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.