Take a stab: Estimating math skills by observing estimation

New research into our ?number sense? may have discovered a link between a …

Quick: How many people did you see on the way home from work the other day? Was it more or less than the day before? Regardless of your answer, it's likely you can't explain what led you to it. Number approximation is an innate ability in adult and infant humans, primates, any many other animal species. In the wild, it helps the endowed identify plentiful food sources and other evolutionarily useful tasks. But in modern society, the accuracy of an person's "number sense" may hint at more than the ability to discern many from few.

The foundations of the "number sense"—or approximate number system—have been widely studied, but links between innate numeric ability and formal math education are less established. Yet a new study published Sunday in Nature may forge that bond. The paper's authors found a surprisingly tight correlation between a person's power of estimation and his or her ability calculate the power of a number.

"We discovered that a child's ability to quickly estimate how many things are in a group significantly correlates with that child's performance in school math for every single year, reaching all the way back to when he or she was in kindergarten," said Justin Halberda, assistant professor of psychological and brain sciences at Johns Hopkins University and the study's lead author.

"This study makes a new and important contribution," said Hilary Barth, an assistant professor of psychology at Wesleyan University who was not affiliated with the study.

"Researchers frequently suggest that approximate number abilities are related to children's symbolic math skills," Barth told Ars. Such "direct evidence of this link is rare."

The new study, she added, also found the correlation between approximate number abilities and math skills to be independent of overall intelligence. The study took advantage of a cohort of children that the researchers had been tracking as they progressed through school, administering tests that included assessments of math skills (the details of the cohort have been published).

Halberda and his co-authors tested the "number sense" of 64 ninth graders by flashing groups of dots on a computer screen. Each flicker displayed 10 to 32 dots of two colors. After viewing the random pattern for the briefest of moments—200 milliseconds to prevent counting—subjects reported which color they felt was more numerous.

Approximate number sense, they found, ran the gamut. Some subjects could tell the difference between nine yellow dots and 10 blue ones, where others could barely distinguish four blue dots from six yellow ones. To normalize the results, Halberda and his colleagues converted the raw scores into Weber fractions.

In the 19th Century, Ernst Weber observed that people think two stimuli are identical in intensity if the differences between them are small enough. For example, when standing on a street corner, you may think one car traveling at 24 miles per hour is passing you at the same speed as another traveling at 27 mph. Yet when the speed differential crosses a certain threshold, say 6 mph, you notice it—what Weber called a "just noticeable difference." Weber fractions enumerate the phenomenon, and lower Weber fractions indicate lower "just noticeable differences." Subjects able to tell nine dots from 10 scored a Weber fraction of 0.11, while those unable to move beyond a 2:3 ratio rated 0.5.

Correlation and causation

Halberda and his co-authors speculate that the strength of a person's approximate number system may affect his or her math scores in one of two ways. As "number sense" is present even in infancy, a more developed "sense" may lead to higher math scores. On the other hand, they say, previous studies have shown people with more math education to have better developed approximate number systems.

Alan Schoenfeld, a professor of education at the University of California, Berkeley who was not affiliated with this study, said classroom and sociological differences have confounded many previous studies.

"When you're testing 14 year olds, society has had a lot of time to influence them," Schoenfeld told Ars.

The authors were cautious to acknowledge possible social effects in their paper, he added, but was hesitant to draw concrete conclusions himself.

"Correlation is one thing, causation is another," he said.

Nature, 2008. DOI: 10.1038/nature07246

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