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NSF Graduate Research Fellowship Program
Start on your essays early. You need time to write and rewrite drafts. You will probably rewrite your personal essay more times than you have rewritten anything else in your life; I went through thirteen drafts of my personal essay. There are many resources available to help you with your essays. This being said, before you look at sample essays, write a few drafts. Reflect on your own research experiences and other influential anecdotes in your own life instead of trying to model your essay after someone else’s.
Your essays should be very personal, and a lot of thought is required to convey what you want to say; copying wording from others is not beneficial. Also do not compare polished essays with your own first draft; these are essays that have been rewritten many times and doubtlessly been proofread by several other people. Reading other essays before you have written and rewritten anything can be very discouraging.
Make sure all three essays address broader impacts and intellectual merit; at the end of the day, these are the only things that the reviewers judge. Have several other people proofread your essays (especially if English is not your first language).
Broader Impacts and Intellectual Merit
Do not just list items on your cv. Describe to the reader why an experience was meaningful. In your essays for the NSF GRF you should market yourself. Do not be modest or assume that the reviewer is reading in between the lines. That being said, do not lie about your accomplishments or over-exaggerate your contribution.
You have limited space in your essays and quality is better than quantity (especially in the previous research essay). It is important to be aware that most winners of the NSF GRP have three or more research experiences, attend conferences, and have good letters of recommendation.
You will be competing against students within your own discipline and with the same level of education. You can apply your last year of undergraduate study and the first two years of graduate school.
Secure letter writers early, but only after you have gone through several major revisions of your essays (they will want to read your essays and cv before writing letters). Remind letter writers about the deadline often (you will be disqualified if your letters are late), and make sure they know how to use Fastlane.
There is not a lot of information for mathematics applicants in particular; looking at applications in related fields is beneficial. I used some good theoretical computer science resources (included below).
This site has a lot of basic information about the fellowship:
Some more tips for applying:
I found this information very useful (especially the suggested outline for the personal statement):
Here is the blog of a winner of an NSF Graduate Research Fellowship in 2006, Philip Guo (computer science), with a lot of advice. The blog also has links to some actual essays:
Here is the blog of a winner of an NSF Graduate Research Fellowship in 2008, Jean Yang (computer science). It has links to her essays: