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How to study maths quickly and efficiently?

  1. Oct 17, 2012 #1
    This is my first semester taking courses by distance, and although I'm not a fan of cramming, managing time has been an issue. I have two tests in the near, but not super-near, future and I'd like to get through these chapters as quickly as possible whilst actually learning and not just speed-reading. The problem is that I do not like to take anything in math for granted; if I skip a proof or proceed through a section without a very clear understanding it bothers me -- and with good reason.

    One way that I've sped things up is by placing a time limit on proving corollaries/theorems; if I can't figure out the proof in the allotted time, I will pretend that "This proof is beyond the scope of this course" is written where the proof should be and content myself with "someone has proven it somewhere." After all, it's not like I'm breaking new ground.

    I did some planning today and I have to cover pages at about 20 minutes per page. Can I do it, or is my "math problem" a riddle even Oedipus, with Euclid's help, couldn't solve?
  2. jcsd
  3. Oct 17, 2012 #2
    It depends on your ability to grasp abstract concepts and arguments, your ability to formulate these independently, the level of your course and the density of your text.

    For example, I have two books in front of me. One is an introductory abstract algebra book. The other is an advanced book about topology or representation theory. I will probably spend less than 10 minutes per page on the first book, but may need a day or two to swallow one page of the second.

    You cannot rush mathematical understanding. It comes from patient contemplation.
  4. Oct 17, 2012 #3


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    One typically does not study Mathematics quickly. Twenty minutes per page as you stated seems about right. Study-time can easily reach about 15 hours per week. One should ideally study every day for about 2 hours per day, maybe more if you can deal with the added time per day.

    Math course by "distance learning": What course is yours exactly?
  5. Oct 17, 2012 #4
    It is completely dependent on what course your taking, page by page analysis of a math textbook has always been a waste of time for me.

    So what math is it?
  6. Oct 18, 2012 #5
    There are two courses, both graduate-level. One is Algebra and Galois Theory, and the other is Analysis (right now, specifically, measure theory and topology).

    "Distance" = I do not attend courses; instead, I self-study with the course materials and show up for the final. I picked it beca
  7. Oct 18, 2012 #6
    Sorry, overactive Return key. To continue: because of my afternoon job and because I'm a little behind and need more time to digest the material.
  8. Oct 18, 2012 #7
    :confused: A careful analysis of the textbook is the only way to learn math. How do you learn math? Just look at the formulas and memorize them??

    But the proofs in topology, analysis and galois theory are awesome and insanely interesting. Why would anybody want to skip them???? Furthermore, they contain important ideas and techniques that are useful.
  9. Oct 18, 2012 #8
    Ah, but micromass -- you've returned to the reason for my original question.
  10. Oct 18, 2012 #9
    This sounds like me. I normally just study axioms and methods. I also look at examples from notes/books.
  11. Oct 18, 2012 #10
    No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem. I don't think I have met many people who do it the sit and read method though but then again I am not a graduate student.

    I read Physics books, but never math books.
  12. Oct 18, 2012 #11


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    So you just skip over all the proofs and examples? I have never seen anyone use this method before and probably for good reason.
  13. Oct 18, 2012 #12
    Some constructive advice: that is an awful method to learn math. I suppose engineers can get away with just reading the methods, but you're a math major!!!
  14. Oct 18, 2012 #13

    George Jones

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    For what math courses have you done this?
  15. Oct 18, 2012 #14
    Everything up through Differential equations, just to add, I have never even taken a trig or Algebra 2 class. I think the formal introduction to applicable concepts is far over rated, for higher level proofs this might not be the case but for undergraduate math an under, I show as proof math average right now is about a B+ and that is only because I had to take Calc II over the summer, it was crammed into five weeks.
  16. Oct 18, 2012 #15
    Most of the time I don't even take a peek.
  17. Oct 18, 2012 #16

    George Jones

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    I don't know what this means. Humour me; please give an explicit list of courss.
  18. Oct 18, 2012 #17
    Well, if you just want to apply the concepts to real-life situations, then I'm sure this approach could work. But this is the complete wrong mentality for a math major. Don't you feel the slightest curiosity as to why properties are true? For example, the fundamental theorem of calculus:

    [tex]\int_a^b f(x)dx=F(b)-F(a)[/tex]

    Aren't you interested why this extraordinary theorem holds true?
    Or the rules of derivatives, things like [itex]\frac{dx^2}{dx}=2x[/itex]. Do you just memorize these facts and accept them without justification??

    If so, I think you should quit your math major.
  19. Oct 18, 2012 #18
    Consider the analogy of tourism. A walking tourist will learn a lot about a little. A flying tourist will learn a little about a lot. You need both. For an overview perhaps you need to read about the history of your mathematical topic, before slowing down and looking at the detail. So, a variable speed approach is best?
  20. Oct 18, 2012 #19


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    That may have worked for regular calc 1 - 3 and DiffyQs because those classes tend to be quite easy but it won't work for the upper division classes. Actually if you had taken honors calc 1 or 2 (the spivak or apostol courses) you would have noticed your method failing there as well.
  21. Oct 18, 2012 #20
    Basic addition, Basic division, Basic multiplication, Long division... you asked of humor... more seriously, Calc I, Calc II (some multivariable but not much), Analytical Statistics and Dif. Eq.
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