Strategies for Designing Challenging Yet Feasible Quiz Problems for Upper-Level Math Courses

Question

In the past semester, I taught two 7-week courses: discrete math and algorithms designs (which is essentially still math) for undergraduate CS students. I implemented weekly 25-minute quizzes containing three problems based on the week's content. (This is due to the observation that the majority of students do assignment via Google or ChatGPT, or copy from each other.)

My objective was to challenge students while ensuring the problems were solvable within the given time limit. However, I encountered several issues:

1. Cheat Sheet Utilization: I allowed cheat sheets, but some students crammed many solutions of exercises on them. This necessitated the creation of original problems, which varied significantly in difficulty. Crafting these problems often required a substantial amount of preparation time.

2. Complexity of Upper-Level Content: The challenge is more pronounced in upper-level courses, which involve intricate proofs, as opposed to lower-level courses with more computational tasks. At times, a difficult problem would stump the entire class, while other questions felt too trivial.

Given these challenges, I would appreciate your insights on the following:

• How can I develop quiz problems that are appropriately challenging for upper-level math students but still achievable within a 25-minute quiz?
• Or maybe I should simply go back to assignment rely solely on one midterm and one final exam as the only true assessment? (One can argue that force students who are not interested in the topic to study by using grades as sticks is futile in the end.)

Answer 1

I get a lot of clarity for designing assessments by writing out a list of learning objectives for the course. To get started, I write out three lists:

• Things I want my students to have seen (only need to show up in lecture, I don't care if students can replicate it)
• Things I want my students to have done (with reference and help, so on homeworks, in-class assignments)
• Key take-away skills (things students should be able to do on their own, so on in-class assessment)

In general, but particularly for in-class assessment, I go through the list of relevant skills and identify the simplest problem that will assess whether students have proficiency in that skill (as per Gerald Edgar's comment, although I would simplify further if your text has complicated questions).

It helps to get really specific about your goals for the course:

• Do you want your students to have a specific proof memorized? A good approach here is to give students a list of ~10 proofs you want them to know, pick a random 4 of them for the assessment and require that students prove 3 out of the 4 of them. Obviously reference sheets are no good in this case.
• Do you want your students to be able to be able to write out a proof by induction given another proof by induction as reference? Reference sheets are okay, but as Dave L Renfro suggests in the comments, giving your students your own reference proof gives you more control over the situation.
• Do you want your students to be able to be able to come up with a proof by induction not given any reference? Pick the easiest proof by induction you can think of.
• Do you want to use quizzes to enhance the effectiveness of homework? As Aeryk suggests in the comments, tell students in advance that you will pick a random homework problem to be the quiz. You won't be able to tell if they came up with the solution on their own, but this improves the chances students will actually get something from the homework.