October 11 Day 1312: What are the Odds?

I am teaching Foundations of Mathematical Reasoning this quarter and it’s all about reading proofs, writing proofs, and revising proofs. It is one of my favorite classes to teach because of the topic and the people. We genuinely build community. Partly because as the students are struggling to develop their proof writing chops, they bond over late nights, too dense readings, and where to even begin.

Today was a workshop providing peer review on their own writing. I adapted the Liberating Structure 25-10 Crowdsourcing for the activity.

Set-Up: Each student wrote a condensed proof of the same mathematical proposition and brought it with them to class. We anonymized the submissions by covering names with sticky notes.

Preliminary discussion: I asked ChatGPT to prove the same claim and we analyzed its output together. (Lesson Learned: Chat GPT is a tool like any other tool and we need to be aware of its limitations. The organization was beautiful–it had a clear statement of the claim, began the proof by stating assumptions, and ended with a strong conclusion. What we found interesting was that its proof called upon a ‘known result’ that restated the original claim without providing further justification– assuming the truth of the conclusion without supporting it. The students were shocked.)

Passing Process: Next, students milled about the room, exchanging proof papers. They could glance at the proof they received before exchanging it again. After two or three exchanges, they were asked to read the proof in their hands and score it according to the following rubric:

1-I’m not following and cannot judge.
2-I’m not convinced by the argument.
3-Mathematically it seems solid but it can be hard to follow in places.
4-Crystal clear. Don’t change a thing.

We had a total of three rounds of passing-then-read-and-score. What was the point of this portion of the exercise?

Students got exposed to several approaches and writing styles, displaying varying degrees of success before they were asked to provide more substantive feedback on one such proof. Plus the scores provided informal confirmation from the crowd on the existence or nonexistence of problems to be addressed.

Written Peer Review: After completing the passing process, each student wrote feedback on last proof that they received. In particular they were asked to discuss the clarity of the writing, the correctness of the argument, and identify the strongest aspect of the work.

Pairwise Discussion: The activity concluded with a verbal discussion of the feedback between author and peer reviewer to identify areas to improve in the assigned rewrite.

Often in peer reviews, students are paired to make giving and receiving feedback easier. The determination of who reviewed whom today was completely random; I wasn’t sure what to expect, except that nobody would be reviewing themself. To facilitate the pairwise discussion, we rearranged locations so authors would be close to their reviewers. Amazingly the chain of student A who reviewed student B who reviewed student C who reviewed student D…, created a Hamilton cycle that travelled through all the students in the room.

One of the student then asked the title question “What are the odds (of this happening)?” What a great question. Let’s compute them. I had 19 students participating in class today. There are 18! =6,402,373,705,728,000 ways to arrange 19 students in a single cycle. And there are !19=44,750,731,559,645,106 ways to arrange 19 students in cycles of length at least 2, the so-called derangements. The chance of getting a Hamilton cycle today was 18!/!19 = 0.14306…… So about 14%. I think that is cool.