To rephrase the question I mentioned yesterday, what exactly happens when we have a problem before us, we think about it, and then arrive at a solution? Can we take snapshots of what happens in between the question and the answer? Maybe something like the think aloud protocol can help, and I'm interested in trying that out at some point.

On the surface, at least, what's happening is paradoxical. How is it that one moment I have only a vague, cloudy mental picture of something, and a few minutes later, without referring to information outside of myself, I have an answer? In Meno, an early dialog written around 380 B.C., the question is expressed like this:

One "cannot search for what he knows - since he knows it, there is no need to search - nor for what he does not know, for he does not know what to look for" [80e, translated by G.M.A. Grube]

The latter case is especially interesting. What happens when we think about a problem, and somehow, even though we may not be aware of what is happening, we see it more clearly?

It may be, in my case, that I don't see anything more clearly, I just see it differently. If I wonder how to proportionally reduce a rectangle, for example, and I arrive at a formula that works, I don't have to understand how it works. A more obvious example is a child who memorizes the multiplication table and learns algorithms such as long multiplication. He might become very adept at finding the answer to complicated equations without having an inkling of what he is doing or why he is doing it. How many students stop trying to learn a concept once they learn the procedure or algorithm for solving a problem? How many engineers, for that matter, learn just enough to get a job done? It's easy to confuse "I know how to solve it" with "I understand the answer."