Intro to Proof in Four Weeks Part III: Teaching and Learning Activities

I've already written about the role this course plays in our curriculum and my plans for learning outcomes, feedback, and assessment. Here, I will discuss what my plans were for teaching and learning activities. Later, I'll tell you how much practice lined up with these plans.

Two hours per day, five days per week, for four weeks seems like an eternity from the perspective of lecturing. When I first started teaching, I wrote out impeccable lecture notes and went into my classroom to drone on for 50 or 80 minutes. Slowly, I started incorporating active learning to break things up, and then I brought in clickers, which created a lot of student discussion and sense-making in my classrooms. (Although I didn't realize it at the time, this was really where I started moving into an inquiry-based approach.) Today, if I stand in front of my students and talk for 10 minutes, there are klaxons sounding in my head telling me to shut up and let the students talk or work or even just think for an extended period. Thus, the idea of having two hours of class is no longer intimidating. In fact, I was so excited about having that amount of time every day for student presentations and work that I was unwilling to give up any of it for quizzes or tests!

My course design was based around a modified-Moore method course. Since I was going to use Carol Schumacher's Chapter Zero (CZ) for my course text, I knew that there would be a lot of good starter problems in the text as well as a variety of difficulty levels of problems I would want the students to present. Before the term started, I managed to lay out a plan for the first 60% of the term (12 class meetings). I categorized the items on each day's plan as

  • oral presentations (Basically, I would call on a student to quickly summarize their answer to an exercise or one-sentence proof of an easy theorem.),
  • written presentations (Students writing up their proofs fully and then presenting them to the class.),
  • group and class discussions (Typically starter problems and examples that would help the students grapple with new definitions before trying to prove things about these things, with the idea being that students would discuss in groups and then we'd discuss as a class.),
  • things for me to present/clarify, and
  • harder theorems for the groups to actively work on proving during class time (instead of coming to class with a proof prepared to present).

In drafting these plans, I leaned heavily on term calendars from Carol Schumacher's 14-week course and the Bates short course, since I wasn't always sure which topics were going to take a long time without some guidance.

Even with two hours, that's a lot of things to try to fit into each class period, and most days had something planned for every category. Fortunately, I knew that if I was judicious in making presenter assignments, I could avoid disrupting a group too much. Thus, the small group discussions of starter problems could take place while presenters were writing up their proofs. My plan was that we would then shift gears to listen to the student presentations and then have a full class discussion of the starter problems to make sure that we were all ready to move on to the next topic.

Another consideration that came in early in the planning stages was trying to build in time for student growth when there wasn't a lot of time. Having taught mathematical induction many times, I knew that matching the number of hours in class that Carol would devote to the subject was not going to give the students enough time to process the ideas. Fortunately, I realized we could easily move on to relations while still working through induction, so I planned a couple of split days where we'd dip back to induction for half the class and then move forward with relations for the other half of the class. I also decided to punt a bit on resolving exactly which parts of CZ I wanted the students to do until we got into the course. My main goal was to get up through functions. (Thus making the topics covered logic, set theory, induction, relations, partial orders, equivalence relations/partitions, and functions. Anything else would be a bonus.) Partially, I was worried about things taking more time than budgeted, and I didn't want to feel stressed about getting through to a certain point. The other part of me just legitimately couldn't decide what topic to conclude with. There's always modular arithmetic as a lovely example of equivalence relations. There's also the mind-bending aspects of cardinality, and since I had a lot of math minors in my class, I knew not everyone would get to see that stuff in real analysis. I was also tempted to jump over the axioms for the real numbers and try to treat sequence convergence in a more intuitive way. This last idea was largely motivated by comments from a colleague who's taught real analysis the last two years, since he always has had issues with students being able to get through those ideas when he needs them fairly early in the course. Punting seemed a good way to avoid a decision and allow myself the flexibility of choosing a topic that fit the amount of time I had left when we reached the end of the pre-planned part of the course.

Oh, I was also supposed to try to teach them \(\LaTeX\), wasn't I? Well, that turned into mostly self-teaching. We took 20–30 minutes in each of the first two class periods for the students to work with \(\LaTeX\) on their laptops in our computer lab and then they helped each other and got help from me in office hours. (I also made a couple of quick screencasts to show them the key features of their editor and how to react to an error message. Credit for this and starting on day one goes to Paul Humke, as I was originally going to wait until day three. Paul told me about his approach, and I'm so glad I followed it. In particular, by starting with \(\LaTeX\) on day one, I had something good to give them for homework on the first day!) I won't really say much more about this than to say that it worked remarkably well. I made most of the first couple of homeworks consist of \(\LaTeX\) problems where they needed to figure out how to typeset something verbatim. I also made sure to carefully choose the things I had them typeset so that by the time they were doing real mathematics in Chapter 2 of CZ, they'd have acquired the relevant \(\LaTeX\) skills (or at least be adept at looking things up). Combined with marking up PDFs and allowing revisions of homework, I found that almost everyone in the class was a suitably-skilled \(\LaTeX\) user by the end of four weeks. Unlike past courses where I was pointing out variables not in math mode for half the class on the last assignment, there were very few instances of missing math mode come the last homework in this class. 

Up next: How did the course actually start off?