Concrete examples don't help students learn math, study finds

In the research presented in Science, the researchers did several separate experiments that examined how well undergraduate students learned a simple mathematical concept under different conditions. The concept involved basic mathematical properties such as commutativity and associativity – the fact that you can change the order of elements without changing the results. For instance, 3+2 and 2+3 both equal 5.

In the various experiments, some students learned these principles using generic symbols, in which combinations of two or more symbols resulted in a predictable resulting symbol.

Others were presented with one or more concrete examples that involved this same concept. In one concrete example, students viewed three images of measuring cups with varying levels of liquid. Participants were told they needed to determine the remaining amount when different cups of liquid were combined.

Two other concrete examples were used in various experiments – one involving how many slices of pizza in a pizza pie were overcooked, and one involving how many tennis balls were in a container.

After learning this math concept using the concrete examples or abstract, generic symbols, the students took a multiple-choice quiz demonstrating that they learned the principles involved. And in all cases, the study showed that most undergraduate students picked up the knowledge easily.

However, the true test came later when the researchers asked these students to apply the same principles in a totally different setting, which was described to them as a children’s game from another country. The rules of this game followed the principles which they had just learned. The researchers calculated how well the participants did on a multiple choice test involving the rules of that children’s game.

In the first experiment, involving 80 students, some participants were given one concrete example before testing on the children’s game, while some were given two or three examples. One group only learned the generic symbols.

When tested on the children’s game, the group that learned the generic symbols got nearly 80 percent of the questions right. Those who learned one, two or even three concrete examples did no better than chance in selecting the right answers.

“They were just guessing,” Kaminski said.

In a second experiment, the researchers gave 20 participants two concrete examples and explained how they were alike. Surprisingly, this still did not help students apply the concept any better and they still did no better than chance when tested later about the game.