What is important to keep in mind in grading proof-based courses?

Question

I am an undergraduate grader at my institution where I have been entrusted with grading a section of an undergraduate analysis course; it's usual for this role to be offered exclusively to graduate students (who usually serve it as a secondary function while TAing) but my university was forced to add another section, creating an opening. I am passionate regarding the subject matter and proficient in it at around a lower-graduate level, but I have limited experience grading, particularly in proof-based courses. With this in mind, I was wondering if anyone had anyone had any tips regarding grading analysis, particularly in the way of maintaining equitable grading.

The course covers all the "platitudes" of analysis, i.e. $\epsilon$ \ $\delta$ proofs and the essential topological aspects of $\mathbb{R}^n$. We do however also include some more unorthodox material for a first course, like the Banach fixed-point theorem and some superficial covering of metric spaces in general. Any guidance from anyone who has graded something like this would be appreciated!

Answer 1

To grade faster and provide more targeted feedback, it could be helpful to review some lists of mistakes that students commonly make during proofs. For example:

• Taxonomy of bad proofs
• https://www.mathcs.bethel.edu/~gossett/DiscreteMathWithProof/CommonErrorsInProofs.pdf
• https://personal.math.ubc.ca/~cytryn/teaching/scienceOneF10W11/handouts/OS.proof.5invalid.html

This way, when you see something weird going on in a student's proof, you can (more often) quickly recognize and articulate the issue, and spend less time being confused about what the heck they're trying to do.

Answer 2

My dealing with proofs, as a student and an undergrad TA myself, left me feeling that a lot of students have one common problem with proof-based courses. They have trouble figuring out exactly which things are considered "obvious", and which they need to prove. (And figuring out exactly what to say, to constitute an acceptable proof of a fact that seems obvious.)

Therefore, to a grader of proofs, I would say: Try to be as clear as possible about what's missing, especially early in the semester. Especially if some things are intuitively obvious, but still have to be written out. (And if some steps are extraneous, maybe note that too, even if it's not a deduction.) Obviously a professor/TA has more opportunity to converse with the students interactively about these issues, but your comments will be the only ones that the students definitely see.

Answer 3

This answer is specific to your situation:

Focus on the things the professor thinks are important.

In general, in a course like this, one could focus on writing, on conceptual understanding, on logic, on many aspects of proof. (Or all at once, which probably many would prefer!) But in reality that is overwhelming for students to actually process as feedback.

For myself, in a class like this I give the most feedback on proper use of definitions, not missing subtle topological/metric issues, and so forth - the things I think make real analysis important. That would include quantifiers. If someone is using this course more as an introduction to proof course, then focusing more in feedback on the structure of the proofs, correct logical connectives, and writing might be seen as more important. Creativity or writing style may feature more highly for another instructor. And no two people are alike on this, so focusing on one more than the other, without other guidance, may indeed lead to perceived (if not actual) inequities.

So my advice is that once you've graded a representative sample of not-perfect, but not-awful, exercises, ask the professor(s) for five minutes to just check over that you didn't go beyond some unwritten, but important, marker in their eyes in one of those categories. Within just a few minutes you should get a pretty good sense of whether you've focused too much e.g. on trivial typos (or not enough), according to the expectations you, as an employee, are given.

Answer 4

Help with the most important problem

The students are likely to make many different mistakes, and maybe in addition the handwriting is messy and they are using a proof by contradiction where that is not actually necessary and so on.

Giving them feedback on all of this at the same is not useful. You need to identify what is the single worst obstacle to proceeding and developing mathematical maturity, and give feedback precisely on that. Maybe some other detail if really necessary.

So typically you would be ignoring lots of minor problems and instead telling them to be very clear about what the claim is, what are the assumptions they make, and how they use the assumptions to prove the claim. Before that has caught on, problems with fractions or unclear writing are bad, but not the central contents.

However

Most teachers do not know how to give feedback, and most students do not know what to make of the feedback they are given. This is a skill issue on both sides. So do not be all too surprised if the feedback you give does not have much of an effect on what the students actually do. This is a notoriously hard problem in university education in general.

Answer 5

At my university we teach a course in Discrete Mathematics that acts as an introduction to proof writing for undergraduate students. Three things of note.

1. We use Ed Scheinerman's Mathematics: A Discrete Introduction as a text, one that I think is very well suited for this course. Among the many helpful elements of the text, the author provides proof templates that are useful to students.
2. To grade proofs we draw from Brown and Michel's paper Assessing Proofs with Rubrics: the RVF Method. Having this rubric proves useful for both graders and students alike.
3. As an additional resource we share with students Appendix A from Dana Ernst's Elements of Style. This also proves useful to all parties.

Keeping all of the things mentioned in the latter two notes above is a good start for anyone new to grading proofs, in my opinion.

Answer 6

Among the most helpful tips I've ever gotten is to keep the grading extremely simple. Specifically, advice on this site by Joel David Hamkins to use a 5-point rubric for grading:

Should students be given partial scores when they gave an incomplete proof by contradiction?