I would expect your first proposal to lead to a serious shortage of math and science teachers unless paired with a commensurate increase in incentives, and possibly with other systemic changes.

If the school administrators that I've talked to are to be trusted, there's already a shortage severe enough to pretty much guarantee employment for people with even a modest math or science background (read: a few classes, or a major in a tangentially related field) who have teaching credentials. That's probably not a situation where you want to be introducing stricter qualifications.