want three things and four things they can pick four. If you're a very smart animal like a chimp you can be trained to understand what the numeral 3 means and what the numeral 4 means and then pick the numeral 4 over the numeral 3. They can do some extremely primitive calculation - extremely primitive.

The problem is, if you want to say that these are ancestral abilities to our own, is to show that animals use homologous brain areas to our numerical brain and so far nobody knows which bit of the brain animals use to do these tasks. So until we know that, we won't know whether it's just in general, kind of handy to make numerical distinctions or whether actually some ancestral creature got this idea and then the rest of us inherited it.

Robyn Williams: Why is it important to know?

Brian Butterworth: Well, we want to know whether our genome contains information to enables it to build brain areas that are specific to mathematics. If it does then that will explain why some people are born without this ability. Also it will help us actually to understand some archaeology. There are lots of interesting arrays of dots in Palaeolithic caves in Europe particularly and no one knows what they mean. My guess is that our Palaeolithic ancestors 20,000 years ago in the depths of the Ice Age were counting things. We don't know what they were counting, they might have been counting bison they might have been just counting the number of people who turned up for a party, but I leave that questions to the archaeologists.

Robyn Williams: Bet they were doing deals.

Brian Butterworth: I wouldn't be surprised.

Robyn Williams: And neither would I.

Brian Butterworth is Professor of Cognitive Neuropsychology at University College, London. His book is called "The Mathematical Brain" and it's published by Macmillan.





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The strange case of the former bank manager who is now able to read only number words as the result of a neurodegenerative condition.


Brian Butterworth: What's fascinating about him is that he can only read one sort of word. He can only read number words and it's never been reported before and that's why we're so interested in it. But it's theoretically very important because the standard theory of reading goes something like this: that when you see a string of letters you do two things simultaneously; you try and find the sound of that string of letters. You might do that by taking each letter and finding the sound of each letter and putting them together. You might try and recognise it as a whole word and try and find the sound of the whole word but, at the same time, you're trying to interpret the meaning of that string of letters, you're trying to find the meaning of that word and these two processes converge to give you the pronunciation of the word.

Now the problem with the theory up to now is that it's never really been tested properly. We don't know whether you can pronounce the word accurately just going via the meaning. So for example, if you see a letter string like t-r-e-e- you get the meaning of a tree and then you have to say what that meaning is. Now what we want to know is: is that route via meaning sufficient to give you just "tree", and we haven't been able to do that up to now.

This case is very interesting because this man can't use the route via sound alone. We know this because if you give him a very simple string of letters he hasn't seen before, like y-i-t- he can't say it. He used to be able to say it but he can't say it anymore. Even if the letter string sounds like a real word, like y-o-t- he still can't say it. Now the other really interesting thing is that he can't say words that he doesn't understand. So he can't say y-a-c-h-t says yacht either, he can only say words that he does understand and if he does understand them then he says them accurately. And we know that he has a very precise understanding of numerical concepts because he can calculate accurately. So he knows the difference between 1 and 2, and 3 and 4, and 13 and 15 and he can read all these words with perfect accuracy, but words he doesn't understand, and these are very common words like "take" and "give", and yet he can't read because he doesn't understand them.

The reason why he doesn't understand them is because this unfortunate neurodegenerative condition has affected the parts of the brain that deal with language and other forms of general knowledge, which is in his case in the left temporal lobe, but has spared the left parietal lobe which is where numerical ideas are.

Robyn Williams: It's very interesting you gave the example of "tree" just then. It's very close to "three' and I would have thought that one little letter might be something that could help him on his way. Are you telling me that he can read "three" but he can't read "tree"?

Brian Butterworth: That's exactly what I'm saying. Now all the previous cases of reading problems due to brain damage have been complicated by the fact that the patient either has very good word meanings in general, in which case we've now shown that this route via meanings is sufficient for reading words, or they've got some of the letter to sound route still intact, and so although they might not really understand "tree" they can maybe sound it out a bit and then get to the pronunciation. But this man unfortunately can't get any of the letter sound relationship. He can say number words OK and he can say a few other sorts of words reasonably well, but not very many. So his language is very affected too.

Robyn Williams: Now Brian, this is clearly significant because it's in the journal "Nature" this month, "Neuroscience", and you're a distinguished researcher in this field, but for innocent bystanders like me, if you're looking at one case of a person who's got a very unusual impairment how do you know that this is a window on the general brain?

Brian Butterworth: There are two answers to this. First of all, it would be impossible to do neuropsychology if you couldn't make generalisations of single cases; therefore you must be able to make generalisations of single cases. The other, perhaps more sensible account goes something like this: that we have no reason to believe that I H's brain was any different from yours or mine prior to his illness and so his brain is likely to be organised at least for things that are rather basic in a similar way to yours and mine. So we can get a clue as to the organisation of what we would call higher mental functions like calculation and language from these single cases, and it's actually proved very effective as a way of investigating brain functions since the time of Brocca in the 1860's. We can confirm some of these single case findings by looking at functional brain imaging, seeing what the brain is doing while it's doing it to see if the areas that we've identified in these single cases are the areas that seem to be important for a particular function.

So for example, I believe in the case if I H that the temporal lobes are involved in language and that the parietal lobes are involved in number. We've done experiments and other people have done experiments using functional imaging of normal people, people who do speak and calculate normally. And we find that the temporal lobes are very active during speech production and speech comprehension and the parietal lobes are active during calculation, so we do have independent grounds for thinking that the brain of I H is not at all unrepresentative of humanity in general. I mean, one point that I should make is that before his illness I H was a bank manager. Now we assume that bank managers have normally organised brains.

Robyn Williams: Sometimes too organised.

Brian Butterworth: Sometimes. He was also and indeed still is an addicted gambler, so he's obviously had a lot of experience with numbers. So it may be that his number areas are larger and better developed than mine are for example. However, one of the consequences of his illness is that he's had to change his preferred mode of gambling. It used to be horses, but in order to bet on horses you have to be able to read the name of the horse. On the other hand, if he goes to the dogs there are only six dogs in every race and you usually identify each dog but its number. So you can go up and say $10 on No.3 and that he does perfectly well and he still continues to lose money in the normal way.

Robyn Williams: Poor man. However, here is a very interesting example just to recap. He knows numbers, therefore he is able to read something he understands and therefore he has this facility, but he can't read non-number words. What does this tell you about how we might approach reading in the first place? What does this tell you that might be of assistance to treating people who've got reading difficulties at all?

Brian Butterworth: As I said, the theory says and the theory now seems better supported on the basis of this case, that when we see a letter string we approach it in two separate ways; at least normal people are trying to find the sounds of the individual letters and trying to find the meaning of the letter string. Now some people, either through brain injury or through some genetic abnormality known as dyslexia, find it difficult to read by one or other system. And I think that one of the implications of this is that you try and identify which system it is that's affected in the individual and then try and exploit the intact system.

Robyn Williams: And so you've got a test, have you now?

Brian Butterworth: We've got tests and these tests have been available for quite a long time. It's just that you very rarely get cases as clear as this, so this tells us that we've been on the right track.

Robyn Williams: And what in the practical sense - how do you apply this; having done the diagnosis on say a dyslexic patient, how do you actually say right, now try this?

Brian Butterworth: Well, what you do is you try and encourage the patient to, instead of sounding out letters if he's the type of patient that I H is, to try and sound out the whole word. And you might give them meaningful hints as to what those words are. So if it's "tree" you might say, well think of what you see in the country.

Robyn Williams: Cows!

Brian Butterworth: Right.

Robyn Williams: Except in England, there's not many left.

Brian Butterworth: Yes, that's not a surefire way of doing it, but certainly most children with dyslexia are going to have intact semantics so they can use this route via meaning if that's the route that's preserved or that's developed reasonably normally in their case. But the case of I H is interesting from another point of view, which is that we've been interested for quite a long time in how language and numerical abilities are organised in the brain and it's these very clear cases that can somehow tell you about that. For example, we reported patients who have got very good language but very impaired numerical abilities. So there was a patient that we reported some years ago who, if you spoke to her, you wouldn't know there was anything wrong with her, but because of her brain injury she wasn't able to do calculations anymore. She couldn't in fact deal with numbers above 4 at all. Before her stroke she'd had care of the books of the hotel that her family ran, so she was pretty good a numbers prior to her injury.

Now people might say, well of course, if you've got brain damage what's going to go first; Is it the stuff that you found hard? And so we find language easy because we use it all the time but numbers are a bit more difficult because you have to learn them at school. Cases like I H say, well that can't be quite right, because here we've got somebody whose language is gone, language of course which he's used everyday, in his dreams even, but his numbers - he's very practised with numbers but nevertheless he's not using numbers as much as language. So numbers are still going to be, on those grounds, slightly more difficult than language, yet it's his numerical abilities that are preserved.

And so we now have evidence what neuropsychologists like to call a double dissociation between language and number. One type of patient has good language and terrible numbers and this other type of patient, I H is the best reported example so far, has got terrible language but very good numbers. So we can see that in the brain these two processes are independent. And in fact, we can look to the evolution of these two processes and we can say, well maybe they have evolved independently as well, because we know that infants, even in the first week of life, are sensitive to the number of things that they see. So if you show a baby of one week two objects, another two objects, another two objects, another two objects, the baby loses interest. But then when you show the baby three objects the baby suddenly starts to look.

Robyn Williams: Numbers, numbers.

Brian Butterworth: Numbers, numbers, numbers, that's right. So you're born with this numerical ability and it seems that the brain circuits that you're born with are in the parietal lobes and in the case of I H those seem to have survived whereas unfortunately his neurodegenerative condition has affected his language very severely.




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Neuropsychologist Brian Butterworth describes the brain's innate ability to process numbers and explains why some students, nevertheless, have trouble understanding mathematics.


Marcia D'Arcangelo: One of the most fascinating ideas in your book "What Counts: How Every Brain Is Hardwired for Math" (Free Press, 1999) is that we are born with a sense of numbers. What exactly is this number sense?

Brian Butterworth: The number sense is having a sense of the manyness, or numerosity, of a collection of things. We believe that babies are born with a kind of start-up kit for learning about numbers that is coded in the genome. Even in the first week of life, babies are sensitive to changes in the number of things that they're looking at, and at six months they can do very simple addition and subtraction. Then, with this start-up kit, they build all the cultural tools - the number words, the counting practices, and the arithmetical procedures and facts that they learn from parents and from school.

Marcia D'Arcangelo: How do we know that babies can add and subtract?

Brian Butterworth: If you show a baby a doll, cover it with a screen, and show a second doll being placed behind the screen, the baby will expect there to be two dolls when the screen is removed. If there is a different number of dolls - more or fewer - then the baby will look longer than if there are two. This "violation of expectation" experiment was carried out by Karen Wynn in 1992 at the University of Arizona.

Marcia D'Arcangelo: Your book suggests that even prehistoric man had a mathematical brain, as you call it.

Brian Butterworth: Yes, and other species may have different versions of mathematical brains as well. Chimps, for example, can learn to do sophisticated numerical tasks. Ten years ago, David Washburn showed that chimpanzees trained to understand numerals - 1, 2, 3 - can also be trained to select the larger of two numerals presented to them. More recently, researchers trained monkeys to select the larger of two arrays of objects. Perhaps the best example of wild animals using numbers is the Serengeti lions. Lions defend their territories against intruders, but they will attack only if they outnumber the intruders. They have to figure out how many lions versus intruders there are.

Marcia D'Arcangelo: Are you saying that the number sense may be a natural skill linked to survival for animals and related to survival for humans - even in today's society?

Brian Butterworth: In many ways, yes. For example, a study by the Basic Skills Agency here in Great Britain looked at the effects of poor numeracy and poor literacy on getting, keeping, and being promoted in a job. The research showed that poor literacy is something of a handicap - but poor numeracy is even more of a handicap. Although the "why" wasn't investigated, one can imagine that mistakes with numbers lead to financial errors. In that sense, numerical skills are certainly an important survival skill in a numerate society.

Marcia D'Arcangelo: Even so, people aren't afraid to say that they're not good at math, yet they wouldn't think of admitting that they couldn't read.

Brian Butterworth: Well, it may not be true in the United States, but in England, literacy and orality are class markers. For example, children at school are taught to say "we were" rather than "we was" because "we was" is the dialect of working-class people, and "we were" is the dialect of the rulers of the country. It's important to try and top the linguistic manners of your socalled betters. This class distinction, however, doesn't apply quite so much to arithmetic. People are more prepared to say "my arithmetic isn't good" rather than say "my grammar isn't good" or "my spelling isn't good."

Marcia D'Arcangelo: Does your research reveal any gender differences with respect to mathematical ability?

Brian Butterworth: We find that women are far more willing to say that they are not very good at math than men are. Our research shows, however, that there is no difference on average in the public performance and examination of women and men in this country. The Third International Mathematics and Science Survey showed very similar mean performances by boys and girls on mathematics tests by 8 - 9 and 13 - 14-year-old schoolchildren. Actually, in the last set of results, girls scored slightly better than boys did.

Marcia D'Arcangelo: Perhaps many people are better at math than they think they are. But if our brains are hardwired for math, why do so many students have such difficulty with it?

Brian Butterworth: Not being good at mathematics can have two main causes. The first is genetic. A minority of people may be born with a condition that makes it difficult for them to learn mathematics; that is, they are born with dyscalculia or are born with dyslexia, which also can have a consequence for mathematics learning. A far more likely cause is that they were taught badly. That means taught in a way that left them failing to understand what they were doing. Thus, everything else that they learned that was based upon what they didn't understand was going to be very fragile. So, they avoided mathematics.

Marcia D'Arcangelo: You mentioned dyslexia. Do we use the same parts of the brain that we use for learning to read to learn mathematics?

Brian Butterworth: The parts of the brain that process words are different from the parts of the brain that process numbers. We store words in two areas, Wernicke's area in the left temporal lobe, at least in most right-handers; and Broca's area, in the left frontal lobe. Numbers are stored in the parietal lobe - not that far away, but far enough to be a separate system. No part of the brain is specialized at birth for reading because reading is a very recent skill for which the brain adapts the language areas. The brain, however, does seem to have evolved special circuits for numbers. There's an important difference between those two types of learning. Mathematics is built on a specific innate basis, and reading is not. It's quite important for teachers to remember that when children are learning mathematics, they are using distinctly different brain areas than they use when learning to read.

Marcia D'Arcangelo: What is dyscalculia?

Brian Butterworth: Dyscalculia is a condition a child is born with that affects the ability to acquire the usual arithmetical skills. Dyscalculic students may show difficulty understanding even simple number concepts and, as a consequence, will have problems learning the standard number facts and procedures. Even when dyscalculic students can produce the correct answer or the correct method, they may do so mechanically and without confidence because they lack an intuitive grasp of numbers that the rest of us possess. Dyscalculia is rather like a dyslexia for numbers - but unlike dyslexia, little is currently known about its prevalence, causes, or treatment. Dyscalculia often appears in conjunction with other learning difficulties - including dyslexia, dyspraxia, and attention deficit disorders - but most dyscalculic students will have cognitive and language abilities in the normal range and may indeed excel in nonmathematical subjects.

Marcia D'Arcangelo: How is dyscalculia diagnosed? Are there tests for it?

Brian Butterworth: Generally, discrepancies between mathematics learning and other cognitive functions, such as reading or I.Q., are taken as diagnostic. We are currently looking at qualitative differences between dyscalculics and other children. By March 2002, we plan to have a fully standardized test battery for this.

Marcia D'Arcangelo: How might a classroom teacher recognize when a student might have this difficulty?

Brian Butterworth: Dyscalculic students seem to have an impaired sense of number size. This may affect tasks involving estimating numbers in a collection and comparing numbers. Dyscalculic students can usually learn the sequence of counting words but may have difficulty navigating back and forth, especially in 2s, 3s, or more. They may also find it especially difficult to translate between number words whose powers of 10 are expressed by new names, such as "ten," "hundred," or "thousand" and numerals whose powers of 10 are expressed by the same numerals but in terms of place value, such as 10, 100, and 1,000. These students may be competent at reading and writing numbers, though some dyscalculic students have problems with numbers over 1,000, even in 6th grade.

Marcia D'Arcangelo: How common among the student population is this condition?

Brian Butterworth: We are not really sure. Using discrepancy criteria, estimates vary between 3 percent and 6 percent.

Marcia D'Arcangelo: Are we as far along in understanding, diagnosing, and treating dyscalculia as we are with dyslexia?

Brian Butterworth: No. We are about 20 years behind in terms of research and, more particularly, in terms of recognition by parents, teachers, education authorities, and the learners themselves.

Marcia D'Arcangelo: If the brain uses different brain areas for reading and math, then why might dyslexics have difficulty with math as well as with reading?

Brian Butterworth: There are many different language systems and visual systems that need to be coordinated to become a skilled reader. We don't really know why dyslexics have a much higher rate of problems with arithmetic than do children in general. One possibility is that they have trouble mapping from symbols onto their meanings. On the other hand, you find many extremely severe dyslexics who have no trouble with mathematics at all.

Marcia D'Arcangelo: What do we now know about how the brain normally processes numbers?

Brian Butterworth: Research shows that we think about numbers as displayed in a line in our head, a kind of mental representation of numbers. Now, when you ask people if they have a number line, most are not conscious of it. Perhaps only 15 percent of people are conscious of having a number line. In most people, this number line seems to go from left to right.

We suspect that when we can map the parietal lobes with great precision, we will see that separate areas do the separate arithmetical operations - addition, subtraction, multiplication, and division. Each of these operations can be selectively affected by brain damage without the others being affected.

We seem to have separate circuits in the brain, probably all in the left parietal lobe, for facts of these separate operations. There are, of course, procedures that we use when we don't have these facts. For example, we don't learn subtraction facts in school. We typically have to work these out on our own - even quite simple ones like 9 - 3.

Marcia D'Arcangelo: Some would argue that we do learn subtraction facts.

French children certainly are taught to recite subtraction facts, as I learned multiplication facts. Of course, some facts are learned, but, in general, we have a different approach to subtraction and to division, by which a procedure turns these problems into additions and multiplications, respectively. These procedures depend on the facts of addition and multiplication. To solve the problem 9 - 3, you might say, "What do I have to add to 3 in order to get 9?" We suspect that the prefrontal cortex tells us what procedure to apply, and the parietal lobe, which stores the procedure, carries it out.

Marcia D'Arcangelo: Do math facts and procedures represent different kinds of memories?

Brian Butterworth: In general, yes. The brain seems to organize its memories in at least two distinct systems. One is called declarative, and the other is procedural. When we learn the capital of Texas, we store that fact in our declarative memory. When we learn how to ride a bike, that's procedural. Now, much of what we know in terms of procedures is implicit, non-conscious. Our declarative knowledge is explicit, or conscious. We can't explain how we ride a bike, at least not very satisfactorily. But we can say what the capital of Texas is.

Most of the facts we store in declarative memory are arbitrary. For example, there is nothing inevitable about Austin being the capital of Texas, but there is something rather inevitable about 3 + 4 being equal to 7. The facts that are stored in numerical declarative memory are systematically organized, with a clear rationale, whereas the facts of geography are much less organized.

Although many procedures are unconscious, such as playing a good backhand shot in tennis, many of the procedures we use in arithmetic are learned quite consciously. We've learned explicitly how to borrow or carry in multiplication or division and can often recall and explain these procedures explicitly.

Marcia D'Arcangelo: Are you saying that the brain stores numerical declarative and procedural information in different areas from those in which it stores nonnumerical declarative and procedural information?

Brian Butterworth: That seems to be the case. Much of declarative memory is stored in the temporal lobes, quite near where the language area is, and many of the procedures that you find in ordinary procedural memory are in the frontal lobes. Declarative memory for numbers, however, seems to be in the parietal lobe with other number concepts. The procedures seem to be there as well. When the parietal lobe is damaged, you not only potentially lose arithmetical facts, such as the sum of 4 + 3, but you also may lose a knowledge of arithmetical procedures. The history of mathematics has been in part the history of making new tools, such as calculus, to solve problems. There was a time when only Newton and a few others knew how to solve problems of moving bodies, but now most 18-year-olds can do it because they have the tools, the procedures.

Marcia D'Arcangelo: There seems to be a relationship between expressing what you know verbally and gaining a better understanding of it. Does brain research offer any insights into why this might be so?

Brian Butterworth: Trying to express something can help you understand it better. We know from a number of studies that language and numbers occupy different regions of the brain. It's nevertheless the case that some of what we know about numbers is stored linguistically. For example, if we learned our multiplication tables by rote, they lodged in the language part of our brain as a kind of poem or perhaps even a nonsense verse, depending on how well we understood them.

There is an important transmission component of learning about numbers through language. Some of our knowledge remains linguistic, but most of what we know about numbers goes to the parietal lobes and is stored in a numerical rather than a linguistic way. Trying to explain mathematical ideas does help us understand what we might know implicitly. For example, children of about 5-years-old know that 5 + 3 is the same as 3 + 5. They probably have never expressed this as a formal rule - that addition is commutative, or N + M = M + N. If you ask them to explain why it is that they think that those two problems are the same, however, they might be able to formulate the concept in such a way that it will help them fix in their mind that it doesn't matter in which order they perform additions.

Marcia D'Arcangelo: How might knowing about the brain's natural ability to process numbers help teachers teach math facts, procedures, and concepts?

Brian Butterworth: The fundamental principle that must guide the teaching of mathematics is that children have to understand what they're doing. The work has to be meaningful for them. Children come to school from different backgrounds and with different information about numbers. It is important for teachers to adjust the way that they teach to fit the skills that students already have. Otherwise, students are going to start getting left behind, even in the first year of school. Learning mathematics is a cumulative process, and if you fail to understand one stage, then anything that is built upon that stage is going to be rather fragile.

Marcia D'Arcangelo: Traditionally, schools have emphasized drill and practice for learning mathematics, assuming that understanding would come as a natural result of learning facts and skills. Is this a good practice?

Brian Butterworth: We don't know whether fluency with number facts actually leads to a better understanding of the number system, but we do know that good understanding makes you much better with number facts. Try to make sure that students understand what they're doing before you start drilling with number facts. Going over the same thing again and again gets information from short-term memory into long-term memory, but you have to rehearse reflectively. Recent studies of musicians suggest that what's really important in becoming a truly excellent musician is reflective practice. This means playing a Bach cantata over and over again, but not just in a rote way. The musician must rehearse in a reflective way, thinking about how the parts of that piece are connected together and what those parts and what the whole means.

Marcia D'Arcangelo: If the number sense is natural for us, why are some number concepts so difficult to understand?

Brian Butterworth: My view is that the kinds of numbers that are easy to learn about are those that correspond most closely to innate numerical concepts, the idea of a collection and its numerosity. Whole numbers are quite easy to learn about for just that reason. Now, you can't have negative collections. It doesn't make any sense. So you have to think about negative numbers in a rather different way. You have to think about moving along a number line, and that is complicated. Fractions, too, are hard because they don't fit easily into our natural concept of whole numbers.

Marcia D'Arcangelo: You've emphasized the importance of keeping mathematics meaningful. What can teachers do to promote understanding?

Brian Butterworth: The nature of mathematics is that you can come to the right answer in many different ways. Encouraging different approaches to the same problem helps students grasp this idea. Using a range of examples is also important because numbers are abstract. They don't apply to particular things. Anything can be counted. You can count eggs. You can count jumps. You can count things you can't even see, like wishes. Using numbers in a variety of contexts helps students understand the abstract nature of the number concept.

You can make mathematics meaningful by actively engaging students with numbers in different ways. Manipulatives, for example, make use of the innate number sense, the numerosity of collections. Many manipulatives are little collections that can be put together. This process helps students understand that operations on the collections have an effect on the numerosity of the union of those collections. That is the basis of addition.

Because mathematics is a cumulative subject, it is also essential for teachers and students to identify and correct misconceptions as soon as possible. This means spending quite a lot of time with each child. Certainly, in England, where we have class sizes of 20 to 30, this is asking rather a lot of the teacher, but it is absolutely vital to know what each child does and does not understand.

Marcia D'Arcangelo: Despite the best efforts of caring teachers, many students seem to develop a fear or dislike of mathematics. Why might this be so?

Brian Butterworth: When understanding breaks down in mathematics, students do feel as though they're swimming in a sea of incomprehension. Because this drowning sensation is so anxiety-provoking, they avoid situations that give rise to it. This leads to more anxiety, and they get worse and worse. It's a vicious circle. To make sure that students don't fall into this sea of incomprehension and drown, you have to make sure that they understand what they're doing at each stage.

Marcia D'Arcangelo: Educators are eager to learn about neuroscience in hopes that it will lead to improved ways to help students learn. What do you think will come of educators' interest in neuroscientific developments?

Brian Butterworth: Education and neuroscience are just starting out on a great adventure. The more we talk to each other, the more we will begin to understand the kinds of problems that we're each interested in and seek common solutions. Perhaps most of the interest will be in the area of special education needs - teaching mathematics to dyscalculics, to dyslexics, and to others who have inherited disorders that make it hard for them to learn. In the future, we might be able to move to more general theories of how the brain learns abstract concepts. And then, I think the great adventure will really have taken off.

Brian Butterworth is Professor of Cognitive Neuropsychology, Institute of Cognitive Neuroscience, University College-London, Alexandra House, 17 Queen Square, London, UK, WC1N 3AR; 020-7679-1150; b.butterworth@ucl.ac.uk. Marcia D'Arcangelo (mdarcang@ascd.org) interviewed Dr. Butterworth about his work when she developed ASCD's The Brain and Mathematics video series.





New Scientist, 1999
Brain Connection, 2000
Britannica & Australasian Scientist, 1999
The Science Show, 2000
The Science Show, 2001
Educational Leadership, 2001
Plus Magazine, 2002
Audio Interviews  


One day old, and already a mathematician When was your very first mathematical thought? At age four? Three? Two? It may surprise you, but it was certainly earlier still - in fact, you were born a mathematician...

In the last few years, researchers have become accomplished at finding out what goes on in the minds of tiny children, even new-born babies. This is done either by watching their gaze (looking away indicates familiarity or boredom, staring intently indicates surprise or interest) or by giving them a dummy (the more they suck, the more interested they are). This means that we can tell what expectations babies have in different situations, and when those expectations are violated. What we have learnt is that, amazingly, we all come into this world ready-supplied with basic mathematical understanding.

"We are born with a core sense of cardinal number", says neuropsychologist Brian Butterworth, author of The Mathematical Brain, reviewed in this issue of "Plus". "We understand that sets have a cardinality, that is, that collections have a number associated with them and it doesn't really matter what the members of that set are. Infants, even in the first week of life, notice when the number of things that they're looking at changes.

Was that a 2 or a 3?

"If you show babies two things and then another two things - you can change what the things are and vary lots of the visual features of these two things, so it's not that you're showing them the same thing over and over again - they gradually lose interest and start to look away for longer and longer periods. Then you show them a set with threeness, and they become interested again, and then you show them more sets with threeness and they lose interest, and then you show them a set with twoness and they gain interest again."

Impressive as this ability is, newborn babies are even more mathematically accomplished. They have arithmetical expectations, says Butterworth. "If you show a baby that you're hiding one thing behind a screen, and then you show the baby that you're hiding another thing behind the screen, the baby will expect there to be two things behind the screen, and will be surprised if this expectation is violated." So even before babies can focus their eyes, they are surprised to see a sum with the wrong answer!

Numbers on the brain

These core abilities, which Butterworth calls the "number module", may be the foundation of everything we learn about mathematics later in our lives. He speculates on this in The mathematical brain - "but I have to stress that it is speculation, because what we need to know is whether babies use the same bit of brain as adults. Adults use the left parietal lobe for this ability to recognise small cardinalities. If babies use the same bit of brain, then the course of learning more advanced mathematics builds on this core. If it's a different bit of brain, it's back to the drawing board."

The notion that children have no mathematical abilities whatsoever until they are old enough to have elements of logical reasoning (four or five years old) is very influential, and was held by the famous educationalist Piaget. Clearly this view isn't correct, but according to Butterworth, some of the mathematical abilities Piaget studied may have deeper aspects that children don't achieve until they're four or five. However, he thinks that "these abilities, such as one-to-one correspondence, are built on a basis which is innately specified. Manipulating sets really does need the achievement of some kind of logical abilities that babies don't have. So maybe Piaget was right in a way, but if he was working today he would see that the child has more going for it when it gets to four or five than simply transitive reasoning, class inclusion, these very general logical ideas, it's also got an primitive idea of cardinality."

Natural born mathematicians As a neuropsychologist, Butterworth has seen many patients with bizarre deficits caused by brain damage. In fact, some of the earliest clues to the existence of the number module came from such patients. "I came across patients who seemed to be perfectly alright in every other respect except their mathematical ability", he says. "Something happened to their brain, as the result of a stroke usually, and afterwards they seemed to be unable to do mathematics. This is a condition known as acalculia. A lot of people thought that mathematics was just language and we thought that if this was so then how could it be that Mrs G. speaks perfectly well, reasons okay, but can't count above 4? So we started to investigate in a bit more detail, and kept our eyes open for patients with other similar kinds of problems and that's really how we got started.

"Recently I've been seeing patients who have terribly disordered language but whose maths is still perfectly good, for example, one guy who has an incredibly striking dissociation. He is unable to understand the simplest words. If you ask him 'what is this?' he can't say 'watch', and if you ask him to point to a watch, he can't do that either. But he is still able to do long multiplication and long division, and to understand the principles behind these operations."

Adults thinking about mathematics tend to think about it as something logical, which of course it is, it has its own structure, but it doesn't develop according to that structure in our minds. You might think that you would have to have the concept of zero before developing thinking about sets and cardinalities, but what neuropsychology shows is that this isn't so. The number module isn't something we develop according to some logically consistent scheme, instead it's inbuilt - instinctive, in fact. "The child's acquisition of mathematical ideas actually seems to recapitulate the history of mathematics", says Butterworth. "But it doesn't recapitulate the logic of mathematics. For example, in the history of mathematics, the concept of zero is rather late. In the Frege-Russell construction of numbers it's rather early! So I would say that we can reinterpret the history of mathematics in the light of the child's development. We could say that some ideas are very easy, rather straightforward extensions of what the individual was born with, and some ideas are rather more complicated, because they're not so natural. Ideas like probability for example, are not very natural. We're very bad at probability, which of course is why insurance companies and banks are rich! You don't really get a mathematical theory of probability until the seventeenth century. That just reflects that ideas of probability are very difficult."

Using my hands teaches me maths

Interestingly, and suggestively, there is evidence that early mathematical development is related to certain physical skills. We all start to count on our fingers, and only later do most (but by no means all!) of us abandon our fingers in favour of mental calculation. Butterworth and his colleagues have just started a project looking at people with dyspraxia. "This means they have difficulty in controlling their bodily movements", he explains. "There are degrees of it, mostly dyspraxics are just a bit clumsy. They tend to have particularly poor finger dexterity, and we want to know, what's their maths like? We have anecdotal evidence that these people are worse at maths than the average, both as children and as adults. But we don't know why that is. It might have to do with their manual dexterity or lack of it, or it might have to do with something else. There might be a common cause for a whole range of different difficulties. We want to know if the kinds of difficulties they have are the sorts you would expect them to have if they had problems counting on their fingers when they were little."

One particularly interesting case, Butterworth says, concerns a woman with a very rare genetic disorder, who was born with neither hands nor feet. She reportedly says that, when doing mental arithmetic, she puts her "imaginary hands" on an imaginary table in front of her and uses them to do the calculation. So it seems that the connection between our hands and our number ability is deeper than we might think at first glance. It's interesting to speculate that hands might be a crucial part of what raises human mathematical ability so far above that of other animals, many of whom are also able to distinguish small cardinalities, but who never develop anything further based on that ability.

Putting in the hours

So far we've only talked about the most basic mathematics - arithmetic and an inbuilt notion of cardinal number. What about more advanced, or adult, mathematical ability? The evidence seems to explain how things can go very wrong - via brain damage or physical problems with dexterity - but what about when things go very right? How come some people are so good at mathematics, and so creative?

In Western culture, the most prevalent theory about talent is that it is innate. When someone is outstandingly good at something, we describe them as "gifted", and say they are "naturals". This idea is not so common in other societies, where hard work is seen as the primary reason why some people excel.

According to Butterworth, all the evidence supports the hard work theory. He goes so far as to say that the only "statistically significant" indicator of mathematical excellence is the number of hours put in. This seems to suggest that anyone could be a superb mathematician if they are willing to put in the hours - but the truth is slightly more nuanced. The crucial word here is "willing". Butterworth says that "anybody who is a good mathematician is slightly obsessed with maths - or more slightly obsessed - and they put a lot of hours into thinking about it. So they are unusual in that respect. But they may be no more unusual than anybody who is very good at what they do, because they have to have a certain obsessiveness or otherwise they're not going to be able to put in the hours to get to this level of expertise. This is true of musicians, it's probably true of waiters. Now, if you start putting in the hours when you are very young, how are we going to tell whether your adult state has got to do with what your brain was like before you started to put in the hours, or what it was like because you put in the hours?"

Which came first?

Butterworth is slightly impatient with this chicken and egg question - which comes first, zeal or hard work? He says that "if, for whatever reason, you start working hard at mathematics when all your classmates don't, then the teacher is going to favour you, so you're going to get external rewards, and you're going to get the internal rewards of being able to do something rather well that your mates aren't so good at, and so you'll start off a virtuous circle of external rewards, internal rewards, you work a bit harder, you get even farther ahead of your classmates, who aren't actually putting in the time. So it wouldn't be surprising that if random people who for some reason select to pursue maths on the whole get rewarded because they are going to be better than their peers."

There are particular cases which give great weight to what we might call the "zeal theory of excellence". Butterworth describes the recent case of Rüdiger Gamm, a German who started to teach himself to become a prodigious calculator in his twenties, because he wanted to win a prize on a TV game show. He won the prize, and became very famous in Germany as a calculator. "He can do wonderful things, because he spent four hours a day since he was twenty working on it, learning new tricks, learning the table of cubes and cube roots, and to the power of four and fourth roots and so on. He learned all the tricks he could find, and worked out tricks for himself."

All that maths has tired me out

So the picture of mathematical ability and its provenance is a nuanced one. Newborn babies, commonly thought to be incapable of anything but eating, sleeping and crying, are actually budding mathematicians. We arrive in this world hardwired with basic number abilities, and very probably everything we learn later in life about mathematics builds on this fundamental core. For some of us, maths will always be difficult, possibly because innate clumsiness made it hard for us to do sums on our hands when we were small. But for the rest of us, how good we end up at maths is mostly to do with how hard we try at it - and that depends on how much we enjoy it.

Helen Joyce, editor of "Plus", interviewed Brian Butterworth, Professor of Cognitive Neuropsychology at University College, London and founding editor of the academic journal "Mathematical Cognition". He has taught at Cambridge and held visiting appointments at the universities of Melbourne, Padua and Trieste, MIT and the Max Planck Institute at Nijmegen. He is currently working with colleagues on the neuropsychology and the genetics of mathematical abilities.

Plus Magazine: Issue 19: April, 2002

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