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: How does the brain process math, and what are the stages of calculation?
BB: 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 G’del, 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!