Babbling Blueberry

There’s a study about learning theory published in this week’s issue of Science. “The Advantage of Abstract Examples in Learning Math.” There’s also an article about it in the NYTimes. “Study Suggests Math Teachers Scrap Balls and Slices.” Even though this study is getting a lot of press (as of this afternoon, it’s the top article on the the NYTimes’ list of most e-mailed articles), I think the study is really flawed. So I want to publish my own thoughts on this article before people go and change how they teach math based on this one study.

The study’s goal is to determine whether teaching using abstract ideas or with concrete examples is better for helping students recognize and deal with novel situations and problems. I agree that it is an interesting goal. They teach some undergraduates a set of “mathematical” rules and operations using either 1) only abstract symbols, 2) one of various concrete examples, or 3) a concrete example followed by the abstract symbols. They then give the subjects a new situation involving a children’s game that uses what the authors say are the same rules and operations and determine which students are best able to figure out the novel situation. They conclude that teaching with only the abstract ideas was the most effective by their measure, using one of the concrete examples was the worst, and using a concrete example followed by the abstract ideas was better than using only the concrete example, but not as good teaching only in the abstract.

All these conclusions depend on what abstract and concrete situations they use, so let’s examine that.

The game situation used to evaluate the learning of the subjects much more closely resembles the abstract/symbolic example in that the task is to memorize a set of symbols and rules for combining them.

The concrete example, on the other hand, involves a “real world” process that combines two volumes of liquid and the operation is based on what happens when you combine two liquids. (*See explanation below if you’re interested.) The concrete example makes sense, but it is inappropriate and unhelpful for helping students accomplish the evaluation task. The objects in the evaluation task don’t have different volumes, so the concrete learners are not only required to determine the rules of the operation, they also have to determine how the various volumes map to abstract symbols. The researchers are requiring the learners with the concrete example to learn more information and solve a harder problem. Plus, the evaluation situation doesn’t look much like the concrete training example. So I’m not surprised that these subjects had a harder time with the evaluation task.

Teachers use concrete examples to provide more meaning and context to the abstract math. For example, if you’re teaching addition, the concrete examples you use might involve fruit in a basket and determining how much fruit you have after adding additional fruit. This example would provide a real world situation that students will recognize and it gives something for the students to visualize or imagine when they’re solving “pure” addition problems. Good concrete examples are related to the abstract concepts that you are trying to teach. The “concrete” learning situation modeled in this study is analogous to teaching addition by having students determine the total amount of fruit in a basket after adding additional fruit, but first, you have to slice up all the fruit and rearrange the slices.

All this study shows is that if your testing situation is more like what subject group A has been trained on and less like what subject group B has been trained on, group A will perform better on the evaluation. If, instead of an evaluation based on symbols, the researchers tested their subjects with a situation involving pie slices or stacking blocks or filling baskets with apples, I am pretty confident that the “concrete” subjects would have performed better. But a study like what I am suggesting, even though it better reflects real teaching methods and learning goals, would have not gone against “conventional wisdom” and would not have gotten published in Science and the New York Times.

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*The operation in the concrete example is: combine the volume of liquid in two containers (that will fill 0 or 1 full containers and there will be some remaining) and take the remaining volume. If the remainder is 0, the answer is a full container.

Described more abstractly, if we are looking for the answer (r), and we are given the volumes of the starting containers (v_A, v_B), and the total possible volume of the container (v_Total):

r = (v_A + v_B) mod v_Total; if r = 0, set r = v_Total