I'm completely clueless on the process, but on track to graduate in two years, so I have a few questions about what I should do.

1) What's the difference between a Master's degree in Mathematics and a Ph.D in Mathematics? I have always wanted to teach mathematics at the college level, and so my family has always joked that I'd be in school forever to get a Ph.D. What is the difference between the two, and how do I go about choosing which is right for me? I don't really know the difference between undergraduate and graduate school, except that the latter is supposed to be much more difficult, much deeper, and is the starting point for research for many people.

2) What do I need to go to graduate school? At the moment, I have a 3.6 GPA, which I guess is alright, but I don't know if it's good enough or if there's anything else I need. I have awful test anxiety, and so my GPA in mathematics specifically is probably a lot worse than my overall GPA boosted by my minor and general electives. I am working on getting an undergraduate research opportunity to help put weight behind those numbers. Is this an appropriate things to do in undergraduate, and is it going to help me go further?

3) I have to come to realize just how expansive mathematics is through my time on MSE. There are tons and tons of fields of mathematics with all kinds of interesting problems. At the moment I'm really fascinated with differential equations because that's what Stony Brook researches at the graduate level, so their undergraduate course is like an introduction. If I decide I want to do that, how can I find other schools with similar focuses? If not, how can I figure out what I do want to study? It seems I'll never experience even the surface of some of these fields that are particularly deep.

Sorry for the long post, but thanks for any insight any of you can offer.

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What country are you in? (I assume the USA?) Do you know where you would like to go to graduate school? (The answer will depend on this). –  Thomas May 20 '14 at 15:52
Yes. I attend Stony Brook University in New York, and their undergraduate course in differential equations and chaotic/dynamic systems has sparked my interest greatly. The university focuses on exactly that at the graduate level. At the moment I would consider attending the graduate school here, and am also looking for other schools that offer such programs. I'm also keeping an open mind though, I have a good 40 credits to earn in mathematics (including 2 graduate courses if all goes well) before my degree, so I may find other things I like too. –  Alfred Yerger May 20 '14 at 21:20
Stony Brook does not focus on exactly differential equations. They have good research groups in several areas of geometry too. You should aim not to stay at the same place for graduate school as you are as an undergraduate. This is not a criticism of Stony Brook; I'm sure the faculty there would also agree that it's best to move to a different university for graduate work. –  KCd May 20 '14 at 23:36
I'd also suggest you look for other graduate schools with a focus on differential equations. It's never healthy to stay in the same school for undergrad as well as grad school. –  Bolt64 May 21 '14 at 2:54
One difference is that you get to see how things are elsewhere: new faculty, new environment. You can't think you're going to get a job and spend your whole life at Stony Brook. Finishing your college degree is a natural first time to step out into a wider world. You should read the part of "Surely You're Joking, Mr. Feynman" where Feynman writes about deciding between staying at MIT for grad school (his original desire) vs. moving to Princeton (what the MIT faculty told him to do, and what he did do). –  KCd May 21 '14 at 5:12

Here are some brief comments on your questions.

1) For most master's programs you have to pay tuition yourself (or take loans). For Ph.D. programs in math you do not; the department covers your tuition in exchange for you doing some type of teaching and/or grading. This is true generally of graduate programs in math and the sciences, and it's a compelling non-academic reason to prefer a Ph.D. program if you can get admitted to one.

2) You should have strong grades in a range of upper-level math courses. It's the upper-level math grades that people look at. Nobody will care how you did in history or biology courses. Similarly, the GRE math subject test is important while the GRE general test is not (the level of math on the general test is a joke for math majors, at the level of the SAT).

3) Don't feel committed to knowing what you want to work on in graduate school before you get there. That means two things: (1) If you think that you love differential equations and write in your application that this is what you want to do, nobody is going to hold you to that when you arrive in graduate school. Unlike lab sciences, where an incoming graduate student is attached to a particular professor's lab early on, in math graduate school you take courses and explore for a little bit, so if you switch interests nobody will hold it against you (it happens all the time, since there are many areas of math at the graduate level that you'd never see as an undergraduate). (2) If you don't want to commit in your application to having a definite area of interest, you don't need to, and could simply write about the areas you currently find interesting and why.

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It is not uncommon to have a assistantship even though you are "just" studying for a Master's. –  Thomas May 21 '14 at 19:06

I can only give you a definitve answer to 1), 2) and 3) seem to be dependent on the country/university/college/etc. that you are visiting. Since I'm from Europe, I cannot give you complete answers for 2) and 3)...

Now to the answers:

[tl;dr]: M.Sc. $<$ Ph.D. (regarding ability, know-how, experience and future job possibilities)

2) Here I cannot really give you an answer, since I do not study in the USA. But clearly you will need to complete undergraduate school. Which qualifications you will nedd to attend graduate school in mathematics, I cannot tell... Yet doing some research projects on the undergraduate level can never be wrong, as long as you still have enough time to complete what you need to complete. It is especially useful for gaining insight into your prospective field or method of research, which might reveal that something else would suit you better, or that you are exactly where you need to be.

3) Basically, as with your other questions, you should do a consultation on your future career. I'm sure your undergraduate school will have some sort of advisory service. They will most certainly know exactly what you'll need and where you ought to go. Last but not least: Take your time choosing your field of study! It can be a quite painful quest, and you should still be able to change fields of study/research later on (as long as they are at least somewhat connected, so with math you would have a good starting point to get into physics, chemistry, biology, even things like psychology, medicine, etc.)

I hope that clears out some of your questions. Also, you might want to consider editing your question, since you may not give enough information (your school for a start...) to get satisfying answers.

Anyway, I hope I could help!

Sincerely, SDV

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Just to point out that (1) you do not necessarily work with a research group when you do your Ph.D. Most often the funding (in the US) comes from a graduate assistantship. (2) most often a Ph.D. will not give you a "leading position" in the industry. (3) Taking 3 to 4 years on a Master's thesis in the US would be considered a long time. Also, many don't even write a thesis. Many universities don't actually require this (which is too bad IMO). –  Thomas May 21 '14 at 19:09

From what I understand you want to continue your studies in mathematics. You have realized that one can so a Master degree and a Ph.D. degree, but you want to know what the difference is and what it requires. You will get different answers from different people because people have difference backgrounds and different philosophies. People do mathematics for different reasons. So the following is just my take on all of this.

While some of the answers might depend on your geographic location, some of the general ideas should be the same all over.