Writing Mathematics Well

Workshop: Writing Mathematics Well

The importance of communicating mathematics clearly and effectively is evident in the many ways in which mathematicians must write, whether to produce technical reports, expository articles, book reviews, essays, referee's reports, grant proposals, research papers, evaluations, or slides for oral presentations. With a focus on exposition, this workshop offers tips for improving writing skills, from grammar and usage to organization and manuscript or slide preparation. It also suggests how participants can contribute to the public understanding of mathematics.

1 or 2 hours

Writing Words

"The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order that you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation. That's all there is to it."

Paul R. Halmos in How to Write Mathematics, American Mathematical Society, 1973.

MAA DVD: I Want to Be a Mathematician: A Conversation with Paul Halmos

Writing Tips

  1. You must have something to say.
  2. You should be excited about what you are writing, and that excitement should show.
  3. Picture your reader. Know your audience.
  4. Get the attention of your readers immediately. Use snappy titles, arresting first sentences, and lucid initial paragraphs.
  5. Keep your title short and include key words to make it informative. Steer clear of symbols.
  6. The first sentence matters. You need to begin your article in a way that pulls in the reader.
  7. Tell your readers in plain English what you are going to write about. The first paragraph of the introduction should be comprehensible to any mathematician.
  8. Have a clear sense of your article and its structure before you begin writing.
  9. Introduce one idea at a time.
  10. Use transitions. An article has to flow.
  11. Make liberal use of examples. Use analogies.
  12. Link ideas to what you assume is familiar to the reader.
  13. Read your work out loud.
  14. Ask others to read your drafts.
  15. Don't overwrite. Avoid clichés. Avoid jargon. Avoid "obviously," "clearly," and "trivially," but give the reader some guidance on the difficulty of different passages.
  16. Omit needless words.
Put every word in every sentence under the microscope.
    What does it add to the sentence?
    Will the sentence lose its meaning if the word is omitted?
    Can the thought be expressed in fewer words?

Audio and slides from presentation on "Communicating Mathematics" at the Fields Institute, Toronto, Oct. 1, 2010.

Grammar, Usage, and Style

Exercise: Eschew Verbosity (Keep It Simple!)

Exercise: Editing

Exercise: Word Noise



Knuth, D.E., T. Larrabee, and P.M. Roberts. 1989. Mathematical Writing. Mathematical Association of America.

Gillman, L. 1987. Writing Mathematics Well: A Manual for Authors. Mathematical Association of America.

Strunk, W., Jr., and White, E.B. 2000. The Elements of Style, 4th ed. Prentice Hall.

Additional References

Alley, M. 1996. The Craft of Scientific Writing, 3rd ed. Springer.

Barrass, R. 2002. Scientists Must Write: A Guide to Better Writing for Scientists, Engineers and Students. Routledge.

Blum, D., Knudson, M., and Henig,  R.M. 2006.  A Field Guide for Science Writers.  Oxford University Press.

Editors of the American Heritage dictionaries. 2004. 100 Words Almost Everyone Confuses and Misuses. Houghton Mifflin.

Gowers, T., editor. 2008. The Princeton Companion to Mathematics. Princeton University Press.

Higham, N.  1998.  Handbook of Writing for the Mathematical Sciences.  Society for Industrial and Applied Mathematics.

Krantz, S.G. 1997. A Primer of Mathematical Writing: Being a Disquisition on Having Your Ideas Recorded, Typeset, Published, Read, and Appreciated. American Mathematical Society.

Steenrod, N.E., P.R. Halmos, M.M. Schiffer, and J.R. Dieudonné. 1973. How to Write Mathematics. American Mathematical Society.

Truss, L. 2003. Eats, Shoots & Leaves: The Zero Tolerance Approach to Punctuation. Gotham Books.

Online Writing Resources

The American Heritage Book of English Usage (1996)

The Elements of Style by William Strunk, Jr. (1918)

Best Online Language Tools for Word Nerds

Communicating Science: Tools for Scientists and Engineers (AAAS)

Case Study

"Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem Solving" (excerpt)

"Lost in America: Top Math Talent'" (American Mathematical Society news release)

"U.S. culture derails girl math whizzes" (University of Wisconsin-Madison press release)

"U.S. Culture Blamed for Lack of Girl Math Experts" (University of Texas at Dallas press release)

Titu Andreescu, Joseph A. Gallian, Jonathan M. Kane,and Janet E. Mertz. Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem SolvingNotices of the American Mathematical Society, November 2008, pp. 1248-1260.

"Math Skills Suffer in U.S., Study Finds," New York Times, Oct. 10, 2008.

"Numbers don't add up for U.S. girls," Science News, Oct. 10, 2008. 

How Math Becomes News

Several case studies illustrate how significant or crowd-pleasing developments in mathematics can gain media attention, in print, online, and elsewhere. Learn how public relations efforts to publicize mathematics can garner public attention. Get a glimpse of how and why major journals, such as Science and Nature, control much of science coverage in the media.


EurekAlert! (AAAS)


Proceedings of the National Academy of Sciences (PNAS)

PNAS in the News



ACM TechNews

D.W. Farmer and S. Koutsoliotas. 2008. The E8 News Story. MAA FOCUS (February):4-11.

American Institute of Mathematics

Mathematicians Map E8 (AIM)

"Top 100 Science Stories of 2007" (Discover, Dec. 12, 2007)

Making Waves (AIM)