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Doing undergraduate math exercises mentally

  1. Sep 15, 2014 #1

    I was wondering, is it a bad idea to solve a math problem mentally instead of writing it down? I recently got a copy of "Understanding Analysis" and try to read it cover to cover using this method. Should I? It resembles blindfold chess in a way.
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  3. Sep 15, 2014 #2


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    If you can do it and if you are sure you cover all the critical details (!), well, why not.
  4. Sep 15, 2014 #3
    It's not a 'bad' idea, but there are other parts of your brain that activate when you write down with pen and paper. I think it was N. Tesla who could do integral calculus in his mind and his teachers thought he was cheating on his tests because he would get the answers right without showing work!
  5. Sep 16, 2014 #4


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    This might or might not relate well to the level of mathematics you intend:

    Physical exercise can stimulate some intellectual mental activity, and this can include a tendency onto whatever mathematics you are currently studying. The type of exercise depends just on how the person is. For some, maybe running; or maybe jogging; for others, maybe weights; or maybe a long walk.
  6. Sep 16, 2014 #5


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    I don't like the idea to work through the whole book this way. This certainly won't help you get better at writing proofs for example. You will sometimes need to write down a proof, and then read through it to see if there's any room for improvement. Then you repeat the procedure until you're satisfied. When I started doing this, I was surprised by how many times I could go through a proof and improve something each time.

    But I do like the idea to do the mental thing some of the time. I like to lie on my back with my eyes closed when I try to find a way to prove something. Sometimes I end up falling asleep, but it's still a good method for me.
  7. Sep 16, 2014 #6
    I loved the comment about falling asleep but still being good for you. Oh god, how true is that.
    Regarding reading the book you really need to try out for yourself. I know I couldn't do that myself. I read textbooks on maths or physics in places like buses etc. pretty often. However, when dealing with lots of technical details I just need to "get dirty" with writing down formulas and algebraic manipulations. You can choose to skip those details or just skim them. However when doing that (especially if you do it more often than not) you risk that you are under impression that you understood what you read but you really didn't. It is quite miserable to read through half the book being confident that it was all easy and then realize that you can't solve problems from first chapter - and that is not unlikely, especially if you are rather new to the subject.

    Oh and another possbility is that you actually can fill all those technical details in without pen and paper. I for sure wish that I had that good memory and focus.

    I'd say just try it and see for yourself, but once in a while make yourself "a test" with pen and paper. If it then turns out that you can really write down and use what you've learned before then it was probably good method for you.
  8. Sep 16, 2014 #7


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    If you can, sure. However, most problems will require deep thinking now.
  9. Sep 16, 2014 #8


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    I second this. It is a learned skill to write down a rigorous proof and to communicate your logic to others. It is also hard to review your logic if you only did it in your head. And things will get tough enough that you will want to review. Finally, don't confuse a proof and real understanding for "in your head intuition". That being said, it is nice to be able to understand things without writing every step down. Exercising your mind that way can be a good thing.
  10. Sep 16, 2014 #9


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    Sounds like a bad idea to me. Writing skills are extremely important, it is very good to develope them at every chance you get. There is no real reason for doing advanced math in your head only, sounds like a waste of time. You would be way better off in the long run to complete a notebook containing all solutions to the problems in the book. This could potentially be useful.
  11. Sep 17, 2014 #10
    Sounds like a good idea, as long as it's not all the time. The main issue I see is that if you did it too often, you wouldn't learn how to write proofs. But it could be a good short-cut, if not over-used. There isn't enough time to learn everything 100% thoroughly in math. That's something you eventually learn. You should also beware of some misunderstandings that can creep in if you aren't thorough enough. Sometimes, I think I understand something, and then it turns out, I don't quite get it when I try to write it down. So, it's a bit of a balancing act.
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