Britannica & Australasian Science: October 1999

A large company of chimps travels through the forest, headed by their fearless leader Brutus. To increase
their chance of finding food they break up into several bands, but keep in touch by "pant-hooting" and drumming
on resonant trees.

When it is Brutus doing the drumming, however, the other chimps
treat the number of drum beats as instructions.
One beat means "change direction." Two beats means "rest"--always for
between 55 and 65 minutes. A beat on one tree and two on another combines
these, instructing the chimps to change direction and then rest. Most
remarkably of all, four beats instructs the other bands to rest for two hours.

These observations were made on animals in the wild, not ones that had been
patiently trained by humans. They appear to provide proof that some animals
are able to count and measure time. The evidence is particularly remarkable
when one considers that counting is not believed to be automatic in humans,
but has been passed down the generations from some "ancient Einstein." This
observation, recorded by the zoologist Christophe Boesch, is one of the many
extraordinary anecdotes and facts used by cognitive neuropsychologist Brian
Butterworth, of University College London, to support his theory of a
"mathematical brain." According to the theory almost all humans, and many
animals, have an in-built ability to do mathematics.

Butterworth, promoting his book The Mathematical Brain , cites evidence from
diverse fields to support his theory, such as the body counting systems used in
the New Guinea highlands. Here each number is associated with a body part,
so that the word for "three" might be the same word used to denote the middle
finger. The attachments vary across the thousands of different communities in
the area.

Among the Yupno the little finger of the left hand is one, the right big toe is 20
and the left ear is 21. When only one man is present--women are forbidden to
count in public--numbers stop at 33 for the penis. With more men present it is
possible to go to higher numbers.

Isolated communities in New Guinea have similar solutions to keep track of
possessions, but it seems that most, if not all, have invented this for themselves
rather than copied from their neighbours. Butterworth concludes this on the
basis of the diversity of counting patterns--some go left hand first and then
right hand, while others cover the whole left side before crossing to the right.
Some can go as high as 68 by including extra body parts. There does not seem
to be a geographical pattern to this as one would expect if trade had enabled
counting to diffuse outwards from the community that invented it.

Numerical skills appear to be concentrated in a particular section of the brain,
the inferior lobule of the left parietal lobe. Within this it seems that different
subsections control distinct mathematical abilities. Patients who have suffered
strokes or other forms of brain damage can lose the ability to tell which
number is larger, or subtract or multiply. Most remarkably, some patients lose
one ability while others are largely unaffected, suggesting that different parts
of the brain have been compromised.

**Adding Cells**

Butterworth believes that almost everyone is capable of being good at
mathematics if they are taught well. He believes ability at math, like many
other skills, fundamentally comes down to practice. For example, Braille
readers develop far more brain cells in the areas associated with messages
from their "reading" finger than those who do not read Braille.

In other words, using the sections of the brain that are good at math causes
them to expand, and therefore get better at mathematical processing. Thus we
should not be surprised that Albert Einstein's left parietal lobe hosted an
unusually high concentration of brain cells.

Butterworth admits that practice doesn't always make perfect when it comes to
math. For some, a defect in the relevant area of the brain means that no matter
how much they practice they will never catch up with the rest of the
community. However, Butterworth believes that this group accounts for only 4
percent of the population, far fewer than those who consider themselves to be
inherently bad with numbers.

If 96 percent of the population is born with the same mathematical potential,
why do some reach genius level while others struggle to calculate the change
on a simple transaction? According to Butterworth something, usually early in
life, stimulates some people to think about math more. They solve
mathematical puzzles in their heads and the relevant brain areas expand.
Once these people reach school age they find they are good at math, and
consequently enjoy it. This leads them to do more, and they become even
more gifted.

In an interview with Australasian Science Butterworth argued that what
mattered was not the time spent studying for the exam but the total time spent
on mathematical thoughts. Thus those who sail through their math education
with little study might be spending much of their free time solving
mathematical puzzles plucked from observations.

Good mathematicians, Butterworth argues, love math and do it at every
opportunity. In interview Butterworth retreated slightly from his claim that
mathematical ability is entirely a function of practice. He admitted it was
possible that some people are born with brains better suited to this area.
However, he stressed: "Practice is vital. There is currently no evidence that
anything else matters." He remains uncertain about why some people gain a
desire to practice math more, but describes this as "individuation, not ability."
The implications of Butterworth's theory are profound. Even if we reject the
argument that ability is entirely based on exercise he makes a powerful case
that activity is more important than variations in natural talent. If he is right we
need to radically rethink our school curricula.

If engaging in mathematical puzzles and games is an effective way to increase
ability then it stands to reason that making math fun should be the priority for
teachers, at least in the early years. Yet Butterworth documents repeated
attempts by the previous conservative government in Britain to enforce more
rote learning and less activity likely to stimulate children towards engaging in
math out of hours.

What is more, Butterworth alleges that nowhere in the reports recommending
more drill is there the slightest evidence that this works, and that a major
report contained just one sentence on the importance of enabling students to
understand what they are doing. He attributes this to a slavish subservience to
the prejudices of politicians. In interview he noted that many of the English
educators setting policy in this area were actually telling the politicians what
they wanted to hear, but doing something different.

Butterworth also notes that an inquiry into math education was specifically
banned from allowing the use of calculators in primary school. While he
maintains an open mind on whether early exposure to calculators is beneficial
he would welcome some research on this topic, rather than an order based on
"common sense."

Butterworth contrasts this approach with the way Chinese students are taught
multiplication. Instead of having to learn a complete table up to 9 x 9 it is
explained that 3 x 5 is the same as 5 x 3. Consequently their five-times table
starts at 5 x 5 = 25. This cuts the memory load from 81 facts to 36 (the
one-times table is also deleted), and also creates the perception that
multiplication is about logic rather than abstract facts. Butterworth believes
this contributes to the fact that 12-year-old children in Shanghai perform better
at math than 17-year-olds in America.

Just how ingrained the rote learning system is can be observed from
Butterworth's point that he had to learn tables up to 12 x 12 because he went to
school before decimal currency. Yet long after shillings and pence were
abolished in Australia children in many schools are still forced to recite tables
up to 12.

**Getting Math Right**

While Butterworth believes that some mathematical skills are ingrained within
us, getting to more complex concepts is much harder. For example, our
number system uses 10 as a base, with all larger numbers defined in relation to
that base. Thus 128 is 1 x10 x10 + 2 x10 + 8. A base-5 system would write
this as 1003: 1 x 5 x 5 x 5 + 3 (the last number represents the units, the second
last represents the number of fives, the next represents the number of 25s,
etc.). In theory any number other than 1 can be used as a base.

Westerners tend to think of a base-10 number system as "natural", but many
other cultures have come up with a wide range of bases, most famously the
Sumerian base 60, from which we get our hours. Furthermore, many of the
cultures Butterworth deals with use a mixed base-5 and base-20 system,
skipping over 10 as just another number. Then there is the Base-12 Society, a
group dedicated to the profoundly fruitless task of converting the entire world
to a number system that, having more factors, should be easier to learn.

Base-10 is not entirely unchallenged within Western culture. As Butterworth
points out, we say "eleven", not "tenty-one", and the French and Danish are
heavily influenced by base-20 counting. More profoundly the concept of zero,
so obvious to those brought up with it, was totally alien to most ancient
cultures, even those that developed quite advanced mathematics. The
difficulties Archimedes must have gone through to calculate the number of
grains of sand in the universe without the use of a zero for place counting are
as staggering as the numbers he came up with. The power of numbers has been
recognised by some authorities who have attempted to prevent further
development. In 1299 the Guild of Florentine Bankers forbade the use Arabic
numerals, demanding instead that numbers be written out using letters.

Yet despite all the obstacles in its path, the study of mathematics has
flourished.

Stephen Luntz

©, Australasian Science, 1999.