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.