As a student in mathematics and physics, I’m part of two different worlds. On the one hand, proofs and abstraction come from the side of mathematics. On the other hand, physics is where concrete examples and applications are the norm. In physics (at least, within the scope of undergraduate education), we only care about the mathematical tools that we can apply to a given problem.

Learning mathematics in school and doing mathematics in general are not the same thing.

In mathematics, the terms “necessary” and “sufficient” have technical meanings. These terms come about when looking at two statements P and Q. If we say that P is sufficient for Q, then that means if P is true, Q automatically has to be true (P implies Q). On the other hand, if P is only necessary for Q, having P be true doesn’t mean Q has to be true (but the other way works, so Q implies P). If we have the P is both necessary and sufficient for Q, that means having one gives us the other for free. They are tied together and are inseparable.

Educators have unrealistic views about memorization.