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A study by researchers from France and MIT published in the May 6 issue of Science indicates that learning the multiplication table may be more akin to memorizing a laundry list than exercising mathematical skills.
Meanwhile, learning to approximate how numbers relate to each other seems to be tied to intuition about space.
The results may have implications for how math is taught, particularly to bilingual children. The research suggests that performing arithmetic may require bilingual individuals to mentally translate a problem before solving it, slowing down their response time.
Through separate studies involving behavioral experiments and brain-imaging techniques, the researchers found that a distinctly different part of the brain is used to come up with an exact sum, such as 54 plus 78, than to estimate which of two numbers is closer to the right answer. Developing the latter skill may be more important for budding mathematicians.
For years, mathematicians, including Einstein, have said that they rely more on mental signs and images than words. The results reported by Stanislas Dehaene of Service Hospitalier Frederic Joliot in France and Elizabeth S. Spelke, professor of brain and cognitive sciences at MIT, show that exact arithmetic uses a part of the brain usually active during verbal memory tasks.
Meanwhile, puzzling out approximations may be more like the kind of brain exercise that many mathematicians use to arrive at new insights. Approximating seems to require a more spatial tool, such as a mental number line. This spatial tool, which some call number sense, may be the most important source of mathematical intuition, the researchers say, although this intuition probably also results from interplay between the two brain systems involved.
Dehaene is the author of The Number Sense (Oxford University Press, 1997), which argues that a specific set of brain circuits underlies people's number sense.
In addition to shedding light on how mathematicians' brains work, the researchers' results may have implications for math education. If the results of these studies on adults also apply to children, the studies imply that children who are drilled in rote arithmetic are learning skills far removed from those that enrich mathematical intuition, Professor Spelke said.
"Down the road, educators may look harder at the importance of developing children's number sense" -- for example, their ability to determine a ballpark answer rather than a specific answer, she said. Number sense is considered by some to be a higher-level understanding of mathematics than rote problem-solving.
In the behavioral study Professor Spelke led, individuals fluent in Russian and English were taught a set of addition problems, some requiring exact answers and some requiring estimates, in one of their two languages. They were tested in both languages.
After training sessions, all test subjects got the right answer to exact math problems (such as "57 plus 43") faster when they were tested in the language in which they received the training, regardless of whether the training occurred in their first or second language. This suggests that this knowledge is stored in the brain in a form that depends on a specific language.
In contrast, for approximate addition, performance was equivalent in the two languages, providing evidence that this knowledge was stored in a form independent of language, the researchers wrote.
The study sounds a warning to those involved in bilingual education. Bilingual people may be slower at arriving at sums because they take the time to mentally revert to their native language when asked to perform exact arithmetic.
"If it turns out that when we learn specific arithmetic facts, what we learn is tied to a specific language, a child could be at a disadvantage learning these skills in a language he or she will not use as an adult. Our results suggest that the switch will not be easy," Professor Spelke said.
The brain-imaging evidence collected by Dr. Dehaene's team shows that approximate calculations take place in the brain's large-scale network involved in visual, spatial and analogical mental transformations. Rote arithmetic takes place in an area usually reserved for verbal tasks. This part of the brain, while not a primary language area, is activated when subjects have to remember verbal material.
What's more, the two kinds of math problems were instantaneously assigned by the brain to their respective areas, suggesting that the calculation itself, not just the decision to perform it, is completed by specific circuits depending on whether an exact or approximate result is required.
A rudimentary ability to approximate develops very early in humans, Professor Spelke said, with even very young infants able to distinguish an array containing many objects from one containing fewer objects. One of her future research goals, she said, is to determine whether the learned ability to approximate is tied to this early-developing skill.
To Professor Spelke, the study reported in Science is an important first step toward more fully understanding higher-order brain function. "We're still at an early point, but we may someday be able to use this approach to shed light on what seemed to be quite ineffable experiences, like moments of mathematical insight," she said.
Other collaborators on the Science paper are Philippe Pinel and Ruxandra Stanescu, also from the French institute, and MIT alumna Sanna Tsivkin (SM 1998).
This work is supported by the National Institutes of Health and the Foundation pour la Recherche Medicale in France.
A version of this article appeared in the May 12, 1999 issue of MIT Tech Talk (Volume 43, Number 30).