From the time that you finish taking courses in graduate school, until you have to evaluate other people’s teaching (for tenure and promotion, say), you could get away with not watching anyone else teach. Of course, we see talks at conferences, and maybe think about teaching quite a bit, go to workshops, etc. But it is not that hard to go through this period of your life without being in any classroom but your own. Sadly, this is also the period of your life in which your teaching is under the most scrutiny. From the moment you get hired for a postdoc or tenure-track job, you will have to prove to others that your teaching is worthy of letters in the first case, and of keeping your job (i.e. getting tenure) in the second. In any case, you are still learning how to teach, and while a lot of this is “learning by doing”, it is surprising how little we see others teaching. I have been fortunate to have had a few chances to see my peers, friends, and colleagues teach, and I wanted to share some of the lessons I learned through that, and also to encourage you, dear readers, to do the same.
Michael Gagliardo teaching an IBL geometry class.
First of all, I am lucky that some of my closest friends in graduate school went on to be very involved with pedagogy and the scholarship of teaching and learning (SOTL). This has given me a close network of people the I can talk to about all matters related to mathematics education. But, as people that I care about personally, I get a chance to visit them every once in a while, and when I do I like to visit their classrooms. Two such friends are Michael Gagliardo and Elizabeth Thoren, both based in Southern California, and whom I like to think of as the Brad and Angelina of Inquiry-Based Learning.
I was happy to visit Elizabeth’s class at UC Santa Barbara a few years ago, in which she was teaching a combo of linear algebra and differential equations. She had oodles of patience, and was great at being very supportive of students but at the same time not giving the answers outright. This is one of the biggest challenges for me when I am teaching IBL-style, since I have trouble holding my tongue and letting people figure things out on their own. It comes from wanting to help, but as all experienced IBL people will tell you, you help more by teaching people to find their own answers, rather than by giving them. Part of the challenge is also that you worry that the students will get frustrated and lose motivation, and this is where watching other people do it really helps. Once you see that people can lead a class without talking too much, where the students are presenting very good solutions, and where there are good discussions between the students about the solutions, you see that it can be done, and it can be done very well.
More recently, in April, I was able to watch Mike teach a geometry class at Cal Lutheran. It was also IBL but slightly different from the traditional “proofs by students at the board” thing that I’ve seen (and done) before. Mike assigned a reading and some problems, but he started the class by recalling the reading and giving a sort of mini-lecture (with really great slides!). At some point, the lecture slowed down and at each problem (or proof) he would stop and ask the students to work on it. They would work in groups and discuss the problems, and Mike would walk around and give advice, listen to their solutions, and guide their problem-solving. The students didn’t present too much (but I wasn’t able to stay the whole class, since I was giving a talk afterwards and needed to prepare) but they did participate very actively. Another of Mike’s colleagues was watching the class because she wanted to learn more about IBL, and I also thought that was very cool. This is someone with tenure, but she still wants to be up to date and learn about other teaching styles. How cool is that? Anyway, Mike’s style definitely gave me ideas on how I could teach some of my IBL courses a little differently, and I really appreciated the opportunity to watch yet another person teach a course in this style so effectively. Talk about a power couple, right?
Something slightly different that I participated in this year was a program, organized by Bates, called Teaching Triangles. The idea was to get people (volunteer faculty) into groups of three, so that each person was actually in a different department. Then we would all watch each other teach, and use this as an opportunity to reflect about our own teaching. That point was made very clear: we are not watching to critique or give feedback to the instructor, but rather to see what we could incorporate into our own teaching. I loved that idea, so of course I signed up! I got teamed up with someone from Sociology and someone from Politics. The first thing I will say is that their lectures were very well prepared, both had slides, and videos, and the students were very focused on taking notes (even though neither was writing anything on the board!). But my biggest take-away was on how they gave the students pre-work for the lecture. No one came in not knowing what was going to be talked about in class. Everyone had at least one reading to do for both classes. Benjamin Moodie (the Sociology professor) even had students write up something on the class wiki about their thoughts on the reading. He would read all of these before class so he could then “cold call” on people, knowing more or less what they were going to say. How wonderful is that? And also, this is when I learned that what we call a “flipped classroom” in math is really only a regular classroom in the humanities. This is what I’ve tried to have my students do in Calculus for years, with much griping and grumbling on their part. What I learned is that they are actually used to doing this in other disciplines, so it shouldn’t be so hard to convince them to do it for math. I also really like the idea of having them write something up before class (I was doing this with webwork, but maybe I can do something slightly less time-consuming) that I can use as a starting point for the discussion. I was also very amused at all of our attempts at comedy (some which worked, some not so much). It is nice to see that other people ham it up as much as you do. At the end, after we had all watched each other, we met up for breakfast (payed for by Bates) and discussed our thoughts and what we had learned. It was a great experience, and if there is something like this in your school, I highly recommend trying it. If there isn’t, I recommend trying to start it.
Francesco Pappalardi teaches all of algebraic number theory in a record 7.5 hours during the CIMPA summer school.
Much more recently, as evidenced in my latest blog post, I was able to see some pretty well-established, famous, great mathematicians teach intensive courses to advanced participants at the ICTP-CIMPA summer school in Benin. It was pretty cool to see material that I already knew (although not all of the topics as well as others) presented in this condensed form. Everyone was so good at presenting a big picture of what the topic was about, and engaging the participants, allowing them to present solutions to the class (IBL again! sort of), and then discussing solutions with the participants informally. Each person had their own style, but everyone was able to engage and inspire the students. I learned from them that in such a short amount of time you have to tell a cohesive story, but sometimes you also have to weigh the importance of showing a proof versus covering more of the material. In a class like that, your goal is maybe not for people to learn everything deeply, but rather to give an idea of what the subject is about, learn the basic definitions, and show why it is interesting. Again, I was really impressed at my colleagues, and even though they said I did a good job, I still feel like I have much to learn from them.
Anyway, this Fall I will be getting evaluated for tenure, and I’m happy that I’ve seen so many people teach in the past year. It has given me great insight into what works, and ideas on how to improve my own teaching. Like I said at the beginning of this post, I highly recommend watching other people teach. If you have any thoughts on this topic or any ideas on how one could visit other classes, please share in the comments section below.