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Studying Attempting to prove each theorem in a book

  1. Oct 13, 2018 at 10:03 AM #1
    I am seeking advice on how to effectively and efficiently learn mathematics textbooks. Currently, I adopt the style of trying to prove theorems in the book before reading the provided proof. I have had good success in this; I noticed a considerable gap in experience between me and my peers in writing proofs as I also kept a journal on general "meta-cognitive" techniques when constructing proofs to help with my consistency. There is still a lot to be learned as I am far from being able to prove all the theorems in the text - I believe this to be impossible. Although, I am happy with my progress after almost 2 years I have been practicing this, but I feel like I am going too slow and it would take me far longer to reach "research level" mathematics. Do you people think this is style of learning is worthwhile and helpful in the long run? Why do you think this method is beneficial or not beneficial?
     
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  3. Oct 13, 2018 at 10:38 AM #2
    I was a physics major, not a math major, but I always learn more from doing derivations than shot gunning equations; so I probably spend more time on that than others. I would say it could be beneficial, but it could be a time sink and could lead to the point of paralysis by analysis. I would say it isn't beneficial if your insistence on getting through proofs yourself stops you from doing your real assignments for class.
     
  4. Oct 13, 2018 at 7:32 PM #3
    Do you now do research? If so, has doing derivations yourself had any benefits?
     
  5. Oct 14, 2018 at 9:15 AM #4
    I wouldn't say I did them all by myself but I learned the physical principles and mathematical considerations behind the derivations such that I can recreate them without looking at notes and such and that is helpful when learning new physics but again it can be a paralysis by analysis situation. I'm a modeling and simulation engineer which does involve plenty of research, and in the real world more often than not an answer that's 50% right but on time is more valuable than an answer that's 100% right and late, so you have to balance.
     
  6. Oct 14, 2018 at 9:25 AM #5

    PeroK

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    If you are doing mathematical research, an answer that's 50% right is not much good!
     
  7. Oct 14, 2018 at 9:29 AM #6
    I'm not a math researcher though.
     
  8. Oct 14, 2018 at 9:31 AM #7

    PeroK

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    Well, exactly. So your advice to the OP, who is asking about maths research, is misplaced.
     
  9. Oct 14, 2018 at 9:33 AM #8
    Not as much as you might think, but as an engineering researcher the goals are different which I prefaced my advise saying my background is physics and not math.
     
  10. Oct 14, 2018 at 11:34 AM #9

    marcusl

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    50% wrong on engineering solutions would get you fired from my employer.
     
  11. Oct 14, 2018 at 11:45 AM #10
    You guys are taking the 50% vs 100% thing a little too literally, it's a general principle I've encountered vs a hard and fast rule; you obviously want to be as right as you can, but sometimes as right as you can be doesn't mean 100%. This is also probably taking away from the OP's thread.
     
    Last edited: Oct 14, 2018 at 11:54 AM
  12. Oct 14, 2018 at 1:53 PM #11

    verty

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    Terrell, do you not find it hard to stay motivated when you are staring down difficult theorems? I'm thinking of something like the Heine-Borel theorem, where the proof is not really obvious. Or do you think to yourself, I couldn't have easily solved that, so it doesn't matter that it was harder?

    Because if you have done it for two years, it must be successful. I mean, perhaps it won't take that long to reach the harder math.
     
  13. Oct 14, 2018 at 3:37 PM #12

    mathwonk

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    This is a highly recommended technique for learning to do research and for actually learning to understand material deeply. This method forces you to understand the idea behind an argument rather than just remember the trick.

    You can see in a recent exchange posted in the homework help section about smooth injective maps, I resisted the temptation to read the argument linked to by another poster and eventually generated the proof myself. This made it more enlightening to me and forced me to observe a clue in the problem that led to the successful method of attack.

    it also caused me to rethink other similar problems in other areas, where one deduces a strong conclusion about a map from just a weak hypothesis, namely injectivity.
     
  14. Oct 15, 2018 at 4:29 AM #13
    No. When I see a difficult theorem, I always wonder how far I could go before I get stuck and trying it on my own helps me see the motivation of the proof provided. The only thing not motivating for me are exams. Exams always make me not want to study at all, but I still do and it's really hard to stay motivated. Also, exams leaves less time proving theorems and forces me to just blindly use it - I hate it.
    I do not know why, but I still feel bad not being able to prove it, but not as bad as having a brain fart when failing to solve a very easy problem, unfortunately it happens.
    One of the root problems I have when studying is not knowing when to give up. I find it tough to move on. I feel like if I don't solve a problem on my own, I miss a key learning experience that would help me be successful down the road. How true is this.
     
    Last edited: Oct 15, 2018 at 4:41 AM
  15. Oct 15, 2018 at 4:33 AM #14
    Yes, I agree. But, how do I know when to 'give up'. For instance, trying to prove Sylow's Theorems. Also, in what particular ways does this method help in doing research? Thanks!
     
    Last edited: Oct 15, 2018 at 5:20 AM
  16. Oct 15, 2018 at 5:35 AM #15

    FactChecker

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    The theorems in a textbook were not meant to be research efforts. IMHO, it is better to look for common approaches and techniques in their proofs than to make up your own proofs.
     
  17. Oct 15, 2018 at 5:42 AM #16
    I cannot disagree with you since the definitions and theorems are already 'nice'. However, do you think there is any benefit in reproducing your own proofs, especially to beginners like me?
     
  18. Oct 15, 2018 at 6:07 AM #17

    FactChecker

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    I think it would be better to study their proofs and then to try your own proofs in any exercises in the book. The first are designed to allow you to understand the concepts and the latter are designed to allow you to practice specific learned techniques. After you have done that, it may be beneficial to look back at their proofs and see if you can do those proofs on your own.
     
  19. Oct 16, 2018 at 8:51 PM #18

    mathwonk

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    there is no rule about when to give up and look at the proof. I was tempted several times to do that in the case mentioned above, but I was really happy when I did think of the right idea without looking. So now that I am retired and have all the time I want, I tend to not look and just keep trying for a long time. Mainly because I want to achieve the result of thinking of ideas on my own, not of just learning somone else's idea. I.e. what is your goal? learning how someone else solved a problem or succeeding in solving it yourself?

    Solving a problem on your own, or generating a proof, is an example of research, hence strengthens your ability to do research. ai found this helpful myself in my career as a researcher, and I have heard the same from others. Indeed I think this is a standard view of professionals. Perhaps one of the earliest I read saying this was Oscar Zariski, one the founders of my subject, and another later in my career as a student, was Maurice Auslander, great algebraist. He used to say, if you think up your own proof, you will often find that your own idea will actually prove miore than you are shooting for, and you will have aklready generalized the theorem, thus having done some actual research just while learning to unbderstand someone else's.

    Indeed this is not easy, and many students will find it challenging, but it is the preferred method of professionals. Even if you do not succeed in proving a theorem on your own, you will ususally succeed at least in thinking of some part of the idea to begin the proof. Then when you do have to read the proof, you do not need to read that part you have thought of yourself.
     
  20. Oct 17, 2018 at 7:34 AM #19
    It does help me practice digesting a problem by talking myself through it. So in a way it also help me get into a habit of thinking more clearly. I realized that I need to "habitualize" techniques I discovered myself or have learned from others so challenging theorems coaxes me to good habits. Although, am I correct in thinking that this is only the "proving-lemmas" part of research?

    Yes, I agree. It gives motivation to the ideas right away a lot of times. Thus, making it easier to read through.
     
  21. Oct 17, 2018 at 10:14 AM #20

    mathwonk

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    As to when to give up, there were two theorems that arose in the previous discussion I spoke about, the Borsuk-Ulam theorem, which answered the question posed there, and the Jordan curve theorem, which could also have done so and went further. I succeeded in solving the Borsuk Ulam theorem after being tipped off that a proof occurred in an elementary book, which encouraged me it should be doable. I have not yet solved the much harder Jordan curve theorem in an elementary way, which occurs in another book I have and looks much harder and longer. I did generate some ideas, and have not yet looked closely at the proof in the book, but may do so soon. I already looked closely enough to see that some ideas I had were relevant, but the trick used in the book was not one I had thought of or even quite understand yet. This proof is more daunting to me since it is famous for having been proved incorrectly by many people even famous ones. But maybe I will try a little more.

    Even if you do not succeed in proving even part of a theorem you will often at least identify what are the key things that need to be proved, whereas if you just read a proof the easy things will often look indistinguishable from the hard ones. So you almost always learn something by trying it sincerely.
     
    Last edited: Oct 17, 2018 at 10:59 AM
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