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In Abstract: Avoid Concrete Examples When Teaching Math

New study indicates that extraneous information in word problems may cover up mathematical concepts
chalkboard blackboard math symbols



© ISTOCKPHOTO/BART COENDERS

Real-world math problems: for many of us, they were the bane of our existence during high school. A train would leave New York City at a given time headed south at some speed. Another would leave Atlanta headed north at a different clip. It was up to students to determine when and where the two coaches might pass one another.

But could all of that information have been for naught? New research published in Science suggests that attempts by math teachers to make the subject easier to grasp by providing such practical examples may actually have made it tougher to learn.

Scientists at The Ohio State University (O.S.U.) found that for all the imagery of conductors, tracks, passengers, engines and cabooses—or what have you—this problem is really just about different rates of change: How much ground one train can cover in a set amount of time compared with the other? And so, asked the researchers, why not just use symbols and numerals to cut to the chase?

"The primary goal of learning math is this ability to transfer that mathematical knowledge," says study co-author Jennifer Kaminski, a researcher at O.S.U.'s Center for Cognitive Science. "Concrete examples might not be the best to promote transfer; they present a lot of extraneous information."

For their study, Kaminski and her colleagues taught 80 undergraduate students—split into four 20-person groups—a  new mathematical system (based on several simple arithmetic concepts) in different ways.

One group was taught using generic symbols such as circles and diamonds. The other groups were taught using practical scenarios such as combining liquids in measuring cups.

The researchers then tested their grasp of the concept by seeing how well they could apply it to an unrelated situation, in this case a children's game. The results: students who learned using symbols on average scored 80 percent; the others scored between 40 and 50 percent, according to Kaminski.

"When you present things with a lot of extraneous details, it's an automatic pull over to [those] more attention-grabbing details," rather than the underlying mathematical concepts, Kaminski says.

Keith Holyoak, a psychologist at the University of California, Los Angeles, says, however, that the findings are not that black and white, noting that in some instances the students who learned by example scored as high or better than the symbols-only bunch. "If they're trying to make the conclusion that abstract is always better than concrete training, I would disagree," he says. "Good problem solvers or better abstracters are better at dropping away what is superficial."

Kaminski hopes this new work will encourage other researchers to review teaching methods that might help make the next generation become more mathematically literate. "If you're a believer in concrete," she says, "then demonstrate that it's better and that students apply that knowledge outside the math classroom."

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