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How to build a better foundation in mathematics?

  1. Dec 24, 2008 #1
    Not sure if this is the right place to post (new here) but here goes.

    I'm a college freshman in the US and am pursuing a major in math. I feel that I have a rather weak background in math: I took the same garbage courses in high school as everyone else, never really paid too much attention, and mostly BS'ed my way through BC Calculus without learning the subject very well. I somehow pulled off a 5 on the AP exam, but I would tend to discount that as either a fluke or a clerical error by the College Board. Last semester I took an introductory linear algebra course and did frightfully badly in it; the professor generously awarded me a B. Next semester I'm taking Intro to Abstract Math along with Multivariable Calculus. Anyways, I'm now wondering what I need to do to develop a better mathematical foundation. I'm willing to start as basic as I need to, and I have some time before the next semester begins. Any tips? Advice on where to go? Free/online resources are definitely a huge bonus over printed material.
  2. jcsd
  3. Dec 24, 2008 #2
    Multivariable Calculus should be relatively easy if you actually know your single-variable calculus well.

    I would pick up a decent calculus book (I hear "Stewart's" is very reputable) and review differentiation, integration and all of the Critical theorems that are mentioned and I would think you would be fine.

    Oh, and by review I do not mean to just read through; I mean to read through and then take pencil to paper and do as many problems from each chapter as time will allow you.

    Hope that helps. (Also, I am an engineering student with a strong interest in Math; since you are pure math, any advice from a math major should automatically supersede mine.)

  4. Dec 24, 2008 #3
    Ah, but I don't. Besides that, I want to develop a stronger foundation in mathematics generally, not just manage to get by in next semester's classes.

    I "borrowed" my dad's ancient calc book (Ellis and Gulick) that he used back in the late 70s. Presumably calculus hasn't changed too much since then, so that's probably what I'll use to review my calc. Any resources added to that would probably have to be from an online source; I know there are several calculus books available for free over the internet, but I'm not sure which ones are good or useful.

    Thanks for the advice.
  5. Dec 24, 2008 #4


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  6. Dec 24, 2008 #5
    To strengthen your mathematical foundation, you really just have to do problems. But none of us really know where your weak points are... That's something that only you know. At my college, Linear Algebra is mostly facts and simple calculation. There's not much in that course that I think builds upon previous knowledge, as a lot of it is just vector algebra and knowledge about what the matrices/system of equations reflects.

    I think you might be being a bit hard on yourself. I doubt very much that you're as bad at math as you're implying. But mathematical success is largely dependent on practice. I hate doing homework problems and I hate doing "busy work," but that's really the key to success. You need to familiarize yourself with the operations and work on your intuition while building your problem-solving. Most of math isn't memorization, so there's nothing to reteach yourself. It's just building the problem-solving into your brain.

    Maybe if you give us more information on where you think you're weak, we could give more specific advice? You should definitely do a lot of single variable calculus to practice your math, as a lot of calculus requires a wide breadth of mathematical knowledge (Especially integration, as it requires you know trigonometric identities, how to do partial fractions, algebra, and simple calculation).
  7. Dec 24, 2008 #6
    Practice, repetition, etc. are important, but I've found the best way to shore up weak points is obsessive review. Don't try to just read through and do everything in a couple days, as you'll forget most of it and be terribly bored. Study a little at a time, but come back to the concepts you've already looked at as well. It's really all about habits.

    For example, when I'm reading through a book and looking at proofs, I usually wait a few days after I've read a proof once or twice, and see if I can work it out in my head. This ensures that I've internalized the methods (and general problem-solving framework) as well as the theorem.

    By the way, I studied introductory linear algebra on my own, but being the fool that I am, I failed to follow my own advice. Rank, nullity, and all that other fun stuff is rather vague and fuzzy in my head. On the contrary, I can tell you that multi-variable calculus tends to stick much better (with less effort) if you've mastered single-variable calculus; most of the concepts in multi-variate are simply logical extensions of the single-variable stuff.
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