[fghvbnvbnfe]

That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point.

Checking is the smart thing to do then, but ideally what is important is the understanding rather than just the diligence to make sure a guess isn't trivially shown incorrect.

[sdenton4]

I strongly disagree. Brains are fallible; a quick check at the end lets you unit test your understanding. I find the ability to identify one's own mistakes and correct them to be a sign of an excellent student.

[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...

[Kenji]

>That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point.

You have no idea how much understanding I gained by doing such 'quick checks', sometimes even during exams. Just a little sanity check to be sure you're still on track.

Besides, inserting 1 for both x and y is an actual mathematical proof that the equation does not hold! It's always a fantastic thing to strengthen your understanding with formal proofs that you came up with by yourself.

[fghvbnvbnfe]

I've always done those quick checks myself because otherwise I'm prone to making errors. They give me a quick answer of "no, you can't do this" but I never found they actually improved my understanding of anything. It just told me I can't do the thing I wanted to do and it was time to move to the next idea. It's a valid proof, but not one that really ever helped me get a grasp on what's happening.

Mostly it just means I'm in trouble and need to be really careful and do more checks because I'm working with things I don't understand fully.

For example, I've never needed to do that sort of check with adding fractions. I have enough understanding of what those numbers represent and how they work together that I've just never thought I might want to add numerators and denominators separately.