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]: