Tevian Dray

(a revised version of a 1998 essay)

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.

Writing mathematics well requires a blend of mathematical knowledge with traditional writing skills such as spelling, grammar, and usage.


The following overview is adapted from the instructions for a writing course based on non-Euclidean geometry, but can easily be adapted to other contexts.

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.

Ground Rules

Again, the following ground rules are adapted from the instructions for a writing course based on non-Euclidean geometry, but can easily be adapted to other contexts.

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.

Technical Details

General Advice
  • Use complete sentences! Always!
  • Don't copy the problem; restate it in your own words. (A reasonable goal is to be able to pick up your written work five years from now and still understand it—without the use of any additional references.)
  • An abstract is useful, summarizing the main conclusions in a few sentences.
  • Lengthy derivations or proofs that would interrupt the flow of the narrative can be included as appendices.
  • Your introduction is a good place for a background paragraph or two that discusses previous work (with citations to appropriate sources).
  • Short equations, such as the statement that $y=x^2$, should normally be inline, that is, contained within a sentence as though they were words.
  • Longer equations, such as \begin{equation} \int\limits_0^\infty e^{-x^2} \,dx = \frac{\sqrt\pi}{2} , \label{test} \end{equation} should be displayed, that is, set off from the rest of the paragraph. This also applies to particularly important equations, such as \begin{equation} E = mc^2 . \end{equation}
  • All equations should be grammatically correct parts of sentences, even if they are displayed.
  • Displayed equations should be, well, equations; they require an equals sign or some other mathematical verb, and can not be merely an isolated mathematical expression.
  • It is therefore an error to display a long sequence of computations as separate equations, unconnected by words. Such equations should be connected by short phrases, such as “so that”, etc.
  • All displayed equations should be numbered, so that they can be referred to (“see Equation (\ref{test})”)—not only by you, but by others who read your work. (Some journal styles do not permit the numbering of equations you don't actually reference yourself.)
  • Unlike equations, figures are not part of the flow of the text.
  • Figures can be displayed as floats, either at the top (or bottom) of the page, above (or below) all other text.
  • Figures can also be displayed at the left or right margin in the middle of a page, with text flowing around them.
  • Finally, figures can be collected onto separate pages at the end of a manuscript.
  • All figures should have captions, and labels such as “Figure 1”.
  • All figures should be referred to explicitly in the text (“see Figure 1”), and explained in words.
  • Tables are treated similarly to figures.
  • There is a difference between a hyphen (“-”), used between words, a dash (“—”), used within sentences, a minus sign (“−”), used in equations, and a short dash (“–“), used in numerical ranges. (Short dashes are called “en-dashes”, long dashes are “em-dashes”. The spacing around minus signs (“$3-2$”) is different from that used with en-dashes (“2–3”).)
  • Avoid the use of pronouns such as “it”, “this”, “that” wherever possible. Good mathematics writing requires precision; don't make your reader guess what these words refer to.
  • Similarly, avoid the use of “fluff”, such as starting a sentence with “Note that”.
  • Mathematical precision also requires you to select the correct verb. For example, integrals are evaluated (not solved).
  • Many mathematical terms also have common meanings. Avoid using such terms unless you intend them in their mathematical sense. For example, a problem is complicated (not complex).

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.


Below are some online definitions of Euclidean geometry. These definitions are taken out of context, and are not intended to be mathematical in the first place, so it is a bit unfair to critique them according to our standards for good mathematical writing. We do so nonetheless.

Encyclopedia Brittanica
Oswego (NY) City School District Regents Exam Prep Center

An online sample of good mathematical writing in the context of non-Euclidean geometry can be found here; a PDF version can be found here. (This manuscript was later published as Pi Mu Epsilon Journal 11, 78–96 (2000).)

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.


Below are some of the criteria that can be used to assess mathematical writing.

  • Correct computation.
  • Correct justification.
  • Use of more than one method.
  • Use of multiple representations.
  • Making conjectures and/or generalizations.
  • Exploring additional consequences.
  • Originality.

  • Use of valid mathematical language and symbols.
  • Correct use of language.
  • Effective organization, such as an introduction and a conclusion.
  • Use of appropriate figures and diagrams.
  • Use of appropriately documented references.
  • Clear communication of mathematical thinking and reasoning.

Overview,   Ground Rules,   Technical Details,   Examples,   Rubric.