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I'm not sure how well this question fits into MSE, but no other community (probably except mathoverflow) has as many mathematicians as MSE who might be able to answer this question.

In the departments of mathematics in Russia (and probably other post-Soviet countries) it is very common to have oral theoretical-oriented exams as opposed to written exams where one is offered to solve problems. For example, for a course in analysis, the students are expected to be able to reproduce proofs of all theorems in, say, Chapters 1-9 of Baby Rudin. At the same time, my experience has shown that many students are not able to apply e.g. the Implicit Function Theorem and they just learn proofs by heart. I wonder what's the point of making the students memorize proofs? How is the knowledge of the cumbersome proof of the Morse Lemma or the Rank Theorem (see the analysis syllabus below) supposed to be helpful? Of course, there are theorems whose proofs only test the knowledge of definitions, and it just impossible not to know their proofs for someone who understands what's going on. But what's the point of making everyone memorize conceptually hard proofs instead of testing the understanding?

Note that I do not deny the fact that one should read and understand proofs of whatever theorem that they are studying. Below I'm giving examples of theorems which the students are supposed to know (with proofs of course), restricting myself to the sophomore year. But the same is expected from freshmen. Mind you that there is no introductory "Calculus" courses, the first "analytical" course is real analysis (covering for example Zorich's book). I'm saying this to emphasize that the students who come from ordinary high schools have a hard time understanding the basics, and yet they are required to memorize complicated proofs as opposed to being able to work with the theorems.

I do realize that writing proofs is an essential part of a mathematician's job, but I find it much more reasonable to learn how to do proofs by solving proof-oriented problems as opposed to memorizing proofs of theorems that took the humanity hundreds of years to come up with.

Here are some examples (lists of theorems whose proofs the students (sophomores) are supposed to know), in Russian [but you can use Google Translate]:

Analysis (Google Translate)

Topology

Smooth Manifolds

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closed as primarily opinion-based by Somos, Will Jagy, John Omielan, Lord Shark the Unknown, Thomas Shelby Jun 25 at 4:54

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I think you are mistaken. "Solving problem" is only a tiny part of written exams. Regurgitating proofs or standard bookworks is a very big part. $\endgroup$ – user10354138 Jun 25 at 0:23
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    $\begingroup$ In answer to your statement "I wonder what's the point of making the students memorize proofs?" Well, I think they are supposed to come up with a proof somehow. Memorizing it becomes an option. If the subject is presented in careful sequence, many proofs would be no different from ordinary problems. I don't say this is a good or bad thing though. If the students are studying to be academically sound, they should know the proofs. However, in an Engineering course, they don't need to. $\endgroup$ – NoChance Jun 25 at 0:29
  • $\begingroup$ I think it depends on the proof that you are talking about. Some proofs have a very ingenious trick ingrained in them that one might argue has to be memorized (anything along the lines of adding a term to both sides of the equation I would consider pretty ingenious, as its never obvious when that should be done). However, some proofs are straight forward, and in general, if you understand the material you can use it to prove certain aspects of the mathematics. I'm not sure if those proofs are too complicated for students to understand. $\endgroup$ – Kraig Jun 25 at 1:24
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    $\begingroup$ you seem to separate understanding from memorization. While they are not the same, think of a proof that you believe you understand, and sit down and try to write the proof in detail. If you get stuck, then you have not understood it. You would be able to pinpoint what exactly place in the proof you have not understood well. At this point, the only option might be to memorize this step. Each such steps carries a relevant idea with it, which you may later recognize and use in a different context. You cannot use it if you did not memorize it, you do not "understand" without memorization. $\endgroup$ – Mirko Jun 25 at 2:15
  • $\begingroup$ I'm pretty sure this is not geographically isolated, and that there are certain teachers in all countries who do this. Maybe you could ask at matheducators.stackexchange. $\endgroup$ – rschwieb Jun 25 at 12:58