What advice would you give students applying for graduate school in mathematics?

I work in the mathematics department of a university that has a new, small, mathematics bachelor's degree program, and no graduate mathematics programs. Some of our students are looking for advice on how to successfully apply for graduate school. I don't know what to tell them. Everyone in my department (myself included) applied to graduate school so long ago that I think our advice may not be relevant. I don't think anyone else who works for my university knows more than my department colleagues do.

What are good things for students at a small school to do to get into graduate school in mathematics? In particular, what are some things that might not be obvious to mathematics professors who've been out of graduate school for a few decades?

EDIT: In response to this question being put on hold for being "too broad", here is my modified question: What advice would you give students applying for graduate school in mathematics in 2014 that you would not have given them a decade or two ago? In other words, how would the advice differ? I am thinking both about what students have to do as undergrads and about the application process itself (which I assume is done online nowadays).

Answer

I personally had the experience of applying to graduate school from a very small college that has not produced many math Ph.D.s (though oddly enough, I'm not the only 2002 BA graduate working at an R1 university in North America) not so far in the misty depths of time. Of course, I only have a couple of data points, but let me try to give my thoughts.

I think one thing is you've really got to get the basic stuff right: take the hardest classes available to you from the selections you have (and you should note in your letters that the student did this!) and ace them. Do well on the general and subject GREs. I assume you could have told them those.

I think the biggest non-obvious thing is that it's essential to seek out experiences outside your small program. If you're at a large school with its own graduate program, probably you have all the resources you'll need at your fingertips, but in a smaller program, I don't think you really have the ability to fully prepare someone for graduate school. Study abroad is good for this (I went to BSM in Budapest, but there are other programs). So are REU's (I did the one at LSU). On a smaller scale, you might be able to take more advanced courses at a nearby college (for example, students at Smith and Amherst can take graduate courses at UMass), or do a guest semester somewhere in the US (like at the Penn State MASS program).

These will, of course, be generally enriching experiences, but they also help by giving some real points of comparison. Graduate schools know what an A at BSM means (where they might not know it for your school); a professor who supervised you at an REU can speak with authority about having supervised many talented young people, and having some experience with which of them succeed in grad school.

Another possibility is working at summer mathematics program like PROMYS or Canada/USA Mathcamp (there are many other) though I think you should give some weight to activities like REU which are more likely to result in letters.

EDIT: I'll just note that what I've written about is still advice from the Web 1.0 world, but it's not so clear to me what Web 2.0 has changed about the admission process. I think it has made communicating with other people going through the process easier (for example, you can see in essentially real time which schools are sending out acceptances and rejections. This seems more likely to drive you insane than help though). I think for students at a smaller school this can be a boon (for example, the applicant profiles in this thread could be helpful for understanding where they might get in, though it's worth a reminder to take things with a grain of salt).

I think one thing I didn't know was that it's very reasonable to contact the Director of Graduate Studies at programs that interest you if you have questions. Don't be a pest about it (only email if you have a real question), but communicating with applicants is part of the job, though my DGS may not appreciate me telling the internet that (sorry, Tom).

The Web, of course, has also changed research about grad schools a lot. In theory, you can know a lot more about individual professors now than it would have been easy to figure out even 10, but especially 20 years ago. It's not super clear to me that this will help very much though. I generally feel like researching individual professors before starting grad school is a mug's game, since you're so likely to shift interests. I think it's much wiser to choose based on the program, and then worry about getting an advisor after a year or so of grad school when you know a bit more.