Study Suggests Math Teachers Scrap Balls and Slices is a New York Times article about research at Ohio State University that may have profound implications for the teaching of mathematics to young children. Unfortunately, just what those implications may be is completely obscure to me, as it is, I daresay, to the researchers.
I've written before (here, and here) about my belief that many children can learn abstract math concepts much earlier than is commonly supposed, and do not need to be limited to practical, real-world mathematics until they reach Piaget's magical age of abstract thinking. So I suppose I should have been pleased to read about a study that purports to show that children actually learn mathematical concepts better through abstract symbols than they do through practical examples. Not so; I'm also a fan of practical math, and believe both to be important. In any case, it's hard to draw any conclusions one way or the other from this research—or at least from the Times report.
In the experiment, students were taught a mathematical concept either through abstract symbols or through concrete examples such as combining liquids in measuring cups. They were then asked to apply that knowledge to figure out the rules of a game. Those who had learned the concept abstractly did better at that application than those who had learned through the concrete examples, and also better than those who had been taught first the concrete examples and then the abstract ones. Supposedly the real-world examples actually obscured the underlying math.
I'm going to break the post here, because some of you will want to look at the puzzle yourselves before reading about my experiences. I strongly encourage you to read the article, but if you don't do that, at least follow the link and click on the link in the upper left under "Multimedia" that will show you the problems. (It's a javascript pop-up, and I can't figure out how to insert it here without offending someone's copyright.) Then click the "more" link after this paragraph to continue reading this post.I quarrel with the application of this research to any conclusion other than "We need to study this further." For one thing, The laboratory rats in question were college students, whose brains might reasonably be expected to have already become capable of abstract thought. It would be risky to apply anything learned from them directly to six-year-olds. For another, the "concrete" examples appear—thought it's hard to tell for sure from the article—to be abstractions themselves. Thinking about a measuring cup filled 1/3 full of liquid is more similar to thinking about a circle than it is to pouring liquid from an actual, physical cup.
Believing my own brain to be reasonably competent at both mathematics and abstract reasoning, I was shocked at how difficult I found the problem. I couldn't get my mind around what they were trying to show. What finally gave me the necessary clue was actually the measuring cup example, but thought about abstractly, as numbers. Thinking about its physical properties was, indeed, distracting from the underlying mathematical concept, but the abstract symbols were no more helpful. When I finally thought about the problem in terms of numbers alone, then answer was obvious, and to my mind much more easily explained abstractly—though I do still have a problem with the third example in the measuring cup system, so I may be missing something even though I got the correct answers. I'll be very interested to hear what other people think. Especially my reader from Oswego. :)
I wouldn't change the way I taught anyone mathematics based on this study alone, but I would be very interested in hearing the results of a similar study done with other demographics: not only children of different ages, but also adults of various occupations and levels of education.
