Years ago, I was listening in on a group of college physics students who were working on some fairly standard torque-balancing problems. They had been given situations like the one below and they had to decide whether the situations were balanced or not.
The group had an interesting strategy that I call the “equal exchange” strategy. For example, students would take the “two blocks at the 1-notch” and replace it with “one block at the 2-notch”, because that was an equal exchange. For this situation, the strategy quickly reveals the answer, because each side now has “2 blocks and the 2-notch”, as shown below.
In working with the other graduate TAs and the professor running the prep session the week before, no one had used or even mentioned this strategy, neither as a strategy they would use or that students might use. All of us simply summed and compared the torques, by writing out 2*2 = 2*1+1*2. And we did the same for nearly every situation.
For me as a novice teacher, I was intrigued by what the students were doing. To me, it was thrilling to witness these students, all by themselves, inventing a novel way to solve the problem that I had never considered. Part of this thrill was that students were doing something different from me, but a large part of the thrill was wrapped up in me knowing that what they were doing was valid, despite being different.
Here’s something interesting to think about. For me, the physics knowledge I had to use to evaluate the validity of the students’ strategy was, in some ways, special to the task of my teaching, because it wasn’t the same physics knowledge I used to solve the problem. I (along with all the physics graduate TAs) summed the torques in order to compare the net torque numerically. The students’ strategy involved getting the situations to be visually comparable. The fact that I could see our strategies as being related and both valid is a kind of content knowledge that I needed to adequately assess what the students were doing.
Of course, some of the problems students had to work on were much harder than the situation above. So, the strategy to get all the blocks in one place can get a lot more complicated. Take for example, this situation:
In this situation, the number of moves not only goes up, but you have to do some more daunting proportional reasoning. As these students got to ever more complicated situations, the students were taking a lot longer than the other groups, and making more mistakes.
The question, for me as a teacher then, was, “At what point, if ever, should I step in to help them to discover other, perhaps more efficient, strategies?”
First, it’s helpful to reflect on some things. First, recognizing why their strategy was becoming increasingly difficult required that I have a particular mastery of the physics content and the physics reasoning. Recall that to solve the problem myself, I didn’t need to consider proportional reasoning or multi-step problem solving, because I just had to sum the torques. But now in this moment, in order to assess students’ progress moving forward, I had to be able to think about the physics concepts and problem-solving strategies in a particular way that was different than before. I had to be able to project the students’ problem-solving strategy into the future and into different problems in and make hypotheses about where it might lead them.
As a teacher, I could have chosen to engage students in developing their strategy, by helping them to be careful with proportional reasoning or with planning out more effective moves; or I could have chosen to nudge them toward my more efficient strategy. Given different goals and constraints, there is no right answer about what to do. But, for a me to make an informed decision, I had to be in the position of listening and making sense of what the students were doing. In order to be in that position, I had to have a unique mastery of the physics content and reasoning that, I’d argue, went well beyond being able to solve the problem myself.
Seeing other Connections
Looking back on this moment, other question for me as a teacher are these: “What does their strategy imply about what they are likely understanding well? What does this strategy imply about what students might not yet understand?”
To me, the students strategy shows me that they are likely making sense of Torque as Mass x Distance. They understand that idea well enough to know that there are variety of ways to get an equal torque by changing the mass and distance. In particular, most of their reasoning fell along the lines of, “if you triple the mass, you better third the distance.”
But their strategy also hints that may not be having the opportunity to develop other important ideas. For example, they might not be learning that torques are summative (e.g., Net Torque = Sum of Individual Torques). If it’s important for students to learn this, a goal could be for me to make sure that this group is provided with an opportunity to learn that idea as well. It’s not just a matter of them learning a more efficient strategy, it’s about the opportunity to make contact with important physics ideas that they might not using their strategy alone.The Big Picture
This example highlights for me that the role of content knowledge in teaching is wide and varied. The content knowledge I mention here is often referred to as specialized content knowledge. It's the content knowledge needed to evaluate a student solution that you may have never seen or thought about before. It's the content knowledge needed to project a problem-solving strategy into the future. It's the content knowledge needed to relate problem solving strategies with important conceptual knowledge. The reasons why this is content knowledge is that it need not have anything to do with students. An expert could have proposed these strategies in a journal of physics, and it could then be my job to evaluate the validity of that approach, or to see how that strategy would play out in a variety of situations, or to see what concepts are embedded within that approach. Some of that content knowledge is, in some ways, unique to teachers and teaching; because the range and variety of alternative solutions that teachers face are unique due to the fact that they are dealing with students. Thus, some of the content knowledge that teachers need to evaluate those solutions is unique to their tasks of teaching.
A big question for researchers is, "What kinds of content knowledge do teachers need for teaching? And where do teachers develop that knowledge?"
For me, I have developed a lot of that content knowledge by paying attention to students, by listening and reflecting on what they are doing. And I have further honed this knowledge by actively seeking out and reflecting on potential connections among what students are doing and the disciplinary knowledge and skills of physics. To be sure, I will continue to develop and refine this knowledge as I continue to teach in ways that allow me to listen and reflect on what students are doing. For this reason, how I arrange my classroom teaching in ways that allow me to listen to students is extremely important.
I hope this helps other to understand my concern of misconceptions listening
, in that it provides less opportunities for teachers to develop the knowledge that furthers their teaching along.