We often think that naive conceptions of force arise, in part, to living in a frictionful world. And while I think that’s true, I also think it’s just at important to recognize that we basically never ever push or pull things with anything approximating a constant force. Our world is full of impulsive forces, forces that rise and fall in intensity. That would be true with or without friction.
There are lots of cases that are super impulsive like hitting a baseball, kicking a ball, or slamming a door, but even seemingly more constant pushes are not. Pushing a shopping cart, you push and the cart gets away from you, lessening the force you exert on it. It may get away from you enough that you lose contact, but more likely you’ve learned to lessen the force in just such a way as to push only hard enough to get the cart up to a speed, the speed you are comfortable, and then you just exert enough force to maintain a speed (against friction).
It makes me think about how foreign the concept of “constant” force really is. We often like to say that a force is simply a push or a pull, but I’d argue that a constant force is not anything like the pushes and pulls we experience in our lives. It makes me curious to spend more time helping students explore the notion of constant force by learning about just how impossible it is to accomplish. And to realize that the situations textbooks commit ask about are so weird as to almost be absurd.
Uniform Circular Motion was first introduced by thinking how it “feels” to be a rider on a swing carousel.
Students described what it feels like differently:
- You feel weightless, like you are not that heavy, almost floating
- You feel heavier than normal, feeling pressed into the seat.
- You feel like you are being thrown outward.
- You are leaning in, or tilting into, to not be thrown out.
Next we had a clicker question about where a rider would go if at a specific point the cables suddenly broke. Choices were varied, about 1/3 saying it would go off tangential, 1/3 radial, and 1/3 something between tangential and radial. There were very few who felt it would continue curving along the circle at all (e.g., circular impetus). We had some discussion about each of the answers, and why. Some felt the velocity (or force) would throw you out, others felt like velocity (or force) would be tangential, and yet others felt like the two would compromise to curve slightly outward while going around.
We next did a review of what we already know about forces and motion. Students were asked to use meterstick to exert forces on the hover puck in order to
- to get the hover puck speeding up
- to get the hover puck moving with constant speed
- to get hover puck to slow down.
I summarized these ideas at the board with motion diagrams and force vectors. We then did a clicker question about what the hover puck would do with a constant force that starts 90 degrees from the velocity. This one was hard, for students. Besides solving projectile motion problems (before forces), we hadn’t really talked about this. After discussion, we watched a simulation and tried it out with the hover pucks to see that it curves while speeding up. I helped them link this observation to motion diagrams and force diagrams for free-fall in 2D. I told students this was a good review of what we currently know about forces– we know how forces can maintain velocity, speed up, slow down, and turn while changing speed. BUT we didn’t know how to turn without changing speed, and that’s what we needed to investgiate
Next students were tasked with exploring out how to get the hover puck to move pretty much in a circle at pretty much constant speed using the meter sticks. Students were asked to discuss a few questions: how getting the puck initially started moving in a circle was different than keeping it moving in a circle, and questions to direct them to think about how they would describe the direction of the force. Then we gathered consensus around the following video of me doing it:
— Brian Frank (@brianwfrank) April 3, 2017
I then ended up introducing the demonstration where the hover puck moves also moves in a circle in using a string. I had originally wanted them to do this (which may have been better), but I/we were feeling the need to pick of some momentum with class. We watched the demo in class, and then I showed them these pictures, which helped identify what is meant by “uniform” circular motion.
I then drew a “top view” motion diagram that showed both the velocity vectors and the forces we were exerting. I put forth the idea that we had now seen 2 situations where an inward force was needed to keep an object moving in a circle – the inward pull of the tension from the string and the inward pushes of the meter stick.
[Note: Old me would have definitely been inclined to think that we are pretty good at this point. But I know better, and know better how else to keep us moving along]
Then was a clicker question about, “Why is a inward force needed?” I had four statements
A. An inward force is needed to balance the force that the puck experiences outward.
B. An inward force is all that is needed to turn the puck around the circle.
C. In addition to an inward force, a force around the circle is needed to keep it moving as it goes around.
D. There are actually three forces: One force that keeps the puck going around the circle, and two forces that are balanced to keep the hover puck from being thrown out of the circle.
I gave students time to think and talk with their small group. We were widely and almost evenly split between all choices. We spent a little time talking in whole class, to clarify what was meant by each idea. Instead of having them talk in small groups, I had them get together with people with similar answers. They had some time to chat, and then elected a spokesperson to make the initial presentation. I allowed questions aimed at clarifying, and added clarifying comments myself, but didn’t allow back-and-forth until each group had a chance to present. Then, we started talking more freely with ideas bouncing around.
- The inward only group focused on explaining that all you are doing is tapping inward, so that’s the only force, and that while they agree there’s a velocity around the circle, that doesn’t count as a force. They agreed that you initially needed some velocity around the circle to get it started, but the question was about “keeping it in the circle”, and they didn’t think it needed any force around the circle to do that.
- The inward + force around the circle went next and said that a force was needed to get it started and to keep it going. They came back to the spinny ride, saying that the ropes have two jobs, to pull you in and pull you around the circle. There were a few who said after hearing group 1, they weren’t sure whether it was just velocity of force around the circle.
- The next group talked about the in and out forces needing to balance. During discussion it became clear that some in the group thought the “balance” force was the velocity, that made it want to go out, and others thought there really was some outward force. Through this conversation, it became clearer to us all that issues around whether something was a force (or velocity), and whether or not these were pointing around the circle or outward from the circle was an crucial difference among the ideas (both within and across groups).
- The last group, made a compelling argument for why an inward force was needed. Without an outward force to balance the inward force, the inward force would “win” and make the circle spiral inward. We referred to this as the “death spiral”. To maintain a circle, the argument goes, a balance of forces would be needed. This was pretty convincing to lots of folks.
During the more open conversations time here are some of the ideas that came up:
A. A big question came to identifying what the outward force was. We’ve been pretty picky all semester about identifying forces. Someone eventually suggested that it might be the force pair. I helped flesh out what the student meant by this, by identifying the forces specifically. But I initially didn’t press for the implications of that idea. I wanted that idea alone to be important. Yeah, what is that force? Maybe it could be this force, the force pair. I probably could have asked, “Do people have others ideas about what the outward force could be?” to further value that line of reasoning.
B. Another idea was that a force can’t be needed around the path of the circle, because a force around the circle would mean that force and velocity were parallel. Our rules state that when force and velocity are in the same direction, speeding up occurs. Through discussion and re-voicing, I helped them to articulate a possible new rule that they were seeming to suggest: “If force stays 90 degrees from velocity, all you do is change direction, without changing speed.”
C. Someone argued that they don’t think that an inward force necessarily means a spiral inward. They were trying to think of what would make something spiral in or out, and they came up with situations where spiraling in or out would be associated with a change in speed. People gave examples of twirling keys with a lanyard, and letting it swing around your finger vs. wrap around your finger. Since we were talking about uniform speed, not spiraling should happen unless you speed or slow the object.
D. There were also more conversation about getting it started vs. keeping, that were helpful, but I can’t quite remember, and ‘m sure there were other ideas that I’m forgetting.
Anyway, we took a break, and I chatted with some more students. During break, I talked with several students about the “force pair” idea, and helped connect that to our previous learning about how force pairs shouldn’t be on the same FBD. So that if the outward force was the force pair, it shouldn’t be included in the diagram that shows forces that act on the puck (or rider). I also talked with the two students who brought up the spiral in idea and the counter proposal.
After break, I tried summarizing some of the big ideas, and helped the whole class in on two ideas:
- That if the outward force is the force pair, then it’s a force on another object, and thus shouldn’t be included, and
- that there are two different ideas that I see as similar. I told them that the “inward” only group thinks that what the puck will do without a force is “go straight” and that an inward force is needed to “bend” the puck so that it moves around with a constant spiral (not spiraling in or out). The “balanced force” idea seems to focus on the fact that the puck is already moving in a circle, so that any extra force would “bend” the puck into a tighter circle. I emphasized that both groups agreed that inwards forces cause “bending in”, but they disagreed about the detail.
There was a bit more conversation after I introduced those ideas, and students had some time to be back at their groups to rethink.
I then introduced a new experiment that might help us decide. I showed them a metal ball going around inside of a metal ring. I told them that I could suddenly pull the ring upward, and that it would be like “breaking” the cable on the ride.
I introduced what we were about to do as very different than what we did before. Before, we were shopping around for ideas, experiences, arguments, and that it was kind of OK to change our mind. But now I wanted us to stop letting our minds think whatever we want. I want to find out not “what we think”, but what our different ideas imply. I asked students to go back to their groups and draw 3 diagrams:
A free-body Diagram (just before the ring is pulled away), based on whether you think the forces are “inward only”, “balance in and out”, or “also around the circle”
A free-body Diagram (just after the ring is pulled away), based on what forces should disappear when the ring disappears.
We agreed that each diagram should be very clear about whether something is a force or a velocity, but that it was OK to not specify exactly what the outward force might be, since we were still not sure. We also agreed that the object of interest for our diagrams was the ball, and that only forces that act on the ball should be included.
The last diagram was to show a path about where the ball goes. I spent a lot of time telling them that this was not where “you think” the ball should go, but rather what the diagram has to say about where the ball should go based on a rules:
- No Net Force -> Constant Velocity
- Force in Direction of Velocity –> Speeding Up
- Constant Forces in 2D –> Curved trajectory
Groups split up and worked on their diagrams, and then we circulated around looking at different diagrams. Several groups went in the direction of “inward only”, one group did balanced in and out, and another group did balanced in and out (with maybe force in the direction of motion).
We concluded that a balanced forces implies that the loss of ring would leave the ball with only an outward force… and that this would mean the ball curves slightly outward as it leaves the circle.
We concluded that the inward forces only idea implies that ball will go off tangentially, moving not only straight but with constant velocity. This is because since the only force was the inward normal force from the ring, once this force is gone, the particle is subject to zero net force.
I sent students off to make the observations at their tables.. Because it’s hard to see, students spend a fair amount of time on this. In small groups, I talked about “confirmation bias” and whether or not they think it was possible to see whatever you want, like those that want to see it as straight can see it that, and those that don’t, can claim they see a slight curve.
Good thing we have the slow motion camera:
— Brian Frank (@brianwfrank) April 2, 2017
I did some summarizing here (should have made them do more of that work), and connected what we were learning to textbook notes and diagrams. This is also something I should have them do. I should have had them open their textbooks to the passages and diagrams, and ask them to do this work. This is like in my previous post about discourse, where I took too much responsibility for discourse 2 and 3, where I should have been the one to support them in engaging in the discourse.
The way I did ask them to engage somewhat in the 3rd type of discourse was by getting practice with applying the ideas, which is better than nothing. We did a few clicker questions about identifying the “force” that plays the role of the centripetal force. We did free body diagrams for penny on a turntable, the swingers on the ride, and then finally free body diagram at the bottom of pendulum swing.
Students did fairly well with the first two. I emphasized how in the first case we might think of just one force (friction) as playing the role of a (net) centripetal force, and how in the second we needed to think of the just the horizontal component of tension as playing that role. But the third one was a really interesting trouble spot.
In the pendulum swing, students had to decide which of the two individual forces was larger, weight or tension. Maybe one or two students said weight would be stronger than tension (probably focused on “outward throw” still), but most groups picked that tension and weight would be equal. What was interesting is that many of the people who said that tension and weight would be equal were the also the ones who were most adamant for the inner force only in the previous activity. I thought that was interesting because now they were saying that balances forces were needed. Some of these students even argued that if the T were greater than the weight than the mass would like “rise up” at the bottom (making a smaller circle). Essentially, the spiral in argument resurfaced, but not from the same students. It made me wonder if “having the right” idea the whole time made them more vulnerable to being tricked later.
Anyway, the situation is hard for several reasons (one it’s not uniform circular motion, vertical forces are involved, and there are two forces in the radial direction). I think a lot of students reasoning was actually trying to borrow from “independence of motions in projectile motion”… Something like Like the velocity is horizontal, and so it should have no impact on the forces vertical. Thus the vertical forces should be balanced. It’s an interesting case of saying “no velocity vertical” means “no vertical force”.. which is “force~velocity” reasoning just resurfacing with knowledge of components.
In discussion, however, groups were able to be pretty convincing that the tension should be larger than the weight. Groups did a good job of making it clear that what we had talked about, seen, and learned today was the basis for saying that the forces cannot be balanced if it’s moving in a circle, and that rather a surplus of force pointing upward toward the center of the circle was needed. I’m glad the class did this (without my help), but I could have done a better of job of pushing questions in small groups, like… “How did you apply what we learned today to help inform your answer?” or “How is your answer consistent with what we learned about the forces necessary to keep an object moving in a circle?” I might have even asked a question like, “So any time you have velocity that is purely horizontal, the forces vertically must be balanced?” The conversation was good, but I definitely see ways I could have been a better at “pressing for disciplinary connections” toward the end of the lesson.
We ended the day by observing the force sensor data. With five minutes left in class, I asked students to take some time to reflect, write, summarize what they learned today. This is also a practice I need to use more, using the last 5 minutes for reflection. All and all a good day. On Friday, we get more quantitative with “how much force exactly is needed” –> this will build off nicely from the spiral in /out question… we want just the right amount of force to keep the circle (not let it widen or tighten)… We will investigate what factors are involved.
In teaching of physics this semester, we did an activity of sorting many question into three categories of discourse:
- Eliciting students’ initial ideas (questions that help us identify our /their thinking)
- Supporting changes to their thinking (questions that make one consider how what one is doing relates to our ideas and other activities)
- Pressing for disciplinary connections (questions that direct students to draw explicit connections between what they are currently doing (and/or what they have done) with specific concepts and/or practices from the discipline.
This categorization is basically from the Ambitious Science Teaching folks, but we’ve been adopting this framework to describe the instructional flow to most instructional frameworks we have talked about, whether it’s
- Elicit Confront Resolve (Described here by Wenning with the additional steps of identify and reinforce)
- Bridging Analogies (Described here by Clement), flow is Anchor –> Bridges –> Target
- Learning Cycle in ISLE (e.g., observation experiments –> testing experiments)
- Invention Tasks (i.e., based on preparation for future learning), and here as well.
as well as others.
One of points I’ve been trying to drive home is how these curricular structures have certain design “features” that make them effective, but ultimately it’s about the discourse that happens during instruction– what talking and thinking are students engaged in during each phase. For the appropriate discourse to happen, teacher talk moves need to shift from “eliciting” to “supporting change” to “pressing for disciplinary” connections. Previously, we had also talked about discourse at the level of “talk moves”- probing, re-voicing, pressing for reasoning, etc.
Our activity today was to try to link the two: what sort of talk moves support what kind of instructional discourse?
Here are the list of questions we considered.
- You said_____, but why do you think that?
- What connections are you seeing between ____ and ____?
- What do you think will happen when____
- Can say more about that?
- How is your thinking about __ different now after ____?
- Who else has ideas about what might happen?
- Do you agree or disagree, and why?
- Who would like to add on to what ___ said ?
- So you seem to be saying ___?
- What do you all think about that idea?
- How did you come up with that?
- Can you describe what happened when___?
- What evidence supports the idea that ___?
- Is there a specific example you are thinking about?
- What does this passage mean in your own words?
- How do we know that ____ ?
- Can you say why you agree with ____
- How can you check your answer?
- How are you making sense of that?
- How is this observation different than your prediction?
- What do you think these results imply about ____?
- What can you say now that you couldn’t before?
- Does this agree or disagree with your prediction?
- How is this situation different than ___?
- What reasoning justifies this ___?
- What assumptions did you have to make in order to ____?
- What does this observation tell us about ___?
- What are some tools we have used to ____ ?
- What are some things you notice?
- What does this tell us about ___?
- Do you think this supports or refutes the idea about __?
- Explain to us what your thinking when you say ___
- What made this particular situation difficult?
- What do you think causes that to happen?
- What do you think that tells us about the data?
- What do the rest of you think?
- How have your ideas changed at all?
- What makes you think that will happen?
- Does the explanation here describe what you discussed?
- What patterns did we notice?
- How did you decide to ___?
- How did you know you had made a mistake?
- How did you reach that conclusion?
- What did you notice happening when ___?
- Can you tell us how you came up with that?
- How do you think ___ applies to this scenario?
- How might we revise our thinking after seeing ___
- Can you tell us why you think that’s not longer true?
- How does your work here reflect what we’ve learned about ______?
For each, we talked about the context or contexts in which it might make sense to ask this question, and worked toward a consensus model of what type of discourse this question would most likely support. We talked a lot in class about how traditional instruction spends too much time in the 3rd kind of discourse, skipping the 1st and 2nd kinds. We also talked about how unsuccessful inquiry often fails to make the turn into the 3rd type of discourse, or perhaps fails to even do much of the 2nd (students just go through activities without ever thinking about them).
It was a good day. I feel like I learned a lot and so did the students.
Here is the gist of circular motion:
1. Elicit initial thinking about UCM through carnival swing ride. Where would rider go if the chain broke? Students asked to record thinking on a sheet.
2. Suggest that this a question about forces and motion. So Then review our ideas about forces and motion in 1D through clicker questions, using physics aviary. Also do 2D constant force.
3. Ss explore how to get hover puck to move in a circle with meter stick. We discuss our observations, then together observe string used to make go in a circle.
4. Ss asked to draw motion diagram for circular motion in both cases, with force vectors added. What’s similar about what meter stick and string exerted forces? Establish a rule: a force pointing toward the center of the circle is needed to keep an object going ina circle.
5. Why? Clicker questions asks why an inner force is needed:
A. Inner force balances the force trying to throw the hover puck out.
B. Inner force is all that is needed to keep the puck turning around the circle.
C. A force in the direction of motion is needed to keep the puck moving, while an inner force prevents it from drifting away.
Call these these the “inner only”, “balanced in and out” and “around the circle” rules.
6. Testing experiment. Ball rolling in a metal ring. The ring is suddenly removed. Goal is to predict what each rule would predict. Ss reason through a before during after diagram. In the before, Ss asked to draw FBD before ring pulled; in the during, Ss argue about how forces change when pulled away (and why); and for the after, students use rules from part 2 to argue where all should go.
7. Observe outcome of experiment and help Ss to see how consistent with inner only rule.
8. Introduce textbook description. Practice with a few clicker questions. First couple about identifying what force is the center pointing force.
9. Harder question is about FBD for bottom of pendulum swing. Debate. Observe. Refine rule to be about Fnet needs to be center pointing.
10. Return to original question. Ss asked to rethink and discuss how thinking now is different than their thinking before. Even if their answer hasn’t changed, their thinking is likely different.
10. Revisit the pendulum swing. Ask question about what changes could be made to make the tension force larger! Likely they will say higher mass, faster swing, may need help with tighter circle. Will press for other contexts in which ideas make sense. Curious what other ideas they will have.
11. Ss are asked to design an experiment to test how Fnet depends on either mass, speed, or radius. Not sure what I will actually ask Ss to do. I think Force mass or force vs Radius. Easier to carry out and either proportional or inverse.
12. Some magic happens depending on what Lab they do vs what Lab data I show them. Introduce Textbook ideas about quantitative uniform circular motion.
13. Some practice clicker questions or ranking tasks.
14. Problem solving in context of pendulum swing. Given this photogatedata, what is force sensor reading? Compare prediction to obsevartion.
In doing Newton’s 3rd Law, here was the process that seemed to work pretty well, but will need a little tweeking:
1st : Introduce idea of multiple objects of interest, and multiple free-body diagrams. Worked through drawing FBDS for case of pushing two unequal mass blocks on a frictionless surface.
2nd: Introduced the idea of a force pair, show the example in the Free-body diagram, and talk about the property of forces pairs: They are “two” perspectives on the same “interaction”; they are forces that act between two different objects; for contact force pairs, they occur at a single location (the boundary between where two object meets), and they always occur on different free-body diagrams
3rd: We did one example clicker question about identifying force pairs (I would do a few more next time) or even have them work whiteboards. It helped having a list of properties of force pairs, because we used them to argue about it.
4th: I told them that we already know the rules about how single forces acting on one object behave (they superpose to generate a net force, which causes acceleration). Now we needed to determine the rules for how force pairs behave.
5th: I introduced, diagrammed, and demonstrated three cases where the force pairs are equal: “Tug-o-War” Scenario where no one wins (linked to carts with hooks, and pulled on equal mass carts equally hard on both sides with identical rubberbands ). Collision scenario where two equal mass carts head on collide with equal speed with equal stiffness bumpers. And a “pushed together” scenario where equal mass carts are pushed together at constant speed. For each situation, the force pair was identified and then measured using force probes. We saw that in each case, the force pairs were identical in magnitude.
6th: We brainstormed changes that we could make that would make the force pairs unequal. Students suggested a tug-o-war where one side wins, collisions with unequal speeds, or unequal masses, or unequally stiff bumpers, and “push together” scenario with unequal mass as well as speeding up.
7th: Every group had to try out at least one variation from each of the three categories (tug, collision, push together)… Some groups already kinda guessed what they would see, but many groups were honestly (and pleasantly) surprised. I let a lot of groups really show me what they had found and shared in their cool findings. Given the different place of each group, my conversations were very different. I talked with some groups about what they were observing, and what that implied. With other groups we talked explicitly about Newton’s 3rd Law, and when it did (and didn’t apply, haha!). With other groups, we talked about how could it be that the forces were the same (tug-o-war). At the end, I did a lot of the work of synthesizing (at the whole-class level) our findings due to time constraints.
8th: We ended, by watching Frank’s Newton’s 3rd law Video, and talked about the squishing as evidence for force pair equality. We could have spent more time here, but also given them more explicit tasks. Need to rework how to engage this task for sure.
From there we took a dive into problem-solving. If I did it again, I probably wouldn’t do it right now. I would do more scenario representing and reasoning. The problem we did was pretty hard, too–two blocks in an accelerating elevator. Students worked well through it, it was definitely productive struggle. But there was a lot of struggle, still, around normal force and weight. That doesn’t surprise me. Vertical stuff is just hard, and to pile on vertical toughness with the newness of Newton’s 3rd Law and multiple objects is just a bit much. They still did pretty good.
Part of what I learned they needed a little more scaffolding on managing just thinking about 2 (or 3) net force vectors simultaneously–that each net force gets separately attached to the free body diagram for that object. Some of it was just labeling issues, but it was also more than that. I’ll have to think more about what other scaffolding they need. But all and all, the Newton’s 3rd Law day was a good day.
My approach here was very much “elicit confront resolve”… And I know there can be some criticisms of such an approach, but what matters is how we as a class frame what we are doing. Students aren’t perceive in my class the activity as “tricked you” (you were wrong!), because they have learned to have a response that’s more “so cool” — being pleasantly surprised by circumstances when nature is different than you expect. The activity is also just fun because we are brainstorming and testing our ideas… I’m not presenting a specific situation for them to be wrong about. They are proposing scenarios, that they themselves are tricked by. That feels more playful, I think. It’s harder to feel like you got suckered, when you are the one who proposed the trick in the first place! Anyway, I’ve just been thinking about this more… about how a lot of “elicit confront resolve” criticisms are criticisms not of the instructional technique, but of an instructional framing that is commonly developed around them. It can unproductive for students to frame it as “guessing”, or “being tricked”, “or always being proved wrong”, or “intuition is never right”, but that’s definitely not the only framing that can happen. Anyway, this last rant was a little unrelated, but it’s been on my mind. I also did very elicit confront resolve tasks for static friction, and that went really well too. With students really engaging their ideas, having a combination of being right and wrong, that led to being intrigued curiously by being wrong (rather than feeling stupid about being wrong). blah blah blah blah…. It’s Friday and Just wanted to keep typing my thoughts…
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!