Here’s the handout I provide students about paper construction. (Additional technical details are in Appendix 1.) Of course, it is in the many conversations we have throughout the semester and multiple revisions that all these points begin to make sense to them.

**Notes on the Writing of Expository Mathematics Papers**

- Your paper is a narrative. It should tell a story, whether of a particular problem, result, or theory. So, include some history, tell us who did what when, share why your topic is important or interesting, and give us a beginning, a middle [where the proof(s) go], and an end!
- The proofs you will find in your research will be correct (usually!) but not necessarily as clearly explicated as an expository paper for your peer group (or me) requires. It is your job to provide written or mathematical clarifications, simplifications, added pictures, and other commentary to make the argument as clear as possible. One goal should not be the short, elegant, yet unenlightening proof, but rather a (possibly) longer, lucid, and revealing proof.
- You may likely find that figures, diagrams, graphs, and visual explanations of most kinds are sorely lacking in your sources. Or worse, in geometry sources you will often find a single massive detailed and frankly almost unreadable picture associated with a theorem or proof. In your explication of the proof expand the number of such figures, letting them evolve from the simple to the complex. As the proof proceeds, delete auxiliary aspects of figures once they have served their purpose, repeat pictures as needed so that your reader does not have to turn back several pages to follow your visual argument, and end with the completed result as your dramatic finish.
- Pictures should be near what they illustrate and should only be numbered if you plan to refer to them from some other page. If not, phrases like “see above” or “see below” suffice. Keep it simple.
- Use the terminology and conventions of modern mathematics (in English) correctly. For instance, the words “expression” and “equation” are not equivalent, “definitions” are not “assumptions”, “equal” is not “congruent”, and so on. Also, explain terms or abbreviations new to the reader, who, remember, has only been with you since your first page.
- Your paper should be as self-contained as possible. If you omit arguments and linkages, or “wave your hands” too much, your reader will justifiably abandon any attempt to follow your argument.
- Use standard conclusory terms with variety. For instance, choose from words or phrases such as
*thus*,*so*,*therefore*,*hence*,*whence*,*then*,*it follows that*,*we see that*,*we have*, etc., when stating the many conclusions that appear in a mathematical argument. Otherwise, you risk putting your reader to sleep with repetition. - If you draw from different sources, as you probably will, you often must unify the notation, sketches, and even the style. The reader should not have to switch for no reason from one set of variables or expressions to another set which symbolize the same ideas. This is sloppy writing, and only obscures your mathematical argument. Translate sources into a single language, unless of course it is those very differences that are the subject of your paper.
- Theorems, Definitions, Lemmas, Proofs, Constructions, Claims, Proofs of Claims, etc. should be indented and set off from the rest of your writing.
- I know that mathematical typography can be tedious on standard word processors, so do the best you can. Microsoft Equation Editor is often enough, but if you know other software feel free to use it. Just let me know. End your proofs with some sort of symbol, such as the black box █.
- Work with original sources (in translation) as much as possible. You’ll find original mathematical authors and other scholars are often not difficult to read, and even when challenging, well worth the effort.
- Consider written Internet sources suspect. Such work has usually not passed through the referee, review, and editorial scrutiny that all scholarly articles and books must. However, Internet sources can provide useful lists of references, and often enlightening illustrations or visualizations. I will be happy to look over anything you find on such sites.
- A paper should usually have at least three references, among them original sources in some form. Quoting actual documents for illustration or flavor is a good idea.
- Do not be too broad. For example, you might restrict to a few well-chosen and thoroughly explicated theorems as the core of your paper. The analysis of mathematics should be the core of your paper. A biographical, historical lead-in and a conclusion are just the frame.
- The level of discourse in your paper should be at least that of a student at this senior undergraduate level of mathematics.
- A few ideas (not complete!) for types of papers:
- Start with a modern statement of a theorem or problem and its proof. Research its history: various early proofs, counterexamples, errors, people and dates, variations, context … .
- Start with a theory, or collection of ideas, and choose representative historical theorems to explicate.
- Start with a person and choose his or her specific important theorems to explicate.

I also include “Advice from Writing Experts” (see** **Appendix 1).