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Improving Reading Fluency with Proofs

  1. Feb 27, 2007 #1
    Any suggestion on how to improve your reading fluency with proofs of theorems?

    It's frustrating to spend over 1 hour to read a proof of a theorem that is under 1 page long (or not understanding the proof altogether). Even when every subtopic within a proof is already known, I find that that one of the main stumbling blocks is forgetting the meaning of symbols and results established within the proof, causing you to spend time backtracking within the proof. But if this is the only problem, then it's just a problem of short-term memory, which I don't think you can do much about. It's definitely more than that though. Drawing pictures always helps to remember what is what, but I don't find it to be enough to be able to read a difficult proof smoothly.

    Perhaps reading shorter and easier proofs of easier theorems for practice? Does doing more exercises help much in this regard?
    Last edited: Feb 27, 2007
  2. jcsd
  3. Feb 27, 2007 #2
    Many proofs involve some degree of mathematical maturity. Don't let yourself get discouraged; struggling, yet trying, to understand a proof, no matter how frustrating it is, is good for you. As you read and read, you will find yourself allot more comfortable. There is no magic trick to get good at it, at least that is what I believe.
  4. Feb 28, 2007 #3
    Ah, the infamous mathematical maturity. The best thing to do if you don't understand a proof is to try to recognize what the author trying to do. Is he doing a proof by contradiction? Some kind of direct proof? So you just follow along and see what he's doing. Grant him all the claims he makes, and if they are true, does the proof work? Then if that works out, check each of his individual claims. Then re-read everything and you should be happy.

    Some mathematicians have a horrible sense of mathematical prose...
  5. Feb 28, 2007 #4
    Good advice, but it's an advice that everyone reading a proof already follows. I wasn't asking how to understand a specific proof, but rather how to become more fluent in reading proofs in general. But perhaps, I guess there is no magic trick for that. Just practice, practice, practice.
  6. Feb 28, 2007 #5
    Considering andytoh is reading differential geometry and differential topology proofs (or is it even more complicated now andytoh?), I assume he possesses the infamous 'mathematical maturity' element. I think your question will be better answered by those closer to your level of mathematics, than those of us still working through undergraduate courses.

    I am glad you mention that you often times forget the meaning of symbols or a result that has emerged previously, because that is the only issue I seem to have, when and if I do have a complication reading through proofs. I thought I might be alone but I am glad that I am not. It is frustrating when your memory lags behind your ability to comprehend and understand something.

    Do they have mathematical thinking books related to higher-level maths that might help? I think someone like mathwonk or matt_graves, and some of the other mathematician's could provide to you better advice.
    Last edited: Feb 28, 2007
  7. Feb 28, 2007 #6
    Who's matt_graves?

  8. Feb 28, 2007 #7
    Good call, who knows? hahah

    I botched the dudes name. I was referring to matt_grime :P he is the homie!
  9. Feb 28, 2007 #8
    I have found one way of improving reading fluency, but its not a perfect cure. And that is to break down the proofs into sections. Every long proof has certain tasks and sometimes is it best to determine the task ahead of time (what is the next step that the proof is try to achieve), then jump to the conclusion after the task. Then understand the next task and so forth. Sometimes this requires jumping to the second last line of the proof to understand what the objective is.

    At this point, I typically have each mini-result underlined and so the skeleton of the proof outlined. At this point I read the fine details that establishes each mini-result. It does seem more clear this way for me, and I find it much easier this way than to read the proof line by line without knowing what the next stage is trying to achieve ahead of time. Unfortunately, some writers do a poor job of paragraphing proofs, or stating what the next task is.
    Last edited: Feb 28, 2007
  10. Feb 28, 2007 #9
    Reading the book "How to Prove it: A Structured Approach" by Velleman, helped me a lot.
  11. Feb 28, 2007 #10
    Many authors prefer to adopt an approach that would not conform to your fragmentation of proofs, though. That is, they will proceed to several assertions that will not necessarily look relevant to you right at the moment, and will only show how these assertions make up the proof at the very end. Also some proofs are so "creative" that it is next to impossible to predict the "next step".
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