Gut Instinct’s Surprising Role in Math, by Natalie Angier, NYT

[Demeter]

http://www.nytimes.com/2008/09/16/science/16angi.html?ref=science&pagewanted=print


You are shopping in a busy supermarket and you’re ready to pay up... You perform a quick visual sweep of the checkout options and immediately start ramming your cart through traffic toward an appealingly unpeopled line halfway across the store. As you wait in line and start reading nutrition labels, you can’t help but calculate that the 529 calories contained in a single slice of your Key lime cheesecake amounts to one-fourth of your recommended daily caloric allowance...One shopping spree, two distinct number systems in play. 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 2,200, 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. ...

This month in the journal Nature, Justin Halberda and Lisa Feigenson of Johns Hopkins University and Michele Mazzocco of the Kennedy Krieger Institute ... 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. (Take a version of the test.)

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..., 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. ...

[zazen]

1. wish they had early childhood/demographic/SES crosstabs

This suggests innate differences that are genetic, but what if some are exposed to lot more visual stimulation (or verbal vs visual) at an early age, for whatever reason? Also, I'd like to see if there's any stat sig difference among sexes (since we hear about that all of the time from other corners). And there's some suggestion that some cultures are better at math due to the way their languages lead them to think (and the formal appearance of their written language.) I guess they can only answer so many questions on a first pass.

[Igel]

2. My 10th grade chemistry teacher taught me one thing.

Considering his general level of competence, that was shocking in and of itself.

It was to use common sense in solving math problems, to estimate the answer ahead of time if you could. After working over a kind of problem to get a feel for it, every such problem in class would get two answers--the first, a quick estimate. Usually he was close; sometimes he was really off. That was ok; the quick answer is a guideline.

My pre-calc teacher, and then my calc 1 teacher, reiterated the lesson--become familiar with the classes of equations you're exposed to, see if they make sense. Map out their properties in a rough way before tackling them--later it becomes automatic, it's an acquired skill. Then, if your exact answer is insane or suspicious, triple check your math.

In the case of the chemistry teacher, he gave us some insane quizzes. 50 problems in 5 minutes, with the instructions including lines like, "Estimate your answer to the nearest gram" and spotting us first 30, then 20, then 10, then 5 correct answers. He sometimes set up the equations (simple though they were) for us. One of the more useful things I learned in high school. He was loathed for this. Still, he pointed out it's a useful kind of job skill: Insurance adjusters, damage assessors, painters, and construction bidders all make rough estimates in real time, and can't take the time to inspect every nook and cranny, to measure everything precisely. Now, perhaps you can't train somebody bad at it to be good at it; but if you have the ability and never practice it, never try to use it, you never learn to use it.

It also kept stupid exact answers in later courses, and in life, to a minimum. "A train is moving at 60 km/hr, the bottom of the window is 2.5 m off the ground. If a 100 kg weight is dropped from the bottom of the window, how long will it take to hit the ground? How far along the track will it have travelled before it hits the ground?" Then you get answers like 3.43 hours, or -3.2 years, or it'll travel 8.9 km. And students think it's not obvious that they should have realized they said absurd things or get partial credit.