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.