Study Suggests Math Teachers Scrap Balls and Slices, from today's New York Times.
The Times article is about a study reported on in today's issue of Science (Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler1. The Advantage of Abstract Examples in Learning Math. Science 25 April 2008: Vol. 320. no. 5875, pp. 454 - 455). Researchers taught the idea of the group Z3 to some students who weren't familiar with it; some learned it "abstractly" (the elements of the group were represented as funny-looking symbols) and some learned it "concretely" (by considering the slices in a pizza with three slices, or thirds of a measuring cup, or tennis balls in a three-ball can). It seems that the ones who learned the "abstract" version more easily picked up the rules of yet another "concrete" version (a children's game) than those who learned the original "concrete" version.
The Science authors claim that this is because "Compared with concrete instantiations, generic instantiations present minimal extraneous information and hence represent mathematical concepts in a manner close to the abstract rules themselves." This seems like the whole point of mathematics -- a lot of what we do as mathematicians is to strip away extraneous details of a problem while retaining those that are actually significant. If you learn about fractions by thinking about slices of pizza, perhaps you will always think that fractions are about pizza. And then whenever you hear about them, you'll think "where's lunch"?