As a mathematics instructor, what is the most valuable (to you) thing that classroom experience taught you about mathematics education?
To be specific, I'm looking for answers not about mathematics itself, or student discipline unrelated to mathematics, but about the learning, teaching, or reasoning of mathematics.
[To split some hairs, there is knowledge of mathematics that falls into this category; the difference between quotative and partitive division is a good example of pedagogical content knowledge. It's something most math users don't need to know about, but it helps teachers understand student conceptions of division. If you learned something like that through classroom experience, it would be a great answer and probably a great story as well.]