I’ve learned a lot last year and this year about how my students solve problems that involve Newton’s 2nd Law. I’ve paid attention to two different aspects of their work:

- their difficulties in understanding and carrying out standard algorithms
- nonstandard algorithms that students employ (spontaneously)

A lot of what I’ve learned centers around the fact that typical instruction on solving Newton’s 2nd Law problems ignores students’ intuitive problem-solving approaches to these types of problems.

**Conceptual Understanding Research:** **Identifying Difficulties and Productive Seeds**

We know from research on student difficulties with conceptual understanding in physics that effective instruction needs to be designed with students’ initial conceptions in mind. We have also progressed as a community that not only focuses on the need to address “misconceptions” but how to build on and even from students’ productive ideas (e.g., anchoring intuitions).

**Problem-solving Research: We know the difficulties but what about the productive seeds?**

I’m not super familiar with all the research on problem-solving in physics. Some of the work I do know focuses on how novices are different than experts, and likely promotes a deficit model of students as problem-solvers. I also know there are some other frameworks for being more descriptive about students’ problem-solving, but this often happens at “general level“, rather than focusing on problem-solving in a particular topic area. And despite being descriptive, such studies still often focused primarily on using the description framework to draw attention to specific difficulties. Which is fine, we need to know difficulties. And, I know there are exceptions to this, where problem-solving in specific areas is the focus, but even then the focus can, at times, be more about the development of general theory.

One exception, that I have really enjoyed for some time is Andrew Heckler’s research on the consequences of prompting students’ to draw free body diagrams. This research now makes even more sense to me in that it does two things well–it shows the unintended consequences that occur when students interact with aspects of the standard algorithm but it show cases non-standard approaches to solving problems (often correctly). The question that has been on mind for a long time, but is becoming more into focus now, is something like:

**Problem-solving Research and Instruction: Call for Supplementary Approach?**

How can problem-solving instruction be more attentive to students’ “intuitive” problem-solving approaches? What does it look like to do this in ways that are not solely focused on deficits students bring? What does it look like to this in ways that are (at least partially) attentive at the level of specific content areas / topics? In other words, what would it look like to build on and from students’ initial approaches, rather than supplant them with our own algorithms. I’m not naively saying, ‘students are already great problem-solvers, they don’t need any help’. But I am saying that students do bring a lot of good stuff to task of solving (force problems specifically), and that more research on what that good stuff is and how it can be used in instruction is needed.

**Where this kinds of approach may have some footholds?**

And I’m not naive enough to think that lots of teachers in the trenches don’t already do this (attend closely to students’ problem-solving and build on it), or that some problem-solving approaches haven’t been developed with the learner more in mind. I think a lot of the work in the Modeling Instruction around interaction diagrams, LOL charts, actually were designed exactly with students in mind. I’ve even written about how these approaches actually turn students misconceptions into correct insights. Kelly O’Shea has been running great workshops on using graphical representations to solve kinematics and forces problems, which can really empower students and move them away from the “plug and chug”. My approaching to teaching problem-solving has been tremendously influenced by all of this, and the research has for a long time promoted the multiple representation approach.

I do love these alternative pedagogical approaches that make use of multiple-representations. My sense is these approaches were developed from #1 rich insight and practitioner knowledge on student difficulties in specific content areas, #2 a commitment to crafting alternative algorithms that emphasized big ideas and relationships (rather than equations), and #3 iterative cycles of use and revision across many different teacher’s classrooms to hone-in effective models. I have personally witnessed the demoralizing aspects of teaching standard algorithms, and the empowering aspects of effective teachers using these pedagogically-focused algorithms.

So what am I saying is missing? I’m still not exactly sure. And I still think that what I ever I think is “missing” is not likely missing from the practice of the teachers I admire so much. It certainly must be the case that teachers who begin using pedagogically-focused algorithms in their classes begin to notice new things about students’ problem-solving, and to notice that students have ways of making sense of problem-solving that are really insightful (and different that one might one expect)… and they begin to adapt their instruction to build on those student-generated approaches.

**Forces Problem-Solving as an Example**

So what’s an example of a student approach that is “common” enough and “productive” enough to serve as an anchor for instruction. I’ve basically written about this specific thing before, and in the cases I had seen then students are still using the standard algorithm (sort of). Now I know these approaches (or even deviations from standard) are pretty common, and that you can see the effect that supporting students in using them has.

Here are a few examples of conversations with students this semester:

- I was working with a student through a standard algorithm. What was really confusing them was how they get the “total tension” (i.e., magnitude) from an the y-component equation of Newton’s 2nd Law. In their mind, the y-component should have given them the y-component of the tension. See in the standard algorithm, you often substitute Ty = T sin(theta), so that you do solve for the magnitude of T. In talking with these students, we ended up back-tracking in the algorithm to solve for Ty first, to actually get a numerical value for Ty. Then, go about using trig to figure out what the total magnitude needed to be to guarantee that Ty would have that value. That process made a lot more sense, but it also gave the students insight into the problem. It was a static equilibrium problem, where Ty was holding all the weight. That the student could “see” a value for Ty and that Ty was holding the weight made a lot of sense. They carried this sense-making with them to many other problems involving equilibrium… I see this as similar to the “two-step” process talked about in the prompting force diagrams paper. The experts algorithm tries to take advantage of every relationships simultaneously, along the way bypassing important insights that students can glean in the process.
- Another group was working problems, back and forth between more standard algorithms and less standard ones, mostly depending on who in their group was taking the lead. In one case, when they worked a standard algorithm, they got the right answer, but were confused about some things. In talking with them, it was clear that they could not see the chunk “T sin(theta)” as a component of a force. To them it was just a bunch of symbols. Like they knew that’s how they calculated it, but their brain couldn’t just look at the equation T sin(theta) – W = 0, and see this as saying the y-component of tension is equal to the weight. This is a bit of why the standard algorithm doesn’t work for students. I see this as related to “disciplined perception“, but also the idea of chunking in long term memory. But it also made me think about how I can scaffold their seeing better. If they are to become experts in the standard algorithm, it’s not enough to use sine and cosine, one needs to see particularly arrangements of algebra as “chunks”.
- Where I saw my helping students build on these approaches pay off was when we were working problems involving dragging a block across a rough surface at at angle. We had a clicker question like this: A block is on a horizontal rough surface, such that it takes a horizontal force F to break static friction. The question was if you now pull with same force at angle will be “more effective”, “less effective”, “similarly effective” in getting the block to budge. We talked about this for a while, and got a lot of the ideas on the table– by pulling up you are lessening the normal force, and thus lessening the grip that surface has… by pulling at an angle, less of your force is horizontal and available for doing the job of breaking it free … others saying that maybe this means it will be equal out… others were talking about how with vectors, we’ve seen the two sides don’t up to the magnitude… so that a 10 N force horizontal is just a 10 N force, but a 10N force at angle, could be like more than 10 N of force… like you could get maybe 9.5 N of horizontal force and 3 N of upward force. It was such a weird statement to say, ” A 10 N force can be more than 10 N”… but it made total sense within the context. We decided to work the problem the following way: 2 groups work 30 degrees, 2 groups work 45 degrees, and 2 groups work 60 degrees. Students worked the problem, mostly not with the standard algorithms, and there were some pretty amazing conversations going on, both within and across groups. Those great conversations, I’m pretty sure, were made possible because of their “solve for the components numerically” approach, rather than “solve for the magnitude directly” approach… It constantly guided their sense-making.

From this I think what I think is missing from the research base is stuff about this–like here are ways that students spontaneously solve physics problems successfully, and here’s how you can leverage those ways to really get students doing sense-making. Here’s the good stuff that they tend to do (and are more likely to understand) that you can build.