This post is primarily for college teachers, but K-12 educators might get something from it too.
This researcher (Tom Bing… here’s his dissertation) has done some very interesting work on the paths that students take as they solve physics problem. He gave one particular example of students who, as they worked a problem, lost sight of the goal of hte problem. They used some very high level mathematical skills and computer programming, but that wasn’t particularly helpful in light of the goal of solving the particular problem. He interprets this in terms of how they thought about what this problem was about.
After all, when we start to solve a physics problem, the first task is to decide what sort of work we have to do to get to the answer. He calls this Framing. Essentially, the frame is the answer to the question, “what is this activity about?”‘ In particular, he’s interested in what kind of math they bring to the fore when trying to go about solving the problem once they’ve framed it. He calls this epistemic. So, he works on epistemic framing.
I’m struck by the idea of framing in this context since there is a fairly controversial science blog called Framing Science which specifically urges scientists to “frame” their research in order to (sort of) sell it to the public and the media. Bing’s work is a completely different context, but this idea of a “frame” is, I think a very compelling one, and one that scientists don’t tend to think about. It’s sort of a journalistic or social science kind of way of thinking about something. How something is framed doesn’t define the content of the item in question (be it scientific research, a math problem, or a conversation) but it does define how that content is interpreted. It puts the activity in question in a particular context, and gives it a particular color or light. I believe this is an overlooked part of education. Bing suggests that someone’s interpretation of a situation affects their actions. For instance, if you walk into a library, that triggers your whole idea of what a library is and what you do there. You have certain expectations of what you’ll find and what sorts of activities you’ll engage in there. It’s the same way in a physics class.
For Physics students, he says, a particular part of what they know about physics is sparked depending on how they understand the problem and what it’s asking. He looks at what they argue, mathematically, in order to probe what their frame is.
In the example he gave, a set of students were trying to solve the expectation value for the problem of the square well in quantum mechanics. They do some sophisticated calculations, but they frame the problem as a calculation problem. They sort of lose sight of what they are doing and even when their calculation doesn’t work, they keep working on it, and don’t go back to the underlying physics. In this case, they mistakenly plug in the limits of integration as going from negative infinity to infinity, thus getting an integral that diverges. Rather than go back to the physics problem to recognize that one only needs to integrate over the width of the well (from 0 to L, because the wave function is zero outside of the well), they continue to use Mathematica and other calculational tools to try to solve the integral in different ways. They are stuck in the frame of “this is a calculational problem.”
Then, one student realizes that the limits should only be over the width of the well. The other students laugh as they realize their mistake and laugh at themselves. There is a complete change in tone in the group, and one student who wasn’t listening to the conversation clues in that something has changed, just by the very tenor of the discussion, even though he hasn’t heard what they have said. Bing notes that this change shows a shift in their thinking and in how they are framing the problem.