The human capacity for mathematics
(May 1999)

A report in Science in early May sheds a fascinating light on how human minds approach mathematics. The evidence of mathematicians themselves has always suggested that there are at least two ways of thinking about mathematics, and this now seems to be confirmed. Albert Einstein reported that numerical ideas came to him in ''images, more or less clear, that 'can reproduce and recombine at will,'' while other mathematicians report that they process mathematics by way of language-related symbols, or verbal representations of numbers.

It now appears that the visual-spatial mode and the linguistic mode of doing mathematics work together. The authors, from France and the USA, believe that this finding may help children who struggle with numbers.

Studies of brain-damaged patients reveal that some can subtract, through a nonverbal quantity-based operation, but cannot multiply, which involves a rote verbal operation, while others can multiply but not subtract. The new study confirms this two-mode theory and locates the point where such mental activity takes place in the brain.

The method used volunteers who are fluent in both Russian and English, who were trained in the mathematics needed to solve certain problems, either in Russian or in English. Then they were tested in one of the two languages.

Where the task involved an exact problem, like deciding if the sum of 53 and 68 is 121 or 127, the volunteers were slower when they were tested in the second language, presumably because the problem used the linguistic mathematical ability. When they were give an approximate mathematical problem, like deciding if 53 plus 68 is closer to 120 or 150, the volunteers showed no language-dependent lag in their answers, suggesting that this task does not involve linguistic mathematical abilities.
This language-based distinction was also demonstrated in other more complex mathematical tasks such as addition in bases other than 10, and approximations of logarithms and square roots. And functional brain imaging techniques showed that exact calculations lit up the volunteers' left frontal lobe, an area of the brain known to make associations between words. Mathematical estimation, on the other hand, involved the left and right parietal lobes, parts of the brain responsible for visual and spatial representations and also for finger control.

Perhaps significantly, finger counting is typically an early stage in a child's learning of exact arithmetic, and the authors point out that both preverbal human infants and monkeys can numerically distinguish among small groups of objects. So perhaps this innately grasped nonverbal sense of quantity, an ability that humans share with other primates, may be a key part to the power which only humans have, the symbolic mode of mathematical thought, the ability that allowed Einstein to capture the whole universe in a single equation.

The findings do not give us a method of selecting children who are ''naturally'' better or worse at mathematics, but they do suggest that the impact of education may be more important than any inherited ability.

Key names: Stanislas Dehaene and Elizabeth Spelke.

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