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There are hard math concepts that I didn't totally 'get' when I took the course decades ago, but coming back later I'd get it.
This is why, as a teacher, I don't buy the idea that some people just can't do proofs, or understand calculus, or see a point to category theory.
Time, increased experience with related math ideas, practice, vocabulary, becoming more a part of "math culture" made everything easier and easier.
Logical reasoning is natural to the human mind-- but so is "going with the gut" --every thinking person can do *both* and you get better at what you practice.
"If A then B, if B then C, therefore since A we know C." is natural reasoning *anyone* can do. And most people do it at least a little... though often in ways that divorce it from logic and mathematics.
The challenge is to help students feel ownership of that logical voice-- not alienation.
@futurebird but if C then not necessarily A
I strongly suspect that *everyone* even those most alienated from mathematics not only understand the concept that:
A->B is not equivalent to B->A
But, I think most people must use this idea to navigate the world, to make arguments and inferences. So, that means there is something about the abstraction, the formalizing, the cultural conventions of mathematics that makes it seem like they don't get it in math.
@futurebird @DrorBedrack I'm pretty sure lots of folks don't get this...
People will get tripped up by it, do lazy sloppy reasoning, everyone can do that... especially when it confirms what you already want to think is true.
But in a pinch, if you try to say to a student "You didn't post the homework therefore you were wasting time and goofing off." They will explain to you the hole in this argument like a lawyer.
@futurebird @DrorBedrack I don't think they can always explain that logically though. They'll just make an argument not appealing to the logical fallacy.