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Studying Your experience of studying mathematics and physics

  1. Jun 16, 2009 #1
    When studying advanced mathematics and physics do you find that your brain is quickly able to understand new concepts and complicated looking math equations on the page?

    Is anyone here actually able to pick up a book, say "Linear Algebra" and read through it understanding it easily like a fiction book without prior experience in the topic?

    Do these new mathematical concepts come quickly to you, or do you have to stare at the page for hours before you understand?

    Regards,
    Adam
     
  2. jcsd
  3. Jun 16, 2009 #2

    Hootenanny

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    I don't know about others here, but I imagine that they feel the same way. Personally, I find it much easier to understand a new concept by either explicitly working through associated problems or applying it in something I'm already doing. More often than not I can understand the 'idea' of say a proof by reading over it, but a lot of the technical details and implications only become apparent after I've worked through it myself.
     
  4. Jun 16, 2009 #3
    I find that understanding the logic as to why a mathematical concept works is often best. That way I don't know have the memorise methods (which is useless), because I understand the sense and logic behind the mathematics I am doing.

    It's good to try and understand the mathematical logic when your learning a new concept. Sometimes it takes a while before you appreciate it but afterwards solving problems is extremely easy.
     
  5. Jun 16, 2009 #4

    diazona

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    Personally, no, I've never been able to do this and probably never will. Even with books in which I have already learned the material, I often still have to go through it somewhat slowly and deliberately and really think about what it's saying in order to get any value out of the book.
     
  6. Jun 16, 2009 #5
    "Staring at the page" or reading it like "fiction"... Are you kidding?

    Even though I have a ph.d. in a field that uses a lot of math, I still like to look at:

    1) how the equation is derived... in essence working through it myself (I still tend to do this on many problems... working from the basics for problems of different symmetries, rather than using text-derivations).

    2) Making sure the units make sense (obviously my field is physics... in something like math this step isn't very pratical unless it's an "applied math" case).

    3) Checking out how the equation would act in certain limits if applicable.

    4) Graphing the equation if applicable.

    5) Working extra examples that apply the concept/equation if applicable... preferably of course examples that approach the same subject from various angles.

    Doing this sort of active engagement (not "staring at the page") is how I get my understanding. Of course the more you've done this sort of engagement in the past, the quicker understanding comes in the present.
     
  7. Jun 16, 2009 #6
    I'm not nearly as qualified as some people here, but I have done a lot of reading, so I feel I could say something meaningful on this subject. What follows simply reflects my experience in essentially reading on my own.

    Reading mathematical books and articles (whether they are in pure math or some field of applied math) is a skill in and of itself.

    First, mathematics almost has a language of its own. I'm not talking about the symbols here, but the English (or another language, of course; I'll talk about English here since I've the most experience with it). There are a variety of phrases that have technical meanings. At first, one has to consciously translate these phrases and expand them to arrive at their true meaning; eventually, one is able to do this subconsciously. As a simple example, I used to have to mentally map out "if and only if" every time I saw it, being extra careful to make sure I got both directions; now I've seen the phrase so much I'm somehow significantly faster at processing its meaning in a given context.

    The growth of this skill has coincided with the growth of my ability to write proofs. I honestly believe that one of the first hurdles that one must overcome when first learning to write proofs is simply learning how to precisely control one's language and how to appropriately use phrases like "if and only if" and other such technical uses of language. I think one of the reasons for this is that people are often first exposed to proofs in calculus classes of some sort (not real analysis but introductory courses), in which the proofs are mostly calculations, not verbal arguments.

    Once one has mastered this skill, I don't think one should have any (or at least not many) troubles reading mathematical articles and books if one has all the prerequisites filled. And this is a bigger point than it sounds like. Nearly without fail I've discovered that when I have to stare out a sentence or paragraph for more than a few minutes it's because I was lacking a piece of knowledge or understanding that the author had assumed.

    In mathematics (and I'm sure it's the same in applied math) it's very easy for this to happen. As a basic example, the implicit function theorem is something that can be interpreted in a variety of ways and stated in a variety of ways. It's easy to learn only one rigid way of thinking of the theorem and then be caught off guard when an author interprets the theorem differently. Or another example could be linear maps between two vector spaces. In linear algebra, it's basically okay to think of it as a black box that takes a vector in the first space and produces a new one in the second space. But in multilinear algebra and related fields, linear maps can interact with other objects in a wide variety of ways, and the sort of rigid interpretation of linear maps promoted by classical linear algebra can be very unwieldy in tensor analysis and differential geometry and multilinear algebra in general.

    As just a little side note, a lot of proofs from textbooks are carefully crafted to be as minimal as possible in order to keep the logic of the proof as crisp and appealing as possible. This usually makes manifest some simple "tricks" that make the proof work. So when reading a textbook I usually try to extract these "tricks" and remember them, which means I'll often re-read proofs several times even if I understood them on the first reading just trying to distill the abstract strategy used by the author.
     
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