As I mentioned in a previous post, I’ve been assigning large-scale collaborative writing projects in my mathematics classes. I’ve had my topology students write a textbook for their class, and this semester I’ve been doing the same in my discrete mathematics class. As I mentioned in that post, the approach has been very successful, but one area that needs work is the editing/revising processes. Some students are very reluctant to edit what has already been written. I think partly they are uneasy about editing the work of others, but partly they do not know how to revise/edit writing. So I’m trying a few new things this semester.
As I have in the past, I will give the students a document that I wrote called “The Nuts and Bolts of Writing Mathematics.” But now I’ve taken that document and turned it into a set of questions that the students should keep in mind while editing the document. (In fact, this checklist should work for any mathematical writing, not just collaborative writing.) You can download the pdf or view the HTML document below. (Note: If you have any suggestions for questions that I should add, please post them in the comments!)
I also plan to model good editing. I will give them a mathematical document (that I wrote) that needs a lot of revising, and we will spend some class time going through it, discussing how to revise it.
Finally, I’ve been reading this book on team writing. It discusses three ways to do team writing: face-to-face (you and I work together on the writing), divided (you do this, I do that), and layered (you look at it first, I look at it next). I think the layered approach will work best for my students. The book breaks up naturally into small parts—theorems and examples. For each problem I will practice the layered approach to writing and revising: Student 1 does the writing, student 2 does the editing, I give feedback, then student 3 does the final editing. (I may add one more layer of student editing before I see it.) I’ve created a Google Docs spreadsheet which gives, for each theorem/example, the names of the student writers and editors and the due dates for their work.
Editing mathematical writing: a checklist
The mathematical argument
- Is the mathematics correct?
- Is the argument easy to follow?
- Is there any extraneous information?
- Is the argument clear?
- Would it benefit from reordering the sentences?
- Is the level of detail appropriate for the audience?
- Are all proofs free of examples?
Mathematical writing
- Is the writing in first person plural? (Use “we,” not “I” or “one.”)
- Is the writing in the present tense? (Write “we find that,” not “we found that.”)
- Is the writing in the active voice? (Avoid “it was shown that.”)
- Are there symbols where there should be symbols and words where there should be words? (Use “x≤ 0,” not “x is ≤ zero” or “x is less than or equal to zero.”)
- Are the connecting words (thus, so, hence, therefore, moreover, furthermore, in addition, consequently, etc.) used appropriately and not repetitively?
- Is the writing free of flowery, imprecise, descriptive, and vague language?
LaTeX
- Are all mathematical expressions written in math mode? (We should see x, not x.)
- Is all text written in text mode? (We should see “such that,” not “suchthat.”)
- Are mathematical functions typeset correctly? (We should see sin(x), not sin(x).)
Figures
- Are the figures of high quality?
- Do they have captions?
- Does the text refer to them?
Mechanics
- Is the spelling, grammar, punctuation, capitalization, subject/verb agreement, sentence structure, etc. correct?
Style
- Does the layout reflect the style of the rest of the document?
Note: do not be afraid to scratch an entire proof and start over. Sometimes completely rewriting an argument is better than trying to fix a poorly written one.
David Richeson: Division by Zero 
I’d love to see the textbook your discrete mathematics students create. And thank you for this list; it looks good.
I think for FERPA reasons I probably can’t distribute the book when it is done. But I hope to make the “skeleton” of the book available after the semester. I began with Dana Ernst’s notes, and have been modifying them to fit my needs.
For FERPA reasons you can distribute the book. Here’s how. I include at the start of every class a paragraph that I ask students to sign. It says, “I give Dr. Gene B. Chase permission to use my work-product in MATH 111 for educational purposes only.” I then promise my students that I will only associate their name with the good examples; I will keep weak examples anonymous. Students appreciate homework solutions that I want to praise written by former students, after they have turned their solutions in.
You can ask your students to sign off more broadly, “I give Dr. David Richeson permission to use my work-product in MATH 100 for any purpose as long as it is not for profit.”
I am not a lawyer; this is not legal advice. :-)
Chris Staecker published his linear algebra Wiki as a book, but only made copies for class members. That’s a more modest solution. Maybe he’ll give you his computer code that did the conversion.
I model what I expect of my students in terms of giving credit. I’m teaching Calculus again this Spring. The rubric that I use for their papers has at the bottom, “Modified from a rubric by Dr. David Richeson, used with permission.” Actually you only gave permission to use it for Calculus at Dickinson College. But it’s easier to ask forgiveness than permission for Calculus at Messiah College too. In any case, this post expands on that in helpful ways and offers public access already.
One of my pet peeves is textbooks and popular journalism which burn the traces of how they got what they are saying. There’s no excuse when that extra bulk could be posted on-line. I’m happy to footnote you.
Thanks Gene. I do print and bind the book and distribute it to the students. I have no problem with that. I’m just hesitant to post the completed book online. FERPA is one reason, but also because I plan to teach these classes in the future I do not want the proofs and answers to the problems available to future students.
While I agree with most of the points in the checklist, I worry that they will be taken as iron clad rules rather than as the stylistic advice which they represent. Certainly I’d hope that there will be a disclaimer somewhere “the intent of this checklist is to allow us to produce a unified and consistent document which matches some of the most common stylistic conventions of mathematical writing” (though perhaps, not in quite such stuffy terms!)
I’d argue fairly passionately that there is, and should be, no absolute standard of “correct” mathematical writing. Each writer should be free to find his or her own voice subject to the overriding consideration of effective communication. So if I were to nitpick about some specific points in the checklist (again remembering that, for beginning writers, I agree with almost all of them):
– Why shouldn’t a proof contain an example? [So long as it contains a proof as well.] If an example would serve to clarify a proof does it make sense to separate it physically in the document and thereby break the flow? Certainly diagrams are considered an essential part of proofs in geometry — and what is a diagram but an example?
– Sometimes writing in the active voice really strains the argument. Furthermore, people take it too far: “We found that x=3 solved the equation.”
– Symbols are overused. Certainly “If $x \leq 0$ then ” but I’d much rather read “Let $n$ be a natural number.” than “Let $n \in \mathbb{N}$.
– I don’t know what qualifies as “imprecise” or “descriptive” language. Consider the following which might appear in a proof by contradiction: “Any counterexample would have to satisfy the following very restrictive conditions: …” Here “very restrictive” is purely descriptive — there’s no change in the logic if it’s omitted. But it can serve as a very useful device to let the reader know that the conditions themselves are almost enough to rule out the existence of a counterexample (as opposed to an argument where the conditions seem none too onerous, but as the proof develops turn out to be).
Finally, a couple of small contributions:
(argument or writing) “Does every part of the document have a reason for being, and a reason for being where it is?”
(mechanics?) “If you use a numbered list, do you ever refer to the numbers?” (If not, use an itemized list instead.)
Thanks for your comments Michael. Just to put this in context, the class I’m teaching is for first-year students. This is their first exposure to mathematical writing. I think much of what you are commenting on are comments for people who are experienced mathematical writers. I think that like most creative arts it is important to learn the rules before you can break the rules. At this level they need the structure of rules for writing. Later, when they are more confident mathematicians and more confident writers they can stray from this rubric. Also, you should follow the link that I gave in the document to the “Nuts and bolts of writing mathematics,” because I elaborate on many of the things that you mention. The purpose of the document in this blog post is just to create an easy 1-page list of questions.
The checklist is a great idea. Here are some further ideas to consider:
Under “The mathematical argument:”
* Is there a clear, brief summary of the “big picture”/overall strategy of the argument?
* Is there a clear, brief passage motivating the section?
Under “Mathematical writing” I would consider adding a list of words/phrases to avoid. For example, “obviously,” “clearly,” and the like are inappropriate, because readers for whom it is not obvious or clear (and there will be many of these) might feel stupid. I would much rather say, “Is this clear? If not, then …” and explain what the reader might do if it is not clear. Alternatively, one can say something along the lines of, “This follows directly from …;” in other words, explain why it’s clear, rather than say it’s clear and hope that readers will know why. The bottom line is that saying something is “obvious” or “clear” doesn’t improve the writing; if it is obvious to the reader, than saying so doesn’t add anything of value, and if it is not obvious to the reader, than saying so also doesn’t add anything of value, and might turn the reader off. There are many other no-no phrases, so it might be of value to start a list and add to it over time as others occur to you.
Under “Figures:” I recall an old newspaper man saying that the purpose of the caption to a photograph in a newspaper is to tell the reader what he or she should be seeing in the photograph, and I believe the same is true of captions to figures in mathematical writing. The caption should identify what in the figure the reader should attend to. So how about a question such as, Does the caption identify and explain what is significant about the figure?
Under “Style:” You might add a question about whether the tone of the writing is appropriate. For example, the discussion before a proof, where strategies are proposed and critiqued might have a much more conversational tone than the body of the proof, which might be much more formal.
Thanks for a very nice checklist!
Those are great suggestions. Thanks, Santo!