One of the reasons why I dislike teaching “algebra-based” physics in a fast pace manner is that for many students algebra is barely in their grasp. This becomes most apparent when we first starting learning about how to solve problems related to Newton’s 2nd Law.
For example, today students were solving a problem, where a 4 kg is being pushed with a 30N. The coefficients of static and kinetic friction are 0.5 and 0.3 respectively. The question is, does the block move, and if so, with what acceleration?
Students all correctly identify that the maximum static friction force is 20N (based on normal force being equal to weight), and so 30N is enough to be in the kinetic regime. They can then calculate the kinetic friction force as 12N. Many of these students can readily say that 18N is the net force acting on the object. Then most can at this point say that a = Fnet / m = 4.5 m/s/s.
Students’ productive and natural problem-solving approach does not look like what we teach them do:
F_net = ma
F – f_k = ma
F – u F_n = ma
F – u mg = ma
a = 1/m (F-umg)
Students of course just simply do this in “chunks”, calculating pieces along the way, comparing values, doing arithmetic calculations. The formal approaches codify these chunks into the algebra and string them together. Logically that’s true, btu students do not experience it that way.
I think that teaching the formal approaches has the promise of being generally powerful (that’s the allure to teach it this way), but in reality it mostly disenfranchises students from their natural problem-solving skills, their ability to apply insight into problems, and to make sense of what they are doing.
What compounds the negative effects of teaching abstract algebra approaches is that it has quite varied effects.
- I do have some strong algebra students and sense-makers who can apply the formal approaches and make sense of them as they do.
- I also have some good “algorithm” followers who can apply formal approaches, but are not making sense of them. They follow the script.
- I also have strong sense-makers who will approach problem from their own intuitive approach, basically ignoring the approaches you are trying to teach them.
- I also have students who cannot follow the algorithms, and feeling disenfranchised from anything resembling sense-making, they cobble together strange algebra.
Having one or two of these different types seems manageable to me, but having all four I find hard to navigate.
Teach. Brian. Teach. 
Thanks for the taxonomy of student approaches as well as well as weaknesses of the standard algorithms. My students tend to either follow the script without making sense of what it means, or feel disenfranchised by the script. I often get stuck there; particularly, with how to respond to the students are are angry that their script-following is no longer earning them preferential treatment from the teacher.
Any suggestions for how to build an orientation toward sense-making, especially among the students who have the proficiency to solve problems without it?
Related question: I sometimes find myself at odds with colleagues teaching co-requisite courses, who find formal approaches to be powerful, beautiful, and as you say highly generalizable, sense-making techniques. The classic example in my situation is loop analysis for solving circuits. The instructor wants to share this power with the students, but don’t seem to notice that for a student who doesn’t have a feel for the sense behind algebra, it can’t be a sense-making tool for learning other things. Something else has to be a sense-making tool for *it*.
I’ve so far failed to connect with my colleagues about this. Any suggestion for having this conversation in the staff room?
I don’t have any universal suggestions for cultivating sense-making (wouldn’t it be nice if I had an algorithm to simply follow!). My experience so far tells me that each student must come to have their own positive emotional experience(s) with sense-making to start to valuing it and to become oriented to it. I can’t logic them to it or drag them to it. All I can do is create an environment where they may are more likely to accidentally start sense-making, and for some reason feel good about it. When that happens I can help point it out. For some students, it seems like one powerful experience can change them. For others, that doesn’t seem to be the case. For some students a few missteps by me can undo progress we’ve but others can withstand my missteps. I certainly have my share or successes and failures with students.
My advice on colleagues is weird, but here’s been my story: I tried very hard when I got here to seek out teaching advice from colleagues on things I didn’t know about or didn’t have much experience. I would try out things they suggested as often as I could, and even if a piece of their advice worked well I made sure to tell them about it, but then also to add WHY I thought it worked well–slipping in my “thoughts about teaching” about why their great teaching tactic worked. Second, when things didn’t work, I went back to them and said curiously, “I tried that thing, and it didn’t go so well, can you tell me more about how you __? What did I do wrong?” Of course, I didn’t do this all the time, but treating them as experts on teaching whose advice I valued opened up the doorway for more challenging conversations. The second thing I tried to do was NOT talk about my teaching, but to showcase something great my students did. “Hey, you have to take a look at this diagram my students made”. I know that’s not the advice you were looking for exactly, but it’s the only advice I have to offer
Hah. Nope, that’s actually very helpful. I don’t like being dragged to a new approach, my students don’t like it… there’s no reason to imagine that it would work in the staff room either. Thanks for the thoughts.