The Mathematical
Brain

Two Authors Say We Have Innate
Ability With Numbers

Commentary by John Allen
Paulos.

Sept. 4 - The new school year looms
ominously for many who claim to lack a
"mathematical brain," and so it may be a
good time to review the findings of Brian
Butterworth and Stanislaus Dehaene,
two cognitive psychologists who have
done much work on the neural basis of
mathematical thinking.

Both their books,
Butterworth's "What Counts:
How Every Brain Is
Hardwired for Math" and
Dehaene's "The Number
Sense: How the Mind
Creates Mathematics" maintain that
there are in the left parietal lobe of the
brain certain specialized circuits that
enable us to do arithmetic.

These circuits, which Butterworth
terms the Number Module, ensure
that all of us can automatically
recognize very small numbers, match
up the objects in small collections,
and tell which of two small collections
is larger. We do these tasks
unthinkingly the way we note colors
without trying to do so. Furthermore,
any numerical achievements beyond
this are a result of our slowly
mastering various representations of
numbers supplied by the surrounding
culture. These include body-parts
(fingers primarily), external aids such
as tallies and abaci, and written symbols such as Roman
or Arabic numerals.

Other cultural tools, laboriously discovered over the
centuries and presented in classrooms this fall, enable us
to master more advanced mathematical notions such as
algebra, probability, and differential equations.

We certainly differ in the extent to which we master these
tools, but we all start with the same basic mathematical
brain, the authors argue.

Experimental Support for Innate Ability

To support their thesis that numerical notions are a part of
our innate neural hardware, Butterworth and Dehaene
describe experiments in which researchers present babies
with white cards that have two black dots on them.

They place the cards a few inches from the babies' eyes
and note how long the babies stare at them. The babies
soon lose interest but resume staring when the
researchers show them cards with three black dots.

After the babies lose interest in these cards, they regain it
only when shown cards with two dots again. The babies
appear to be responding to the change in number since
they seem to disregard changes in the color, size, and
brightness of the dots. Another experiment: When
researchers place two dolls behind a screen in front of
babies, but only one remains when they remove the
screen, the babies are surprised. The researchers elicit a
similar surprise when they place one doll behind the
screen and there are two when they remove the screen.

The babies are not surprised if two dolls turn into two balls
or a single doll turns into a single ball.

The conclusion is that violations of quantity are more
disturbing to babies than changes in identity.

Disorders Can Tell Us More

Victims of disorders in the brain's number module and
their resulting deficits provide more support for claims
about the region. There have been many such cases.

A person has a stroke that damages the left parietal lobe
and, although still articulate, can no longer tell without
counting whether there are two or three dots on a sheet of
paper.

Someone can't say what number lies between two and
four, but has no problem saying what month is between
February and April. Someone else with a tumor in the left
parietal lobe cannot connect the arithmetic facts she
recites in a singsong way to any real-world application of
them.

One patient understands arithmetic procedures but can't
recall any arithmetic facts, while another has the opposite
condition.

Particularly intriguing is Gerstmann's syndrome, which is
characterized by finger agnosia (an inability to identify
particular fingers upon request) and acalculia (an inability
to calculate or do arithmetic).

Butterworth theorizes that during a child's development
the large area of the brain controlling finger movements
becomes linked to the specialized circuits of the Number
Module, and the fingers come to represent numbers.

Role of Education

In arguing for the innateness of some numerical concepts,
both authors take exception to the work of the Swiss
psychologist Jean Piaget. In one of Piaget's famous
experiments, for example, researchers showed very
young children two identical collections and then moved
the objects in one collection farther apart.

The children were likely to say that the spread-out
collection had more objects, and Piaget concluded they
did not yet really understand the notion of quantity. More
recent experiments seem to show that what the children
did not understand was the question they were being
asked.

Dehaene shows that if the same children are asked to
choose between four jelly beans spread apart and five
jelly beans close together, they are very unlikely to go for
the four jelly beans.

Of course, education is still important, and since the
number module is hard-wired in all of us, Butterworth and
Dehaene argue that one of the primary reasons
(sometimes they implausibly appear to be saying the only
reason) for disparities in mathematical achievement is
environmental - better instruction, more exposure to
mathematical tools, motivation for hard work.

The authors note the burden imposed on students by the
cumulative nature of mathematical ideas and the self-perpetuating
nature of different attitudes toward the
subject.

In particular, Butterworth contrasts the virtuous circle of
encouragement, enjoyment, understanding, and good
performance leading to more encouragement with the
vicious circle of discouragement, anxiety, avoidance, and
poor performance leading to more discouragement.
There is much else of interest in both of these books, but
my sense of number tells me I've gone on for long
enough.

Professor of Mathematics at Temple University and
adjunct professor of journalism at Columbia University,
John Allen Paulos is the author of several best-selling
books, including "Innumeracy" and "A Mathematician Reads
the Newspaper". His "Who's Counting?" column on
ABCNEWS.com appears at the beginning of every month.