Advice - MS in mathematics to increase competitiveness for PhD programs? I am a senior maths major (computer science minor) who is pretty worried about the next step in my academic career. First, let me state that I'm about as sure as I can be that I want to get a PhD in mathematics. Unfortunately, I didn't realize what the field entailed, or my passion for it, until I had made many really poor decisions - mostly in the form of bad attendance. For example, I basically just showed up for tests in calc 2, 3, linear algebra, and differential equations, and consequently, my field of potential letters of recommendation is quite small. To make things worse, I come from a party school - I need letters! 

I've had one professor (abstract algebra) offer to write me a letter, and I've taken my advanced calc sequence under a professor who I think could write me a good recommendation (adv calc 2 was a graduate course; had [i think] the highest grade out of about 15 students). I'm also taking topology (graduate level) this semester, and am hoping to impress my way to a third letter.

My GPA is okay - cumulative about 3.61; math is all A's and one C in linear algebra. I've also been working through a few books (Spivak's "Calculus" and "Calculus on Manifolds," and am about to start Birkhoff and Maclane's "Survey of Modern Algebra." Although I love the material, and enjoy learning it, the independent studying probably stems from some feeling of inadequacy due to my past immaturity.

I got a 169/170 Q, 165/170 V on the general GRE. Also, I think I can crack 80% on the subject test, but am not overly confident about this. One glaring hole is that I have done zero research, and have done nothing extracurricular - I literally have nothing "extra" going for me.

My concern is that I've seen the resumes of many people accepted to top universities (PhD track), and I just don't stack up. But if my goal is to become a professor one day, it seems that where I go to school is extremely important. So should I just hope that I can get accepted into a top 30-50 school, or would it be beneficial to consider improving my resume in a solid Masters program so that better schools become available? 

And if a masters is a viable option, what caliber of school would I need to excel at in order to be a competitive applicant for a top 10 PhD program?
If you have a thin B.S. background in math, it is definitely very helpful to get a M.S. in math before (re-) applying to Ph.D. programs. (I say this having been on the Grad Admissions cte, and having been Dir Grad Studies in two different incarnations...) In the U.S., many undergrad programs are really very thin, due to the requirement (otherwise wholesome) of "breadth".

The "undergrad research" episodes in summers, and during the academic year, are good for generating enthusiasm and camaraderie, and especially for getting outside the rigid classroom/textbook atmosphere, but (apart from very broad features) are not at all good indicators of what serious research is or will be like. Those programs are designed to be fun (pizza parties, etc), so "doing research" in that sense is fun for nearly everyone.

So, apply to MS/PhD programs at the top 30-50 schools, and do the best job you can in the standard/required PhD curriculum (usually, there is no distinction between PhD curriculum and MS, except that the latter is designed to accommodate, if necessary, weaker students, perhaps weak enough so that the MS will be their last degree in Mathematics... don't be misled into taking an "easy route"). And, in the course of doing the coursework, don't be a stranger to the instructors of those courses, who will be your letter-writers for either re-application to "better" schools in a year or two, or will be your letter writers if you need to re-apply to that institution itself for the PhD program, for bureaucratic reasons.

Grades in non-math courses, and grades in calculus and lower-division courses don't matter much, although obviously good grades are a not a bad thing. Admissions committees are well-acquainted with the changes people go through around age 20 and so on. The question is not so much what silly things one has done a few years back, but where one is heading now, and what documentable evidence there is for this.

In particular, although self-learning is the most significant long-term way to develop scholarship, it is obviously hard to document. Perhaps the best way is to sign up for courses that appear to re-iterate (serious) content you've already studied. Presumably, you absolutely ace the material and draw the instructor's attention... since self-study beyond "requirements" is, strangely-or-not, extremely rare. Evidence of non-passivity is excellent, if it really proves to be what you feel it is (rather than, say, mere obsessiveness).

So, again, yes, think about "proving yourself" during a year or two of "MS work". No downside, really.