I've been wondering lately what is the best way to study mathematics.
Obviously, we are all different and so there can hardly be one single best way to manage studies. Still, I wonder how others handle these few questions - in hopes that I'll learn new ideas or get to apply the ones I already know in a more effective way.
Please don't feel obliged to answer most of these questions! $ $ I'd be already grateful if you can simply share your own way of tackling one of them.
1- First of all, do you find most mathematics easy? And that's why you study it. Or on the contrary, you find maths difficult, and that's really why you study it (i.e. for the challenge). Should this affect how you approach maths studies?
2- Do you mainly do lots of exercises until you feel you could handle any question on the topic? Or do you just stick to exercises provided in your main textbook (as long as you didn't struggle with any)?
3- Do you simply make sure you understand every proof, or you also make sure you can reproduce even the most difficult ones? How then do you not forget key (unintuitive) steps after, say, a few months? Do you simply take note of original techniques you encounter in those proofs (and systematically revise them), or it really doesn't matter if you can't reproduce hard proofs easily (or at all), so you don't bother with such notes?
4- When studying full-time (i.e. when your time is limited), should understanding of topics (or harder proofs) be completely optional, as long as you can handle each activity/exercise, because you can't afford to 'waste' too much time on a single topic? if not, what is a time effective action to take when you simply can't seem to grasp a concept?
5- When in search of additional exercises, should I do those in the exercise book provided with my module, or should I get exercises from a third-party source in an attempt to get more diverse questions?
6- Is diversity (in techniques and exercises) a good thing? Or it will only prevent me from focusing on the techniques my module is trying to teach me and ultimately assess me on? (Even if my study time is limited?)
7- How long should you spend on a question before throwing in the towel?
8- Finally, how do you revise for exams? Is it a good idea to start my revisions with past papers (as many as I can find), and only revise related topics from the main textbook IF I get stuck on a past paper question? Is this approach more time efficient and also more effective since I'd be familiarising myself more with the type of questions I'll have to solve during the exam?