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