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I'm taking real analysis I, and my professor gives us a previous exam with solution each time before the midterm/exam. I would do the previous exam and check if I get it right. But it's obvious that my professor are not going to give us the same questions.

So, I also read and tried to understand the proofs that are given on the book. Sometimes, I proved the theorems and corollaries by myself. Besides, I would memorize the theorems and corollaries.

For courses like calculus, we have some general steps to solve a type of question. So it's easier to prepare for the exam. But for proof-based courses like real analysis, algebra and topology, etc, how can we know whether we're well-prepared for the exam? How did you study the materials and prepare for the exam when you were an undergraduate student? Exams are designed to test our understanding of the materials, but how can we test ourselves first?

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  • $\begingroup$ That’s why I did, long time ago. But it takes beside of you another fellow student, $\endgroup$ – Michael Hoppe Nov 29 '17 at 17:59
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    $\begingroup$ That’s what we did, long time ago. But it takes beside you another fellow student. One poses a question, say, e.g., to show that the connected sets of the reals are intervals. Now you try to prove it, in all cases, maybe just on your way to college. Don’t work on your own, just participate. Let it be a game. It doesn’t work with any fellow student, but I’m sure you know someone to work that out. — Just play. $\endgroup$ – Michael Hoppe Nov 29 '17 at 18:06
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    $\begingroup$ Understanding in-depth the arguments in the various proofs etc and knowing the important results. You should solve many problems related to the course-material. You should definitely be able to find such problems/questions on MSE. No memorisation should be needed. On the contrary math courses with exam problems in a way that support and encourage memorisation of proofs step-by-step should be considered a crime! $\endgroup$ – Test123 Nov 29 '17 at 18:08
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    $\begingroup$ Most of the time I just needed to memorize how to start a proof. Preparing for an exam I would write all proofs on paper which helped me to remember it during an exam as a picture was in my mind. $\endgroup$ – Vasya Nov 29 '17 at 18:09
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    $\begingroup$ with enough practice you must be able to prove anything non specially pathological related to some material. An intuitive/geometric point of view can help a lot to prove things. $\endgroup$ – Masacroso Nov 29 '17 at 18:20
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Did you get all the problems on the professor's sample exam right? By “right”, I mean, were you able to write perfect proofs, from scratch, without notes or checking against the solutions? If so, stop. You are definitely prepared.

If not, you know what topics you are less than prepared on: the ones that you missed the questions on, or had to check notes for. Find more problems about these topics and attempt those problems. If you have a textbook, it probably has problems in it. Ask your professor or TA about them if you have trouble solving the additional problems.

When organizing the material, you should definitely memorize the definitions and theorems. But see if you can organize them, not in a list, but in a concept map. That will help you see the connections between the ideas, so you know how to get from one to another in a new situation. For instance, you tagged this with linear algebra, and I was able to find some premade linear algebra concept maps online (example). But the real benefit will be when you create your own.

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