Here's some sage advice about what to put in your essay - from my fellow NExT'ers (with their permission):
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I have proof read a number of statements of purpose recently, and I found a few occurrences of:
Math is good. Math is great. That's why I like math.
OK, they were not that bad, and didn't use those words, but they didn't have much more content than that. To combat this, I suggested that the students use very concrete examples of their interests and experiences, so that what they are writing, though it may basically say they like math because math is math, some of their own personality shows through.
Some of my students have come from other countries and cultures through various hardships, and I encourage them to write about how they ended up where they are mathematically, academically and otherwise. These essays have ended up the most interesting and memorable to me.
One of the goals of most REU programs is to expose students to research, so that they might decide whether or not to seek a PhD in mathematics.
If your student is considering applying to PhD programs, it might not go astray to mention it.
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In my REU students, students are requested to state two preferences of possible research areas. So their statement should give some background on their interest in those preferred research areas.
Also, the statement can contain information about their involvement in math-related activities beyond usual classwork --- like tutoring, special class projects, competitions, etc.
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For the past ten years I've been reading the applications for [an REU] and when I read the essays I'm looking for two things:
1. A genuine passion for math, I don't want to use one of my 18 spaces on a kid who isn't driven by a love for math. We see some applicants who appear to be trying to pad a resume, not indulge their love of math.
That's very much the primary desideratum, but also when I read 120-150 such essays over the course of a few days, they all start sounding alike, so
2. I like seeing something of the kid's personality. I'm going to spend an intense four weeks seeing this kid every day; it's nice for me if in addition to being excited about math, she's an interesting person.
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This may or may not be something your student could use in his/her statement, but I always look for students that don't have much of a research opportunity at their school - since in my opinion those are the students that most need an REU opportunity.
