In pondering the possible relations between inventive thought in mathematics and bodily phenomena, the mathematician Jacques Hadamard (1865-1963) sounded a note of despair in his 1945 book __The Psychology of Invention in the Mathematical Field__.

“While mathematicians have not sufficient knowledge of neurology, neurologists cannot be expected to penetrate deeply (as would be necessary) into mathematical studies,” Hadamard wrote. “Will it ever happen that mathematicians will know enough of the subject of the physiology of the brain and that neurologists know enough of mathematical discovery for efficient cooperation to be possible?” he asked.

That gap is beginning to close, particularly with the pioneering work of Stanilas Dehaene, a mathematician turned cognitive neuropsychologist, who works at the Service Hospitalier Frédéric Joliot in Orsay, France.

In the May 7, 1999 __Science__, Dehaene and collaborators at the Massachusetts Institute of Technology present the first hard evidence that two quite different modes of brain activity underlie our inborn capacity for mathematics.

Their study indicates that learning the multiplication table is akin to memorizing a laundry list, whereas learning how numbers relate to each other appears tied to visual intuition about space.

Using brain-imaging techniques, the researchers discovered that calculating an exact sum, such as 53 plus 68, and estimating which of two numbers is closer to the right answer, such as deciding whether 53 plus 68 is closer to 120 or 150, activate different parts of the brain. Performing exact arithmetic uses a region of the brain usually active during verbal memory tasks, and puzzling out approximations exercises the visual side.

In sum, the study suggests that mathematical ability results from the integration of two nonnumerical circuits in the brain, remarks neuroscientist Brian Butterworth of University College London in England in the same issue of __Science__. One is the left frontal lobe, which controls linguistic representations of exact numerical values, and the other involves the parietal lobes, which control visuospatial representations of approximate quantities.

Mathematicians have long reported that they rely more on mental images than words to arrive at new insights. Albert Einstein (1879-1955) once stated, “The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined.”

The findings suggest that understanding relationships among numbers involves some sort of spatial tool, such as visualizing a number line. Such visual aids may be important sources of mathematical intuition. Developing skill in their use could be crucial for budding mathematicians.

At the same time, behavioral experiments with test subjects fluent in Russian and English indicated that learning how to calculate sums mentally in one language may not help much when performing the same summations in a second language. Yet making approximations is equally easy in both languages.

The language-related distinction also showed up when the researchers trained and tested the bilingual volunteers in more complicated mathematical operations, such as addition in a base other than 10 and the approximation of logarithms and square roots.

This is just the beginning, however. “Even the simplest arithmetic is more than just fact retrieval,” Butterworth comments. Moreover, the parietal lobes may serve other mathematical functions as well, including some involvement in exact calculations.

“It is worth noting that the parietal lobes. . .are part of the neural circuit that controls handshapes and finger movements,” he adds. “This raises the possibility that these brain regions contribute to finger counting and finger calculation-an almost universal stage in the learning of exact arithmetic.”

In the end, studies of how the brain functions while doing mathematics may have implications for how mathematics is taught, potentially leading to new, improved ways of teaching arithmetic to children who struggle with numbers.

**References:
**Blakeslee, S. 1999. Brain’s math machine traced to 2 circuits.

__New York Times__(May 11).

Butterworth, B. 1999. A head for figures. __Science__ 284(May 7):928.

Dehaene, S. 1997. __The Number Sense: How the Mind Creates Mathematics__. New York: Oxford University Press.

Dehaene, S., et al. 1999. Sources of mathematical thinking: Behavioral and brain-imaging evidence. __Science__ 284(May 7):970.

Hadamard, J. 1996. __The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field__. Princeton, N.J.: Princeton University Press.

Halber, D. 1999. Different kinds of math use different parts of brain, research finds. __MIT Tech Talk__ (May 12). Available at http://web.mit.edu/newsoffice/tt/1999/may12/math.html.