So you’re thinking in terms of physical models now instead of in terms of chapters, units, or equations. Now what? Is there a high-level ordering of these ideas? How are they connected? Are there common themes?
One activity that I’ve done to help students build higher-level organization into their thinking is concept maps (which I got from Matt Greenwolfe, and which I hope to soon write a post outlining—so hopefully there will be a link here eventually). Concept maps have students put a structure to how the models they’ve built interact (or don’t interact) with each other. They are sort of a system schema for models. But in addition to that work (which we tend to do about twice per year), we also frequently talk about the biggest ideas we have as we go through physics (our fundamental principles).
At the end of my mechanics materials, I’ve long had a unit that returned to momentum and energy to pick up the idea of elastic and inelastic collisions and to try out harder problems that combined the big ideas in mechanics. It’s gone through a lot of names (like Conservation of Momentum, Part 2 and Momentum Transfer Energy Transfer (aka MTET)) and is now drifting toward the name 3 Fundamental Principles: Problem Solving. Here’s this year’s packet cover:
We’ve had three really big ideas this year. Three basic ways that we’ve viewed everything. We’ve been referring to them as fundamental principles. Can you tell me one of them? (Newton’s Laws.) Great. [I write it on the board.] How about another one? [It might take a little talking back and forth, but eventually we get] (Conservation of Energy, Conservation of Momentum.)
Cool. Okay, in a moment, not yet, let’s break into three teams. Let’s say… Newton’s Laws in that back corner, Energy in the other back corner, and Momentum somewhere around the front here. Make sure there are people on each team. Grab a whiteboard for the team and put together a board of the different representations for the principle. Work right on the whiteboard first, because your ideas might change as you go along.
Okay, ready set go. [They organize themselves pretty quickly, and I start cycling around from group to group asking questions and nudging them a bit here and there.]
Newton’s 2nd Law
If you had to pick one of Newton’s Laws to be the banner one, or the one most useful for solving problems, which would it be? (The 2nd Law.)
[The toughest part for every board is getting them to write the principle, not write about the principle or write a title for the principle in the verbal representation. This part takes the most nudging as I cycle through, and the conservation laws come together more easily than N2L. The class whose board I put below actually had the easiest time of putting a verbal statement together, opting to state it as Newton's 1st Law, which worked really well.]
[The big idea graphically is the connection between the slope on a velocity-time graph and the unbalancedness of an FBD or force vector addition diagram. (I wish my students had put more labels on their diagrams here, but I think the intent was to leave them looking generic.)]
[The students usually identify this principle as Newton's Laws collectively, but when it comes up, we talk about how it's really the 2nd Law (which includes the 1st). And how the 3rd Law is really part of the momentum principle.]
Models of N2L:
- Kinematics (Constant Velocity (CVPM), Constant Acceleration (CAPM))
- Balanced and Unbalanced Forces (BFPM, UBFPM)
- Special cases of Unbalanced Forces: Projectile Motion (PMPM), Uniform Circular Motion (CFPM), Universal Gravitation (CFPM), Simple Harmonic Motion (OPM)
Conservation of Momentum
So what’s the big idea? (The total momentum stays the same.) Always? (Unless there’s an unbalanced force.) Any force? (Unbalanced outside force.) That sounds good.
One thing I notice—your equation doesn’t seem to say the same thing as your verbal description.
[The verbal description for each class came together most easily for momentum transfer. The toughest part was instead the equation. In each class, the group started with p = mv for their mathematical representation. After pointing out the inconsistency, they easily moved toward a statement of total initial plus change = total final. Their graphical description (IFF charts, or momentum bar charts with a Force-time graph in the middle to show the ∆p) followed along easily. I liked how this group put the F-t graph in parentheses to show that you only need to draw it if there is a change in total momentum.]
Models of Momentum:
- Momentum Transfer Model (MTM)
Conservation of Energy
What do you have for your verbal description so far? (Energy can neither be created nor destroyed.) So are you saying the total energy of a system never changes? (Right. No, wait. Unless there’s work.) Okay, so that probably needs to be part of the statement, too. You need to find a way to include that “unless”.
[I found it really interesting that the statement "energy can neither be created nor destroyed" showed up in each class, even though it wasn't something that we ever said in class nor something that I ever heard them say often when they talked about energy before. It was a good reminder of how what they've learned before still exists alongside what they are learning now and that they really want to find ways to connect those ideas (and especially to honor their oldest ideas).]
[This group also chose to include very generic diagrams, and I loved that they included the F-∆x graph for finding the work done on the system.]
[I had a good conversation with many groups about choosing just one mathematical representation (the one that communicated the big idea, even though there might be other mathematical components involved, like the formulas for each flavor of energy or each type of force). The group from this class even noted that conversation on their board when they identified the most inclusive statement they could make about energy.]
Models of Energy:
- Energy Transfer Model (ETM)
- Universal Gravitation (escape velocity, etc) (CFPM)
- Simple Harmonic Motion (OPM)
Once the groups had their boards together, we regrouped and put the three boards up front. Students basically stayed in their seats and one person from each group spoke a little about the choices they made. Students wrote down what they wanted from the boards on that front page of the packet. Nothing was new here except the way we were writing it all together or organizing it on paper, so the questions and anxiety of starting something new were pretty minimal.
If students didn’t bring it up themselves, I pointed out that all three of our fundamental principles are pretty similar—each is about how some quantity stays them same unless there are unbalanced outside forces. So what we really have is three sets of glasses for viewing, describing, explaining, and predicting change.
Now, when we start solving tougher problems and/or problems that require more than one big idea, we are ready to add a new mantra to our list of problem solving advice: if you’re getting stuck, switch fundamental principles.