Math discoveries catch kids unawares

Science News, Jan 2, 1999 by B. Bower

Many educators and scientists assume that conscious knowledge is the engine that drives learning. A new study suggests instead that, at least among grade-schoolers, unconscious problem-solving insights often set the stage for academic advances.

Second-graders who practice solving inversion problems--such as 8 10-10 = 8--start out by computing the answers but frequently turn to a more efficient strategy unconsciously. Without realizing they are doing it, they learn to ignore the number that is both added and subtracted, report psychologists Robert S. Siegler of Carnegie Mellon University in Pittsburgh and Elsbeth Stern of the Max Planck Institute for Psychological Research in Berlin.

Children more quickly attain this insight, and become aware of it sooner, when practice sessions include only inversion problems, rather than mix them with other math problems, such as 9 10-3.

However, after becoming aware of the shortcut, kids employ it only part of the time, returning at other times to more time-consuming calculations. In the long run, the child's nurturing of an array of problem-solving tactics allows for adjustments in tougher math challenges, Siegler and Stern propose.

"We find that a strategy can be used to solve a problem without people knowing that they used it," Siegler says. "We need to see if children and adults generate unconscious discoveries on other tasks."

Siegler and Stern studied 31 children, 8 or 9 years old, who attended an after-school center in Munich. In pretests, children said that they solved inversion problems by using computation (such as adding 10 and then subtracting it from 8) or negation (adding 10 to 8, then realizing that 10 - 10 = 0 and skipping to the result)--strategies taking at least 8 seconds per problem.

Children then completed six weekly practice sessions. Each session featured 20 problems, all inversions for some kids and a mix for others.

Nearly all children in both conditions discovered the shortcut strategy for solving inversion problems without realizing it, the scientists report in the December 1998 JOURNAL OF EXPERIMENTAL PSYCHOLOGY: GENERAL. Kids exhibited sudden, sharp drops in solution times, to 4 seconds or less, that are typically achieved by using the shortcut. Nonetheless, they continued to tell researchers that they used calculation or negation to reach correct answers.

Children who solved only inversion problems reported using the shortcut after having employed it unconsciously five or fewer times. Those exposed to a mix of problems took longer to become aware of using the shortcut and often cited a transitional strategy, such as saying "8 10 ... 9, 10, 11 [counting on fingers] ... oh, it's 8!"

Kids given only inversion problems proved more likely to apply the shortcut both to appropriate (8 10 - 8) and inappropriate (8 10 10) variations.

"This study shows that conceptual insights emerge unconsciously during practice," remarks psychologist David C. Geary of the University of Missouri in Columbia. "Learning may often be outside conscious control."