Schools What is Grad school anyway? (and related questions)

[ArcanaNoir]

The transition from undergraduate to graduate school is confusing. I am a math major, prefer pure math. I am entering my senior year for my Bachelor's degree, and it's time to do the grad school thing. My goal is to be a research professor (so, Ph.D.). I was under the impression that next I get a master's, and then after that I go for a Ph.D., however, when I was looking at the website for one of my favorite schools (Brown University) I saw that they do not offer a master's in mathematics, only a Ph.D., and that they recommend a strong undergraduate background in mathematics when considering the program. So what does this imply? Does it imply I should be applying to Ph.D. programs without first getting a master? Does it imply that at this school in particular that I should apply to a Ph.D. without a master, but not necessarily try that at other schools? Is it normal to pursue a Ph.D. without getting a master's degree if one intends to get a Ph.D. in the end? But the credentials of most professors seems to indicate that they received master's degrees as well as Ph.D.s.

I really think I should get a master's first because even though I'm quite smart and highly motivated, whenever I try to overdo things I run the risk of epic fail. I'm really not interested in epic fail at this late stage...(getting my four year degree after 8 years, I do not wish to risk further delays by being hasty).

Can someone please look at the PhD in mathematics at Brown University for me and let me know what's up with why there is no master's, and if I should get a master's before trying to go to Brown. Basically I want to know if it's a standard PhD (so I should not try it before my masters) or if its a slightly longer PhD, so it would be essentially equivalent to getting a masters first just without the degree being awarded.

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Part II
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Funding funding funding... Lots of schools seem ready to offer stipends and assistantships and tuition waivers for PhD students, but not so much for master's students. How do you do the master's thing without drowning in debt? Are certain schools more likely to provide funding? Say, better schools, or worse schools, or big schools, or small schools, who funds more?

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Part III
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Choosing fall-back schools. I got a 167 (out of 170) on the GRE math, and I have a 3.5 overall GPA with 3.97 in mathematics courses. I anticipate a poor score on the GRE math subject exam. :/ I feel like a tremendously average student.
I have a list of the top tier schools I will apply to, that's easy, but how does one choose lower level schools that are basically guarantees to get in?
Things to consider:
1. Even though it won't be a spectacular school, I would still like to have a shot at getting into a top tier school for my PhD, so it has to be somewhat reputable.
2. Tuition/funding is a factor. If I'm going to end up at a second rate school, then I want the cheaper second rate school. Things like available funding or out of state tuition are factors here.
3. It has to be highly likely I will get in. This is not the big-name list, this is the "when all else fails" list.

It should be noted that when I graduated high school I applied to several top-tier schools and one fall-back school. I only got accepted in the fall-back school. This is a serious possibility and must be prepared for.

So how do I choose these fall-back schools? Rankings like News Week and the Princeton review only list the best/most competitive schools. How do you go about looking for a second-rate school? Is it basically just geography? I don't want to necessarily pick a crap school, I can do better than that... But mid-grade.

 

[jtbell]

Does it imply I should be applying to Ph.D. programs without first getting a master? Does it imply that at this school in particular that I should apply to a Ph.D. without a master, but not necessarily try that at other schools? Is it normal to pursue a Ph.D. without getting a master's degree if one intends to get a Ph.D. in the end? But the credentials of most professors seems to indicate that they received master's degrees as well as Ph.D.s.

In the USA, Ph.D. programs in many fields (including physics which is what I'm familiar with) are effectively master's + Ph.D. programs. You enter such a program after finishing a bachelor's degree with the expectation (on your part and the university's part) that you are working towards a Ph.D. You spend the first year or two taking graduate-level coursework while checking out research opportunities and lining up a Ph.D. advisor and the rest of your dissertation committee. After you finish a certain amount of coursework, you can pick up a master's diploma.

Separate master's degree programs (if a university offers them) in these fields are usually "terminal degrees" that do not lead on to a Ph.D. It's not the usual thing to get a master's at one university and then a Ph.D. at another.

 

[ArcanaNoir]

Thanks jtbell, that's helpful.

[edit]
Actually, that is extremely helpful. I am now noticing that professors either do not have a master's listed, or they have a master's and PhD from the same school.

(And my advisor didn't mention this WHY?)

Thanks!