Welcome to Your Child's Brain: How the Mind Grows From Conception to College (37 page)

Chapter 24
GO FIGURE: LEARNING ABOUT MATH

AGES: BIRTH TO EARLY TWENTIES

As Barbie famously said, “Math class is tough!”—and not just for girls but for everyone. Your child’s brain is optimized to provide rapid solutions to everyday problems. That means it is less suited to solving an algebraic equation than to calculating whether it would be a good idea to punch that kid who just insulted him. (Of course this social calculation does involve some numerical ability, since it’s important for your child to determine whether the other kid has more friends available nearby than he does.)

Young babies and many other animals share a brain system that supports this sort of rough number sense. Under the right conditions, this number sense can combine with our species’ ability to create and manipulate symbols to produce formal mathematics, found in some societies but not others. Indeed, math, a seemingly inhospitable place for dandelions to grow, is surprisingly fertile ground.

In the last few decades, our appreciation for babies’ ability to form number-related concepts has expanded tremendously. Infants express surprise by looking longer (see
chapter 1
) if one object goes behind a screen and two come out. If an infant sees a Mickey Mouse doll go behind a screen and then the screen lifts to show a truck, she doesn’t care. If she sees a Mickey emerge along with a second Mickey, now that’s a surprise, as evidenced by her long gaze. This ability to notice an extra object—the twoness of the Mickeys—is a necessary component of numerical concepts.

This ability goes beyond small numbers. When a six-month-old infant sees a series of pictures, each containing a number of objects—dots, faces, anything—he will notice if the number either doubles or decreases by half. This general sense of
numerosity
gets better with age, too. While infants can recognize a 1:2 ratio (for instance, comparing 4 and 8 objects, or comparing 6 and 12) without counting, adults can recognize a more subtle 7:8 ratio.

Numerosity detection, the ability to distinguish between groups of different sizes, is an ability that all humans have.
Subitization
, another universal capacity, refers to the ability to immediately distinguish small numbers without counting. The term comes from the Latin word
subitus
, which means “sudden.” Both numerosity detection and subitization are apparent in other animals—and they involve some of the same brain mechanisms in people.

These abilities, which can be seen in mice, dogs, and even pigeons, provide an
obvious survival advantage: they allow us to estimate the quantity of something, from food sources to possible enemies. For instance, a subgroup of a pride of lions reacts differently to roaring sounds depending on how many lions they hear—and on how many members are in their own group. If they are outnumbered, they call to the rest of the pride for backup. Similarly, chimpanzees avoid conflict with other groups when they are outnumbered.

One reason it took so long to understand young children’s number sense is that early researchers (such as Piaget) asked the wrong questions. If asked “which row has more objects?” children aged three or four will point to a smaller number of clay pellets if that row is spaced out to look longer. Change the pellets to chocolate candies that they can have right away, though, and children do much better. In retrospect, it appears that this research tested for two things: a sense of number and the ability to express it in a clear way. Your three-year-old knows, but evidently she isn’t saying. Aside from her mouth being too full of chocolate to talk, her awareness isn’t accessible to an interviewer’s questions.

Oddly, two-year-olds do fine with either pellets or candies. This result seems to imply that at that age, children have a clear sense of numerosity, but then they lose the abstract sense of it for about a year. What could be happening here? One possibility is that at three or four, children’s brains are in the midst of hooking up their intuitive understanding of quantity with an explicit, later-developing sense of abstract numbers. By five it’s all sorted out, at which point she simply counts the pellets—and perhaps wishes for candies instead.

Grabbing candy bits may seem somewhat primal, as indeed it is. Evidence suggests that chimpanzees can also combine quantities in a mental operation that resembles addition. If a chimpanzee is shown two trays in succession, each with a different number of chocolate bits, he can determine whether the combination of the two trays contains more or less than a third tray. The rudiments of arithmetic are therefore evolutionarily older than our species and are basically one facet of your toddler’s inner ape.

These senses of number involve similar brain regions in people and chimpanzees. Numerical information seems to be represented in the prefrontal and posterior parietal lobes. One key location is the
intraparietal sulcus
, a buried groove in the brain where specific semantic number content (for example,
seventeen
) is represented. When this brain region is damaged, people can give approximate but not exact answers—at about the level of a chimpanzee.

This retained ability for general numerosity has led scientists to suggest that our brains represent numbers in a way that relates to their relative magnitude, as a mental number line. One piece of supporting evidence for this idea is that when we are asked to judge which of two numbers is larger, it takes longer to answer when the two numbers are close (8 versus 7) than when they are far (8 versus 2), as if the closer numbers are actually closer in mental space. Judging between closer numbers produces more activity in the intraparietal junction. You could imagine numbers being stored in some computerlike, digital fashion, where small differences are just as easy to detect as large differences, but instead, brains seem to use a more ordered representation, akin to marks on a ruler.

Can an individual child’s general capacity to handle quantity be trained even before she begins to count?

In monkeys, some neurons of the left and right intraparietal sulcus fire when the animal encounters a particular number of objects—or an approximately similar quantity. In general, these brain regions are part of the same major pathway in the brain for identifying where things are, including how many things are there and where they are going.

The “where” capacity of the parietal cortex (see
chapter 10
) seems to encompass a variety of functions. The posterior parietal cortex becomes active, in both monkeys and people, in conjunction with eye movements. In relation to math, neuroscientists noticed a curious additional capacity of this brain area by having people do simple math while they were lying in a fMRI scanner: the same regions become active when people do mental addition and subtraction problems, even though the eyes are not moving. Nearby parts of the brain with many shared connections to this region are intimately involved in visual functions, such as the abrupt eye movements called saccades, attention, and detecting which way a visual pattern is moving. Thus the way we look at space might be closely tied up with our mental number line. The pattern of activity in the posterior parietal cortex can even be used to predict with middling accuracy whether a person is adding or subtracting.

This seemingly odd overlap in the brain of eye movement commands and basic arithmetic suggests that some aspects of our brain’s ability to process abstraction are built upon our capacities for dealing with the physical world. Many cognitive abilities other than arithmetic seem to be “embodied” in a similar manner. In this way, we are able to think abstractly with brains that evolved for more concrete actions, such as looking for prey or finding a path through a forest.

PRACTICAL TIP: STEREOTYPES AND TEST PERFORMANCE

People’s performance changes a lot if they’re reminded of a stereotype just before an exam—even by checking a box for male or female. Any relevant negative stereotype can impair performance, especially when people believe that the test is designed to reveal differences between groups. Stereotypes can be activated even if test takers are not aware of the reminder, for instance, when African American faces are briefly flashed on a computer screen. Even more curiously, these effects can occur in people who are not members of the stereotyped group: young people walk more slowly after hearing stereotypes about the elderly. This appears to happen because thinking about the stereotype takes up working memory resources that would otherwise be used for the test.

A little effort can minimize this problem. Obviously, teachers shouldn’t expect certain students to perform poorly. Standardized tests should collect demographic information at the end of the answer sheet. The effect also works in reverse: girls do better on a math test after hearing a lecture on famous female mathematicians.

Most people belong to more than one group, so perhaps the most practical approach is to bring a more positive stereotype to the task. For example, a mental rotation task shows consistent sex differences, with men faster and more accurate than women. When reminded of their gender before this test, women got only 64 percent as many correct answers as men. In contrast, when reminded that they were students at a private college, the women got 86 percent as many correct answers as men. The men did better when reminded of their gender, while the women did better when reminded that they were elite students. Thus the gap between men’s and women’s scores was only a third as large when women were reminded of a positive stereotype that fit them as opposed to a negative stereotype. The remaining gap is likely to be a real sex difference: a single shot of testosterone temporarily improves women’s performance on this test.

Stereotyping is a strong brain tendency, unlikely to disappear soon (see
Did you know? Stereotyping and socialization
). Instead, we recommend taking advantage of such brain shortcuts by reminding your children of a stereotype that will improve their performance.

Converting these approximation abilities into the precise representations of formal mathematics requires symbolic representation. This capacity comes with language, which is an elaborate means of representing information efficiently. Parrots, dolphins, macaque monkeys, and chimpanzees can use symbols to represent numbers. For example, two macaques named Abel and Baker were able to pick the larger of two digits to get a larger number of candies. For the most part, animals cannot combine the symbols to add or subtract. One exception is a chimpanzee named Sheba, who after several years of training could perform some simple addition.

Even though people have the mental capacity for arithmetic and mathematics, they don’t always use it. The researchers Stanislas Dehaene and Pierre Pica investigated the Mundurukú, an Amazonian group that lacks arithmetic and has very few words for numbers. A few of the words they do have are precise (
pug ma
= one,
xep xep
= two), but most are approximate (
ebapug
= between three and five,
ebadipdip
= between three and seven). Mundurukú do approximate addition of large groups of objects very well, performing as accurately as Western, numerate adults. But exact calculation of small numbers is beyond them; for instance, if six beans are placed in a jar and four are drawn out, when asked how many are left, they say “zero” or “one” more often than they say “two.”

Formal arithmetic ability is predicted by a child’s earlier capacity for approximate number. This suggests that individual children differ in their general ability to handle quantity even before they begin to count. Can this capacity be trained? Perhaps children could be taught to do approximate number problems to improve their later acquisition of arithmetic skills. Although this idea has not been tested, it presents an intriguing possibility.