by Sarah Hagen
Recently as a graduate student I taught a week-long boot camp for incoming mathematics graduate students at Oregon State University. It was my first foray into teaching under the active learning model and it was a completely transformative experience for me as an instructor. The change in my own attitudes towards teaching and pedagogy were so abrupt, so all-encompassing, that I felt compelled to immediately record my thoughts on the experience.
The purpose of the free, optional boot camp is to ensure that incoming students have a common base of content knowledge and proof techniques. The camp is run by an advanced graduate student, with every aspect at the discretion of the instructor. In designing the boot camp I chose to focus on concepts from real analysis and linear algebra (the subjects of our PhD qualifying exams). Each day I would choose a major idea and orient readings and problems around it. In the mornings we explored the historical development of the idea using guided primary source readings and exercises from the excellent TRIUMPHS projects[1]. The afternoons ran as problem sessions where I posed more sophisticated problems with modern definitions and theorems.
The model of active learning that I chose meant that I did almost no lecturing at all. I assigned readings and exercises to be completed before each session. This meant that the students never came at a definition or theorem completely cold. The readings, exercises, and later problems were discussed first in small groups, and then as a whole class with a student presenting on the board. I believe that the boot camp had a strong positive impact on the students. However, the impact on me was even greater. I am a total convert… an evangelical, born-again, active learning enthusiast. This is my testimonial.
My Own Transformation
I hope never to lecture again. This experience of teaching with active learning was eye opening for me. So many things jumped out at me as reasons to prefer active learning to lecturing. First of all, I noticed that students were much more engaged when their classmates presented on the board than when I presented on the board. The students seemed to feel freer to ask “dumb questions” to their classmates than to me. If they didn’t understand a line in a proof that the student had written they would just ask them to explain more. However, the few times that I wrote on the board I could feel the room tense up and the students’ eyes glaze over (and I pride myself on being an engaging instructor!). The problem is that they felt like what I had to say was gospel, and so they didn’t engage as much or question as much. They just accepted what I wrote, copied it down, and then waited for the next piece of information.
The next thing I noticed is that students often came up with solutions very different than what I had in mind. When there were multiple solutions to the same problem I had students write them all up. We then looked them over, compared them, and discussed the virtues and drawbacks of each attempt. This would never have happened if I had simply written my own proof on the board. Also, it sometimes happened that there would be a subtle error in a proof that made it to the board. These errors prompted great discussions every time. Usually one of the students would notice the error but maybe not know how to fix it. Often some students would still need convincing that there really was an error in the first place. As a class we would discuss the various nuances, and I pretty much never had to swoop in and resolve the mystery. The students would solve it themselves. At the end of these debates I would spend a minute summarizing the debate and emphasizing the main takeaways. When appropriate I would also place the debate in a broader context or relate it to something else we had done that week. These moments allowed me to share my expertise in a way that would not have come out if I had been lecturing.
This led me to the realization that lecturing is a waste of an instructor’s expertise. What is the benefit of having me write down the standard proof of a famous theorem on the board when this is something that the students could find in any textbook? I could perhaps explain the reasoning behind certain moves. But then, if the students have read the proof beforehand, I could still give that explanation, and without wasting time writing out symbols on the board that the students already have in front of them. Even better, I could ask questions or assign exercises that allow the students to discover the subtleties on their own. That is where my experience and knowledge becomes useful. It is in designing the readings, exercises, and discussion questions that best facilitate understanding. It is in answering questions on the fly to help students get beyond some mental block. It is in facilitating thoughtful discussions that bring out the nuances in subtle reasoning. Writing something on the board that the students all have in front of them already is not only a waste of my own expertise, it is a colossal waste of everyone’s time.
Teaching with active learning was way more fun than I had anticipated. I have always enjoyed lecturing, but teaching with active learning was more fun, more engaging, less stressful, and ultimately less work than lecturing. The fun was being able to interact with the students on a more personal level. It was also a fun challenge to meet the students where they were and figure out how to get them on board. Writing proofs on the board can be stressful. Subtle errors easily creep in, and when you are writing on the board you don’t always see the mistakes that you made (even if you would notice them immediately if someone else was writing on the board). With the model of making students write their answers on the board, the stress of presenting is broken down into small pieces and spread out amongst the whole class. Students also gain experience and confidence presenting their work (something that academics are expected to do frequently). And students often enjoy showing off a little bit when they have proved something on their own and they are sometimes quite eager to share.
By the end of the week of boot camp, I trusted the students enough that I could relax a bit on my own preparation. At the beginning of the week I had my own answers to every problem or exercise that I assigned written out in painstaking detail. This was useful as the students were getting accustomed to the format of the class and needed more precise prodding. However, by the end I was no longer worried that no one would be willing to write on the board or that a subtlety would be overlooked if I didn’t micromanage. As a result, by the end I was assigning problems that I knew were doable and that I had an idea of how to do, but that I hadn’t worked out fully on my own. This made class even more fun for me because I got to think along with the class. It was also good for the students because they got to see more into how a relative expert tackles certain problems and why I can do them faster even though I’m not any smarter than them. I would say things like “I’m not sure how that goes, but when I see something like this it always makes me think of…” So my being slightly less prepared actually allowed students to gain new insights into how to problem solve at the graduate level. (Not that I am advocating being under-prepared for class! I am only saying that after a while it was useful to hand the reins over to the class a little more, and that my relaxation of control had its own benefits.)
Conclusion
In the year since the boot camp I have occasionally found myself lecturing (usually by accident), and each time I have deeply regretted it. I can always tell that I’ve started accidentally lecturing by the looks in my students’ faces. They may be following and focused, but they are falling behind. My unease over lecturing is so complete that I now have trouble giving presentations without inserting a significant amount of active learning. This has also worked successfully for me with presentations related to pedagogy as well as with academic presentations to REU students and public outreach. However, I have yet to try this approach in a research talk. My friend and mentor in all things active learning, David Pengelley, has been a great encouragement (bad influence?) in this respect. He is working on incorporating active learning into his research talks. Once he figures out how to do so successfully, perhaps none of us ever need lecture or be lectured to again.
[1] 1https://digitalcommons.ursinus.edu/triumphs/