Sat next to a nice young man on the airplane. He asked what I was studying, or doing, or... I said I was a mathematician. (I still have trouble saying this with a straight face, and during the semester I used to say I was a teacher. Why is it easier to say that? Why do I still sort of feel afraid I'm lying when I say I'm a mathematician? I've had my PhD for several years and am actively -- daily -- involved in research.) He said, "Oh, wow, I've never met, like, a real mathematician!"
He was not from the US. (Inevitably) he told me he'd never been that great at math; I told him that what he did in school bore little relationship to what we as mathematicians do, in that he was trying to find the answer when he did his problems in school while we are often trying to figure out what the right question is. I was working through a series of examples on a topic that's given me some trouble lately. In that I was simply doing a bunch of computations and looking for the right answer for each one, my mathematics and his school math were similar. However, I was looking to get a feel for the general situation and prove a few consequences of my starting hypotheses and explore what would happen if I relaxed those hypotheses. It is this synthesis that seems to separate research mathematics from school math. He asked a good question: if we switched to base 60 and counted like Mayans would the quadratic formula still hold? I said yes. Then I asked him, if we counted like Americans tell time (mod 12), how many solutions to x^2-1=0 would there be? For instance, would 5 or 11 or 13 suddenly be solutions? What would that mean? He seemed a bit disturbed.
I suppose that some of us do construct some homework sets to lead students to a larger realization about a type of problem or a property of limits... how many of them realize that? What does it take to shift one's viewpoint to "doing twelve problems to explore all sides of a bigger question" from "doing twelve problems as fast as possible"?