tl;dr: Sometimes, a “problem” in the classroom is really a symptom of a bigger issue.
Here’s the backstory. Lately, I’ve been struggling with a lack of engagement in whole-group discussions after small-group work. I was growing increasingly frustrated that the majority of the students – especially the students who had a good grasp on the concepts – didn’t volunteer to share their thinking, and that they had no qualms about chatting with their friends when a student (or me) did choose to put forth a question or share an idea. Increasingly, I was feeling like I was losing them, and searched for explanations ranging from summer being around the corner, to deficiencies on the part of my students (kids these days!), to the possibility that I am a terrible teacher for whom there is, frankly, no hope. (During a recent conversation, Jimmy Pai described teaching as a balancing game between overconfidence and despair, and these days I’ve been leaning heavily to the right.)
After a particularly rough class, I decided yesterday that resolving the issue amounted to (temporarily) giving students some type of external motivation to listen and to participate. “What if,” I conjectured, “I gave an exit slip at the end of class, asking students to recap and reflect on a strategy to a class problem that was different from their own?” I was satisfied with throwing this at the wall and seeing if it stuck, until I shared my problem with a fellow teacher. Here’s some of that conversation, which I’d like to share because it caused a substantial shift in my perspective [emphasis mine]:
- As much as possible, allow enough time so that every group can finish the task (give fewer, but meatier problems), and check in often with groups that seem to be making very slow progress.
- Check in with every group at least once. If someone appears to be working by themselves, bring them into the conversation – “Could you explain why you did this?” “What would you do here?”
- Enforce one marker per group. If a student appears to be hogging the marker, ask them hand it off to someone else.
- Always have extension and reflection questions ready for groups that are done quickly. Some go-to questions: “Can you solve it another way? E.g., without using method X?” and “I want you to talk about how you can explain this strategy clearly to someone who did it another way.”
- When you notice an interesting or efficient strategy, stop the action and have the group explain to everyone what they did. In some cases, it may be more appropriate to get two groups to share and compare strategies.
- During whole-group discussion, ask students to compare their strategy with that of another group. “What’s similar? What’s different? Why do you think they did X? How is X represented in your strategy?”
Many of these are well-established strategies for supporting collaborative learning in the classroom, and they’re not new to me, either – but I know that I had been letting some of these slip as I focused on the norms that my students weren’t following.
Today, I started class by saying that our goal for the lesson was twofold: To do mathematics, and to work well together. I had two problems prepared for the entire class, which gave the students and myself a lot of room to stretch our feet (see point 1). The problems were not elaborate: pulled straight from the textbook, they asked students to find all of the missing sides and all of the missing angles for a given triangle. Here’s the first:
There was, however, a catch: Each group was given one constraint among the following:
- Use only sin.
- Use only cos.
- Use only tan.
- Don’t use the Pythagorean theorem.
- Don’t use the fact that the angles in a triangle add up to 180 degrees.
(Nat Banting has referred to such restrictions in similar tasks as enabling constraints, a notion developed by Davis, Sumara, and Luce-Kapler: “‘The common feature of enabling constraints is that they are not prescriptive. They don’t dictate what must be done. Rather, they are expansive, indicating what might be done, in part by indicating what’s not allowed’ (Davis, Sumara, & Luce-Kapler, 2015, p. 219). By restricting what can be done, action orients itself to the possible.” The whole post is definitely worth a read.)
You may have noticed that some of these are impossible. In my discussions with groups, some admitted that they “had to cheat,” which led to great discussions about which trigonometric “tools” can be used in a given situation. As soon as a group finished and I was satisfied that everyone in the group understood, I asked them to check their work by solving the triangle given another constraint (see point 4). Finally, I asked individual groups to find the “easiest” strategy possible.
I circulated around the room, making a point of checking in with every group and bringing students into the conversation when they seemed to be disengaged (see points 2 and 3). At one point, I stopped the action (see point 5) when a group told me that the task they had been given was impossible, and gave a very clear explanation for why. Even better, it turned out (and I hadn’t noticed) that they had made a mistake in calculating one of the sides, and other groups had evidently done the same, because two answers were put forth with confidence by the class. This led to a great, spontaneous whole-group troubleshooting session, which helped to reinforce the fact that the sides in a trigonometric ratio depend on the “vantage point” (angle) you take. I swear – and this makes my heart leap – I haven’t seen so many hands shoot up to share in a long time, and several students even volunteered to go up to the board. We resubmerged, and soon after I set students to work on a different problem (see below) with the same constraints.
Near the end of class, we regrouped again for a brief discussion – notably, not simply a show-and-tell (prompts: “Which constraint did you hate the most? Why?” “Which one do you think was easiest?” “What would you do on a test, if you were pressed for time?”). Finally, I gave students an exit slip – but not as a way (as I had originally envisioned) to test their attention; they were to solve a triangle using any strategy they wished, then to verify using another. Again, my heart leaped when I was marking (!), because 30 out of 32 students destroyed the problem, and the other 2 were well on their way.
Now, I know that this class may have been an anomaly; at least some regression to the mean is likely to follow. I also still have a long way to go in supporting productive collaboration in my math classroom. Nonetheless, this was an incredible learning opportunity for me, demonstrating in a powerful way – in addition to the power of teacher collaboration – that turning a problem upside down and changing your perspective can make all the difference, perhaps revealing that the “problem” in question is really a symptom of a bigger issue.