This list from the JMM 2013 Minicourse Teaching and Assessing Mathematical Communication characterizes effective writing of mathematics majors. The first version is a transcription of the brainstorming notes from the minicourse. The second version is in more freely edited paragraph form.
From lightly-edited notes
Clear differentiation between the hypothesis/assumptions and the conclusion.
Only the stated hypotheses are used to draw the conclusion.
Correct use of notation if it is an established notation.
Answers questions asked using correct mathematical language.
Includes relationships to other work and suggestions for further research.
Topics are ordered in a logical manner.
Paper is broken appropriately into sections/paragraphs.
The title is thoughtful, careful, and brief, giving a good idea of what it’s all about.
The closing summarizes what has been done/achieved/conveyed through the writing, so the reader is not left to determine what conclusions were reached.
An appendix (optional) is used for details that are needed for completeness but are not essential to overall understanding of the topic.
Careful statement and identification of definitions and theorems used to make an argument or draw conclusions.
Proofs follow a logical line; are not haphazard.
Quality of Writing (Text)
Paper clearly describes motivation/set-up of problem; Motivates topic/subject before getting into details; Contains introductory paragraph(s) explaining the problem, the source of the problem, and the approach taken.
There’s a clear statement of the problem to be addressed (or the statement to be proven); The point of the article is stated clearly and succinctly in the abstract or introduction.
Any external supporting material is acknowledged/cited in footnotes or (annotated) bibliography.
Grammar and spelling are correct.
Language is straight-forward and expressed as simply as possible. “Fancy language” and “cuteness” are avoided.
If non-standard notation is defined by the author, it is “good” (easy to use); Choice of variables makes sense and is consistent.
Every variable is defined; All terms are defined clearly before they are used; All acronyms and specialized terms are defined at first instance and a glossary is included if text is long.
If a proof uses something, no matter how trivial, which is important for the proof, it is pointed out. If the proof heavily depends on the fact that 2+3=5, then it is mentioned “since 2+3=5.”
The text is appropriately concise; there are neither too few words, nor too many; the level of detail is appropriate; the student assumes the right amount and doesn’t attempt to write a “complete proof” that proves everything used; equations and steps are explained rather than being merely listed; lots of words, not just equations.
Examples illustrate what’s going on.
Quality of presentation (Graphics)
When a picture is worth a thousand words, use it. (and only then)
Clear, simple caption, with reference to further explanation in the text.
Neat legend if needed.
Clear labels on axes.
No “showmanship” without a purpose.
More freely edited paragraph form
The paper answers the questions asked, using correct mathematical language, notation, and logic. For example, in a proof, the hypothesis and assumptions are clearly differentiated from the conclusion, and the latter follows logically from the former.
The paper is structured appropriately into sections and paragraphs that are ordered logically. For example, a carefully crafted title briefly conveys the main point of the paper; the paper begins by introducing the topic and indicating its relevance to other work; definitions and theorems that are needed to make the paper’s argument are stated carefully where needed; proofs follow a logical flow; the paper’s ending ensures that readers know what conclusions have been reached and suggests further research, if appropriate; and (optional) appendices include any details that are not needed for the audience to follow the paper but that are needed for completeness.
Quality of Writing (Text)
Throughout the paper, context is provided before details so readers know why the details are being presented. For example, the paper begins by introducing the problem to be addressed (or the statement to be proved). This introduction includes motivation for the problem, a clear, succinct statement of the problem, and a summary of the approach taken to address the problem, as well as sufficent groundwork for readers to be able to follow the introduction. Further context is provided as needed throughout the paper. For example, equations are integrated with text so the relevance of each equation is clear to readers.
Any external supporting material is appropriately acknowledged, for example by being cited within the paper and included in an (annotated) bibliography at the end of the paper.
Language conveys content clearly without distracting attention from the content. For example, grammar is correct and the language is simple and straightforward, avoiding “fancy language” and “cuteness.”
Terms, abbreviations, variables, and non-standard notation are introduced before or as they are needed, are used consistently, and their definitions can be easily found on skimming (a glossary may also be needed for longer text). Variables and nonstandard notation are carefully chosen to be helpful for readers. For example, symbol choice and notation structure hint at their meanings (e.g., C for “cost” and subscripts for indices), and the notation is as simple as possible while achieving its purpose.
The level of concision, detail, and explanation is appropriate to the target audience and to the context within the paper. For example, details are omitted if they are relatively unimportant and can be easily supplied by the target audience, while the important steps of proofs are explicitly stated, no matter how trivial they may seem. Examples are crafted to help readers understand what’s going on.
Quality of presentation (Graphics)
Pictures are included iff they are worth a thousand words. Like text, figures are designed to convey content without distracting: no ”showmanship” is used without a purpose. The figure and text work well together; for example, the text references the figure, and a clear simple caption, legend, and axis labels enable the picture to convey meaning to readers who skim while leaving further explanation to the text.