Gut Instinct’s Surprising Role in Math
By NATALIE ANGIER
New York Times, September 15
...Whenever we choose a shorter grocery line over a longer one, or a bustling restaurant over an unpopular one, we rally our approximate number system, an ancient and intuitive sense that we are born with and that we share with many other animals. Rats, pigeons, monkeys, babies—all can tell more from fewer, abundant from stingy. An approximate number sense is essential to brute survival: how else can a bird find the best patch of berries, or two baboons know better than to pick a fight with a gang of six?
When it comes to genuine computation, however, to seeing a self-important number like 529 and panicking when you divide it into 2200, or realizing that, hey, it’s the square of 23! well, that calls for a very different number system, one that is specific, symbolic and highly abstract. By all evidence, scientists say, the capacity to do mathematics, to manipulate representations of numbers and explore the quantitative texture of our world is a uniquely human and very recent skill. People have been at it only for the last few millennia, it’s not universal to all cultures, and it takes years of education to master. Math-making seems the opposite of automatic, which is why scientists long thought it had nothing to do with our ancient, pre-verbal size-em-up ways.
Yet a host of new studies suggests that the two number systems, the bestial and celestial, may be profoundly related, an insight with potentially broad implications for math education. One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific. Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.
...This month in the journal "Nature", Justin Halberda and Lisa Feigenson of Johns Hopkins University and Michele Mazzocco of the Kennedy Krieger Institute in Baltimore described their study of 64 14-year-olds who were tested at length on the discriminating power of their approximate number sense. The teenagers sat at a computer as a series of slides with varying numbers of yellow and blue dots flashed on a screen for 200 milliseconds each—barely as long as an eye blink. After each slide, the students pressed a button indicating whether they thought there had been more yellow dots or blue.
Given the antiquity and ubiquity of the nonverbal number sense, the researchers were impressed by how widely it varied in acuity. There were kids with fine powers of discrimination, able to distinguish ratios on the order of 9 blue dots for every 10 yellows, Dr. Feigenson said. “Others performed at a level comparable to a 9-month-old,” barely able to tell if five yellows outgunned three blues. Comparing the acuity scores with other test results that Dr. Mazzocco had collected from the students over the past 10 years, the researchers found a robust correlation between dot-spotting prowess at age 14 and strong performance on a raft of standardized math tests from kindergarten onward. “We can’t draw causal arrows one way or another,” Dr. Feigenson said, “but your evolutionarily endowed sense of approximation is related to how good you are at formal math.”
The researchers caution that they have no idea yet how the two number systems interact. Brain imaging studies have traced the approximate number sense to a specific neural structure called the intraparietal sulcus, which also helps assess features like an object’s magnitude and distance. Symbolic math, by contrast, operates along a more widely distributed circuitry, activating many of the prefrontal regions of the brain that we associate with being human. Somewhere, local and global must be hooked up to a party line.
Other open questions include how malleable our inborn number sense may be, whether it can be improved with training, and whether those improvements would pay off in a greater appetite and aptitude for math. If children start training with the flashing dot game at age 4, will they be supernumerate by middle school?
(anyone else start to feel a little queasy towards the end ? 
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