Best practices for Proof Revision/ Proof Portfolio?

Question

I'm teaching a class small enough that I'm considering encouraging proof revisions (i.e. students taking a second try on proof based homework problems after getting feedback) for the first time. I'd love advice from instructors who have tried it about how it went and how your course and the revisions were structured. How did you award points? What did you do to balance the increased grading workload? How did you spread out the grading work? Did you think it was useful?

I remember benefiting from a similar class as a student, but I'm not the typical student and as an instructor I can now see some challenges with this approach. Any experiences (positive or negative) are appreciated.

Answer 1

Here's a thing I did stepping in that direction. In a discrete mathematics course, I made use of the discussion forum in the LMS in this way -- require each student weekly to pick a distinct homework problem from that week's section, post an answer to the discussion board, give a specific critique/advice for improvement on another person's post, and edit their initial post incorporating any feedback.

So in some sense I was trying to model the activity of reading and writing other people's (possibly substandard) math writing, and also leverage the students to engage in doing the feedback for me.

This being on Blackboard, there's a math editor for symbolic expressions, and optionally you can directly write LaTeX for those bits. I required students to learn the minimal LaTeX and write symbolic pieces that way.

This all occurred within a week cycle -- e.g., class Mon-Thu, suggested initial post Friday, feedback Saturday, improved solution Sunday. Grading was kept to either 0-1-2 points for both the finalized post-feedback solution, and also the critique to another person's work.

With practice I could get a weekly grading cycle done (along with some feedback of my own), for both the solutions & critiques, in about an hour for a class of up to 25 students. (Data shows for that number of registered students, I'd get about 20 submits weekly; taking 2-5 minutes, average 3, per submission.) That's in line with time grading for my other courses, but it felt much more mentally taxing -- instead of one problem I was grading longitudinally, for every submission I had to parse a unique problem and think about whether the solution & feedback made any sense (with assistance from the instructor solution guide), and any feedback from me had to be customized to that problem (e.g., no copy-paste from a FAQ document).

Answer 2

I taught an abstract algebra course this way.

Each student made an overleaf account which I required to be something like "Firstname-Lastname-MTH358-Spring-2024". They shared this folder with me.

Each homework assignment had a few conceptual questions, a few computations, and a few proofs. I would have them complete most of the homework using pencil/paper. One proof problem per homework was marked as a "portfolio problem": they had to type this up on overleaf.

I would make comments using the overleaf comment system and send them an email when I was finished commenting on their proof. They then had one week to revise and email me to check their revision to update their grade.

What did you do to balance the increased grading workload?

I didn't: I massively overworked myself. Burnout from this kind of experiment was one of the contributing factors which led me to leave the profession. I wouldn't say that burnout was the primary contributing for me, but it is still something to think about carefully.

Did you think it was useful?

Yes, I saw a consistent improvement in the mathematical writing of my students throughout the semester. Some of them thanked me for being "the only professor who actually showed them how to write proofs".

Answer 3

I've done this multiple times for our intro to proofs class. I used to do it all with LaTeX files, etc. but, like the others, have moved to using Overleaf. (I just have them use link sharing to avoid having to pay for a premium account.) I could say a lot about this (in fact I gave a talk at JMM in 2016 about it, slides here.) For brevity I will try to answer your specific questions:

advice from instructors who have tried it about how it went

It went great! Do it! I'm doing it again this semester.

how your course and the revisions were structured... How did you award points?

See the aforementioned slides. Basically problems were reserved by students, submitted solutions were hashed over in class, and then either deemed accepted or needed to be revised. I award regular points for work base on a rubric but I also give "reputation" points for good answers and participation in the critiquing process. I got the idea from... Stack Exchange! More details in my syllabus.

What did you do to balance the increased grading workload? How did you spread out the grading work?

This was the main key to this: Students do far fewer exercises, perhaps as few as one or none in a given section. But it's quality over quantity. Since this course is primarily focused on proof writing, I feel like, if you can write one really good proof in a given concept, that's enough for me to believe you know how to write a proof with those ideas. Anything beyond that is just learning the mathematics concept better as opposed to learning about the process of writing a proof. (Admittedly, this part would need to be altered in a course that actually is about content.) Most of the grading is done live in front of the class so everyone can learn from it. So the extra work is just the bookkeeping, compiling, prepping, etc.

Did you think it was useful?

Yes. The students who knew what they were doing did great. The students who struggled were given the chance to do it until it was right and significantly improved over the semester. In fact, it was the success with revisions in this class that led me to embrace standards-based grading in other courses.