Teaching Math

Modern fads in math teaching tend to lean heavily on so called manipulatives - the theory being, apparently, that the hands are the window to the math brain. Kenneth Chang reports in The New York Times that Ohio State U reseachers have cast doubt on this theory.

An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”
. . .
In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

There are a whole lot of problems with generalizing from a study of college students to first graders, or even high school students, and the researchers are now planning to something similar with younger students.

I don't have a dog in that particular race - I think that I do better reasoning from the particular to the general rather than vice versa, but the confusing effect of extraneous information is also a persuasive idea.

More importantly, for me, is what this study reveals about the crappy state of our educational practice. Why is the answer to this question not known? If millions of kids are being subjected to manipulatives, and if schools are paying billions for said manipulatives, why don't we know if they work or not? Why haven't large scale, multi-method random comparison tests been carried out?

If a textbook publisher, or other vendor of school materials has a teaching method they wish to sell, the country should carry out a large scale comparison test of several methods, in real classrooms, with randomly selected schools and teachers.

There is no excuse for us to let education remain a fashion industry. Many far less vital aspects of society have been subjected to the discipline of the scientific method. It's time for education to join the modern world.