Abstract equations may be the best way to learn math, and the familiar word problems of traveling trains or Sally's apples might be mere distraction. Researchers at Ohio State University performed a randomized, controlled experiment with the two methods.

Students were taught a simple but new set of mathematical rules. Some were taught with concrete examples such as tennis balls in a container, and some learned only with abstract symbols. When they had to apply these rules to a new situation, those who learned only from the concrete method could not transfer the knowledge. Students who learned from abstract symbols fared best. Those who were provided with the symbols after learning with concrete examples were in between.

"They tend to remember the superficial, the two trains passing in the night," said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. "It's really a problem of our attention getting pulled to superficial information."

The research focused on college students, and the researchers have experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a "pervasive assumption" among math educators that concrete examples help more children better understand math. It is unknown how manipulatives help young children learn mathematical concepts, but this study suggests their value should be assessed.

Source: The New York Times, Study Suggests Math Teachers Scrap Balls and Slices, by Kenneth Chang.

See also the Scientific American article, Word Problems Fail Math Students and their podcast, 60-Second Science: Word Problems Fail Math Students