The statement of purpose for graduate school should be between 1-2 pages long. Graduate committee members read your statement as part of your overall package. When a graduate committee member reads your statement they also have a copy of your transcript, perhaps your resume, and GRE scores. The statement should be viewed as a way to present some of your experiences that do not appear on your transcripts or to expand on some item on your resume. If you have poor grades in some courses, you could use the statement to explain why this occurred. If there was a gap in your education, this should also be explained. If you have a research and/or teaching experience then you should use the statement of purpose to give some detail about your research experience. This gives you the opportunity to discuss mathematics so that the reader gets the sense that 1) you have some deeper understanding of a topic and 2) you can communicate mathematical ideas.
Your statement of purpose should not rehash items that can be obtained from your transcripts or your resume. The statement of purpose should convince the reader that you are knowledgeable about undergraduate mathematics and that you have given some thought to your career path towards the doctorate. The statement should also give confidence that you will succeed in a graduate program.
Students will apply to more than one graduate school. Typically students apply to about seven programs. Students should consider writing their statement of purpose and leaving room in the statement to personalize it for each of the institutions where it is to be sent. Students should look up the webpage for the department, look at the research areas and the faculty, and comment on some aspect of that webpage.
In preparation for writing this statement of purpose, students should meet with their advisors and have the advisor look over the statement of purpose before it is sent out. Below are some helpful links when writing your statement of purpose.