Guidelines for Writing Mathematics Projects

Over the years, I have noticed that even students who have experience writing essays or papers for non-mathematics course, are not sure about how to write a project for a mathematics course. These guidelines are intended to give you an idea of what I expect in a mathematics project. They apply to projects for any mathematics course. In addition, I generally provide specific requirements (for example, choice of topics and length) for specific courses.

1. The project should have a cover page (title, course, date, and author), an introduction (that briefly presents the topic of the project and explains how it arises or why it is worth studying), a main body (that may have several sections), a conclusion (that summarizes the content of the paper), and a bibliography (that gives complete citations for all material used in the project). Clearly, the heart of the project is the main body, and its contents will vary depending on the nature of the project.

2. The writing in the project should be grammatically correct, which means that it should have no errors in spelling, punctuation, or basic grammar. Try to model your writing on the (good) writing that you see in texts and papers. It is impossible to achieve grammatical accuracy without proofreading! Please do not submit projects that have not been carefully proofread.

3. There is always a question about how the project should be written. There are three options (other instructors may have fewer):

• You may write the project by hand, but it must be immaculate and easily read.
• You may use a word processor for the text and leave spaces where equations and other mathematical displays can be handwritten.
• You may use a mathematical typesetting package. Some word processors such as WORD have fairly good equation editors that produce acceptable mathematical text. However the professional standard is LaTeX, and it produces by far the most beautiful text. Learning LaTeX requires some work, but if you plan to continue doing mathematics, it is a worthwhile investment, and I am willing to give you plenty of support.

Here is the important consideration in choosing a writing tool. Regardless of what tool you use, you must write mathematics! Students who use computers to write their projects often avoid writing equations and formulas, and instead try to describe the mathematics in words. For example, I often read sine of theta or the integral of f from a to b. This sort of verbal mathematics is unacceptable. Mathematics is a powerful and concise language, and it must be used to its fullest. If your software cannot produce complete mathematical text, then you should write by hand or learn to use more powerful software.

4. Most projects require some visual support in the form of graphs, figures, tables, or other displays. You should not avoid graphics in your project because you do not have adequate tools. Many software packages enable you to draw figures and graphs, and then import them into your document. They are not hard to learn, and I am willing to help. However, if you do not have access to these packages, then neat, carefully hand drawn figures are the next best thing.

5. I don't expect student projects to contain original work; in general, they are what we call survey or expository papers. This means you must use the work of others as it appears in books, journal, papers, or web pages. The first inviolable rule is that when you use the work of others, it must be cited. Footnotes are generally not used for citations in mathematics papers. A more common technique is to place a number, say [10], next to the cited material, which refers to item 10 in the bibliography. It is not necessary to cite every single instance in which you use a result from another source. Often a section can begin by saying, "the following derivation was taken from [10]" or "see [10] for the original development of this idea."

6. At what level should projects be written? Or, to whom should I write the project? The easiest answer is that the project should be written for you. Write it in such a way that it would be understandable if you read it five years from now. Don't write over your head using words and terms you don't understand. At the same time, there is no reason to write for a reader who knows less mathematics than you do.

7. I have left the most difficult point until the last. Here is one way to say it: In writing a project, you must do mathematics, not just reproduce and talk about mathematics. It is common to see projects in which the ideas from the original source are merely reproduced or paraphrased. Even if the ideas are cited, this is not adequate. The goal is to give your own presentation of the original ideas and, if possible, extend them beyond the original presentation. This means that you should

• fill in the gaps in proofs, derivations, or solutions so they are perfectly clear to you;
• give explanations that are not included in the original source, again so that the discussion makes sense to you;
• create and do examples of your own to illustrate important ideas; and
• explain and analyze the assumptions that are used in models and theorems.

Said differently, while based on the work of someone else, your project should reflect your thinking and writing. It should be filled with your own explanations, insights, and interpretations. There should be very little in your project that you do not understand. And if you do encounter something you do not understand, please discuss it with me so we can attempt to clarify it.

8. Finally, I am always available for help on writing. I generally set intermediate target dates for an outline or rough drafts. At a minimum, you should meet these deadlines. But I am always willing to read drafts of your project before it is due. Projects that I have not read in draft form usually do not meet expectations. Please use projects as a way to learn mathematics and improve your writing.

December 2002