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I'd love to learn from anyone's recent experiences teaching online proof based math courses, especially those that have a large group of students who will be working asynchronously. My usual proof based course includes some lecture, with a lot of student interaction and some group work/guided discovery. I'm struggling with how to translate that online.

How have you tried to present proofs online? Have you been able to encourage group work in asynchronous classes? What have you done for online exams which have to be taken in different time zones?

Any advice or links to discussions elsewhere is appreciated. I've seen a lot of discussion of best practices for calc classes and below, but not so much about proof based classes.

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    $\begingroup$ I haven't tried this so I can't vouch (sorry), but one idea is to use some kind of pdf annotation software to do "guided reading" of proofs and let students make comments. One of the activities I would do in these courses is take excerpts of homework submissions from past students and discuss those during class, showing students how to read arguments critically. Seems like this could be somehow done asynchronously. $\endgroup$ – Brendan W. Sullivan Aug 10 at 15:33
  • $\begingroup$ At the recommendation of a colleague, I'm using Overleaf.com this semester for students to work collaboratively on documents. $\endgroup$ – Aeryk Aug 12 at 14:20
  • $\begingroup$ Guided reading like @BrendanW.Sullivan suggests could be done using Perusall. $\endgroup$ – Mathprof Aug 12 at 15:14
  • $\begingroup$ For absolute beginners, you might consider downloading my DC Proof freeware and the accompanying tutorial. Visit my homepage at dcproof.com For more advanced, but still very accessible topics, visit my my math blog at dcproof.wordpress.com $\endgroup$ – Dan Christensen Aug 17 at 19:29
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Have you been able to encourage group work in asynchronous classes?

In linear algebra last term, I used our learning management system (Blackboard) to put students into groups that I created, and assigned each group to complete a set of problems and type up their work. Since we were using Slack for class discussions/chat, I suggested that each group create their own Slack channel to collaborate on the work. I gave them deadlines for rough- and final-drafts, with some number of days in between each of those. [e.g. Assignment posted Sunday night, rough draft due by Friday, final draft due Sunday night.] Since I was also on Slack for questions, I was reading a lot of pre-rough drafts (which is more than I usually get in a face-to-face class).

How have you tried to present proofs online?

As a first online attempt, I made videos, stopping periodically to say "now pause your video to answer this part of the problem". When I do this again, I would like to try a feature of an online homework system that allows a video to automatically pause and a graded question to appear. [e.g. "For this problem, we're going to use mathematical induction. As a reminder, what are the steps for a proof using this method?" Problem opens up with various, jumbled steps for an induction proof, etc.]

What have you done for online exams which have to be taken in different time zones?

Last term, I made exams available for a long period of time (e.g. 8AM-8PM) to accommodate this and other issues.

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  • $\begingroup$ Can I ask how you assigned groups? (At random, by time zone, by ability level, by their preference, etc...) $\endgroup$ – Mathprof Aug 10 at 16:30
  • $\begingroup$ I grouped by ability level. These were almost all students I knew from a previous, face-to-face class, so I knew them pretty well. If they were all new to me, I would probably just do it randomly. [Edit: Since there was a good amount of time for them to work on an assignment, I did not group by time zone.] $\endgroup$ – Nick C Aug 10 at 16:36
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Jennifer Quinn blogged about having oral final exams, at her blog "Math in the time of coronavirus", https://mathinthetimeofcorona.wordpress.com/2020/06/10/june-10-day-94-finals-part-i/. and the more detailed part 2 https://mathinthetimeofcorona.wordpress.com/2020/06/11/june-11-day-95-finals-part-ii/

She gave each student 20 minutes to answer two questions. She gave the class a list of 9 questions. The students could choose one question to answer, and she would choose the second.

In her posts she discusses many aspects of this, including choosing the questions, and how long it took her (vs. how long it would have taken to grade "traditional" paper exams).

I haven't tried this. I only read about it on her blog. But I'm going to try it in one of my classes this fall.

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    $\begingroup$ A colleague did essentially this same thing and found all of the problems posted to Chegg with solutions. It seems like for this to really work, the problems should be given during the oral exam and unique to each student. $\endgroup$ – Nick C Aug 11 at 6:42
  • $\begingroup$ @NickC (1) Even with the types of reflective questions described by Quinn, there were "answers" posted on Chegg?!? (2) Whether or not answers are posted on Chegg, in Quinn's setup it's expected that students prepare ahead, including working with each other. So getting answers is okay, although it would be better for them to do some work, not just copy from Chegg. (3) Part of the point of the oral exam is to ask followup questions like "can you explain that step", "what if instead of x it was y", etc. So that provides some uniqueness for each student—and penalizes if they mindlessly copied. $\endgroup$ – Zach Teitler Aug 11 at 7:49
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    $\begingroup$ My colleague's exam questions were reflective but more traditional, such as those most-commonly chosen by Quinn's students, and there were fewer of them. I think he was troubled that students had the potential of going from clueless to having complete "solutions" via Chegg, though they still had to defend their answers. If the posted answers were packaged with lots of expository material, then becoming expert in his short list of questions gave the appearance of deep knowledge in the topic. Perhaps he just needed a longer list of problems to account for this. $\endgroup$ – Nick C Aug 11 at 13:15
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    $\begingroup$ Thanks for the comments! I guess there’s no silver bullet for cheating, even if Quinn’s (and your colleague’s) ideas might help somewhat. $\endgroup$ – Zach Teitler Aug 11 at 15:19
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    $\begingroup$ I went ahead and gave credit to the top ranked answer here, as I think it was the most comprehensive response, but this response made me think the most, as I think it challenges us to think about not just transitioning existing ideas to another medium, but actually creating new ways to connect with students. Thanks, @ZachTeitler! $\endgroup$ – Mathprof Aug 17 at 13:27
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To complement the other great answers, I'll add a helpful link to an MAA discussion of exactly this topic available here (along with two other helpful videos for online courses) which I recently discovered.

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