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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.)
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    $\begingroup$ More of a comment/suggestion than an answer: Not allowing "cheat sheets" seems like it would fix your problems. Another suggestion is to give a handful of problems as (uncollected) homework beforehand. Make the quiz questions a subset of either exactly the same as the homework or with minor changes if you are concerned about rote memorization of a set of solutions. $\endgroup$
    – Aeryk
    Dec 7, 2023 at 17:15
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    $\begingroup$ Generally, I would recommend against "challenging" quiz problems for undergraduats. Indeed, problems straight from the text should suffice. $\endgroup$ Dec 7, 2023 at 17:26
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    $\begingroup$ Rather than cheat sheets, perhaps include solution strategy outlines and other such hints with the problems. $\endgroup$ Dec 7, 2023 at 18:37
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    $\begingroup$ One approach I'm starting to like is giving 3 problems on a quiz, 2 of which are more or less basic and a third which is intentionally tricky. I include bonus points, so even if the students make a poor grade, it doesn't hurt their grade. Then the students who are very gifted might score some bonus for their work. This way I challenge the strong students at the same time as I don't crush the weak. Solutions are given so in the end they all should be ready for similar problems on the test. This is more or less my typical pattern. Cheat sheet or not seems to be not much difference. $\endgroup$ Dec 8, 2023 at 3:02
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    $\begingroup$ In my experience, allowing cheat sheets can be a good idea in higher years, and the students will even gain some understanding of the material during the process of designing good cheat sheets; but in the first two years of university, if you allow cheat sheets, most students erroneously conclude that they don't need to study at all since they can just refer to the cheat sheet during the exam; and of course all these students fail spectacularly. $\endgroup$
    – Stef
    Dec 8, 2023 at 21:30

1 Answer 1

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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.
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    $\begingroup$ What if my learning objective (at least for some students) is 'I want my students to be able to make progress (on their own, without prior preparation) on a Putnam-level problem on this subject"? $\endgroup$ Dec 8, 2023 at 0:08
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    $\begingroup$ @AlexanderWoo: Then you'll probably face the problem that it's essentially impossible to break down this objective into objectives that are sufficiently small and concrete to be measurable on a single day or in a single week or even within a single course. Which explains quite well why many mathematicians I know find the idea of basing all teaching on 'measurable learning objectives' rather ridiculous. $\endgroup$ Dec 8, 2023 at 20:04
  • $\begingroup$ This principle is very flexible: to assess whether students can make progress on Putnam-level problems, the simplest question to assess that entire goal at once is a Putnam-level problem. This may not go over well with weaker students, though. To break it down further: identify the specific skills and knowledge necessary to make progress on Putnam-level problems by looking at how you teach toward this goal. Putnam-level problem solving isn't a mysterious thing: it's having seen a bunch of stuff before, identifying what of that is relevant, and putting it together, all separately measurable. $\endgroup$
    – TomKern
    Dec 10, 2023 at 5:38
  • $\begingroup$ @TomKern: I've deleted my latest comments to instead ask a separate question about it here. $\endgroup$ Dec 10, 2023 at 18:19

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