One of the changes our new curriculum requires of instructors is a shifted vision of what sophisticated problem-solving can look like.
I’m not sharing this particular photo of student work necessarily because it’s exemplary. Rather, it’s a good boundary case– I see evidence for sophisticated problem-solving and I also see what are often seen as traditional markers of immature problem solving.
I notice that students write the equation for gravitational potential energy as U = gmh. This is non-standard and so stands out to me. When they use this equation, they include units, and their final expression has units appropriate for energy.
I notice that the students do not write an explicit energy conservation equation. Rather they bracket the two gravitational potential energies and find the difference.
It suggests to me that they know that energy differences matter, and that perhaps they are thinking in terms of energy transformation (and perhaps less so explicitly as energy as a constant).
They set this energy difference equal to an expression for kinetic energy. They drop some units during the mathematical work, but include final correct units upon determining the speed.
They later use this speed in an equation to calculate the net force, this too includes units, some that look like they were squeezed in afterward.
This net force value makes its way to an arrow next to a free body diagram. The free body diagram is drawn, with Tension force drawn longer than weight. They then make use of known value for weight and net force to calculate the Tension.
As with energy having no explicit algebraic statement of conservation, the students write no explicit algebraic statement for Newton’s 2nd Law or for the sum of forces.
Traditional markers of sophisticated solutions value explicit algebraic statements of big ideas– energy conservation, Newton’s 2nd Law. We do not see that here. What is also valued at times is prowess at algebraic manipulation. Here we see calculations done piece meal.
Students use equations to calculate intermediate values. How many joules? How fast? What net force? None of these intermediate calculations are big ideas: “potential energy”, “kinetic energy”, and “centripetal force.”
Big ideas are instantiated arithmetically–considering a difference in potential energies to determine a quantity of kinetic energy. Arithmetic reasoning about relationship between individual forces and net force.
One of things I’ve come around to seeing in students’ work is this– I look for evidence that they are organizing their work around the big ideas.
The traditional view mostly looks for that evidence in the limited places, especially for the new learner, and thus often misses sophistication when it appears. And inadvertently, such limited looking can end up encouraging the opposite of one intends. Mindless equation use.
One of the ways to see students work here not as mindless equation Work is this. is the following. It is true that Students do not seem to use equations to express big ideas. Rather, I would suggest that students use equations as a means to get into the world of big ideas. As such, we see that they know how to reason about concepts like forces and energy, and are adept at enacting such reasoning when they have concrete values with which to reason. They sometimes use representations like the FBD to help organize how to do that arithmetic thinking. The equations are a tool that gets them a concrete handle in to the world thinking about forces and energy.
I can see this also in how they use or don’t use equations that are old and more familiar vs new and unfamiliar.
Students actually don’t even write an equation for relating mass and weight, like W = mg. Rather, they just write m = 200 g, and W = 2N. This unit prefix change and calculation is familiar to them , since they learned it months ago. I see their fluency with this and presumed fluency of others as making sense with them not showing this explicitly.
Students write the energy expression for potential energy in their own way, with the “g” as the leading variable. This equation helps remind them of what information is needed and how to put it together. Energy was learned weeks ago, and so has undergone some revisions. They use this equation fluidly with units as they calculate, even converting length prefixes from cm to m without much ado.
Circular motion, however, is our most recent topic and the equation for centripetal force that they write takes on the exact form it was presented to them. They plug numbers in first and go back and add the units later. This makes sense to me with their having less familiarity. It’s like right now This an equation that is strictly for a calculation process, one they have not yet internalized. Yet the result of that process (net force) they seem to what it tells them and how to proceed with that information.
Part of this could be that I’m not pressing students to work at a sufficient level of abstraction. It could be that I’m allowing them (too safely) to work concretely with these big ideas. As I see it, I’m getting them to actually learn the big ideas using skills with which they can actually be thinking about the big ideas. My students can do mathematical sense making, but it more likely to take place with concrete values. Often for new learners, the push for algebraic abstraction suppresses thinking about the big ideas. And so I’m somewhat happy with the balance, but I know that it is also true that I should be looking for fruitful ways to stretch that understanding into uncomfortable territory. We do explore that boundary some, but probably not enough for certain populations of student who need that.
Anyway, those are my thoughts for the evening.