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I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single demonstration, studying the basics every time if I don't remember them. And this makes it impossible for me to prepare the exams, because I can't go on, I fix myself on wanting to derive by myself a theorem and I lose days in it. And so I ask mathematicians if it is always necessary to be able to prove everything, or we must accept what the theorems say and give it for good. If possible I also ask you some advices to help me study, knowing my problem.

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    $\begingroup$ I seem to remember an earlier post here describing a similar situation, but I can't find it. $\endgroup$ – Joel Reyes Noche Feb 12 at 11:30
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    $\begingroup$ What an experienced mathematician does is look for the key idea(s) and try to remember them. Often finding them requires playing with examples (which for a student means doing exercises, reworking examples, posing examples/exercises for oneself). For example, I remember the integration by parts formula (where do the signs go?) as a consequence of integrating the product rule for derivatives and rearranging the result, so as a sort of formal inverse to the Leibniz (product) rule. Remembering that idea one can recover all the details if one needs to do so. $\endgroup$ – Dan Fox Feb 12 at 16:43
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    $\begingroup$ Would you mind changing the title of your question to show what the question is, rather than who the asker/audience is? $\endgroup$ – Jessica B Feb 12 at 19:22
  • $\begingroup$ @JoelReyesNoche Have you tried looking on Academia.SE? It could be there. $\endgroup$ – Jessica B Feb 12 at 19:22
  • $\begingroup$ @JessicaB, I can't seem to find it there either. $\endgroup$ – Joel Reyes Noche Feb 13 at 4:31
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Concentrate on doing the homework problems. They are most important for developing your skills.

Your approach is not time efficient. You need to learn things at a surface level and then come back and learn it deeper. The human brain is not a computer. We are imperfect animals and learn by starting a groove and deepening it over time AND making connections over time to previous knowledge. If you try to do it perfect immediately, you are fighting nature.

Keep a notebook and use it for a parking lot. Record questions, thoughts, ideas as you study something. Often this will set your mind at ease as the concern is recorded at least. Often later on, you will realize the answer or resolve the question. But it won't have stopped you from advancing in the material. Where it does not, at least you have an organized list of questions to discuss with your teacher.

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    $\begingroup$ Hey OP: This answer is a little cynical and does not necessarily reflect the views of everyone here. I think there is a kernel of truth to the advice of focusing on homework problems, because those are chosen to give you important, fundamental practice. However, do not focus on those to the detriment of actually understanding what you are doing. Please continue to work hard to reflect on what you are doing and why it works. Sure, think about being "time efficient", too, but don't lose sight of why you enjoy doing mathematics and get stuck in a homework rut! $\endgroup$ – Brendan W. Sullivan Feb 12 at 18:46
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You will never know everything in mathematics. One place you could read about this is Bill Thurston's 'On Proof and Progress in Mathematics'. He gives a (slightly fictitious) list of many, many ways of understanding the concept of derivative, some of which will almost certainly be beyond you at this point.

It sounds like you need to recognise that your personality is making your studies difficult, and find ways of dealing with that. There is nothing wrong with wanting to understand everything: mathematicians would generally encourage that, and many wish their students took more of that attitude. But you should perhaps try to remember that you don't necessarily need to understand everything now. If you have an exam coming up, you should change your purpose temporarily from 'understanding mathematics' to 'learning what you can to get enough marks in the exam'.

It is also worth considering that the process of going through the motions of something, even if you don't understand it, may in fact help you to come to understand it. As you do examples, you may spot the pattern that you are not noticing from the general description.

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