I’ve been thinking about active learning strategies for teaching algorithms. The course is typically offered by a computer science department, but I think of it as a math course in spirit. So a few weeks ago, I Googled “active learning proof math” and found a paper published in Educational Studies in Mathematics earlier this year: “Collaborating with mathematicians to use active learning in university mathematics courses: the importance of attending to mathematicians’ obligations” by Estrella Johnson, Keith Weber, Timothy Patrick Fukawa-Connelly, Hamidreza Mahmoudian, and Lisa Carbone. (Coincidentally, Weber and Carbone work at Rutgers, which I attended as an undergraduate.)
The paper starts by pointing out a paradox of sorts: Many mathematicians say they believe that active learning is more effective than traditional lecturing while rarely incorporating active learning in their courses. As the authors put it, “This juxtaposition is curious and raises many possible explanations.” For example, maybe mathematicians think that they’re using active learning strategies when they’re really not, or maybe they don’t have enough time to modify their teaching strategies.
The authors’ thesis, based on work by Chazan et al. (2016), is that mathematicians’ perceived obligations prevent them from incorporating active learning strategies. More specifically, the authors believe that many mathematicians think there’s not enough time for active learning because, for various reasons, they feel obligated to cover a certain (large) set of topics:
Content coverage was reflected as an obligation to their institution (e.g., the department chair would be unhappy if different sections of the same course covered different topics), their students (e.g., students would be unprepared for graduate school if relevant content was not covered), and their discipline (e.g., a course that did not cover the Jordan Canonical Form theorems would not be a legitimate linear algebra course)…
What we wish to emphasize is how the obligation to cover content limited the active learning strategies that the mathematicians were willing to use. A primary consideration that was continuously raised was how mathematicians felt constrained by time due to the challenge of covering course content…
To better understand the situation and potential solutions, the authors collaborated with three mathematicians teaching a proof-based linear algebra course at Rutgers.1 Their collective goal was to design active learning strategies that met everyone’s interests and needs.
In the pre-interviews, all three mathematicians expressed confidence in the potential benefits of active learning; when asked about it, Mathematician A even stated, “I think the students learn better. I think that’s been established. I have no doubt about it.” But they also thought that active learning would take too much time. As Mathematician B stated, “We want people to think. Thinking will not take a very short time to actually finish. That is the main problem.”
In response to their concerns, the authors proposed using “exit tickets,” i.e., at the end of class, the mathematicians would ask students to submit a response to a question. The mathematicians, believing that the exit tickets would be more useful if the students had more time to digest the material, instead asked for a response at the beginning of the next class. Even though they thought the activity was beneficial, the experience confirmed their concern that active learning would take too much time. Mathematician C said that providing answers to these “launch activities” took over 20 minutes.
So the education researchers and the mathematicians negotiated and settled on the following guidelines: the question should be in a multiple-choice (or true/false) format, align with the flow of the lecture, be short and easy to understand, and require a response from every student. By asking questions that they had planned to answer anyway, the mathematicians did not need to spend as much time as they did with the exit (or entrance) tickets. Fortunately, they still thought the activity was beneficial, and all three mathematicians said they incorporated, or planned to incorporate, more active learning in their subsequent teaching.
In the end, the sample size was very small, there wasn’t even an experiment, and I don’t think any data was collected, but none of that is the point. Instead, I think the paper nicely captures my general impression of mathematics education: many instructors believe that there are simply too many topics that they need to cover for students to spend time pondering during class. The paper doesn’t imply that these concerns are illegitimate; in fact, it suggests to education researchers that they “interact with mathematicians in a respectful manner and consider their goals and needs.” Balancing breadth (e.g., fast lectures on many topics) and depth (e.g., active learning for fewer topics) is always a challenge, but the paper serves as a reminder that perhaps sometimes, we can improve in one dimension without sacrificing the other.
When I took this course, the set of covered topics was actually quite small. The instructor was frustrated by our (i.e., the students’) proof-writing abilities, so we spent a lot of time on topics that should have been covered in the prerequisite courses.