Numbers and Our Divided Brain
Probing humanity’s uncanny knack for numbers, an international research team has unveiled evidence that the human brain handles mathematics in two distinct ways: One part of the brain does calculating, another does estimating.
The findings not only illuminate the fundamental nature of mathematical reasoning, but may lead to improvements in how basic arithmetic is taught, even to children with severe learning disabilities, according to the scientists based at the Service Hospitalier Frederic Joliot in Orsay, France, and the Massachusetts Institute of Technology.
In a series of behavioral and brain-imaging studies published last week in Science, the researchers pinpointed where in the brain such numerical abilities reside. They were able to show how different modes of thought weave together in the tapestry of mathematical reasoning.
The scientists demonstrated that the regions of the brain involved in exact arithmetical calculations such as addition or multiplication are also involved in joining words to make a sentence.
In contrast, the ability to muse more generally about the approximate relationships between numbers falls to a part of the brain more involved in spatial skills.
Cognitive neuroscientist Brian Butterworth at University College London, author of “The Mathematical Brain,” called the new research “a virtuoso performance.”
The work draws attention to growing efforts to fathom the roles played by neural anatomy and learning in forging such a unique human mental ability.
“The development of higher mathematics is surely one of the crowning cognitive achievements of our species, which no other animal even approaches,” said MIT cognitive psychologist Elizabeth Spelke, who conducted the studies with psychologist Stanislas Dehaene in France. “What gives rise to this achievement?”
To answer that, the scientists probed the biology of numbers.
The ancient followers of the Greek mathematician Pythagoras, who is credited with codifying many of the principles of geometry, were so enthralled by the natural order revealed through arithmetical operations that they believed numbers to be the basic elements of the universe.
Certainly, the new research shows them to be an elemental property of the mind.
By the Numbers
Indeed, the neural circuits that allow human beings to count on their fingers also allow them to balance their checkbooks, calculate the engineering of cathedrals and unravel the numerical fabric of space and time.
“The whole area of mathematical cognition is tremendously important, though it is only just becoming recognized as such, because the ability of humans to use numbers has been one of the keys to raising us, if you will, from the Stone Age to the phone age,” Butterworth said.
Many animals share with humans an inborn instinct for approximate numbers--the ability to intuitively sense quantities such as the “twoness” of one’s hands or the “fiveness” of the fingers on each. But only human beings have taken the additional mental leaps to create the more formal, ordered structures of mathematical calculation. Human infants appear to display an innate number sense as early as 5 months of age.
“This . . . sense might be the foundation on which our ability to understand numbers is based,” said the Paris-based Dehaene, who led the research team.
Until recently, however, researchers who wanted to understand the actual neural processes at work were limited to studies of people whose brains had been damaged in ways that impaired their ability to calculate. The roster of those with number disorders includes stroke patients who can subtract but not multiply, or who can recite strings of numbers flawlessly but cannot add or subtract them.
“I’ve seen patients who lack this number sense, this intuition of quantity,” Dehaene said. “They can’t do 3 minus 1 or tell you what is between 2 and 4. They can learn to recite ‘4 times 7 is 28.’ However, they never really grasp what it means, and they fail to apply this knowledge appropriately.”
To investigate the biological basis of arithmetic, Dehaene, Spelke and their colleagues tested how volunteers fluent in Russian and English solved a series of arithmetic problems designed to pinpoint mathematical modes of thought.
Volunteers who were taught in one language and then tested in another took a full second longer to perform the calculation than when they performed both tasks in the same language.
But that time lag occurred only when the problem involved an exact calculation. When tested on approximate math problems, there was no language-dependent delay.
The distinction persisted even when the bilingual volunteers were tested on complex mathematical operations, such as addition in a number base other than base 10 and the approximation of logarithms and square roots.
Dehaene’s team then used two different brain imaging techniques to track which brain regions were activated in each kind of task.
Exact calculations activated the brain’s left frontal lobe, which is known to make associations between words.
Estimates, however, activated the brain’s left and right parietal lobes, which are known to be involved in spatial reckoning--particularly those areas used in guiding hand and eye movements.
The researchers were especially intrigued by the anatomical link between mental arithmetic and finger control.
“Finger counting is an almost universal stage in learning arithmetic, not just for children but also for our species,” Butterworth said. “Even human cultures that have relatively little in the way of mathematics nonetheless have finger counting.”
New Strategies
The researchers suggested that an understanding of the mental process of arithmetic could lead to improved teaching techniques, with instructional materials that could be designed to take advantage of each mode of mathematical thought.
“Our research may have a strong impact in education,” Dehaene said. “For instance, it clearly indicates that even children with severe language problems can and should learn to develop their nonverbal number sense.”
Spelke said the research also could have implications for teaching students in nonnative languages.
“If it turns out that when we learn specific arithmetic facts, what we learn is tied to a specific language, [then] a child could be at a disadvantage learning these skills in a language he or she will not use as an adult,” she said.
“Our results suggest that the switch will not be easy.”
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A Head for Figures
The human brain handles mathematics in two distinct ways: One part of the brain does calculating, another does estimating, new research shows. The researchers used a series of behavioral experiments and brain imaging techniques to pinpoint brain regions involved in handling numbers.
Sources: Science, Service Hospitalier Frederic Joliot
