LC0a 3.Powerful Ideas

#### 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?

Computer-based microworlds such as we have long advocated, 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.

Publication notes:

• Written in 1981.
• Published in MIT AI Memo 652 and Logo Memo 60, “Some Powerful Ideas,”1982, April.
• Published as a series “Logo Ideas,” in Creative Computing, 1982-83.