Math manipulatives are concrete objects used as props in math education. As mere objects, they are, in and of themselves, not a bad thing. What is terrible is how they are often used in instruction.

At one school that I visited recently, students learned the standard formula for the area of a triangle using a plastic manipulative consisting of pre-cut triangles. The triangles’ pieces could be rearranged to form rectangles. Because rearranging pieces doesn’t change area, the area of the original triangle was equal to the formed rectangle. The rectangle’s length was that of the base of the original triangle and its width was half the height of the original triangle. So the student could deduce that the area of a triangle is half the base times the height.

Is that a good way to teach the area formula for a triangle?

Let’s suppose that a good math education should aim to empower students to more effectively use their minds to solve problems, and, particularly, to solve the yet unsolved and evaluate the question against this supposition.

Note that the pre-cut nature of the triangle manipulative makes the exploration of its area much more guided than the essential problem, which is to figure out a way to compute the area of a triangle. There is an infinitude of ways to cut a triangle into pieces. But it was the designer of the manipulative who chose from among this infinitude of ways the particular cuts that made the manipulative. A key idea for computing the area of a triangle is, therefore, implanted in the manipulative and the important exercise of finding the useful cuts from the infinitude is taken away from the student. This lost exercise is important because it is an exercise that gives the student the chance to develop the skill of coming up with criteria that will cull out the special from the general, and this skill is crucial to creativity.

The seduction of the pre-cut triangle manipulative is that it leads students to some understanding of the standard area formula for a triangle quickly, and that will please many parents because most parents like to see concrete manifestations of understanding. Students will also be pleased because they struggle less and feel like they have advanced faster.

It takes longer, however, to develop the skill of coming up with criteria that enable one to cull out the special from the general and it is much harder to see progress on this skill, not to mention that it is a skill that happens in the mind and as a more abstract skill, it is so much harder to see. Most students educated in the US do not hone this skill because they are not often given exercises to develop it – and this partly explains why Americans are often a minority in elite US math graduate programs.

A very rough analogy might be like going to a home economics class where the lesson of the day is to make chicken soup. In one class, the teacher has the students heat up a can of Campbell’s chicken soup. In another, the teacher tries to have the students concoct a chicken soup out of the raw ingredients that nature provides. After the first class, everybody will be enjoying some chicken soup. But after the second, it’s by no means clear if the class will have succeeded. Probably parents will end up politely tolerating sips while forcing the words “It’s delicious!” out of their mouths. But does that mean that therefore teachers should use Campbell’s soup?

I’m cautious about pushing this analogy further, but to try to make another point, I will push it just a bit more: What would you be thinking if one of the students who made the soup from a Campbell’s chicken soup can said to you, “Yes, I know how to make soup!” And what would you say to that student and his/her proud parents and teacher?

The pre-cut triangle manipulative kills other opportunities to learn beyond developing the skill to select the special from the general. For example, it has the built-in notion of comparing the area to a rectangle, an idea that would likely be better left to students to discover, though it will make the lesson take even longer.

Before you use a manipulative, please consider carefully what opportunities for your students you are destroying and think hard about whether you really want to rob your students of those opportunities.