In high school I was a good maths student and took AP Calculus BC my freshman year and got a 5 and then took Multivariable Calculus, Linear Algebra, Differential Equations, Introduction to Mathematical Proof, Real Analysis, Complex Analysis, and Abstract Algebra through a dual enrollment program at the University of Maryland-College Park and at a local community college and received mostly A's and B's but I struggled quite a bit with a lot of the proofs and problem sets in the more difficult courses. Next year, I will start college I will probably take courses such as abstract algebra and topology my first year and by second year start graduate level math coursework but I am extremely worried that I am ill-prepared for the level of abstraction in these courses and that my college GPA will get destroyed immediately. My questions are:

  • What is the recommended number of math courses per semester at that level?

  • How do I make sure I do not overwhelm myself and burnout?

  • What courses are good preparation for graduate school in maths and how should I pace myself? Should I try to specialize in a specific field of math this early?

  • What is a good academic pathway for someone like me?

  • If my school offers a combined BS-MA degree for math and I could complete it in 4 years, is it worth it? And is it riskier to graduate early from college (in 2-3 years) and then apply directly to a doctorate program? If so, what are the risks?

  • Finally, how do I learn whether or not I will do well in research and whether getting a doctorate degree is a right decision for me? How do people usually find out? What process should I go through before applying?

Any advice/encouragement/experiences would be very much appreciated. Thank you. Is there anyone who has been through a similar situation?


closed as primarily opinion-based by Jim, Rene Schipperus, Aditya Hase, Joonas Ilmavirta, user147263 Nov 23 '14 at 19:36

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ If you're planning to get a doctorate in math, it is a waste of time to get a master's degree. A master's degree is not required for entrance into a doctoral program. $\endgroup$ – Matt Samuel Nov 23 '14 at 17:22
  • 7
    $\begingroup$ @MattS That depends of the university. $\endgroup$ – Cure Nov 23 '14 at 17:22
  • 3
    $\begingroup$ Since you've already taken so many advanced math courses, I'd recommend you get past the minimum undergraduate prerequisites and start taking as many graduate math courses as you can. Since you already took abstract algebra, I wouldn't recommend you take it again, unless the first time around was at a very low level; check what the course will cover and compare to what you already know; if most of it sounds familiar, move on to the next more advanced algebra course. $\endgroup$ – Robert Cardona Nov 23 '14 at 17:26
  • 3
    $\begingroup$ Making sure that you get some kind of research experience (e.g. getting an REU over the summer or working with a professor at your school) is generally (in the U.S.) more valuable then getting a master's degree (as opposed to a BS or BA). $\endgroup$ – Omnomnomnom Nov 23 '14 at 17:28
  • 2
    $\begingroup$ @MattS In some countries is the only choice there is. In my country (Uruguay) get a master degree is mandatory before applying for a Ph.d; France may be another example where I believe you need a bachelor and master degree before applying. $\endgroup$ – Cure Nov 23 '14 at 17:29

On what to take

It depends where you are. If there's a rigorous course for freshman who expect to be math majors, I would recommend it even if you've already seen the material.

This has many benefits:

1) You admit you struggled through some of the college material. Someone advanced enough to be taking college math courses in high school should not be happy moving on from a subject with only a B in it.

2) There's a huge difference between community college math courses and rigorous courses at some universities, especially in the difficulty of the problem sets. You mention proofs and problem sets as one area in which you struggled.

3) You'll get to know the other math majors in your year very well in such a course.

If there is no such course, then I'd look for one course for your first semester (possibly two your second) that will best approximate that sort of entry. Something with challenging enough work but also with enough other students you can get to know. If your peers are going to be the sophomores or the juniors, so be it.

Don't feel you need to rush through things and burn yourself out if that's where you see this going.

On graduating early / masters

The generic advice would be not to graduate early, but there are some circumstances in which it makes sense. Perhaps graduating early would alleviate a financial burden, or perhaps you will run out of mathematical content.

As for a masters, for US graduate schools there isn't really a specific benefit if you'll be continuing on in math. If calling three of your years undergraduate and one of your years masters is strategic for some reason (e.g. you won't have to take that 8th distribution requirement in basket weaving), go ahead. I have no input on whether a masters is useful if you'll be going the non-academic route.

On research experience

All the suggestions to do an REU are excellent.

  • 2
    $\begingroup$ An additional comment: making sure that you take care of any distribution requirements early (e.g. any science or English/writing requirements to graduate) is a good way to ease your way into a potentially heavy load of purely mathematical coursework $\endgroup$ – Omnomnomnom Nov 23 '14 at 19:30