When I taught Honors Multivariable Calculus, I used three writing assignments: one on differentiation, one on change of variables, and one on Stokes' theorem. You can see a full assignment here. Below are explanations of some of the supporting activities.
It is intimidating to have to write several pages on math that you barely understand. Warmup assignments give students them bite-sized tasks that directly contribute to their paper. Here are the ones I used (click the arrow for reasoning behind the assignment):
Propose at least three ideas for your paper.
Benefits of this assignment include:
I can check that papers will be of the scope I'm looking for
By stretching themselves to think of several ideas, students find better topics
Students sometimes have to change topic partway into the writing process, and this way they have backups ready
Turn in a complete (but messy!) draft of your paper.
This is the natural assignment to have before peer review
It is important that the draft is allowed to be messy or have known issues. This makes it easier to get something out.
I have not yet found a satisfactory way to define "complete" and articulate the idea to my students. Given the nature of doing math, I think it is more challenging to produce a "complete" draft in earlier stages of writing than it is in other disciplines.
Identify the part of your paper that you are least confident about, or have the most trouble explaining. Bring it to office hours.
I have found that students (myself included) struggle to identify parts of their mathematical reasoning that have errors. They (or we) turn in papers, thinking they are correct, that in fact have serious problems. I developed this exercise as way to try to address that problem. I have found it to be successful, in that it often does find problems in students' work (I've used it successfully in my own papers too). Of course, it does not find all the serious problems.
Cover letter: write a few sentences summarizing what you learned about math, writing, or life, by doing this assignment.
This is of course a cool-down assignment rather than a warmup. I ask my students to turn it in with their final draft.
Research shows that we learn things much better, and retain them longer, if we articulate what we have learned. It’s called reflection. See, for example, Reflection: a review of the literature by Atkins and Murphy.
Students like to know why they are being asked to do things. So I put the above explanation in my cover letter assignment.
Why do peer review:
Almost all professionals work in teams. We need to know how to talk about complex ideas, how to hash out technical problems with peers using spoken and written words. Peer review is one way to introduce students to technical conversations.
Students learn from being reviewers as well as from being reviewed: I often invite them to read Chamberlain's "Workshop Is Not For You." .
How to do peer review:
Invite students to comment on what they do and do not understand in their peer's work (as opposed to what is or is not correct). I used this worksheet.
More materials can be found on my page about workshops for communicating mathematics.