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I've been considering applying to graduate school of some form or another in Mathematics. During my undergraduate years, I did decently on my undergraduate coursework (mostly A-s, some As, a B) but not stellar, and it wasn't until I started taking graduate courses as a senior that I started buckling down and getting solid As or higher. I also had no research experience at this time.

I eventually want to apply to a Ph. D program in pure mathematics. Since I didn't really click until later in undergraduate years, so I was considering applying to 2-year masters programs in mathematics, and then based on my performance there, decide whether I'm fit to work towards a Ph. D.

My question is, what are the advantages and disadvantages to completing a master's program before applying to a separate Ph. D program?

I know masters degrees are sometimes considered terminal degrees. Would doing well in masters coursework be advantageous in applying to Ph. D programs later on as opposed to immediately after undergrad years? Does strong performance as a masters student make one a more attractive candidate, or do programs have less interest in applicants who already have a masters? Is it wiser to apply for a Ph. D directly? I'm sorry if this question is considered too much of a soft question. Thank you.

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    Welcome to Academia.SE, and you don't need to be so hesitant - these questions are exactly on-topic for this forum! – TCSGrad Feb 28 '12 at 14:47
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I can only speak for my (computer science) department. Students who apply to our PhD program with MS degrees are held to a higher standard than applicants who apply as undergraduates. What we look for in PhD applicants is strong evidence of research potential. Most undergraduates don't have an opportunity to undertake a real research project, but MS students do have that opportunity, by definition. It's much harder for MS applicants without publishable results to be admitted than an undergraduate in the same situation, all else being equal. Grades are much less important (unless they suck, which yours don't).

On the other hand, the fact that your already decent grades improved when you started taking graduate classes is a huge point in your favor. Be sure to get recommendation letters from your instructors in those courses, and hit them up for research opportunities. I recommend applying to both MS and PhD programs; some departments will even let you apply for both simultaneously.

There's a similar effect in NSF's evaluations of graduate fellowship applications. NSF splits the applicant pool into three piles based on the number of years of graduate education: still an undergrad, less than 12 months, and more than 12 months. Expectations are higher for applicants in later piles. In particular, publications are a de facto requirement for students in the more-than-one-year pile, even if their one year of graduate education was in a non-research masters program.

  • do the publications need to be in the same field? For instance, a completely hypothetical TCS student with publications in say mathematical biology. Are those useful? – Artem Kaznatcheev May 12 '12 at 19:21
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    @ArtemKaznatcheev — I think publications in any field is more useful than no publications at all; Sticking with any project long enough to ship it is a good sign. But publications in distant fields might be less useful than publications in close fields. (Speaking personally: I don't know what publication in mathematical biology "means".) – JeffE May 13 '12 at 4:38
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Many PhD programs require you to do a masters on your way to a PhD, so the point in many cases is moot. That being said,

Advantages:

  • You'll get a chance to see whether research is something you like before committing to the PhD process.
  • You'll complete most of the coursework, so when you get to the PhD work you'll be able to more quickly focus on research.

Disadvantages:

  • Depending on the masters, you may be spending money on your degree. Many PhD programs will fund your way (in exchange for you doing awesome research).
  • You'll be required to do a master thesis, which can be comprehensive, and you may come to view it as a waste to do it twice (one for masters, one for PhD).
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    Do not spend money on your MS degree. At least in the sciences and engineering, if they don't want to pay for you, they don't really want you. – JeffE Feb 28 '12 at 12:12
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    Don't want to be the grave digger of this old question, but I think one needs to add here that JeffE's comment is certainly not true for Central Europe - around here, paid master's programmes are almost unheard of. – xLeitix Feb 28 '14 at 16:00
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Other than what eykanal mentioned, you might want to consider the following points:

Pros

  • If you have lackluster UG grades with little/no research experience (in terms of publications/term papers/internships), few top PhD programs would be even willing to look at your application, not matter what are your scores in the GREs. I assume your letters of recommendations (LORs) from faculty would be average at best as well, which would really, really hurt your admission chances. So, a Masters degree would help you rectify that - you can aggressively start pursuing research-based projects with the faculty, and work at a publishable thesis. This would both give you something added to your profile (a publication), and you can get much better LORs from your masters faculty - something that may get you into that top school where your dream adviser works!

  • You can get (some of) your Master's course credit transferred to your PhD program, so that you can start working on your thesis much early, and may even finish faster! However, this really varies from department to department, so make sure you inquire about this before applying!

  • If you are really, really sure of what field you want to research in, you can find the top people in your field (whom you'd want to be advised by during your PhD) - and start corresponding with them during your Masters, asking queries/having discussions about their most recent papers, and using them in your Masters thesis (assuming you'd be working in the same field). That way, when you finally apply, you would stand out in their mind from the other applicants, as someone who is genuinely interested in their research. Mind you, I'm not asking you to fake anything! Be sincere about your passion to work in the same research areas, and they just might be convinced to bat for you during applications review!

Con

  • On the other hand, if you do not fare well in Masters program, most admission committees (ad-coms) would take it as a red flag - as it would bolster the less-than-stellar impression your undergrads created, and hence they might deem you not fit for PhD at all. This might be the case with even the ad-coms that might've been willing to give you the benefit of the doubt when only presented with your undergrad records - so, tread carefully if you happen to embark on a Masters, as your performance there would undergo even greater scrutiny!
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    Thanks for your insights shan23. The con you mention is also a personal red flag for myself. I figure if I can't perform well in a master's program, that's better than applying and then washing out of a Ph.D program. – Morgad Feb 28 '12 at 5:11
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    The con may be a pro in disguise. If you do not fare well in your Masters program, especially in your thesis research, that may legitimately indicate that you are not really suited for research. Better to learn this about yourself earlier than later. – JeffE Feb 28 '12 at 11:46
  • @JeffE how about this qualification : "that may legitimately indicate that you are not really suited for academic research." Academia is a unique beast. – Amy Feb 29 '12 at 2:05
  • Fair point, @Amy. (But then your definition of "academia" has to include my colleagues with PhDs in industry who publish in the same venues as academics.) – JeffE Feb 29 '12 at 9:38
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    @JeffE Of course, but there are lots of opportunities to do fulfilling research that don't involve holding a PhD. I see a lot of people in academia forget this, so I just wanted to add my voice! – Amy Feb 29 '12 at 17:04
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So I'm a first year math PhD student. I did not get my masters before entering university, but I don't think that had a big role either for or against me. I didn't end up applying to any masters programs actually, and I had two major reasons:

  • the primary reason for me was money. None of the programs that I was interested in funded masters math programs. I'm sure that there are such programs, but school is expensive. And I'm poor. So it goes.

  • the secondary reason was inspiration. I'm going to do original research, right? That's the goal. And I'm impatient ad ready for it (at least in spirit). It happens to be the case that we spend our first year here doing a lot of work, quals, etc. It's the second year where things kick into gear, and I'm already straining for it. (However, if I already knew everything, I would have already passed my quals, so it's not unfair treatment or anything).

If I were to give you my personal recommendation, knowing that you wanted to get a math PhD eventually, then I would say apply to both if you are in doubt. What's the worst that can happen? If you are a strong enough candidate to go straight to PhD, then great. Why not, right? And if not, then a little graduate coursework can't hurt your application.

I would also like to mention that, at least in my program, there is no transferral of credits. You come in, pass qualifying exams, and write a dissertation. The transferral of credits would come in the form of you already knowing enough to pass the quals immediately (I did not know them all, for instance - some people knew more and some less). In all likelihood, you'd still burn at least a semester, more likely a year, just like all of us do. So a masters would likely lengthen your studies (at least at my uni).

On the other hand, it is possible that you don't have a big idea of what area of math, or what field of math in particular, you want to work in. This would be a big issue, perhaps. I knew I wanted to do number theory, and I was interested in the work of some of the number theorists at Brown. But a combinatorialist, graph theorist, or many other people would be hopelessly alone here (there is little love for combinatorics here). I suspect this sort of problem could be true in many schools. But if you know what you want to do, then this is no issue.

To end, I wanted to note that it's fine to test the waters, i.e. to see if you're fit for a math PhD, in a math PhD program. I've known people who have gotten a masters a year or two in, decided that was enough, and left the program. I suspect this isn't uncommon (although it would be uncommon where I am), and you'd get paid to do it.

  • Thanks mixedmath! I'm interested in algebraic things, (algebraic number theory, commutative algebra, algebraic geometry) but most of my experience with these subjects came from the graduate courses I took. I feel like I don't really know what real research in those areas entails. When you applied to graduate programs, how familiar were you already with the research field for number theory? I'm curious because in the case of writing a statement of purpose, I'm unsure of how knowledgeable one should be already. – Morgad Mar 5 '12 at 8:30
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This is speaking only from my experience in Public Health and Epidemiology, so its slightly more general, but hopefully it will be helpful. Some of these have already been mentioned by other posters.

Advantages:

  • You have the potential to establish your "research chops", or shore up your grades. If you don't think you have a strong application coming directly out of undergrad, this is probably a plus.
  • If you move programs between your MS and your PhD, you get to see two schools. That's good for both perspective on research (rather than only being exposed to "The University X way of doing things"), and it means an extra set of contacts, though admittedly your network from the Masters university will probably end up being somewhat weaker.
  • You get the "feel" for your potential research topic. Which means that if you don't like it, you can switch. Getting an MS and a PhD in two different but related topics (like say Pure Math and Applied Math/Operations Research/CS, etc.) is pretty run-of-the-mill. Switching PhD programs mid-stream is a much less pleasant experience.
  • In contrast to some of the answers here, in my field at least I'd argue that undergraduate applicants are judged more harshly than their MS-weilding colleagues. Several major programs simply do not accept folks directly from undergrad into their PhD programs, and several others say they do, but in practice without someone pulling for you or an act of God they're not going to do so.

Disadvantages

  • It will probably add time. If you're coming straight from undergraduate, a university's combined MS/PhD track (wherein essentially you get an MS by way of your PhD coursework + a thesis) is likely the "least time" path to a doctorate. While you'll be able to transfer some of your credits from your Masters into a PhD program, some will almost certainly not transfer, and the university may have a core set of courses that they'll want you to take over again. For example, in my program, I'd say having a Masters trims about a year from your time in the PhD program. But most Masters degrees take 2 years...
  • You have to move again. The flip side to getting to see two places worth of stuff is you're going to find yourself hopping across the country not once by twice.
  • There are potentially some cultural problems. Some places view Masters students (not necessarily incorrectly) as transient, so its hard to command a professor's attention as someone worth investing tons of time into, compared to a doctoral student who is going to be around for 5, 6 or more years. This is especially true for professors outside your thesis advisor and maybe the professor who taught a class you did particularly well in.
  • Money. In contrast to @JeffE's assertion (which is absolutely true for doctoral programs) my field doesn't necessarily fund Masters degrees that aren't part of a combined MS/PhD track. It's hard to get faculty to use their limited RA positions on Masters students instead of doctoral students, and because in the field there's actually a very small number of undergraduate programs, there are few TAships available to people who haven't passed through at least some of the coursework. Because the MPH is also heavily used in my field as a terminal/practitioners degree, it's treated somewhat like a professional school - you pay for your degree.
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I cannot speak for Mathematics but for Computer Science the importance of getting Maters degree first is multi-fold which of many have been stated above but one which I feel is also very important. Sometimes in the middle of the PhD program you might feel like taking a break for couple of years and consider doing a job either for relaxing or to get a hands on experience in the industry. Having your Masters degree then will help you find a better job than with undergraduate or god forbid you quit your PhD, you will have Masters degree in your hand.

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In some fields (this is true in my own experience for some humanities and social science disciplines), there are departments whose terminal MA programs are recognized as being "feeders" for more competitive Ph.D programs. So applicants who go through those are not necessarily at a disadvantage—possibly the reverse, especially if one's undergraduate grades or training are lacking in some respect. You will need to find out, though, what is true of the field you are working in, and of course which MA programs, if any, have this reputation.

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