WRITING MATHEMATICS
Tevian Dray
2014
(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.
Overview
The following overview is adapted from the instructions for a writing course
based on nonEuclidean geometry, but can easily be adapted to other contexts.

Choose a topic
For instance, pick a nonEuclidean geometry you find interesting.
Is there some aspect of it which was discussed briefly somewhere without much
detail? Is there some way of changing the rules which intrigues you?
Once you have tentatively chosen a topic, write a few sentences explaining
it. If you are creating your own model, describe exactly what it is. If
there's something missing from a proof, or from the coverage of a topic
in a textbook, or whatever, describe what's missing.

Justify your choice
Why is your topic interesting? Why is it important? Write down a few bullet
points that address these questions.

Make an outline
Now that you have chosen the topic, you should know at least in principle
what geometric model(s) you will be working with. The next step is to decide
what questions to ask about it. So make up a list of questions about your
model. Does it need a distance function? Do you plan to determine what
corresponds to circles?
Select several of these questions (1 is too few; 10 is too many) which
you hope to answer while writing your essay. Divide them into appropriate
categories. Now you're ready for the outline: Start with an introduction,
end with a summary/discussion/conclusion, and put the various (categories
of) problems in the middle. Briefly describe each part. Incorporate the
list of bullet points you made above.

Do the math
Solve the problems. This is the fun part!

Make a rough draft
Write up what you did. You need to include enough detail so that people
can understand it. Most calculations should be given explicitly. Lots of
figures (with suitable captions/descriptions) are a big help. But you also
need to include enough words so that people can understand it; theorems
and proofs may be appropriate, but are certainly not sufficient.

Rewrite as needed
Be a perfectionist. Fix your math mistakes. Fix your grammar mistakes.
Fix your spelling mistakes. Make sure your logic is sound. Make sure your
reader will know at each stage what you're doing. Perhaps some reminders
are needed: “Now we will solve the Dray conjecture”
or “We therefore see that the Dray conjecture is false”.
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 nonEuclidean geometry, but can easily be adapted to
other contexts.

You must do some math
A discussion of the history of nonEuclidean geometry is not appropriate.
A comparison of different (historical) versions of neutral geometry might
be.

Your work must be original
This does not necessarily mean that you must do something nobody's ever
thought of before, although you'll certainly get brownie points if that
is the case. You do need to work through the math yourself, and present
the results in your own words. And you should carefully distinguish your own
contribution from previous efforts, indicating that your work is a
simplification/generalization/application of previous work (which should be
explicitly cited), as the case may be.

References must be cited
You may use whatever references you can find which might be appropriate.
But you must give appropriate credit. A direct quote, for instance the
statement of a postulate or a theorem, should be clearly labeled as such.
A figure which appears elsewhere must be so labeled. It is not appropriate
to make minor changes in text, or to redraw a figure, without giving a
proper reference; this is plagiarism. By all means paraphrase an argument
you find elsewhere. But give credit to the author. And don't fill up the
entire essay this way; that's a book report.
Your references should appear separately at the end of your essay, with a
section heading such as References or Bibliography. Full
publication data must be given, including title, author(s), publisher, and
year. Page numbers may be given if appropriate.

Readability
Your essay should be easy to read. Ask a friend to read it. Tell them
not to worry about the details. Is the argument clear? They should be able
to read the introduction and conclusion and tell you what your essay is
about. Can they?
(Your friend does not need to be a fellow student in the course; this exercise
is about presentation, not content. But it wouldn't hurt to repeat this
process with someone from your target audience, someone you expect will
understand the content.)
Your essay should be easy to read in another sense: Use a word processor!
Get that new printer cartridge you've been thinking about! Use section
headings. Indent your paragraphs. Don't run lengthy equations into the
text—display them neatly on separate lines. (You may handwrite
equations if you can not type special symbols.)

Figures
By all means include lots of figures! These can appear in the text or on
separate pages at the end, and may be handdrawn. Each one should have a
label, such as Figure 1, as well as a caption. You should describe each
figure in the text in enough detail so the reader can figure out why it's
there.

Mathematical content
It's a good idea to get the math right!
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).
 Equations

 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.)
 Figures

 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.
 Miscellaneous

 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 “endashes”, long dashes are
“emdashes”. The spacing around minus signs
(“$32$”) is different from that used with endashes
(“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.
Examples
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.

Wikipedia


Euclidean geometry is a mathematical system attributed to Euclid, which he
described in his textbook on geometry: the Elements.

Euclidean geometry consists in assuming a small set of intuitively appealing
axioms, and deducing many other propositions (theorems) from these.

Euclidean geometry is an axiomatic system, in which all theorems ("true
statements") are derived from a small number of axioms.

Encyclopedia Brittanica


Euclidean geometry is the plane and solid geometry commonly taught in
secondary schools.

Euclidean geometry is the study of plane and solid figures on the basis of
axioms and theorems employed by the Greek mathematician Euclid.

Oswego (NY) City School District Regents Exam Prep Center


Euclidean Geometry is the study of flat space.

Euclidean Geometry (the high school geometry we all know and love) is the
study of geometry based on definitions, undefined terms (point, line and
plane).

Euclidean Geometry (the high school geometry we all know and love) is the
study of geometry based on definitions, undefined terms (point, line and
plane) and the assumptions of the mathematician Euclid.

All of these definitions except possibly the last are descriptive, in
that they don't actually tell us what Euclidean geometry is, only what (some
of) its properties are. That may be good prose, but it is not good
mathematics.

Three of the definitions refer to Euclid's work; an explicit citation should
be given.
(When citing something like Euclid's Elements, good practice is to
cite both the original and a more accessible modern translation; in this
case, the latter could be a reference to an online copy.)

These definitions contain too many undefined terms. What is a
“mathematical system”? What does “consists in
assuming” mean?

These definitions are imprecise. What does it mean to “study”
geometry?

The first Oswego definition is incorrect! The nonEuclidean geometry used in
special relativity is also flat.
(It is important to know the audience for whom you are writing. Yes,
precision matters, but so does communication. Mathematicians have an
unfortunate tendency to sacrifice clarity of exposition for precision.
That style may be acceptable when addressing experts, but not usually
otherwise. In this example, an important consideration would be whether
the intended audience would know enough to be confused by this error.)

In the last two definitions, the parenthetical clause about high school
geometry is fluff.

Finally, note the two different spellings “Euclidean geometry” and
“Euclidean Geometry”. Each usage is correct, but carries a
slightly different meaning.
An online sample of good mathematical writing in the context of nonEuclidean
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.
Rubric
Below are some of the criteria that can be used to assess mathematical
writing.
 Content

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

 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.