<p>Well, Linear Algebra is a natural place to introduce proofs. In fact, I think most linear algebra courses require some proofs, although there is a huge variety in how theoretical it could be (a few simple proofs throughout the semester on one end vs. entirely proof based on the other). They could also take two semesters, first one that’s computation based and then another that would proof based.</p>
<p>If someone knows they’re going to be a math major coming into college, then they’d probably want to take an honors version of the Calculus sequence that would be proof based (like linear algebra, this can vary a lot in just how rigorous). Another example would be discrete math. This is where computer science students usually learn about proofs - no reason math majors can’t as well. In any case, there’s a big gap between purely computational intro calculus and Rudin-style real analysis. </p>
<p>I don’t think many (or even any) schools currently do this, but one solution would be to have something like a 1-credit seminar for potential math majors that would introduce them to proofs. The main problem I have with “intro to proofs” classes is that the stuff you need to know in order to write a correct proof can’t possibly fill up a whole course. So these classes usually have a whole bunch of other stuff packed in that, while interesting, are also quite unnecessary. A short seminar would probably still be more than they “need to know”, but it could include stuff like problem-solving strategies and general mathematical information.</p>