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
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
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).
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:
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? 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. 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.
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.
So, an activity I want to do is have students make “cards” for each of the types of forces.
The card will require some picture but also have information like this is example for weight.
Personality: reliable and down to earth
Special Powers: Can pull from long range
Stats: varies force depending on the mass of victim, using an invisible web (aka gravitational field) to pull with a strength of 9.8 N for every 1.0 kg.
favorite song is Tom Petty’s “free falling”.
Feel free to play along in the comments!
In intro physics, I’ve been teaching N2nd law problem without introducing any formal algebraic solutions. Instead, students have been asked to use multiple representations to organize their thinking and then to reason about net force, and Newton’s 2nd Law using their representations to help them think through their work.
On Friday, students were working through a set of of the challenges, and I had one group struggling with one of the static situations. It was a situation where a 1.0 kg object was partially supported from below via a force sensor and partially supported by a rubber band (suspended above). Students could read the bottom force sensor, and were asked to predict the reading on the top force sensor.
The students had correctly drawn a free-body diagram, but were having a hard time thinking about to go from there. I have an undergraduate physics major in the class, who must have ended up helping the students, because their work had become very an algebraic.
N + T – W = 0
T = W- N
And students got the right answer. When I came over, I recognized that it was likely the undergraduate student had helped them, and that it was likely that there solution didn’t really make sense to them.
So decided to asked them some questions, the first of which was about the net force. It was not obvious to most the students what the answer to that should be. However, one student, who doesn’t often speak up, finally said that the net force must be zero because the object wasn’t moving. I helped them to add that information to their free-body diagram, since in our class, we follow Knight’s procedure of drawing a separate Fnet vector next to your free-body diagram. I pointed to the individual forces on their diagram and added, “So that means, these three individual forces need to come together to act as if there was a total of zero force. This question is essentially asking us to figure out, what must each individual force be doing to end with a result of zero net force.”
I then asked the students about the value of each forces, and asked them add those values to their free-body diagram values as they figured them out, but I left them to work it out, and walked away. When I came back, I asked them to tell me about their new solution and whether it agreed with the first, and you could just immediately tell from tone of voice and body language (especially from one of the students) that their new solution made a whole lot more sense.
To help them summarize their work I asked them a sequence of questions
- Which forces did you determine from a direct measurement? (normal in this case)
- Which forces did you determine indirectly from knowledge you have about how specific forces behave? (weight in this case)
- Which force did you have to reason about? How did the diagram help you to organize your reasoning? (tension in this case)
I then helped them connect their second solution to their first solution. I probably could have asked more questions rather than telling. My main goal was to help them see two features of how they are related– how the direction of arrows related to the algebraic signs, how the net force being zero was related to the RHS of the equation, and how once they rearranged their equation, both solution methods concluded that subtracting the normal force from the weight force would give you the amount of force remaining for the tension to hold up.
So I have two this is my goal for this week: To progress toward algebraic solutions and to progress toward problems that non-orthogonal forces. I’m inclined to start with the vector stuff, and then later get more formal with the algebraic.
Here’s my thinking so far about how to get this started:
I’m going to set up thrre Situations where a 500 gram mass is in static equilibrium. The first one has only vertical forces. The second one has vertical and horizontal forces. The third one has one vertical force, one horizontal force, and one angled force.
I’ve set it up so that each of the tension forces can be read with a scale. I’ve set all the purely horizontal forces to be equal.
I might ask students to draw the FBD first without seeing those measurement values, but I’m not sure yet. Either way, I want students to compare and contrast these situations in terms of how the individual forces “work together” to hold the mass at rest.
I definitely want them to compare and contrast the first two situations. How are they similar? How are they different?
Then I want them to compare the 2nd situation with the 3rd situation. How are they similar how are they different?
I’m really curious how this will stir up students’ thinking, especially about the 3rd situation and its relationship to the 2nd. While we have done vector components (with projectiles), we haven’t yet done so with forces.
Somethings we might think about / question?
- In each case, 5N of vertical support force is needed. And it’s fairly obvious how the first two situations do that. How does the 3rd situation accomplish this
- In each case there is a net horizontal force of zero. This is trivial in the 1st, fairly straightforward in the second, but less obvious in the 3rd
- Students might try ask how the value of the angled force compares to the horizontal and vertical forces… they should note that they don’t add. Students might go to Pythagorean?
- I’m curious if students’ ideas will provide an opportunity to think something like… in the 2nd situation, does this mean we could replace two of the forces with a single force of the strength and angle of the 3rd?
- Will stuff about angles come up ? Will any of that link to trig? Or maybe it will take the form, how will changing the angle of the 3rd force effect the force reading.
- This 3rd situation is easily “seeable” as vector addition, and while we’ve done some of that, I’m doubtful that will come up, but maybe?
- One possibility is we start investigating empirically “How does the angle of the 3rd one effect force reading?”
- Another possibility is that students will have ideas about how this works that they want to test… like, “Will the reading on the angled force always be a Pythagorean result?”
- I’m curious if a good time will occur to introduce a situation where both force are angled (probably the symmetric case)… but I’m also wondering if I should include this as one of my examples…. like the 4th example.
- I’ll have to do some direct instruction, just given the pace of the course, and I think after discussion and some chance to observe more and/or test out ideas, I’ll move to formalizing these ideas under the umbrella of vectors and vector components. Probably some clicker questions and exercises, before some problem solving? Not sure
I have some time to think about this, since our next meeting is a test.
Today in Physics was an assortment of challenges related to mass, weight, and friction, ranging from easy to hard. None of the situations was inherently hard, but what makes them hard is seeing the situation for the problem you need to see it as. Students productively struggled, and along the way really strengthened our understanding of static friction, normal force, net force, and Newton’s 2nd Law.
- An object hangs from a scale. From a spring scale reading, determine unknown mass.
- A mass is suspended from above by a rubberband and below on a surface. The bottom surfaces has a force sensor reading that is shown to students. Students have to predict the force reading for the rubberband above and then check.
- A mass is pulled by rubber bands both from above and below. Again, students can see the force reading from below, and must predict scale reading above.
- An object with known mass rests on a rough surface. Students have a force probe and are asked to determine an estimate for coefficient of static friction. Then they are asked to put an unknown mass on top of the first one, and determine the mass of the unknown.
- Building on task 4, they are also asked to connect the original known mass to a rubber band that pulls up slightly on it (vertically, but not enough to lift it off the rough surface). They can read how much force the rubberband is exerting with a spring scale, and they have to predict how much force it will take to budge the object.
- A Half-Atwoods setup is shown. Students have a 1.0 kg block, moving along horizontal surface, with hanging mass being equal to 500 g. Students are asked to use a force sensor and a motion detector to determine a value for the kinetic friction force, and the coefficient of kinetic friction.
Students were encouraged to work all the stations, but had to turn in three carefully worked out problems: Station 2 or 3, and station 4 or 5, and station 6.
For each problem students had to draw a pictorial representation, which identified the object of interest, its boundaries, and the contact forces. They had to draw a free-body diagram that included a separate Fnet vector. They had to include the readings of any data they used. And then show the work they did to arrive at their predictions, and for the ones you could check, how they compared.
Before we did this, we had a short conversation about mass and weight, and reviewed big ideas from forces so far. Over the course of discussing, reviewing, and circling around during the challenges, I made good use of our “forces” and “teams” analogy.
A Fnet Analogy: This is going to drive the force ontology people crazy:
I’m finding it in class very useful to think of forces drawn on a FBD as showing what individual “players”, and the net force as what the “team accomplishes together”. We’ve been using this analogy to organize our problem-solving without algorithms.
During the review, I talked about we can extend this analogy since we have learned more about the individual players–their behavior and personalities. Weight, for example, is a player who always does the same thing, no matter what anyone else is doing. His job is to pull downward with a force of 9.8 Newtons for every kg. He doesn’t change what he is doing in response to what other team members are doing, or in response to what the motion is. Normal is quite different… the normal force is always adjusting what he is doing, depending on what other forces are doing. It’s useful to think of him as being lazy, he will pick up whatever slack is left over, but he will only do what he has to. Tension and normal are pretty similar in this way, except that one only pushes, the other only pulls, and when multiple tensions and normals are at play, they have to work out a compromise as to who does how much. Finally there is friction, and friction is not so much lazy as “reactive”… Static friction sits around waiting to react to any other player’s efforts to get things moving. But he is only so strong, and can be overpowered. Once static friction is overpowered, he can only rely on a weaker form of himself, kinetic friction, and he works do brings thing to a stop if he can.
I haven’t even started thinking about this beyond just asking, so join in!
Say you have a half-atwoods setup with friction on the horizontal object. You start with a light enough hanging vertical mass such that the tension is not large enough to break static friction. Everything stays put, and so the tension is equal to the weight of the hanging mass.
Let’s say you then keep adding mass until it breaks static friction. The moment you break static friction, two things change:
- The tension in the string lessens, and
- and we switch from static to kinetic.
What are the different possibilities of things that can happen? Under what circumstances do they happen?
“This is a new blog I’ve started for the purpose of reflecting about my experiences playing and learning in the family childcare that my wife runs in …”
I developed 6 static friction stations over the weekend for our introduction to friction in introductory physics. Most of the stations present students with two objects that need to be “budged”, and an accessible visible / tactile way of telling which one is harder to budge. They are asked to predict which one they think will be harder to budge, and to articulate their reasoning.
The first one (above) is 2 identical blocks wrapped on the bottom with sand paper of a different grit. Students tend to know that material should make a difference, but often select the wrong one. The rougher feeling sand paper is not grippier! [Makes me wonder why do we sometimes say that coefficient of friction has to do with the roughness of the surface? Rubber is quite grippy, but not one would describe it as rough. Some rocks surface feels quite rough but are pretty slippery.] I’ve been asking students afterwards to feel the two and tell me which one they think they would rather have on the bottom of their shoe and why?
The second one (above) has identical blocks placed with different surface area. It’s common for students to think that more area will allow it to be more grippy. Students are surprised that it doesn’t seem to make a difference. Later, we’ll address why that might be the case.
The third one (above) has objects of different mass, which students tend to get correct (although not necessarily for the right reason). Mass comes up in two other demos.
The fourth one (above) has two identical masses, one sitting on a surface, but the other identical mass is being pulled up slightly with a rubber band (thus lessening the normal force) Students tend to get this one correct, but getting them to articulate why they think so is a bit of work.
The fifth one compares barely lifting (vertically) to barely budging (horizontally across a grippy surface). It’s a 1.0 kg block, so it takes about ~10N to lift. So far, students mostly think it will take more than 10N. The reasoning I hear has been, to lift it vertically, only the weight is opposing you, the air isn’t offering any resistance… to move it horizontally, you have to contend with both weight and the friction. Students are confounded! After, I try to get students to shop through their minds of objects in their house that they think they could budge, but not lift.. it’s easy to come up with some. This definitely gets us in the stickiness of the different between mass and weight,
The sixth one has a heavy frictionless cart, and a light friction cart. Students pause to think here, quite a bit. In talking with students, it seems they are conflicted. What is most surprising to them is that any force gets the heavy cart to budge!… I’m going to have us watch the “million to one ” video from PSSC, after I think. Five and six together really get at tough ideas, not just about friction, but about mass/weight/N2nd Law, etc…
I’ve made a few changes to the setups from the pictures, including labels for the objects (A and B), and coloring that makes it easy to identify that one that has a friction pad.