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This question will essentially be more of a how-to plea or general help request.

I'm currently studying math and I'm at the point where I've transitioned into upper-division classes, most if not all of which are proof based.

To be blunt, I currently feel discouraged at the prospect of being able to succeed in "upper-level" classes. Last quarter I was able to get decent grades in my classes, but nothing like the success I enjoyed in lower-level math. My discouragement stems from the fact that in the past if I studied I did well. Now I feel as if I'm studying and not doing well at all.

Obviously this is a reflection of either my study methods or frequency with which I review the required material, but either way, I'm starting to become disillusioned with the good ol' saying that if you "practice, practice, practice" then you'll get better.

My question to whoever cares to comment, and believe me I greatly and truly appreciate all advice, is how did you transition into proof based classes and succeed in them? Obviously the material is different for each course, but in general how did you approach absorbing the material? While everyone is different, what are review methods that you found helped you to best understand the material and also retain your understanding?

I don't know if others have this same issue but I find that for proof based classes, since problems tend to all be different, I don't quite retain a sense of how to tackle problems even after completing an assignment whereas for classes such as calculus in which e.g. I had to learn integration, after so many integrals I had a general sense of how to solve them. Any advice on how to remedy this deficiency?

I apologize if this question is inappropriate in any way, but I'd love some perspective from those that have gone through a similar process.

Bob
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    Math is all about proofs, rather than computation. Have you considered a different field that still involves or relates to math? – anomaly Apr 20 '16 at 23:15
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    This is a thoughtful question, but it is inappropriate for this site; questions seeking personal advice are off topic. – MathematicsStudent1122 Apr 20 '16 at 23:15
  • I don't really know how I did it, but I think that grabbing a proper real analysis book helped me quite a bit. When I first learned Calculus, I was disgusted (strong word, I know), by the way it was taught to me. I needed real (heh) arguments, and not just handwaving. When I started reading these kinds of books, I was shocked at how much more difficult they were, it was really challenging, but in a nice way. I found motivation this and also learned A LOT about theorem-proving. Maybe this helps you aswell. – YoTengoUnLCD Apr 20 '16 at 23:16
  • @MathematicsStudent1122 Is "How to transition from calculus-y stuff into proof-based classes?" really an inappropiate question? Come on guys, at least read the description below the close reason. If you read this question carefully, its clear it doesn't fit there. – YoTengoUnLCD Apr 20 '16 at 23:17
  • @YoTengoUnLCD I can understand your point of view. Even still, surely very similar questions have been asked before? – MathematicsStudent1122 Apr 20 '16 at 23:19
  • @YoTengoUnLCD Sadly, this sort of thing definitely is considered off-topic here. If I were in charge maybe there'd be a special section for this sort of thing, but I'm not in charge. And neither of us is paying the bills. Note there are already 3 votes to close. – David C. Ullrich Apr 20 '16 at 23:20
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    Well in an attempt to actually answer the question. Do three things:1)learn basic logic youll need it. 2) Find a good book, (I used Rudin) and read the theorems then try to prove them on your own. If you are stuck read the first line of the proof in the book, and do the rest yourself. If you are stuck, read the second line. For some proof I read the whole thing but for many I read nothing. 3) Solve as many problems as you can find. That is: learn to think and problem solve like a mathematician. – Rene Schipperus Apr 20 '16 at 23:41
  • If this gets closed, you might want to try Academia.SE. I know it is specific to math but there are enough math people there to help you out. I've seen questions of this type there. – M47145 Apr 20 '16 at 23:44
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    @anomaly I respectfully disagree. Math is about proofs and computations. Both are equally important. – Mark Fantini Apr 21 '16 at 04:03
  • @MarkFantini: Hmm, depends on what qualifies as computation. The sort of work one does churning through integrals in a standard introductory class calculations to compute bears little relation to what mathematicians actually do in practice. – anomaly Apr 21 '16 at 05:27
  • @anomaly Those are computations designed to make the beginner comfortable with integration. Indeed it bears little resemblance to what mathematicians do, but then so does most if not all of the curriculum and how it is taught (not the subjects per se). Computations provide the explicit examples we have to address using the theory. This is why they are so important. Utilizing your own examples, the whole theory of integration in calculus does not tell you how to compute some complicated integrals that show up both in calculus or in applications. You need experience with the examples to do so. – Mark Fantini Apr 21 '16 at 15:06
  • @MarkFantini: Indeed, undergrad math bears little resemblance that what mathematicians actually do, and the point of math isn't to prepare students for computing complicated integrals in calculus or applications. For that matter, going through the theory by lots of explicit examples and computations isn't representative of how mathematicians do math either. You almost make it sound like it's impossible to apply the theory without a battery of examples and computations, which certainly isn't the case (as the latter are de-emphasized or absent from higher-level classes). – anomaly Apr 21 '16 at 15:22
  • @anomaly It's certainly not impossible, but I want to make it sound as incomplete. That's all. Just because higher-level classes de-emphasize or omit them doesn't mean it's the right or best way. I don't want to mean theory is not important. I'm just saying that more examples would be more helpful. – Mark Fantini Apr 21 '16 at 15:44

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This is a tough question with no easy answer, so I'll just write down some tips which may be helpful for proof-oriented courses.

  1. Read the proofs in your textbook (and maybe other sources), to the point where you can understand why each proof proves what it claims to prove. This means you understand why each sentence in the proof is correct, and how the paragraph(s) of the proof as a whole proves the claimed statement. When doing this, don't yet think about "How could I possibly come up with this proof?" (although eventually you will need to address this question). The first step to being able to write correct proofs of your own, is to be able to read other people's proofs and determine why they are correct (or incorrect!).

  2. Study basic logic. Concepts like indirect proofs, the meaning of "if and only if", use of quantifiers, etc., should become second nature, so that you immediately recognize them when they are used in a proof. This is independent of whatever the actual content of the class is (linear algebra, group theory,...) Some logic textbooks use non-mathematical examples on familiar subjects for illustration ("If it's raining, then the ground is wet, but not conversely"), which might be helpful for separating logic from the new mathematical concepts in a class. Once you don't have to think about basic logic any more, you can concentrate more on the actual concepts in a class.

  3. You say that in a proof-oriented class, "the problems tend to be all different". If this is the case, you need to be grouping the problems at a higher level. One possible approach is to group together proofs with a common goal. For example, in a linear algebra class, there may be many problems that ask you to prove that something is a subspace. The details of the "something" may look very different from one problem to the next, but there will be common steps in the proofs. (Some textbooks may explicitly point out these common steps.) These common steps will help you get started the next time you are asked to prove that something is a subspace.

  4. When you're doing integration in a calculus class, you can compare your answer against someone else's pretty easily to determine whether they are the same. (Maybe you need to do a little algebra or use some trig identities, to put your answer in the same form.) This is not the case with proofs. It's possible for two different people to come up with quite different proofs, both of which are correct. That's why point 1 above is important. You need to be able to read a proof you wrote and determine whether it's correct.

I hope this is helpful in some way.

Ted
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