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Almost two years ago when I was just starting to learn "university mathematics" I asked this question related to how much memorization should one do when learning math.

At that point I was (as I also said there) mostly self-learnt because I hadn't attended formal lectures yet. Nevertheless, my university experience has definitely been unusual due to the pandemic. All my exams until last semester were online, so they obviously didn't have questions such as "state and prove Theorem X in the lecture notes" because they were open book exams.

But ever since returning to closed book exams questions such as the one I quoted above occur on many of the exams. Some professors give us a list of the results from the lectures that we need to know how to prove, others just say that any result in the lecture notes can come up in such a question. Mind you, some of these theorems are highly nontrivial ones (for instance, on my Abstract Algebra exam one of the possible theorems whose proof we might have been asked to reproduce was the Fundamental Theorem of Algebra and its "most algebraic" proof, i.e. the one using symmetric polynomials, the one we covered in the course, is kind of complicated and long), so most of the students fall back on memorization for such questions (to a greater or less degree; it's obvious that a good student won't just memorize how to prove simple statements simply because they can produce those proofs anytime they are requested to from the top of their head without much effort, but for difficult theorems even good students memorize the proofs to a greater or less degree - see below for further explanations).

I initially thought that this had something to do with the culture of my country (I live in Eastern Europe and here our school system often asks you to memorize stuff), but after a bit of Googling I found that even at top universities in the world such questions do come up on exams (I remember seeing an exam from Cambridge that asked the students to state and prove Cauchy's Theorem in Group Theory - I don't consider this to be a result that one can just prove from the top of their head) and I can't seem to understand the point of asking such questions. I for one try to do the proofs of the theorems in the course as many times as possible and I try to understand the steps. But, in the end, this results in more or less memorizing the proofs because of all the intermediary steps and the computations that I don't really have the time to work out on the spot during the exam due to time constraints. I can do those parts just fine provided I have the time, but in an exam I don't have that time and I end up committing to memory those parts of the proofs. This doesn't seem beneficial to my understanding of the subject and this is why I don't get why such questions seem to appear so frequently on closed book exams (most of my peers seem to be in the same boat as me, so that's why I don't get this questions).

P.S Let me know if this question is better fitted for https://matheducators.stackexchange.com/.

user69503
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    It’s important for students to read and understand the proofs. And ideally not just understand the proofs line by line but grok the proofs — see the big picture strategy, the key idea or ideas at the heart of the proof. And students must learn to write proofs. How can a teacher check that students have been doing this? It seems hard to think of a better idea than asking students to reproduce proofs. You can also ask them to invent proofs for related results that aren’t listed as theorems in the book, but which involve similar ideas. – littleO Sep 02 '22 at 12:33
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    What's the alternative? Asking students to mechanically work through computations of a sort they have seen before (but with different numbers) isn't obviously better. As for asking for proofs of new theorems...well, sure. But it's hard to be creative on demand and proof creation doesn't work well on a timer. At least asking for reproduction of a proof requires that the student read the proofs. – lulu Sep 02 '22 at 13:06
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    Related and helpful: https://math.stackexchange.com/questions/2895291/is-there-much-benefit-in-memorising-proofs-outside-of-an-exam-setting https://math.stackexchange.com/questions/1834814/are-there-any-tricks-to-remembering-proofs-of-mathematical-theorems https://math.stackexchange.com/questions/1745002/should-i-remember-the-proof-of-mathematical-theoremsevery-step You shouldn't memorise proofs word-for-word, although for example, remembering the layout of the simplest version of the proof can help, and you can use a timer to improve the speed at which you regurgitate the proof. – Adam Rubinson Sep 02 '22 at 14:52
  • @littleO thanks for your input. I always thought that asking them to prove similar statements to the ones discussed in the lectures/seminars is better fit, but I can start to see why asking them to prove exactly the statements in the lectures makes some sense. – user69503 Sep 03 '22 at 12:36
  • @lulu thanks for your answer. I agree that mechanical computations are no way better. I was kind of a supporter of proving new theorems on exams before posting this, but your comment actually made me think about all the times when I was asked to do this on my exams and the stress I was under because I had little time to come up with ingenious proofs. – user69503 Sep 03 '22 at 12:48
  • @AdamRubinson thanks for the links. As I said, I don't memorize them word-for-word, but I am sure that a good number of people do and I wonder if this type of questions contributes to their learning or just entices them to have this toxic learning behavior. – user69503 Sep 03 '22 at 13:07
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    "I am sure that a good number of people do"- I doubt it as it takes longer to memorise word-for-word than to just understand the proof and bullet point the main ideas behind the proof. You can't memorise all proofs word-for-word, as once you get to 3rd year undergraduate maths, there are tons of proofs. However, you should "memorise" the important proofs, like any proof with a name like "Heine-Borel Theorem" or "Bolzano-Weierstrass Theorem". Usually it says in your (online) lecture notes if you need to "memorise" a proof for an exam. – Adam Rubinson Sep 03 '22 at 14:49
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    But 90% of proofs in my 3rd year I didn't need to memorise, so it's not that bad. Anyway, trying to understand the proofs is far more useful for exams than trying to memorise them... The point is that memorising proofs is not about memorising word-for-word. It is about understanding the proof. Once you understand both the theorem and the proof of the theorem to a certain depth then you should be able to recall it in an exam. – Adam Rubinson Sep 03 '22 at 14:49
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    The benefits are the same as the ones music students get when they are asked to memorize and reproduce the works of the great masters. – Sam Sep 03 '22 at 17:39

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