Here are some brief notes about graduate fellowship applications. The notes focus on the NSF graduate research fellowship (GRF), which is the most common variety in computer science and many other science and engineering disciplines. However, most of the advice is applicable to any graduate fellowship.
Also see my advice about writing a research statement.
Ensure that all of your application materials are of equal quality. There is a temptation to focus on the research statement, but panel members question who actually wrote the research statement if the other parts are poorly written.
If you are applying during your first year of graduate school, get at least one recommendation letter from faculty at your grad school. It won't be able to say much specific, but the absence of such a letter is likely to be taken as a black mark.
As with applying to grad school, the recommendation letters are very important. They give an external assessment of your abilities and promise. The best way to get a good recommendation letter is to have done research with a faculty member. Since graduate school is about doing research, anyone interested in graduate school is also interested in research and has probably done some research projects — those projects might even be what hooked you on the idea of grad school. So long as you have at least one letter that can discuss your research, the other letters can be from people you took a class from, TAed for, etc.
Use correct grammar. For instance, learn the difference between “that” (the defining or restrictive pronoun) and “which” (the nondefining or nonrestrictive pronoun). Even if you don't consider the distinction important, you don't want to antagonize someone who does.
You may be able to look up previous winners online, or obtain winning application packets from your friends. A better use of your time is getting feedback on your own proposal. Definitely don't look at other essays until after you have fleshed out your own. If you consciously or subconsciously mimic someone else's ideas, your application will be less effective.
The NSF GRF solicitation is often a poorly-written document resulting from negotiations. http://www.nsfgrfp.org/ may be more readable than the solicitation.
For the NSF fellowship, a student may apply up to twice: once as an undergraduate, and once as a graduate student during the first or second year. NSF has a strong preference toward awarding fellowships to undergraduates. There is no clear evidence about whether to apply in your first or second year as a graduate student, though second year may be preferred.
A first-year grad student will be judged more on research promise than a second-year grad student, who needs to have strong letters from faculty at the current institution. A second-year student with a proven track record of success probably has a better chance than a first-year student who has only promise.
If you are enrolled in a joint BS/MS program (for example, UW's CSEM 5th-year-masters program), then you should accurately indicate your status — in particular, do you have a bachelor's degree? Students in BS/MS programs are allowed to apply again once they join a PhD program, even if they did already apply as a grad student during the BS/MS program. This isn't documented on NSF's webpage, but it's what NSF told me when I asked them in October 2016. You might want to double-check with them.
Tell the reviewer what motivated you, what shaped you, what excites you, what your goals are, and how you want to contribute to society. This will give evidence about how you will act in the future and whether you might be successful. Fellowships are given not just to the smartest students, but to those who have the potential to make a difference.
Make sure that the reader finds out what you are all about, what challenges you have faced, and how you have interacted with or helped others. By contrast, it's more boring to read an essay that just says “my research results will change the world and they are the broader impact”.
Make your personal statement a narrative. First, decide what you want the readers to take away from it; a good example is that you have an abiding love for programming languages, research, and teaching. Then, make sure that every anecdote or other bit of information builds upon your main point: about how you came to learn about it, about your reaction, about how you proactively sought out new opportunities, about what you did differently than other people, about your successes, about why you decided to go to graduate school (hint: “to get a PhD” is the wrong answer). Don't include details that do not support your main point.
The personal statement can also show your personal growth by say what you learned or how you changed over time.
Emphasize any leadership role you have had.
Be positive about your previous experiences. Look for the silver lining even if a project didn't turn out as you initially hoped. In what ways was it successful in part? What did you learn? How did it spark your interest or set you up for other experiences or successes? Try not to dwell on negative experiences. (Also recall that you are the sum of your experiences, so you would not be where you are — you would be a different person — if you had not had those experiences.) If there is an obvious negative on your transcript, consider addressing it and turning it into lemonade, instead of leaving a reviewer to wonder and assume the worst.
If you are interested in teaching or other ways to give back to society or to your research community, that can be a useful thing to mention.
You probably have a right to be proud of your accomplishments, but don't sound arrogant.
Since you are applying for funding to do research, it is important to give evidence that you are likely to be successful. For example, tell about your research experience, and especially about your research results. It is still possible to get a fellowship if you have not done research previously, but it is harder. If you have no experience, the committee may wonder whether you are suited to it and whether you will enjoy it; they don't want to spend money on a student who, arriving at graduate school, may be surprised by what research is and may not enjoy it.
There is some controversy regarding whether you should open your essay with a “hook” that doesn't make sense without context, to get the reader's attention. Examples include “I couldn't believe my eyes when I opened the letter” and “It had been a long night” and “Summer is a lonely place in a college town”. I find such an opening gimmicky and annoying — it fits well in a novel, but not in technical writing where the goal is clarity. Other people advise students to start their essays like this to draw the reader's attention. Someone who worked at NSF evaluating essays told me, “I doubt that the opening sentence has much of an influence either way [by comparison with the other advice on this webpage]. If the essay naturally reconnects to the message in the first paragraph, it may work. However, opening sentences like ‘Ever since first grade, I wanted to be/do ...’ usually cause a real yawn.”
Use a descriptive title for your research statement (not just “research statement”!). A descriptive title sets the tone and helps the reader know the context for the document. It's also less boring than “research statement”, which will put the selection committee to sleep after the previous 100 applications.
As with any technical writing, explain the problem first. Don't assume that readers will understand it (your application will be read by a generalist (say, any computer scientist), not by a specialist in your field), and avoid the very common problem of diving right into technical details.
As with any technical writing, be specific, not vague. Repeating generalities or stating only high-level points will not impress readers with your ability to understand, distill, and communicate technical ideas. That is what you are being judged on as much as anything else.
As with any technical writing, show, don't tell. That is, rather than simply making claims, it is better to present evidence that will lead the reader to that conclusion. That makes the reader more likely to believe and remember your point.
Show vision. Feel free to have your essay talk about a big project and vision (that you might be involved in, or that you imagine; it does not have to be something that only you are involved with) and also discuss your particular contribution toward one particular scientific question. That is a way to have your application both show significant potential impact-picture issues and also show concrete contributions.
Ensure that the proposed work will take 3+ years (the duration of the fellowship) to complete. Don't just talk about one piece that you happen to be working on now.
Include at least a few citations to the literature. A complete lack of references is a black mark because it looks unscholarly. You won't have space for many citations, though, so make each one count and don't include any gratuitously.
Broader impact is a major factor in getting fellowships accepted. This wasn't true in the past, but apparently most of the applications now have solid research components so broader impact has become a differentiator. It is essential that your research statement contains explicit section headers titled “intellectual merit” and “broader impacts”. The reviewers have to check a box indicating whether you addressed these issues and how well you did it. Poor broader impacts is the most common reason for denial of a fellowship, so don't give these short shrift.
Better type systems or even better software is not a broader impact. Keeping planes from falling out of the sky is a broader impact. Your broader impacts should be stated in terms that anyone would understand: in terms of our nation's economy, welfare (including health and environment), and defense. Connect the dots to show how your work contributes to those broader impacts; don't stop short.
Educating other people (including K-12 outreach) can be a broader impact, but it should not be the only one. If you say this, then your essay should demonstrate that you know how it works and you aren't just throwing around buzzwords. If the primary field of study is “STEM Education”, then your proposal needs to hold up to education researchers as well as reviewers in the discipline.
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