• Your paper 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 will 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.

• I realize that your software may not allow you to draw the pictures you need. Drawing them by hand will be fine.

• Use the language of mathematics 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 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.

• Theorems, Definitions, Lemmas, Proofs, Constructions, etc. should be indented and set off from the rest of your writing as in the following examples:

Theorem: A pentagon can be constructed with unmarked straightedge and compass.

Proof:

Construction:

Lemma:

Definition: An isometry is a transformation of the plane which preserves the length of segments.

• I know that mathematical typography can be tedious on standard word processors, so do the best you can. If necessary, write out expressions by hand rather than make a valiant effort with your software, only to produce clumsy approximations. End your proofs with some sort of symbol, such as a black box, or a pair of slashes // as I often use at the chalkboard.

• Work with original sources (in translation) as much as possible. You’ll find original mathematical authors and other scholars not difficult to read, and 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 print off such sites.

• From Strunk and White, The Elements of Style, 1972 edition:
  II-10: Use the active voice.
  II-11: Put statements in positive form.
  II-12: Use definite, specific, concrete language: "…deal in particulars and report the details that matter."
  II-13: Omit needless words.
  II-15: Express co-ordinate ideas in similar form. [Extremely important in mathematics.]
  II-16: Keep related words together. [Also, keep related pictures together with related words.]
  V-5: Revise and rewrite.
  V-6: Do not overwrite.
  V-7: Do not overstate.
  V-16: Be clear.

• From [6, p.10]: "…all the best writers whom I know read their work aloud to themselves. Reading your words aloud forces you to make sense of what you have written, and to deliver it as a coherent whole. If you have never tried this technique, then your first experience with it will be a revelation."

• From [6, p.92]: "Envision your reader sitting on a park bench reading your expository [paper], or putting his feet up and drinking a cup of coffee while reading. Do not imagine your reader with a pencil gripped in his fist, slaving away over each detail of your paper."

• From [2, p.8]: "Keep the reader informed of what you are doing and of how things stand. I have always enjoyed reading Sierpinski: first he tells you what he is going to do, then he does it, then he tells you he did it…. If you use a key term or symbol after a long spell without it, recall its definition or refer to where it was introduced. Your readers will be grateful."

• From [2, p.20]: "Write simple unaffected prose. Writing is harder than speaking because your tone of voice isn’t available to help make your point clear. Keep sentences crisp – think of what you want to say and say it. Mathematics is hard enough to read without convoluted writing that makes it harder."

• From [5, p.94]: "All writing benefits from revision. Your first attempt can always be made clearer, more concise, more forceful. Effective revision is a skill that is acquired through practice…Put a draft aside for a few days, so that you can examine it with a fresh mind. Then analyze the draft in different ways. Read it aloud and listen to the rhythm. Read it at high speed. Does the text flow? Are the sentences of varying length? Read at the page level, focusing on the shape and the density of ink on the page…Is there a good balance between equations and text? Does the paper look inviting? Do the key ideas stand out ?"

• From [7, p.90]: "There is a place for rigor in every kind of learning. All of us need to sharpen our thinking, and when we are challenged we should have to become more precise about what it is we are saying, how we are saying it, whether we are marshalling good arguments, and the like."

• From [8, p.11]: "Experienced writers know that they have to get something down on paper (or up on the screen) as fast as they can, just to have a draft that they can revise into a better one… When you revise your early confusion into something clearer you better understand your own ideas. And when you understand your ideas better, you express them more clearly, and when you express them more clearly, you understand them even better…and so it goes."

• From [9, p.31]: "…Care lavished on expression is not some optional embellishment bestowed upon your work; it is the means through which your work begins to exist. Your research turns up raw materials – very raw. Writing and rewriting make them into finished, usable products. Until brought out in full view by the best possible arrangement of words, your results remain incomplete, doubtful, hidden from every mind but your own. And your own, as your first draft shows, is none too clear."

• From [9, p.228]: "Rewriting is nothing but able tinkering…Only an apprenticeship under a vigilant critic will gradually teach a would-be writer how to find and correct all the blunders and obscurities that bespangle every first draft."

• From [9, p.230]: "You discover what you really think by hacking away at your first spontaneous utterance."

1. How to Write Mathematics
   Norman Steenrod, Paul Halmos, Menahem Schiffer, Jean Dieudonne
   American Mathematical Society, 1973, 1981, 64 pages
   ISBN 0-8218-0055-0, QA41.H6
   
2. Writing Mathematics Well: A Manual for Authors
   Leonard Gillman
   Mathematical Association of America, 1987, 49 pages
   ISBN 0-88385-443-0
   
3. Mathematical Writing
   Donald Knuth, Tracy Larrabee, Paul Roberts
   MAA Notes, Number 14
   Mathematical Association of America, 1989, 115 pages
   ISBN 0-88385-063-X
   
4. Using Writing to Teach Mathematics
   Ed. By Andrew Sterrett
   MAA Notes, Number 16
   Mathematical Association of America, 1990, 139 pages
   ISBN 0-88385-066-4
   
5. Handbook of Writing for the Mathematical Sciences
   Nicholas Higham
   Society for Industrial and Applied Mathematics, 1993, 241 pages
   ISBN 0-89871-314-5, QA42.H54
   
6. A Primer of Mathematical Writing
   Steven Krantz
   American Mathematical Society, 1997, 223 pages
   ISBN 0-8218-0635-1, QA42.K73
   
7. Writing in the Teaching and Learning of Mathematics
   John Meier, Thomas Rishel
   MAA Notes, Number 48
   Mathematical Association of America, 1998, 100 pages
   ISBN 0-88385-158-X
   
8. Style: Ten Lessons in Clarity and Grace, 6^th Edition
   Joseph M. Williams
   Addison Wesley Longman, 2000, 309 pages
   ISBN 0-321-02408-7
   
9. The Modern Researcher, 3^rd Edition
   Jacques Barzun, Henry F. Graff
   Harcourt Brace Jovanovich, 1977, 378 pages