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Testing Math Journey/Qualifying Exams

  1. May 6, 2009 #1
    Well, my first post here couldn't be more appropriate I suppose given the situation I'm currently in...

    Let me outline the story...

    I graduated in December 08' with a combined mathematics and adolescence education degree to teach high school. After a bit of conversation with some of my math professors, I attended the MAA conferences in Washington in January to get a feel for what the mathematical community is all about. In any case, I presented a poster at the undergraduate poster session on research I conducted in my junior and senior year. To be honest, I wasn't entirely convinced that I should continue with my goal of one day becoming a college professor and mathematician because I felt that I wasn't gifted enough with the skills that I see many others possess in this field.

    After attending a few different seminars for undergraduates regarding graduate school, I became convinced that I needed to go after it, and go after it with everything I have. After applying to five different schools, I'm headed next semester to the University at Buffalo straight from a B.A. to a Ph.D, or so I think...

    From your experiences (those of you who are graduate students and/or professors), was this a wise decision to jump straight into a Ph.D program with no Masters degree?

    I was also provided a teaching assistantship worth 16,500 with a full out of state tuition scholarship (although I live in the state), and I'm wondering if I made the decision to take a Masters then jump into the Doctoral once that is complete, would my assistantship and scholarship vanish? I'm asking if anyone knows, because I'd rather ask here than walk into the graduate director's office after he's practically bent over backwards for me with doubts of his judgments.

    Lastly... Qualifying Exams are in August, and I've just begun studying for them. The topics are Analysis and Algebra, and I'm wondering if anyone has any tips for studying for these dreaded exams? Remember, I've been out of a mathematics oriented class for about a year now, so what suggestions do you have regarding this upcoming exam...
  2. jcsd
  3. May 7, 2009 #2
    I haven't personally taken the qual exams but I bet there some good books out there, probably even at the library (rather than buying them as they may be pricey). They would probably give you some good practice materials.

    As for the question about your teaching assistanship, I think that is a question better directed to the school you are going to. You could just shoot them an email and say you have been considering doing a masters first before the PHD and just are looking for some guidance on the issue and also have questions concerning the funding.

    In general though, I thought the first year or so of the PHD was fairly equivalent to the masters program in that you don't jump right into research for your PHD in your first year, rather first you take some graduate courses and maybe get assigned to an adviser.
  4. May 7, 2009 #3


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    The main advantage of doing a master's degree first is that it allows you to get into graduate school withouth the huge committment. Sometimes, when you've just finished undergrad, it's hard to know exactly what field you want to pursue. You should have some inclinations at that point, but jumping into a 4+ year project is a big leap. The master's program allows you to take some graduate level courses, do some research, and see if that field is really for you. This disadvantage is that it can take longer to eventually finish the Ph.D if you do a master's first.

    As for qualifying exams:
    - start early and make review part of your regular routine from now until the exam
    - talk to students who have successfully gone through it in the past and try to find out as much as you can about the types and style of questions that have been asked - what format they follow, if there seem to be any recurring themes, etc.
    - make your own call on what areas of knowledge are the most important and try to imagine what questions could be asked to test those in an exam
    - avoid people who talk about how nasty they believe it's going to be or tell horror stories about it
    - prepare a schedule of topics to cover and get your advisor's advice on how much time you should spend on each topic (and if there's anything you've missed)
  5. May 7, 2009 #4


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    In my field (physics) students are often admitted as PhD students with a BA/BS, but are expected to take the masters coursework on the way to getting the PhD. Some schools will grant you a masters at that point, others don't. So I would expect you'll be taking the masters courses, even if you are a PhD student.

    I spent about 4 months solid studying for my qualifying exams (my university makes you take them after you've finished the masters coursework, before you can go on to the PhD research). Call up the math dept office and see if they have copies of old qualifying exams lying around - many will make up copies for students studying for the exams so you can see what's been asked in previous years. That can be invaluable.
  6. May 7, 2009 #5
    Fortunately, I have older copies of these exams, which is nice...
    The test is primarily based on Algebra and Analysis. I'm incredibly strong in Linear Algebra, which is practically half of the Algebra section. The Analysis part is where I'm most likely going to spend most of my time, because I'm not incredibly solid on doing proofs...

    So, I suppose a follow up question would be this: Does anyone have suggestions for getting really good at proofs? For me, I have a natural ability to see patterns, but organizing the patterns into a way that makes logical sense is rather difficult for me.
  7. May 7, 2009 #6
    Well here's my 2 cents on what I did in my Real Analysis course granted other schools and courses may be a lot different. I found it hard to get used to doing proofs in RA at first. In my opinion it was a lot different than say 'proving' things in high school geometry.

    What helped me was to sketch outlines of proofs for a lot of the theorems. Ie. To prove X theorem I know you need to show x,y,z and then you use ABC theorem to show this and that.
    I'm not sure if that helps you or not but it did for me, rather than trying to just memorize what proof was given in the book or what was in my notes I would make outlines of the proofs. Often for our tests that was all that was required due to time constraints. You can't be expected to do full proofs of a semester's worth of Analysis in only an hour or hour and a half.

    That being said, on the Qualifying exam you will need to show a lot of knowledge but at the same time I don't see them expecting you to walk them through step by step the proof for every theorem covered in their Analysis class.

    Its been a while since I took that course but I find the things i remember using most were Nested Interval Theorem, Bolzano-Weirstrass, Int/Mean Value Theorem, basic definitions of convergence, Cauchy sequences, differentiability/continuity.

    Maybe that helps, maybe not. I can't give you too much help on the side of Complex Analysis, though I did better in that course It was much less focused on the proof side.
  8. May 7, 2009 #7
    Thanks for the advice...

    From the courses I've taken in Undergraduate, which were about a year ago, I recall using the theorems that you stated the most. The one thing I'm concerned with the most are the obscure theorems that you either know or don't know, i.e. Darboux's Theorem.

    Although, it would be logical to get the most important stuff first, then go back for the other things that aren't used as frequently... hmmmm... =)

    Something that a friend of mine suggested was to take the concepts tested on the exam and organize it into a miniature syllabus for the summer, so I'll have stuff to study every single day. I'm more inclined to lean in this direction because of my Adolescence Ed. degree. We'll see how that works out, but I'm feeling pretty optimistic.

    Also, I received clarification regarding this Qualifying Exam, as well as my degree...

    I'm enrolled initially in the Masters program, with acceptance into the Ph.D program pending the 30 credits for Masters and passing the first Qual. Exam. Therefore, I technically have two years to study for this thing if I don't do well the first time out, so we'll see. The graduate secretary told me today that in the last 10 years, there has only been one domestic student who has passed the exam for the first time, fresh out of undergraduate.

    I'll be the second...
  9. May 8, 2009 #8


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    Darboux isn't exactly an obscure theorem. It's a lot easier to use Darboux for a lot of integrability proofs instead of the Reimann sums definition. Read the chapter on integration from Spivak's Calculus on Manifolds to see it used for integrability proofs for functions from R^n -> R.

    Also, most students planning on entering a math Ph.D program in the US apply to them during their senior year of undergrad.
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