That's an American problem though. In most of Europe you need a masters degree to teach highschool and that involves at least an undergrad level of understanding the subjects you will teach.
E.g. in Hungary I had a university CS professor that originally wanted to be a highschool teacher and a highschool physics teacher that originally wanted to be researcher. Their choice of degree didn't determine which outcome they got. The researcher and teacher curriculum had an 80%+ overlap.
> If you can come up with a way to do math without reasoning, that would be, in a sense, even more interesting than AI.
Logic is just syntactic manipulation of formulas. By the early 90s logical reasoning was pretty much solved with classical AI (the last building block being constraint logic programming).