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It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important. But I honestly have no idea what I am supposed to be getting from this. It honestly seems useless to me. I know it isn't. I know it's a rite of passage to prove xyz theorem. But is this really worth sometimes two class periods?

I honestly don't know what I'm supposed to be learning. These proofs require unique logic a lot of the time, and this logic is difficult to transfer to other problems.

In lower div classes, it would be informal explanations of things followed by a lot of examples. I'm not saying that that's the way to teach functional analysis for example. But at least it made sense to me. I know what I was supposed to be learning.

I can go through all the details myself, I go to lecture for motivation/intuition/things you can't normally get from a textbook. The proof of Riesz representation theorem for example can be found in any text. Why can't I just read it on my own? It's much easier to read proofs like this on your own rather than in class.

10Sometimes you are just learning what a proof of a non-trivial (and non-computational) fact looks like. Most students will not read the proof on their own. Many of the remainder won't recognize what parts of the proof are important. Sometimes the lecturer just really likes the proof. – Adam – 2015-12-07T04:49:28.903

4Perhaps you should ask yourself what you expect to get from these graduate/upper div math classes. Presumably, you expect to get an education. What kind of education do you want? What do you expect to do with it? Why are you taking graduate/upper div math classes? – Scott – 2015-12-07T06:17:09.063

3@Scott The same as most people taking these classes, to learn skills necessary for mathematical research. – Zachary Selk – 2015-12-07T06:22:43.090

2Doesn't that answer your question, then? If you plan to do mathematical research, aren't you going to need to know how to prove things that aren't obvious? – Scott – 2015-12-07T06:27:22.843

5@Scott That doesn't really answer my question, no. – Zachary Selk – 2015-12-07T06:29:29.107

"These proofs require unique logic a lot of the time, and this logic is difficult to transfer to other problems." OK you need to look at the bigger prospective here. Apart from long proofs, apart from mathematics what you learn from these is to develop your logic, develop you mind to handle situations/problems that haven't had come through your life. – Jinandra – 2015-12-07T17:39:37.070

Have you asked yourself what else you think you should be learning in lecture time? One way to view it is as analogous to learning to play music. A music student starts by learning to play and understand the works of "great" musicians. Then slowly the student composes their own works, reproduces other's works in their own style, and synthesizes the existing body of music with their own point of view. Rarely, if ever, does a musician spring up with no knowledge of what came before. So it is with mathematics (and probably many other fields). What are you learning? Math. – Todd Wilcox – 2015-12-07T20:08:40.810

5Funny thing happened to me recently. I remember taking a topology course as a beginning grad student and being completely lost by the proof of Urysohn's lemma. I spent the rest of my Ph.d. studying things which do not involve a lot of point set topology. Sat in on a friend's class toward the end where prof was proving Urysohn's lemma, and the theorem and proof now seem completely obvious! The students in the class were all lost. So clearly I have gained

somethingthrough the process of going through graduate school... – Steven Gubkin – 2015-12-07T20:26:34.28710Sometimes (and it's more often than not), you have to understand the proof to understand the theorem. Why the assumptions are necessary? Where they are used? What would happen if we took them out or assumed something else? If you can't answer these questions (at least partially, it happens that these might be some very hard problems), your intuition is lacking (of course, that does not mean being able to answer is enough). – dtldarek – 2015-12-08T00:58:57.917

I have felt just like the OP but my best math teachers provided me with enough of a framework for "why we learn proofs" that I felt like sticking with it. Thinking in this mathematical 'proof' style is mentally exhausting for me, but some of my teachers actually managed to turn it into something interesting. More like an interesting puzzle or a riddle, and less like reading the phone book. If you don't like puzzles, and convoluted thinking, then maybe Math's not right for ya. Or maybe your teachers have simply been all of the dull-as-dishwater type. – Warren P – 2015-12-08T21:50:42.817