[fghvbnvbnfe]

I agree with you in general, but not when only talking about the mathematical understanding.

If you want to add fractions by individually adding the numerators and denominators and the only reason you don't is one of these checks, you have some kind of fundamental lack of understanding about fractions. It's great that you are wise enough to verify, but that's irrelevant to your mathematical understanding.

[sdenton4]

Misunderstanding also exists, though, at many stages of the game. There's a pair of famous books - counterexamples in topology and another on analysis - dedicated entirely to counterexamples which dispel some common mis understandings.

And these misunderstanding are not limited to undergrads; often they stand in the way of progress at the edge off research, simply because we don't actually know what's possible. Research level math is often guided by folklore: important conjectures and shadows of what might be. And when the folklore is wrong, we go down the wrong track until someone finds a counterexample...