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I am finishing my undergraduate degree and one thing I've noticed is how little weight has been placed upon the ability to read proofs, in basically all of my math courses. In first year calculus you are shown the proofs for things like the limit of sin(x)/x at 0, but in my experience there is no incentive for you to understand them. This pattern continued even in more advanced undergraduate courses on foundations and real analysis. As one example, the professor spent an entire lecture proving the schroeder-bernstein theorem, and very few students made an effort to understand it (they certainly weren't motivated to do so through grades). Generally speaking, my classes have followed a format where the professor will prove theorems for a significant portion of the lecture time but tests are designed with applications and proof-writing in mind and certainly most proofs done by the professor are far too hard for a student to recreate independently, so there is no incentive to learn the details of the more complicated proofs.

This seems unusual to me, considering the format of most courses requires you to understand the arguments backing up a particular proposition. Is this true of most university programs? Should a greater emphasis be placed upon learning how to read complicated proofs?

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My own experience both as student and as teacher is completely different. As a student I was expected to produce proofs of results that I’d not previously seen. My first topology course was even taught modified Moore method: we were proving things on our own almost from Day $1$. As a teacher I expected my students to do so in such courses as real analysis and abstract algebra. Of course the results that I asked them to prove were simpler than some of what we covered in class, or I broke them into manageable pieces, but I expected them to do math. –  Brian M. Scott Dec 15 '13 at 23:28
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@BrianM.Scott I should have been more clear, but I see a distinction between learning proof-reading and proof-writing. On a typical exam I wasn't expected to follow a long, complicated mathematical argument, or recognize and be capable of explaining every step of one, instead I applied results proven in class on short proofs of my own. Using my example, assume you are teaching the Schroeder-Bernstein theorem. My professors would have expected us to be able to use the result in a proof, but there was no grade incentive for students to understand the proof itself. –  user115980 Dec 16 '13 at 0:48

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When I was doing my undergrad your experience is the same as mine, so I guess things are kind of similar in undergrad programs. The motivated student can understand and do the proofs of his/her own but the exams are designed to only know whether the student understood the material covered in the class and whether (s)he could apply them for a given problem.

That being said, in my transition to grad studies I have noticed a significant difference. Here more emphasis is given on proving theorems. Sometimes they even give a theorem that is not covered in the class to prove in the exam. Most of the time the proofs that we have to carry out in exams aren't so difficult compared to what is done in lectures, but yet I feel it quite tiresome and difficult with regard to the very little experience I have had in proving theorems in my undergrad days.

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