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?