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In my discrete math course, I am often finding myself spending way too much time attempting a single math proof.

I am starting to think that reading as many proof solutions as possible is a better approach to practice. I guess this would help me recognize more proof patterns on the exam.

Did anyone get significantly better at writing eloquent proofs by simply reading and absorbing as many proofs? Or did the process of thinking from ground up in new proof situations help you develop your proof writing skills?

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    $\begingroup$ It doesn't strike me as an either-or situation. Sometimes we get stuck on attempting a proof, and later an idea will occur to us that opens the way to a successful attack. Of course reading proofs may "seed" our mind with ideas (perhaps of what to avoid!), and reading proofs is a useful skill in its own right. But just reading proofs will not likely get you ready to write them. $\endgroup$
    – hardmath
    Dec 20 '15 at 15:09
  • $\begingroup$ @hardmath Yes, I see what you mean. How long should I be trying to prove something before I move on? Sometimes a proof is just so frustrating and I want to peek at the answer. Are you suggesting that I should just skip it and come back at a later time? Because I feel like so much time is wasted just starting at the problem, when I can just use that time to read multiple other proof solutions. $\endgroup$
    – Azurite
    Dec 20 '15 at 15:17
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    $\begingroup$ Of course I'm no longer in school, so I have the luxury of waiting until I have an idea before attempting to write up a proof! As advice in taking an exam in mathematics, particularly ones that ask for proofs, I would say read over all the problems, assuming a reasonable amount of time is given, and do triage on them: which ones are clearly doable, which ones seem hopeless at first glance, and which ones is there a bit of an idea that needs more time to puzzle out. All your efforts and tactics will pay off in the long run, so keep trying proofs and reading those "model" proofs. $\endgroup$
    – hardmath
    Dec 20 '15 at 15:28
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The problem with reading a proof before attempting them yourself, is you have no muscle-memory to help you remember them. It's like learning to swim from a book.

The ideal process is something along the lines of. . .

  1. Attempt the proof on your own. If the proof is an exercise from a textbook, you can be fairly certain you already have the tools to complete the proof at your disposal -- there is no 'flash of inspiration' required. How long to do this depends on how much progress you seem to be making. If you are making absolutely no progress and the problem completely stumps you, a good idea is to go back to the logic of earlier proofs, and see if anything might apply to your current problem.

  2. If you are unsuccessful, then read the solution or complete proof. Often your attempt will read like a messier, incomplete version of the given solution. Skip to where your attempts end and try to see what stopped you from finishing. This way you will understand which parts of the exercise are hard, which parts are easy, and you will be able to pick out what was missing with your approach; better plumb the depths of your understanding; and see what you have to learn, and what you already understand. Don't bother memorising the parts of the given proof that you already gotten in some form. You already know them, albeit in a slightly different form.

  3. Come back a while later and try the same exercise again. If you can do it, there is no need to consult the sample solution to make sure you have 'the correct' proof.

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    $\begingroup$ I was not thinking of actually memorizing a whole proof or part of the proof. Just the idea of scanning over several proof solutions in a short time, and just get in the rhythm of seeing and quickly understanding proofs. Once I attempt a proof, I feel obliged to finish most of the times, and then when I truly get stuck, I feel like I spent a lot of time just thinking of ways that are way off track from solution manual. $\endgroup$
    – Azurite
    Dec 20 '15 at 15:27
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    $\begingroup$ Don't think of that as wasted time. Think of it as time figuring out 'what doesn't work and why'. Which is the younger brother of 'why we are doing THIS and not THAT'. $\endgroup$
    – Daron
    Dec 20 '15 at 15:30
  • $\begingroup$ The reason I started to think this way is because I often see instructors or students simple come up with a proof by saying something like 'I just have a feeling from seeing other proofs', its like they had it embedded in them. And most of the times their intuition just leads to a perfect proof, without them putting in some mental exercise before hand or completly have an array of proofs memorized line by line in their head. $\endgroup$
    – Azurite
    Dec 20 '15 at 15:35
  • $\begingroup$ If you will permit me to come back to the swimming metaphor, once you have properly learnt to swim, no concentration is needed to repeat the behaviour. I would expect these students/instructors have indeed put in mental exercise, not immediately before writing out their proof, but back when they were first digesting the theory. $\endgroup$
    – Daron
    Dec 20 '15 at 15:39
  • $\begingroup$ I am thinking that proofs are kind of like the swimming analogy, but with proofs its kind of like theres the breathstroke, and then theres a slight varieation of the breath stroke w/ a little tweak maybe a little more rotation of the hips, maybe called the breath strike. With proof problems, a single little clever move (maybe algebraically) could be the point for me not getting the proof. I feel like practicing a general proof technique requires practice for sure, but in terms of knowing little 'tricks' in proofs to solve a clever problem, breadth comes in handy. $\endgroup$
    – Azurite
    Dec 20 '15 at 15:53

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