Last year, when we were going through the first year of piloting new curriculum, I lamented about the struggles students had with the algorithms we were (and often do) teach to solve Newton’s 2nd Law problems, and was wondering about non-standard algorithms, like these. Here’s how I’ve avoided the pitfall this year:
1. Qualitative Representations and Reasoning Galore (before getting Quantitative): In kinematics, students always had to draw qualitative position vs time and velocity vs. time graphs before writing down any equations, and I almost never let students use an equation that didn’t describe a graph that they had drawn. In addition, I was a stickler for drawing good motion diagrams (velocity vectors and acceleration vectors drawn using the appropriate color coding scheme). This helped to frame class as primarily about sense-making qualitatively. We spent a lot of time making qualitative predictions of graphs and testing against motion detector data, and then practiced using those graphs to solve problems. This was present in all activities, labs, clicker questions, and problems. This has carried over to forces quite naturally.
2. Slower Pace in Forces: Last year, there was too big jump from “what are forces” and “N2nd Law” to solving complex problems involving statics/dynamics all with vector components to worry about. With that escalation of complexity, we fell back on “monkey see, monkey do”, which undermined students’ natural problem-solving sense. This semester, so far, we have only done force problems where students have had to worry about either horizontal forces or vertical forces– not components. Last year we jumped right into hard problems…. But this slowed pace has meant that I have not yet worked an example problem for students to see an expert solving problems with forces, and students are, I think, showing a lot of competency to solve challenging 1D problems involving Newton’s 2nd Law. Instead of doing example problems, after our intro to forces, we did a lot of FBD clicker questions and practice, Net Force Ranking tasks, acceleration ranking tasks. And I did model and scaffold expectations for doing a good job of representing the forces. But mostly I just equipped them with skills for representing clearly, and for thinking about what net force is, and how it is related to the individual forces.
3. Forces Always with Kinematics: So far, I have refused to divorce solving force problems from motion considerations. Their first force questions began with “Given these forces”, what distance and speed will the object achieve, and “Given this motion detector data, what can we say about the unknown force?” and finally, “Given this force vs. time data for the elevator”, what can you say about the mass, speed, and distance of the object. I can’t describe how important this has been to (1) the class feeling continuous in our course of study, and (2) in helping students’ developing confidence. The problems students are solving a hard (not because they are hard forces problems), but hard because they require students to apply newly learned knowledge and integrate that with skills they have been mastering over the past 5 weeks.
A Caveat about what my students can and can’t do: If an instructor in my department saw my students’ work, they would probably be impressed by my students’ ability to represent the motion and forces of some of these complex scenarios, but they would likely be scared at the lack of “clean neat algebra work” involving Newton’s 2nd Law. They would see my students as “hodge-podging” their way through through the arithmetic. And it’s kind of true, but I see this as productively grappling with thinking about net force and how individual forces come together to create a net force, and how net force causes an acceleration. And It doesn’t mean my students don’t struggle with the math– they have struggled with deciding whether to add or subtract some of the force values they are having to think about. For example, a group last week, had subtracted two force numbers (that they actually needed to add). When they got their answer for the unknown force, they recalculated the net force, and what acceleration that would cause, and found that it didn’t match the acceleration given in the problem. They did this checking without prompting, and they called me over for help, because they had an approach that they though should work, but knew that something was amiss. To me, that’s what productive struggle looks like, and students in that group were really primed to start talking with me about what they did and what didn’t make sense. Someone could argue that my students would not have made such a silly add/subtract error if I had just taught them to write Fnet = T1 – T2. But I think they would have gotten the right answer without ever having thought about it, or if they had made an error, they never would have stopped to think about whether it was sensible.
So what now? Wait until we need a better algorithm.
Right now, My goal is to not introduce /model an algebraic algorithm until my students come across a problem that breaks them. My guess is that it’s going to be either statics problems with non-orthogonal forces, or interacting systems (where systems of equations are needed). I’m not really sure when this sort of arithmetic, step-by-step, sense-making approach will break down, but I’m pretty sure that it will break naturally, or that at least I can put us in situations that it’s increasingly likely to break down. So I’m happy Right now for them to be gaining confidence, learning to think about Newton’s 2nd Law, and solving problems through the process of representing and thinking (rather than following an algorithm). At some point, I figure I am going to model solving a problem in a manner my colleagues would recognize as legitimate, but it will only be after we encounter a problem that really demands it.