The sheer volume of advice available “out there” on writing mathematics can be quite intimidating. Here’s a (nonexhaustive) list of interesting pieces. Many of these were found by undergraduate researcher Artur Araujo.
Writing mathematics well (audience is mathematicians)
- Terry Tao’s blog contains several posts dedicated to this topic:
- A fun video of Serre’s lecture about how to write badly
- How to Write Mathematics, by Steenrod, Halmos, Schiffer, and Dieudonne
This book is a collection of four essays by Norman E. Steenrod, Paul R. Halmos, Menahem M. Schiffer, and Jean A. Dieudonne. It was originally published in 1973 by the American Mathematical Society.
Paul Halmos’ paper on “How to Write Mathematics” is here and is summarized in Ivars Peterson’s blog post Paul Halmos on Writing Mathematics.
- L. Gillman, Writing Mathematics Well: A Manual for Authors, The Mathematical Association of America, 1987.
A short manual, with many good tips. The appendix is available here: “The Use of Symbols: A Case Study.”
- J.S. Milne has a webpage of sarcastic Tips for Writers.
- A Primer of Mathematical Writing: Being a Disquisition on Having Your Ideas Recorded, Typeset, Published, Read & Appreciated, by Steven G. Krantz
Published in 1996 by the American Mathematical Society
- Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham,
Published in 1998 by SIAM
- I Want to Be a Mathematician: A Conversation with Paul Halmos, by George Csicsery
Halmos discusses writing, teaching, and research (44-min film ) Trailer
- “Some Hints on Mathematical Style,” by David Goss
This 4-page collection of “hints” is based on advice the author received from J.-P. Serre as well as on experience publishing a conference proceedings and editing the Journal of Number Theory.
- “Advice for amateur mathematicians on writing and publishing papers” by Henry Cohn
Contains helpful advice that professional mathematicians typically learn from their advisors.
- “Three Sins of Authors in Computer Science and Math” by Jonathan Shewchuk
This online essay is written in an irreverent style that may help to make the advice memorable. Advice includes
- “Get to the point!”
- Rather than listing a paper’s contents in a paragraph at the end of the introduction, “I submit that the references to each section of the paper should have been folded into the introduction, each appearing in its logical place.”
- “Conclusions should synthesize the results of your paper and separate what is significant from what is not.”
- R. P. Boas, “Can We Make Mathematics Intelligible?” American Mathematical Monthly 88, 1981, pp. 727-731.
This 5-page article by an editor of the American Mathematical Monthly uses humor to (somewhat scathingly) point out common pitfalls with mathematical writing. Addresses aspects of terminology, symbolism, proofs, rigor, enthusiasm, skills, and lectures.
- Mathematics Into Type, by Ellen Swanson and Arlene O’Sean, AMS, updated 1999.
This book is directed less to authors than to editors who have little or no mathematical background.
General resources (not specific to mathematics)
- W. Strunk, Jr., & E.B. White, The Elements of Style, Macmillan, 4th Ed., 1999. Available online
- The Chicago Manual of Style, 15th Ed., Univ. of Chicago Press, 2003.
- R.W. Burchfield (Editor), H. W. Fowler, The new Fowler’s modern English usage, Oxford Univ. Press, 1996.
Leonard Gillman writes of the classic 1965 edition, “The peerless classic, revered for the wit and artistry of its writing as much as for its pungent advice. Must be sampled slowly; ideal for browsing: pick out any page and enjoy the language…” (Quoted from the bibliography of Writing Mathematics Well: A Manual for Authors, by Leonard Gillman, The MAA, 1987.)
Resources for students
- Undergraduate Guide to Writing Mathematics (unpublished) by Stephen B. Maurer
With patience and good sense, Stephen Maurer explains for undergraduates myriad aspects of how to write as a mathematician. This book addresses taking notes, writing reports, giving presentations, and revising. Particularly valuable is the second half of the book, which provides advice on diverse topics such as proofs, definitions, word choice, style, when and how to punctuate displays, and when and how to use citations, examples, figures, footnotes, humor, etc. The book also includes a glossary of mathematical terms and an extensive collection of writing exercises.
- Maurer, “Advice for Undergraduates on Special Aspects of Writing Mathematics,” PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 1935-4053, Volume 1, Issue 1, 1991, pp. 9-28.
- This brief handout by Olivier Bernardi addresses not only the characteristics of a good math term paper but also the process of writing one.
- “Maximum Overhang” by Paterson et al. received an MAA writing award in 2011. This pdf of the article is annotated to point out to students how to write a mathematics paper. The annotations address the structure and content of an introduction, how to integrate equations, text, and figures, how to guide the audience through the content, how to cite, etc. The article addresses the question of how far a stack of blocks can extend from the edge of a table. It was published in the American Mathematical Monthly 116, December 2009.
- John M. Lee’s “Some Remarks on Writing Mathematical Proofs”
This 6-page handout targeted to undergraduates provides guidance on audience, clarity, motivation, precision, level of detail, use of formulas, etc.
- Dave Richeson’s “The nuts and bolts of writing mathematics” provides guidance to students in Discrete Mathematics. A link to a pdf for students is provided.
- Steven Kleiman’s “Writing a Math Phase Two Paper”
Guidance for how to write a math paper, written for a now defunct writing requirement but with much good general guidance. Topics including organization, use of language, and presentation of mathematics (e.g., symbols and formal arguments). A short example of mathematical writing is included. Because the guidance is written in the form of a journal article, the .tex file can act as a template for students to use to create their own papers.
- Haynes Miller’s notes on writing mathematics
These 8-page notes address the parts of a paper; mathematical style for attribution, citation, and internal references; the distinction between belief and proof; and a miscellany of 15 common errors. These errors range from grammatical issues such as articles and comma splices to errors of precision, notational consistency, and sequencing of information. Examples are included.
- Writing a Research Paper in Mathematics, by Ashley Reiter
This webpage summarizes ideas presented in the book How to Write Mathematics, by Steenrod, Halmos, Schiffer, and Dieudonne, and includes thoughts from the author’s experience as well. The site addresses introductions, structuring a paper, formal and informal exposition, writing a proof, and includes a list of 14 specific recommendations ranging from the small scale (“…never use “any” in mathematical writing. Replace it by ‘each’ or ‘every’…”) to the large (“There should be a clear definition of the problem at hand all the way through.”)
- Krantz, Steven G., “How to write your first paper.” Notices Amer. Math. Soc. 54 (2007), no. 11, 1507–1511.
- Language of Mathematics (unpublished), by Warren W. Esty
This textbook is written to an audience of non-math majors and elementary ed students. From Chapter 1, “The goal is for you to become fluent in the symbolic language of mathematics so you can efficiently read, write, learn, and think mathematical thoughts.” (Math Forum abstract)
- See also this website’s page for English learners and global English as well as the pages on General Principles of communicating mathematics.
- See also the materials for teachers below, some of which contain materials for students
General resources (not specific to mathematics)
- The Purdue Online Writing Lab (OWL) is a comprehensive resource for students and educators about writing (not specific to mathematics). From the home page: “We offer over 200 free resources including: Writing and Teaching Writing, Research, Grammar and Mechanics, Style Guides, ESL (English as a Second Language), Job Search and Professional Writing.
- Grounds For Argument “guides writers through interactive lessons that introduce discrete principles of effective writing. Writers can use individual lessons for one–stop help on a specific issue [or] group lessons into a personal writing course.” Includes a public discussion forum for writers.
- Kit Eaton’s English Grammar Aids for Both Native Speakers and Students is an article in The New York Times App Smart column reviewing a few apps for learning grammar. Jan 2015
Resources for teachers
- Patrick Bahls, Student Writing in the Quantitative Disciplines: A Guide for College Faculty, Jossey-Bass, 2012. (link goes to publisher’s webpage)
- Annalisa Crannell’s “Writing in Math” website. Includes writing exercises for calculus, a checklist for writing and grading essays, and links to other useful materials and webpages.
- Writing in Introduction to Euclidean Geometry with Joel Weiners, University of Hawaii, Manoa
Sequencing of assignments, sample problems, instruction, sample drafts & instructor responses
- Writing in Mathematics at Columbia College
Includes departmental communication goals and objectives, descriptions of assignments, a chart of the types of assignments in each course, and writing tips from professors.
- D. E. Knuth, T Larrabee and P. M. Roberts, Mathematical Writing, MAA Notes #14, Mathematical Association of America, Washington, D.C., 1989.
Note from a semester-long course on mathematical writing, taught at Stanford University in 1987.
- R. Gerver Writing Math Research Papers: A Guide for Students and Instructors, Key Curriculum Press 2004
This book is written to secondary school students, guiding them as they write (beginning by expanding class notes), do research, keep a research journal, and write up & present their research. The appendix provides guidance for instructors.
- “Communicating Mathematics,” a course taught by Ivars Peterson at East Tennessee State University
- “Writing for a Math Class” at the website Platonic Realms
This webpage briefly addresses why to write in a math class and how to design and grade assignments. Sample assignments include a personal essay, a biographical sketch of a mathematician, daily notebooks, and 3-5 page research/thesis papers on infinity, large numbers, overpopulation, and logical fallacies. The site also includes a list of formatting requirements for students along with an annotated paper illustrating these formatting requirements.
- Writing in Mathematics at Marquette University
- “A Two-Genie Strategy Helps Students Write Weekly Papers,” by R. G. Montgomery, PRIMUS, Vol 4, No 4, 1994, pp. 347-358.
Abstract: “Mathematical Perspectives” is a seminar required of beginning math majors at Southern Oregon State…The talks are complemented by reading assignments, and therein lies an instructional challenge: insuring attentive reading by participants in a Pass/No-Pass one-credit seminar which has no examinations. Short written papers do the trick…But, the challenge is now is to elicit papers which are well-focused, easy to evaluate, and amiable to an instructor’s comments. This article tells how two playful genies, Alf and Bet, bottle this writing challenge.
General resources (not specific to mathematics)
- The Writing Lexicon, by Kerry Walk
In two pages, Kerry Walk summarizes some communication terminology students may have learned in their other courses: “thesis,” “motive,” “structure,” “key words,” “methodology,” “evidence” or “data,” “analysis,” “sources,” “orienting,” “citations,” “conventions,” and “mechanics.” These terms are also used in the general resources listed on this page. Information related to these concepts can be found at the following locations on MathDL Mathematical Communication.
- Information related to “thesis,” “motive,” “structure,” and “analysis” can be found at Focusing and structuring longer communication (see for example the annotated introductions).
- Information related to “evidence” can be found at the pages about rigor and proofs (evidence can also be supplied by examples and data).
- Information related to “sources” and “citations” can be found at Using sources.
- Information related to “orienting” can be found at Guiding the audience through the content.
- Information related to “mechanics” can be found at Wording and punctuation and notation.
- Information related to “conventions” can be found throughout the site.
- Bean, John, Engaging Ideas: The Professor’s Guide to Integrating Writing, Critical Thinking, and Active learning in the Classroom, 2nd ed., Jossey Bass, 2011.
Review by Mary Beth Culp, The Quarterly, Vol. 21, No. 2, 1999
- The Purdue Online Writing Lab (OWL) is a comprehensive online resource for students and educators about writing (not specific to mathematics). From the home page: “We offer over 200 free resources including: Writing and Teaching Writing, Research, Grammar and Mechanics, Style Guides, ESL (English as a Second Language), Job Search and Professional Writing.”
- The WAC Clearinghouse at Colorado State
An clearinghouse for online “journals, books, and other resources for teachers who use writing in their courses.” Includes An Introduction to Writing Across the Curriculum.
“An inventory of publications in writing studies, including post-secondary composition, rhetoric, technical writing, ESL, and discourse analysis”
- Grounds For Argument, University of Virginia (based on the Little Red Schoolhouse)
“This site guides writers through interactive lessons that introduce discrete principles of effective writing. Writers can use individual lessons for one–stop help on a specific issue, group lessons into a personal writing course, or follow a prescribed series as part of a writing class. Teachers can establish class accounts with customized curricula and class–specific exercises. The site also provides several public forums for discussion, including a writers’ forum, a teachers’ forum, and individual class blogs. …For a complete list of existing and planned lessons, see the curriculum map.”
- Electronic Communication Across the Curriculum, edited by Donna Reiss, Dickie Selfe, and Art Young
- George Mason University’s Writing Across the Curriculum website includes learning modules & resources and a helpful blog, The Writing Campus.
- Additional Writing Across the Curriculum sites include (but are NOT limited to) Boise State University,Harvey Mudd,University of Toronto,Colorado School of Mines,Colorado State University,University of Minnesota,University of Maryland University College (Faculty Resources and Online Guide to Writing and Research for students),Massachusetts Institute of Technology (resources from Writing, Rhetoric, and Professional Communication and resources from the Writing and Communication Center),Harvard University,McCallie School (Resources for Secondary Education),Monash University,University of Richmond (Writers Web and Resource binder).
Research on communication and its pedagogy can be found in the General Principles of Mathematical Communication area of this website.
Resources for writing proofs are available on the page Types of proof & proof-writing strategies.
Resources for using writing to help students learn mathematics are available on the page Writing to learn.
See also the sidebar of this page for files that have been uploaded here as resources.