This semester I decided to take a (maybe temporary) break from lecturing. Don’t get me wrong, I love lecturing, and I love going to good lectures. However, there have always been a few aspects of my own lecture classes that are not satisfying. If I take enough time to explain some topics very carefully, we don’t have enough time for other topics I want to cover. Many students in any lecture are focused entirely on copying down my blackboard work and are not actively engaged with the material. If I do group work in the classroom or try to incorporate interesting activities, we don’t get through enough material and I end up rushed later. If I assign reading to make up the difference, students will only rarely complete the reading and then often complain a lot about it. Also, articles like this one and this one also got me thinking that there might be good philosophical/ethical reasons to look beyond lecturing. So this semester I decided to go all the way and turn it around.
What instead? The best choice for me, and these classes, seemed to be something very structured, that made use of existing resources as much as possible. The plan:
- Assign reading before each class.
- Give students a reading assignment and check their notes for each section.
- In class, discuss the reading assignment and take questions on the material.
- Work in groups on homework assignments from the text.
I am currently teaching two classes this way. The first is Foundations, Villanova’s bridge to higher mathematics course. Second is Modern Algebra, required for all math majors, but not a prerequisite for many other courses in the catalog. In both cases, the students already have some mathematical maturity, so reading a book was not as distasteful to them as it might have been to a Calculus I class.
Another reason that this was a good semester to try something different is that my good friend and colleague Katie Haymaker is also teaching Foundations, and we both wanted to take a very hands-on approach to the course. As Katie says, “for me, one really important aspect of this format is that students are practicing proving things during class. I feel like intro to proof classes in particular suffer from the osmosis issue – students think that because they saw you prove something and understand every step, that they understand how to do it. But then when they are asked to prove something themselves, they freeze up and have no idea how to start. This is why I much prefer this format in the proof-heavy classes, but why I don’t feel compelled to do things this way in Calc 2 (in addition to the fact that no one wants to curl up and read Stewart after a long day of classes).” We have been planning our Foundations classes together and comparing notes on what worked and didn’t. This has been incredibly helpful. I can also get Katie to talk to me about my Algebra class, since she is teaching that class next year and is now considering how to organize that.
For Foundations, Katie and I selected Book of Proof by Richard Hammack, which we really like. All the students have reported that it is very readable, it has a lot of exercises in each section, and the odd problems have solutions, which provide many extra examples for the students. Also, it is available as a low-cost paperback or as a free pdf from the author’s website. Most of my students bought the paperback, but they always have access to the book online. For Modern Algebra, I chose A Book of Abstract Algebra, by Charles Pinter. (I apparently like these direct titles; titles that let you know “I’m a book! About this!”) Pinter’s book is also extremely readable and has many exercises. A few more solutions and examples would be great, but there are a few full solutions in the back of the book. My students have reported that they like the book. Both of these books are divided up into reasonably short sections, something I think is important for this model. The students can read an entire section or chapter, then answer questions relevant to just this material before we move on.
The reading assignments are generally short, and I aim for them to be relatively easy, given the reading. Sometimes I overshoot and ask more challenging questions, but it seems to work out since we go over the reading assignments in class. Each class I also check for 1-2 pages of notes on each section. The reading questions and notes are worth 10% of the students’ final grades. Weekly homework is worth 30%, tests are worth 30%, and the rest of the grades come from: presentations (10%), a proof portfolio (10%), and weekly-ish blog entries (10%). I am not sure now if I would distribute the points the same way (more on that another week).
The blog posts were inspired by a blog post (meta!). One of the main goals I had for the course was to help my students become great at both communicating mathematics and communicating about mathematics. Writing proofs is important, and we put a lot of energy into that, but I think it’s almost as important that the students can have a satisfying conversation about mathematics with either a mathematical peer or a non-mathematician. The blog posts ask them to practice explaining mathematical ideas in everyday terms, or reflect on especially difficult concepts in the course.
I got the idea for a proof portfolio from Chris Rasmussen at Wesleyan University. The assignment asks students to choose their ten best proofs from the whole semester, revise them to make them absolutely impeccable, and then submit the portfolio in the last week of class. I used proof portfolios in a previous number theory/intro to proofs course, and was really impressed with what the students submitted.
Teaching this way has been a roller coaster for me. I really like it, and think it is working great in many ways. However, it is stressful to ask students to invest in something they initially find foreign and annoying. If the students don’t buy in, it can’t work, and I often feel stressed that I am not really in control of this aspect of the class. It has required a huge investment from me, too, in developing new materials, lots of grading, and the straight effort of organization. That said, I think that teaching this way was a good choice. My students are taking responsibility for their own learning in a real way. They are submitting solid proofs and becoming clearer communicators. I just gave the students a mid-semester feedback survey, and their responses indicate that they find the general structure of the course really effective. Hooray!
As the semester winds down, I will report back with more in-depth discussion of the components of the course, what has worked and what hasn’t. In the mean time, please send me some good teaching vibes, and tell me what has and hasn’t worked for you in the comments!