For example, say I plan on studying Calculus. Calculus itself has many theorems and, as such, it has many proofs one should learn during his/her math career. But when is it recommended go and learn this proofs? I see three options:
- Go through the proof as soon you stumble upon a theorem/rule and learn how to apply it to some problems.
- Learn how to apply everything you learn from a subject such as Calculus I and then, after that, you go and learn every proof in Calculus I. And repeat the same for II and III.
- Like 2, but instead you learn Calculus I, II and III in a row and then you learn the proofs.
It seems option 1 would be the most rigorous, I wonder if it carries more benefits other than actually understanding why things work as soon as you see them working.
My recommendation . . .
While learning Calc 1,2,3, learn the concepts well, including the heart of the reasons why things are true, but go light on the proofs, else you'll take away the "joy of Calculus".
If you're heading towards Math Major, definitely start doing easy, fun proofs from other areas of Math, with independent study.
High School level contest exams provide a wealth of challenging problems, and working on them can promote the development of both problem solving skills and proof skills.
Also, echoing imranfat's comment, master precalculus. It's the base.