The October 2 issue of Nature, the weekly journal of science, contains an interesting study about something called the "approximate number system." (The headline, in catchy Nature form, is "Individual differences in non-verbal number acuity correlate with maths achievement.") ANS is an innate counting sense, present in animals, infants, and others who've had no formal training in math. Some people have a much better ANS acuity than others. Those that have this acuity wind up doing better on standardized math tests, even controlling for IQ.
To prove this idea, scientists at Johns Hopkins and elsewhere gave 64 14-year-old children a test to gauge the innate numerical sense ANS measures. They showed them a scatter plot containing dots in 2 colors. They asked which color was more prevalent. They changed up the ratio, and found that when there were twice as many purple as yellow dots, almost everyone could tell, but when there were only 25% more purple dots than yellow, only about 60% of the children got this right -- not too much better than guessing. But, of course, some children were far more accurate on these close ratios than others. The researchers then looked at the children's scores on standard math tests going back through kindergarten, and found that those who could gauge the ratios easily had higher scores.
This wasn't just because the kids were "smarter." They had previously given all the children an IQ test, and found that, controlling for IQ, kids with higher ANS acuity still did better on math tests. They also gave the kids a rapid color reaction test to make sure they weren't just testing reaction times. Again, the kids with the best ANS scores did best on math tests.
The result? "Individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense," the authors wrote. They said that much more research would be needed to know if ANS could be taught and, if it could, how this would affect student achievement.
I always like studies that use the scientific method to get at the obvious. We all know that some kids are just naturally more number-oriented than others. Some highly intelligent adults panic when asked to make change; others can look at a room and see instantly that there aren't enough chairs.
That said, this is all fine for student achievement tests. But if there's anything I've learned over the years through trying to write about math and mathematicians, it's that at some point, numbers have little to do with it. Indeed, I remember one day in my freshman math class in college looking at the chalkboard and realizing there wasn't a single number up there! This was somewhat disappointing to me, as I really liked numbers. But as my older brother once wrote as a math graduate student at Princeton, "Despite what the general public thinks, math isn't just about being able to multiply numbers in your head quickly, or memorizing thirty digits of pi (I know good mathematicians who struggle to calculate 15% tips in restaurants). We've got computers and calculators for that. Mathematics is about finding structures and truths in the world of patterns, and explaining why they're there."
I suspect that what happens is that kids who have both a good ANS acuity and a high IQ wind up finding early math easy and fun. So they become more interested in it, and are encouraged along. The ranks of top mathematicians are then pulled from this group of folks who have spent more time studying math and playing with it than other people. Ultimately, though, it's not the quick counting that will help them -- it's the ability to see patterns and draw inferences.