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Math major

  1. Feb 11, 2008 #1
    I am planning on becoming a Math major if my plans to attend a different school does not go through. I really do love math, I find it very fun to learn and the outcome/solution to some problems amaze me. My main problem is that I have huge gaps and they'ved continued to build since I am now in Calculus III. I started studying math just last summer, and it was the first time I ever took Calculus, or more importantly, took a math class seriously. I have been tutoring math for the past year, courses from Algebra to Calculus II, though only a few Calculus students come now and then. I have a decision to make on whether to take summer courses, or to use the time during the summer to continue working as a tutor to practice what I've learned and catch up on what I did not learn.

    I am already self-studying material that I did not learn or learn well enough to feel satisfied, such as Geometry. A few things that really hurt me is not having been even exposed to mathematical induction, sequences & series, and simple proof writing.

    Books I am currently using for self-studying:

    Geometry, Peter Selby
    Algebra & Functions and Graphs & Trigonometry, I. M. Gelfand
    Precalculus, Cohen
    Calculus, James Stewart

    I also have Spivak's Calculus book, but obviously that book is too hard for me. I can get by doing some of his problems, but not the ones that ask me to prove stuff. I am very application driven, I love being able to apply the tools I've learned, but I also realize that as being a math major, I will have to face courses that are abstract and demand me to prove and disprove things. I'm a good student and highly disciplined. I'm willing to spend days/weeks on a problem before I give in and ask for help or look at my solution manual.

    When I am self-studying, I re-write the proofs and even do the derivations of the formulas myself. I really want to learn math, and it has really helped me to do well in other courses. I have learned to analyze my answers more thoroughly, and overall, think logically. Math has boosted my confidence, I make less errors each day whether tutoring or doing my homework, and I know this skill is truly priceless and it will definitely carry my far into my professional career.

    Am I ready to take on upper level math courses?
    Last edited: Feb 11, 2008
  2. jcsd
  3. Feb 11, 2008 #2
    Hi roco,

    I'm in my third year of math undergrad and I've learned a couple of things this year regarding upper level Math courses. I am also interested in applied math, so I usually don't take the abstract courses (I still need to take some but I'll leave them for next year). I still need to prepare for lectures by pre-reading the material just as last year but now I have to read the topic from multiple sources such as texts from the library and attempt the problems as well. This way when the prof blazes through the proof, you don't get lost because in your mind he's filling in the blanks that you were not able to get. You should also try to play around with programs such as Matlab, and Maple. There are lots of tutorials on the web. It's one thing to know what to do but it's another thing to set up a model so it can be used by other students or people in your group.

    I would also suggest taking a course or two during summer so you can take more electives (from other depts) during the school year. Be warned: taking courses ESP math, is a lot of hard work in the summer. You cover a week in one day.

    The class sizes in upper level courses are quite smaller than the previous years since you are not usually sharing classes with engineers and physics students. With this in mind, you have to make an extra effort to get on the prof's radar which means asking meaningful questions and going to office hours.

    I don't know about your school but at mine the profs usually have their website and a link for the course they are teaching. On this site, they usually put their course materials such as readings, sample exams, hw problems/solns during the term. Even though you might not be taking a particular course with them, it's a good idea to to visit the sites around exam time so you can download all their material. This provides an excellent framework for self studying because you have problem sets, some handouts, a course outline and etc. I find this much more effective than just reading the text.

    Hope this helps, I'm sure the senior members have better advice to give.
  4. Feb 13, 2008 #3
    HOw can one be a math tutor if he/she is not good enough in it? At least to handle the calculus part, concerning proofs also!

    P.S> Don't take it personally though. Because maybe i will start a new thread on this tutoring thing.
  5. Feb 13, 2008 #4
    No, not even the applied stuff. Just try taking a "good" upper level PDE course (the PDE course designed for engineers is extremely boring). The point is that in both pure and applied math you need the "analysis" thinking style...

    Basically, if you get stuck with Spivak, perhaps you should take a break. Do something else, like download C# express and "coding4fun" at http://www.microsoft.com/express/vcsharp/Default.aspx.

    Or try studying metric spaces. See if you are able to do the proofs there. Take a break, do something else.. If (real) calc is hard for you, see if you can prove La'Grange's theorem.

    If all else fails, look at a book like "Foundations of Analysis" by Landau, and spend a month constructing the real number system from the natural numbers (well you will probably not get that deep in the book -- but at least chapter 1), and write out a proof that 2+2 = 4.

    Landau's book should give you an idea of what "mathematical induction" is.

    But ultimately, Spivak's calc contains a lot of what prerequisites to any (pure/applied) math is... that is unless all you want to do is pass actuarial exams, which is another option for a math major..

    P.S. If Spivak looks "too big" hence "too intimidating", see my thread "Calculus, how to prove it": https://www.physicsforums.com/showthread.php?t=214916.

    The basic idea I tried to convey is that calculus is what you make it to be -- and on the problem of "rigor", the point is that you don't need to prove a 600 page book just to know you are in the right direction.. Just prove a subset, of the important calc theorems, and you can just not even worry about exp/log/sin/cos when doing so.
    Last edited: Feb 13, 2008
  6. Feb 13, 2008 #5

    I graduated with a BS in math and you know what? I HAVE NEVER studied series. I took the AP Calc AB exam and skipped out of Calc I and II. I mostly stuck to pure math like Algebra and logic, so I never really needed to ever use hardcore series techniques. I even took the required advanced calc and complex analysis classes and passed them with flying colors since we didn't even need to use series for most of it. I mean you do learn a little bit by just doing some problems here and there.

    Usually most universities offer a transition course that introduces students on how to write proofs. You will learn induction then.
    Last edited: Feb 13, 2008
  7. Feb 14, 2008 #6
    I'd get comfortable with proofs. Prove silly stuff to yourself, understand logic inside and out, heck, go back to your high school geometry book and re-prove all the two-column proofs. Prove that the nth derivative of 1/(1-x) is n!/(1-x)^(n+1) by induction, prove that the square root of two is irrational by contradiction. But in order to be a successful math major, you have to be able to think (and argue) logically.

    We had a few girls where I used to work say, "I loved being a math major, but I hated proofs." I don't even know what that means after you've taken higher level classes. Did they like the atmosphere in the math building? Because there's nothing but proofs after a certain point. Even if you do applied, you'll have to take a real pde course (not engineering pde) where it's all proofs. Maybe some numerical analysis courses are lighter; I haven't taken any of those.
  8. Feb 14, 2008 #7
    Your response is definitely appreciated! Thanks for advice.

    I can tutor Algebra-Calculus II since I understand the basics that are required of me. I can pretty much solve most of the problems in Stewart's book, and my biggest obstacle at the moment is transitioning to being able to solve the harder problems in Spivak's book.

    Thanks rudinreader! Ever since I've attempted Spivak's book, it has somewhat shot down my confidence, but it also driving me to get better. I think I'm being too hard on myself, just b/c I can't solve all the problems, doesn't mean I'm not on the right track. Also, I am unable to find Landau's book at my library. Have you heard of the book An Introduction to mathematical reasoning : numbers, sets, and functions by Eccles?

    Lol, hey it's good to know that not having studied a particular topic such as one that's as important as series hasn't prevented you from doing well in higher level math courses.

    I don't mind proofs at all, I even enjoy reading them, but proving is a whole different story. I proved a more than half of the problems in Spivak's first few chapters. It just got too rigorous after that, I'm not going to give up though. I know my weak points, and the best way to tackle them is going back to precalculus material.
    Last edited: Feb 14, 2008
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