When an undergraduate student progresses through their first research project, the new skills they are required to develop along the way can be daunting. In mathematics, students need to learn how to: ask good questions, make and test conjectures, find research relevant to their questions, present their findings in a professional setting, and write up their results. This last aspect is often de-emphasized, left for the very end of the project, to be completed in a mad dash. The results are often sloppy at best, completely disjointed and incomprehensible at worst. I know; I have personally struggled with this issue over the years. 

In my first experience as an undergraduate research mentor, I had an unusually strong, well-prepared group of students. They were self-motivated to write up their results early and often, and they produced two (yes, two!) publishable papers which were submitted to journals during the last few days of our summer research program, with relatively little guidance from me. I thought this is what all students would do naturally, so I did very little mentoring of the writing process with my next groups of students. As a consequence, one of my next groups never actually wrote up their results. Another group of students wrote up results, but when I went to edit the writing to elevate it to a publishable level, it took me (literally!) years of hair-pulling revising to reach this goal. And improving this group’s writing was by far the least favorite part of my job at the time. Something had to change.

After these experiences, I began to request regular drafts of written results early in the research process. I would ask students to hand me their papers, dutifully mark them up with red ink, and hand them back to the students for revising. However, while my students would attempt to incorporate some of my suggested edits, their revised drafts did not significantly improve. Students would misunderstand my comments and would even ignore suggestions I had painstakingly scribbled in their paper draft’s narrow margins.

One summer, I tried something new. During the first week of the research process, I asked my students to start writing up their results in a shared Overleaf document using the typesetting software \(\LaTeX\). I then requested that they meet with me weekly for a joint editing session; I booked a room with a projector each week for this one- to two-hour session. When we met, one of the students would project their paper for all of us to see. They would then plant themselves at the computer, ready to implement the edits we’d discuss in real time. Other students might view the paper on their laptops to help with edits or just to simply follow along and make suggestions.

With this new process in place, I’d lead us through their writing, analyzing each line. If a line introduced a definition that wasn’t worded quite right, I might suggest, “In our field, the way someone would typically phrase this is…” If a line introduced a (horrific) new notation, we might discuss what notations might be more natural and what kinds of notation might confuse or disengage a reader. Often, I would know what my students were trying to say, but their wording would have a different meaning. I might point out, “What this sentence actually says is… Is this what you meant? If not, what did you mean to say? How can we change this sentence to get that idea across?” We would also step back and talk about the structure of the paper, reorganizing material as necessary. 

Here’s the final result, coauthored with four undergraduates. A significant portion of the body of this paper was written using this editing technique:

During our editing sessions, my students weren’t just considering a static set of suggestions, scribbled in red ink.

During our editing sessions, my students weren’t just considering a static set of suggestions, scribbled in red ink. They were thinking about various aspects of mathematical writing. They were being educated on writing norms in our field. They were suggesting their own edits instead of blindly replacing their phrasing with my phrasing. The resulting papers have been significantly higher quality. Instead of taking me years of revisions to get a student paper into a publishable form, it would take me months, or sometimes even weeks. Now, I cannot imagine doing undergraduate research without this live editing process.