Of all the different types of text you will have to write in the academic world, the statement of purpose is one of the most difficult, not least because it is about you. We spend our time trying to eliminate ourselves from other academic writing, research proposals or from term papers. Now you have to write a paper about yourself. Like any other academic genre, a statement of purpose has a logical structure and development, and its purpose is to simultaneously show why you are the best candidate for a given course or grant, and why this course or grant is the most suitable one for you.
1. Math Highlights
Remember some concrete moments and activities in your life that turned you towards mathematics, ignited some passion for math, or expanded your view on mathematics. Make a (chronological) list of them. Include some key words how and why these events influenced you. Decide on one or two that you may want to describe in some more detail.
2. Look Ahead
Think what intrigues you about mathematics that you still want to learn. The idea is not to come up with a program of study for future years – which is likely to appear very odd anyway not having had extensive exposure to graduate mathematics already. Rather give specific examples of things you read about or things you learned the beginnings of and wanted to understand more about. Those should be personal and specific and help underscore your eagerness to move further into math. At all times be yourself and honest with what you can understand. It is OK to say you are undecided and are excited about a broad range of mathematics.
3. Specific Mathematics
Choose a specific piece of mathematics that you can talk about that fascinates you in particular. Try to describe the math fact in a short paragraph (you may assume a mathematician is reading this). Try to figure out what is fascinating you about this math fact.
4. Evolution of Interest
Think about how these key events can be put together to tell s story about yourself and how your relation about mathematics evolved. Keep early things (childhood etc) very brief – emphasis should be things that happen after high school.
5. Career Clarifications
Is you career path standard and obvious from your CV? The answer is probably YES if you went straight from high school to college and stayed at the same college/university. For more complicated career paths that may not be clear from the CV you can summarize in a narrative and explain some choices (worked for a while between schools, transferred schools due to recommendations and interests, etc)
Be careful with explaining poor and trying to excuse performance during some time, which can come across as whining. You might state basic facts (medical or family issues, over-enrollment, work) but then leave longer explanations to a professor writing a letter for you. Make sure this is explained as a temporary issue and emphasize that a lot of positive things came later.
What are instances, you think, that particularly demonstrate your mathematics strengths and talent – what are you proud of having accomplished. Think how you can turn the focus of the reader on these points without outright bragging about them – that is, thread them in the stories above, explain that those things gave you confidence, talks about something you were persistent at (instead of saying you are persistent), etc.
7. Research Graduate Programs
Look on the web and talk to people about graduate programs, their specific mathematical strengths and general qualities. Get an idea what type of program would be a good fit for you interests and personality. Find out what the type of research is done in general. Look up individual professors, the research groups, publications, and seminars. Browse other resources to find out in which area(s) some of the mathematical keyword that you come across come up and what they very roughly mean.
8. Expressing Specific Interest in Programs
Can you use what you found to express a more specific interest in the program or in particular faculty. This can be very effective but also has its pitfalls:
Make sure you do not get ahead of yourself but stay honest with what you understand. It will look very strange if you say that you want to work with Professor X on sheaf cohomology when it is highly unlikely from your course background that you have understanding of what that is. You can say that you are fascinated by the idea of using higher algebraic objects to describe geometric spaces and it seems to you that Professor X is doing great things in this direction. As with anything else be honest, be yourself, and show enthusiasm about mathematics.
Make sure professor or area you refer to are accurate and active in the program you write to. Look whether someone had published in recent years, if there are seminars or other activities in the area, and your interests really relate to what is done. Ask a professor from your program to help if you have doubts.