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.
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.