Galois Theory

I am lucky enough to be teaching a course on Galois Theory that only has five students in it. The set-up of the course is this: we are working out of Pinter (which I really like due to its ease of reading, great problems, and price of $12; I don’t like that he defines subgroups/subrings/etc in a weird way and is sloppy with defining variables), and students present problems from the textbook to each other.

This will be obvious to people who do IBL, but: holy cow do you get a sense of what students understand and what they don’t. A large part of this is the small class size, but this IBL-like format helps. What students understand (or not) is often not what I would expect.

Because I only have five students, I am doing oral exams. The format is this: I give them four problems the week prior to the exam. Students are allow to work together to figure out the answers. They come to the oral exam having written one of the four problems up nicely in LaTeX. The oral exam starts by the student choosing one of the remaining problems and explaining it. I then (randomly) pick one of the two remaining problems. The session ends with me presenting a problem they haven’t seen before (I try to make these easy enough that the student should know what to do immediately). We will do these exams four times during the semester, and they take 30 minutes per student.

Once again: this testing format makes it crystal clear to me what (and how well) students understand things, and it is clear how I should adjust the in-class work based on the information I get from the exams.

I am wondering if this could scale: could I give a class of 25 students, say, 4 questions, have them write up one, and then have them write the solutions to a subset of the remaining problems? I like that this is a learning opportunity for the students, since they get to learn from each other in an exam situation, but they are still individually accountable. However, I am wondering how much is lost if the exams aren’t oral.


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10 Responses to “Galois Theory”

  1. Andy "SuperFly" Rundquist Says:

    In recent years I’ve been doing 10 minute oral exams, with more per student if the class is small. In 10 minutes you can look over their work and ask probing, deeper questions, rather than have them solve new things. For that it seems to me that really do need more time.

    • bretbenesh Says:


      Please tell me more! Are these the ones that you do in class every couple of weeks (ish)? If I remember correctly, these are public oral exams, right?

      What work are you looking over? Is this like a take-home exam? Bret

  2. Andy "SuperFly" Rundquist Says:

    I don’t do them public any more because they were still too nervous for that (and I could use my whole calendar this way). For a class of ~15 I tended to do between 6 and 9 with each student throughout the semester. Sometimes I’ll put in a bunch of google appointment slots in one week because I want everyone done before, say, midterms. Other times I’ll just pepper them throughout a month or so because I just want to see them and I don’t really care when.

    I’ve been having students develop portfolios ( and now the oral exams have me randomly picking one of the ~75 problems they have to work throughout the term and they flip to that page. I then stare at it for a minute or so and start to ask probing questions. I tend to focus on things like negative signs because I want to make sure they understand the physics about why it has to be negative.

    • bretbenesh Says:

      Cool. Do you try to make sure that your 6–9 meetings cover different types of topics, or do you just assume that students will mostly be tested on most of the course material by the end of the term? (Or is this unimportant to you?)

      • Andy "SuperFly" Rundquist Says:

        Often I’ll do 3 in a week for a student and I make sure those 3 are different. Otherwise I tell them it’ll be completely random of those that we’ve spent some class time on.

      • bretbenesh Says:

        Nice. I might try this next semester in Real Analysis.

  3. TJ Says:

    I have used (much simpler) oral exams in the past. I liked them, but they were intense for me and for the students.

    generally, I like this plan. “Your Assessment Strategy should line up with your pedagogy” … and this lines it up pretty clearly with daily work and the expectations you have.

    I wanna hear more about how this structure works out. Then I can think about stealing/adapting it in the future.

  4. bretbenesh Says:

    “Your Assessment Strategy should line up with your pedagogy”

    This is brilliant. I must remember this.

    I will give you an update on the end of the semester.

  5. Dana Ernst Says:

    I’ve been tinkering with oral exams in my classes for the past few semesters, mostly in my grad classes. The format that I’ve settled on is a 30-minute exam for each student, which they can schedule during a 2-week window. Sometimes I might have 20 students in a grad class, so this amounts to doing at least an extra 10 hours of work over those two weeks. However, I’d much rather do 10 hours of oral exams over 5 hours of paper grading. I really enjoy the oral exams and it’s great practice for our grad students that have to take oral exams in three subjects at the end of their program. For the most part, the questions I ask them are exactly or related to previous homework questions or theorems we discussed in class.

    • bretbenesh Says:

      >However, I’d much rather do 10 hours of oral exams over 5 hours of paper grading.

      This is a good way of thinking about it. With larger class sizes, you could cut down to, say, 20 minutes, too.

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