From the book What Counts (US edition of The Mathematical Brain)
New Scientist Opinion Interview
Alison Motluk · New Scientist · 3 July 1999
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.
The Number Module
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.
From Words to Numbers
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 master’s 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.
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 words for numbers. And ingenious experiments have shown that even tiny babies notice when 2 objects become 1 or 3.
What About Prodigies?
So if we’re all born with this basic number faculty, where do mathematical prodigies fit in? Butterworth is emphatic that there is no evidence that the brains of prodigies are actually different from the rest of ours. He suspects that these people just have a zeal for numbers that developed early in life — “maybe things went well in their very first encounter with numbers, and then they were off.” From that nugget of early enthusiasm, he says, “a sort of virtuous circle developed” — one in which the more they liked maths, the more they learned, and the more they learned, the more they liked it.
When Things Go Wrong
So we’re all born numerate and any of us has the potential to be a prodigy. But some people are hopeless at maths. What has gone wrong? Of course, simple lack of interest and effort is probably the most common explanation. But research is increasingly identifying people who genuinely have a disability in the number region. Butterworth draws a parallel with dyslexia. Just as dyslexics seem normal in many ways, he says, “dyscalculics” have normal or high IQs but seem unable to understand the most elementary aspects of number. One particular case that interests him is Charles, a young graduate, who though successful in many walks of life finds it harder than a five-year-old would to tell which is the larger of 5 and 3.
Are You a Little Bit Dyscalculic?
What makes the area so complex is that there are many different degrees of dyscalculia. A few people seem completely unable to apprehend the meaning of even very small quantities, but for many it is a subtler problem. They understand what numbers are in principle, say, but find it much harder than other people to work with sums, even though they are perfectly intelligent. Butterworth is now working to develop a screening test, which he hopes will eventually lead to people being properly diagnosed and helped.
What everybody wants to know, he says, is “are the genes different?” For the first time, proper large-scale twin studies are under way which may answer — at least in part — this question within a few years. “Our hypothesis is that there are specific genes — but it may be wrong.” For now, the most that can be said is that dyscalculia, like dyslexia, “runs in families”.
Plus Magazine Interview
Helen Joyce · Plus Magazine, Issue 19 · April 2002
Born to Count
According to Butterworth, there is strong evidence that the number sense is innate. He says: “It all depends on what you mean by innate. What I mean by it is there are brain circuits that are specialised for recognising what I call ‘numerosities’ — the number of objects in a collection — and for doing things with numerosities: comparing them, and very simple forms of adding. This is what I call the ‘number module’.”
“The evidence that this is innate comes from a number of lines. One is that even very young babies notice number. They are sensitive to changes in the number of objects that they see, and they are sensitive to the relationship between the number of sounds that they hear and the number of objects that they see. Evidence from brain imaging shows that there is a particular area of the brain that is specialised for numbers, and this seems to be universal across all cultures.”
Historical Development
If mathematical skills are innate, to what extent does mathematical development in infants and children recapitulate the history of mathematics? This was one question for Butterworth. “You can see the child’s development as recapitulating the history of mathematics,” he says. “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.”
Numbers and Fingers
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.”
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.
Mathematical Excellence
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.”
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.”
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.”
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.
Originally published in Plus Magazine, Issue 19, April 2002.