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Abstract versus concrete in mathematics education

In the New York Times, April 25 2008 Kenneth Chang wrote an article reporting on research done by Jennifer Kaminski, Vladimir Sloutsky, and Andrew Heckler at Ohio State University and reported in Science. I've quoted the core of that article 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.”

Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.
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.

I find the results of this study partly obvious and partly misleading. It is very obvious that when trying to teach a concept, giving examples with too many extraneous details will confuse the student. Imagine trying to devise checkers strategy when each of your checkers was a different shape. It would be very hard to avoid coming up with strategies that had nothing to do with the rules of the game, like: "a square checker should always jump over a triangular checker". Some degree of abstraction is necessary in order to allow the transfer of knowledge from one domain to another.

The study was done with undergraduate college students. I find it hard to believe that the researchers will replicate their results with children in grades K – 12, given what is known about the differences in learning with age. I also question the usefulness to practice of studies that report the statistical distribution of learning results from two teaching methods (say an "abstract" method and a "concrete" method). It is well established that different students have different learning styles, and that these styles may be determined, either by testing or informally by trained teachers. A teaching method that works well for a student with a strong verbal-procedural learning style may work poorly for one who has a visual-kinesthetic learning style. If one style is predominant in a population and the abstract style works better for that style, the results will be better on average, but it might be a great mistake to teach all children using that method, in effect writing off the students with the minority style.

The system the researchers used is isomorphic to the cyclic group with 3 elements, or addition modulo 3. There was a concrete model where the elements were measuring cups filled 1/3 full, 2/3 full, and 3/3 full, which I found the easiest to understand. The abstract symbolic model where the elements were circles, diamonds, and an irregular figure seemed much more mysterious and indeed Richard Weiss pointed out to me that it is possible to construct two non-isomorphic addition tables that fit the abstract model. See the New York Times article for more detail.

One reason given by proponents of multiple concrete representations is motivation, related to the idea that students must see mathematics as relevant to their lives before they will invest effort in learning it. I've always felt this is dubious, because in most textbooks or classrooms the supposedly real-life scenarios seem cooked up. I feel there are two ways to enhance motivation through curriculum design: (1) Go all out. Get students interested in some activity that excites them and requires real mathematics, such as the design of computer games, building a robot or a racecar, etc. The problem with this approach is that it fails to address the testing mania which grips our country. Students who learn mathematics this way are going to acquire skills in a non-standard order, and will not know some topics that they need to pass high-stakes tests. (2) Teach mathematics through problem solving. Eliminate, as much as possible, phony word problems. Instead tap children's innate curiosity and competitiveness with questions like: "Can 2467432 be a perfect square (no calculators allowed)?"