They say they have established that two very different brain functions are involved - as previous studies of brain-damaged patients have hinted.

One is a non-verbal, visual and spatial sense of quantity - the other has to do with symbols related to language.

Mathematicians themselves have suggested that this might be so. Albert Einstein said numerical ideas came to him more or less as images that he could combine at will - whereas others have said they rely on verbal representations of numbers when thinking about problems.

The team of French and American researchers were led by cognitive neuroscientist Stanislas Dehaene of the National Institute of Health and Medical Research (Inserm), in Paris, and cognitive psychologist Elizabeth Spelke of the Massachusetts Institute of Technology.

Bilingual volunteers

Their findings, published in the journal Science, not only confirm the theory but locate where in the brain the mental activity takes place.

They worked with volunteers who were fluent in two languages - English and Russian.

Precision is linked to word association functions

They asked them to solve a series of problems after first teaching them the necessary maths. One group was taught in Russian, the other in English.

If they learned in English and were tested in Russian -- or the other way round -- the volunteers needed as much as a second more to solve problems involving exact calculation: Does 53 plus 68 equal 121 or 127?

When they were tested on an approximation - is 53 plus 68 closer to 120 or 150? -- there was no time lag.

"I was amazed that the dissociation could be so sharp," said Stanislas Dehaene.

Brain tracking

"After all, we presented our subjects with tasks that are superficially extremely similar. Our brains really solve these two tasks in quite different ways."

The researchers moved on to train and test the volunteers in more complex mathematical operations, such as addition in a base other than 10 and the approximation of logarithms and square roots. The distinction remained.

They then used brain imaging techniques to track which parts of the brain were operating in each kind of task.

Exact calculations lit up the volunteers' left frontal lobe, an area of the brain known to make associations between words. Estimation involved the left and right parietal lobes - responsible for visual and spatial representations.

The parietal lobes are also responsible for finger control - and counting on the fingers is something children almost everywhere do early on in learning exact arithmetic.

Nurture not nature

Dr Dehaene says the findings cannot be used to say which children are "naturally" better or worse at mathematics.

He says many studies have indicated that the impact of education is probably much greater than any initial difference in innate ability.

That holds out the prospect that the results could lead to improved teaching methods, and that even children with severe language problems should be able to improve their number sense.

Dr Dehaene said he would put some of the $1m McDonnell Foundation grant he was awarded in January towards the study of children and adults with dyscalculia - difficulty with numbers.