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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:

  1. Go through the proof as soon you stumble upon a theorem/rule and learn how to apply it to some problems.
  2. 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.
  3. 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.

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  • $\begingroup$ Before you start with Calculus, make sure you are very proficient in pre-calculus. That is what you should focus on first. When it comes to your three options, my advice is to hang between 2 and 3. As far as 1 is concerned, you should understand the theorems in terms of what they mean and what they do (direct application). Proofs can be studied only if they are relatively easy, otherwise no. The more difficult proofs will be understood when you mature more in math. Remember that theorems also come with examples. They are worth a pot of gold $\endgroup$
    – imranfat
    Apr 19 '17 at 1:20
  • $\begingroup$ I think the first step should be to understand what the idea behind the math is in as concrete a manner as possible. I learned tons of math before I met some people who really understood it. For example, I mentioned exponential functions to someone, and they said something like 'Oh, babies having babies.' They understood exponential functions in a visceral way that I didn't. Ask questions like 'What is calculus?' 'What is the derivative?' 'Why is it important?' 'Who invented it and why?' After you understand things in a more global way, then learn the manipulation and proof. $\endgroup$
    – Arby
    Apr 19 '17 at 1:21
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    $\begingroup$ In my opinion, you should learn to derive the formulas from scratch as you are learning calculus. It's easy to derive the formulas of calculus (once you learn how to do it). And even before calculus, you should be learning to derive formulas as you go. For example, when you learn the quadratic formula, you also learn how to derive it. When you learn the pythagorean theorem, you also learn how to draw a picture that explains why the formula is true. You should never just be memorizing things, you should understand as much as possible. I think options 2 and 3 are pretty bad. $\endgroup$
    – littleO
    Apr 19 '17 at 1:31
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    $\begingroup$ Generally I think math majors will essentially learn calculus two or maybe three times: Once at a high school / beginning college level, and then again when in a "real analysis" course. In real analysis, you will re-derive results from calculus using rigorous proofs. But at the basic calculus level, most students will see a few proofs and derivations but without as much detail, and without having to prove much themselves. $\endgroup$ Apr 19 '17 at 2:08
  • $\begingroup$ There are actually two different versions of rule 1: The first version is to prove everything rigorously, the second is to prove most of the theorems, making exceptions when the proofs are too complicated or when the theorem can be given compelling intuitive justification. Most calculus books that are "non-rigorous" actually fall into this second category. To answer your question, there are tremendous benefits to learning calculus with proofs, because it teaches you how to think about mathematical problems in the best way. But.. $\endgroup$
    – user49640
    Apr 19 '17 at 19:38
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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.

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  • $\begingroup$ No, that's fine -- just don't burn out on the hard ones (or the dreary ones) -- those can wait for courses such as Elementary Analysis. $\endgroup$
    – quasi
    Apr 19 '17 at 2:00
  • $\begingroup$ Nice post. If you find joy in the math, you will want to do more. You'll also be motivated to get through some of the more difficult and dry areas because you know they are helpful in continuing your fun. $\endgroup$
    – Arby
    Apr 19 '17 at 2:31

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