[tt] Gut Instinct’s Surprising Role in Math

[Eugen Leitl]

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

September 16, 2008 Basics

Gut Instinct’s Surprising Role in Math

By NATALIE ANGIER

You are shopping in a busy supermarket and you’re ready to pay up and go
home. 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 and will take you 90 minutes on
the elliptical to burn off and you’d better just stick the thing behind this
stack of Soap Opera Digests and hope a clerk finds it before it melts.

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.

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.

“When mathematicians and physicists are left alone in a room, one of the
games they’ll play is called a Fermi problem, in which they try to figure out
the approximate answer to an arbitrary problem,” said Rebecca Saxe, a
cognitive neuroscientist at the Massachusetts Institute of Technology who is
married to a physicist. “They’ll ask, how many piano tuners are there in
Chicago, or what contribution to the ocean’s temperature do fish make, and
they’ll try to come up with a plausible answer.”

“What this suggests to me,” she added, “is that the people whom we think of
as being the most involved in the symbolic part of math intuitively know that
they have to practice those other, nonsymbolic, approximating skills.”

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. (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
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?

Dr. Halberda, who happens to be Dr. Feigenson’s spouse, relishes the work’s
philosophical implications. “What’s interesting and surprising in our results
is that the same system we spend years trying to acquire in school, and that
we use to send a man to the moon, and that has inspired the likes of Plato,
Einstein and Stephen Hawking, has something in common with what a rat is
doing when it’s out hunting for food,” he said. “I find that deeply moving.”

Behind every great leap of our computational mind lies the pitter-patter of
rats’ feet, the little squeak of rodent kind.