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I know that discrete Mathematics grows harder and harder with the progress. But sometimes my brain just seems to stop working even with the easiest material. (e.g. the mechanics behind rewriting linear maps into matrices)

As an undergraduate Mathematics newcomer, I would like to ask:

  1. Is this a common problem or just for me?

  2. What drives you through these difficulties if such status last desperately long?

  3. Is there an easy fix for this kind of issue?

Great thanks.

李智修
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  • the question that you must answer is: why you get stuck? For what reason? If I dont know how to do something generally is because I have a lack of knowledge on the level of (abstract) theory or some algebraic identity. – Masacroso Feb 06 '18 at 02:32
  • Work on a different problem for a while then come back to it. Also, review the sections in the book where the material's presented. Make sure you know the exact specific technical definitions of each new concept. Often just writing down the definitions makes the proof write itself. – user4894 Feb 06 '18 at 02:36
  • @Masacroso But the difficulty of writing linear maps into matrices is so low! That I even doubt whether I had my brain damaged somewhere. Anyway, I will try to review definitions and theorems more. Thanks for the reply anyway. :) – 李智修 Feb 06 '18 at 03:30
  • @user4894 OK, noted. Thanks for the reply. :) – 李智修 Feb 06 '18 at 03:32

1 Answers1

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First, there are no "easy" fixes for when you're stuck. If math were easy, more people would do it and the best would just focus on harder problems, where they'd get "stuck" too. Think of Andy Wiles working on Fermat's Last Theorem: seven (or more) years, alone, in the evening, being stuck nearly every moment.

Nevertheless, here some techniques I use when I get stuck:

  • Work on the simplest version of the problem (typically the smallest, where one can write out terms)... look for limiting cases
  • Write a computer program to perform the calculation (explicitly add a large number of terms, say)
  • Try to visualize the problem (as some graph, or plot, or ...)
  • Change the problem to make it easier (reduce the dimensionality, the constraints, etc.) and then try to work back to the original problem
  • Re-read the textbook or relevant research papers
  • Explain the problem to someone else. (This has proven remarkably helpful!)
  • When truly lost, ask for help (online or with a faculty member)... but don't ask for the answer!

I would also recommend two books:

  • How to solve it by George Polya
  • Solving mathematical problems by Terence Tao
David G. Stork
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    +1 for the simple process of Explain the problem to someone else. In the computer science world, we call it "rubber ducking", because you don't necessarily need to explain it to someone. Just something will often do. – Alec Feb 06 '18 at 05:59
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    “Explain the problem to someone else,” is great advice. And if you don’t have someone around that you can explain to I often find it helpful to type up a question here on MSE. Writing a good detailed question really makes you think through every step of the problem and forces you to hone in on exactly where your confusion is. Often times, you can answer your own question before you finish typing it up. – wgrenard Feb 06 '18 at 06:23
  • @David G. Stork great thanks. Your answer is really helpful. – 李智修 Feb 06 '18 at 08:39
  • @Alec yeah I too likes the explaining one the most, it should force me to investigate the problem carefully. Thanks for the reply. – 李智修 Feb 06 '18 at 08:46
  • @wgrenard True, type up questions on MSE will make me think more. Maybe I should do it more often. Thanks for your reply. :D – 李智修 Feb 06 '18 at 08:48