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What's a Powerful Idea?

Everybody knows what a grapefruit is and how to cut one in half. When you cut a grapefruit "properly," perpendicularly to the core, the cut face shows a pattern like a wheel. A little energy and perseverance are all you need to dig out and enjoy the juicy meat from between the spokes. But there must have been a time when you didn't know which way to cut a grapefruit. Did you find out how the hard way? What happens if you divide the grapefruit the other way, along the core? It is nearly impossible to eat the still buried meat, for the tough skin of the sections is an intact obstacle. The grapefruit looks pretty much uniform on the outside of its skin, but when you look inside you can see there is a very specific and important organization that you must understand if you want to get at the meat.

This very simple, very concrete situation provides a useful way to look at many other very troublesome problems. It points up the issue that how you analyze a problem, whether your analysis goes "with the grain" or "against the grain," can make a world of difference in how hard the problem is to solve. The intrinsic character that gives such an example power in thought is its simplicity, in the sense that given the perspective of the idea (embodied in the concrete example), the primary conclusions that can be drawn are obvious without long chains of arguments. Such ideas are elegant in the mathematician's sense. The extrinsic root of power is the example's fruitfulness, how well it can serve in helping you understand other problem situations by analogy.

The more powerful ideas you have the better. If you have only one way of looking at a situation, you are a prisoner of a limited point of view. If you can interpret a situation in terms of several possible models or representations, you can compare the fit of each to judge which is the most appropriate. As a situation changes, some alternative model may come to fit the situation better than the one originally best. Could it be that the flexibility of mind we ascribe to "smart" people derives directly from their having a well developed stock of such powerful ideas?

The computer-based microworlds described in this section present different representations embedded in activities that some children will enjoy. From playing with such microworlds, those children will better understand "what's what" later, whether they face problems in a more formal environment or solve problems of their own posing.

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 | Learning and Computing | Education | Computing | Psychology | Artificial Intelligence |