Brilliant: The Science of Smart

Why Guessing Is Undervalued

Being able to estimate may be more important than doing quadratic equations

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Quick, take a guess: About how many feet high is an eight-story building? Approximately how many tons does the average pickup truck weigh? About how many oranges must be squeezed to yield a gallon of juice?

Maybe you gave these your best shot — or maybe you skimmed right over them, certain that such empty conjecture isn’t worth your time. If you fall into the second group, you may want to reconsider. The science of learning is demonstrating that the ability to make accurate estimates is closely tied to the ability to understand and solve problems. Estimation, this research shows, is not an act of wild speculation but a highly sophisticated and valuable skill that, some experts say, is often given short shrift in the curriculum. “Too much mathematical rigor teaches rigor mortis,” says Sanjoy Mahajan, an associate professor of applied science and engineering at Olin College. Many math textbooks, he notes, “teach how to solve exactly stated problems exactly, whereas life often hands us partly defined problems needing only moderately accurate solutions.”

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Everyone, even people without formal mathematical training, possesses a basic capacity to estimate. This aptitude appears astonishingly early in life: babies are already able to discriminate between different-sized sets of objects at six months of age. But it’s also the case that there are pronounced individual differences in the ability to estimate, and that these differences are linked to a more general facility with arithmetic. Especially in children, it appears that one leads to the other: strong estimation skills lay a solid foundation for learning more math as students grow older. In a 2004 article published in the journal Child Development, for example, psychologists from Carnegie Mellon University reported the results of an experiment in which they showed a group of elementary-school pupils a line with a 0 at one end and a 100 at the other. The researchers asked the children to indicate where they thought various numbers would fall on the line. The more accurately a child estimated, the higher was that child’s score on a math achievement exam.

Other researchers have examined the strategies used by people who are skilled at estimating and explored how such techniques could be taught to all. Their first finding: good estimators possess a clear mental number line — one in which numbers are evenly spaced, or linear, rather than a logarithmic one in which numbers crowd closer together as they get bigger. Most schoolchildren start out with the latter understanding, shedding it as they grow more experienced with numbers. Surprisingly, one of the best ways to give kids such experience is to play board games with them. Flicking the spinner or rolling the dice in a game like Chutes and Ladders, then counting out the number of spaces to move their tokens, gives them helpful cues as they construct the number line that they carry around in their heads. And, in fact, an intervention program employing board games, led by professor of education Sharon Griffin of Clark University in Massachusetts, produced large and lasting improvements in children’s math performance.

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Another strategy used by good estimators is to compare an unfamiliar quantity to one they know well: a football field is the length of 60 Dads, stretched out head to foot. Parents and teachers can help kids acquire a large and flexible store of mental benchmarks by remarking on the dimensions they encounter in everyday life: how many miles from home to school, how many pounds a basket of apples. Children benefit, too, from hearing the range of others’ estimates — so try having each member of the family guess how long it will take to get to Grandma’s house, or having each student estimate how many inches of rain fell last month. This open-ended approach will give kids a familiarity with the way math works in the real world — and tools to help solve real-world problems. How often does life hand us such problems? Professor Barbara Reys, co-director of the Center for the Study of Mathematics Curriculum at the University of Missouri, puts the proportion of mathematical applications that call for approximation, rather than exact computation, at 80%. Of course, that’s an estimate — but it sounds to me like a pretty good guess.