Today, I didn’t have time to revise the sample problem I’m supposed to do in physics. So I worked the sample problem as given. We were doing standing waves. In the particular problem, there was a string with both ends fixed and we were told it was vibrating in the second harmonic. In this case, the wavelength is equal to the length of the string.

After my sample problem, students were given a problem where the string was vibrating in its fifth harmonic. A fourth of the class did it correctly, drawing it out and concluding that the wavelength must be 2/5 the string length. Half the class did it wrong saying that the wavelength was equal to the string length. And a third of the class said the wavelength was 5/2 of the string length. As I was walking around, I asked a student why they thought so many people were making these mistakes.

She responded without hesitation, “The sample problem was poorly designed. You shouldn’t have given us an example in which the wavelength and the string length were equal. That makes it easy for everyone to think that’s what you are supposed to do every time. Plus dividing by 1 or multiplying 1 gives you the same result, so it’s easy for people to mix it up. Next year, you should use a different harmonic to set up the problem.”

I wish my future physics teachers knew had to unpack a sample problem like that, and see how it might lead to over-generalizations and misinterpretations.

One of the things I’m doing this year is trying to provide more opportunities for deliberate practice (with feedback), while trying to support students in building and articulating explicit strategies for solving problems (not me just telling them strategies), and selecting class problems to work on that can be solved with current ideas but also put us in contact with puzzles that later ideas will help to resolve or bring insight into.

Here is an example from earlier this week:

This week my students are going to be evaluated (by another instructor) on their understanding of projectile motion and Newton’s laws (including circular motion).* Afterwards, we are going to be diving into energy. So, how do I give them more deliberate practice and feedback with projectile motion and forces while also putting us in contact with energy puzzles? Here is what I tried:

First, I asked students to predict which of two ramps would result in a block of ice having more speed at the bottom. Both ramps were from same height, but one was shallow one is quite steep. No numbers were given. They think and vote peer instruction style. We were pretty much evenly split between all possibilities, so they discuss in small groups and then I collect arguments at the front board. The arguments were basically the following:

• The steeper ramp has a greater acceleration, so it will be faster.
• The shallow ramp will provide more time for the ball to speed up, so it will be faster.
• The opposite effects of acceleration and time will balance out so that they take the same speed.

We’ve gotten pretty good at doing this, so I can mostly stand to the side and just write down arguments and do some re-voicing. After hearing the arguments, I have them revote. There were some shifts, but still not near any consensus.

I now tell them that I want to help settle this by applying some of the skills we’ve learned over the past couple of weeks. I add some information to the scenario. Block of ice has a mass of 25 kg. The ramps are angled 30 degrees and 60 degrees. The height is 5m. I split the class in half, half the groups work the 30 degree problem while the others work the 60 degree problem.

Before sending them off to work the problem, however, I tell them to talk strategy with their group–what will you need to figure out to answer the question, what skills and ideas might be useful, what might you do first, second, etc? They talk for a minute or two, and then we collect strategy ideas at the board. They say most of the things they need to–drawing free body diagrams, using Newton’s laws to find acceleration, finding the length of the ramp using trig, using kinematics ideas / equations to determine the final velocity, etc. Now they are off, and the board is there to help remind them of things they can try if they get stuck.

Doing this together makes me free to monitor for progress rather than helping students get started. I’m checking free-body diagrams for bizarre combinations of Normal and Weight fores, if and how they are finding components, whether they are using a rotated coordinate system and using that consistently with forces and kinematics, etc. I point out things that they are doing which are very “physics-y”, like drawing careful diagrams with labels, starting from big ideas rather than launching into equations, etc. If groups finish early, I ask them to solve for other things that came up in our arguments. For example, I might ask students to solve for the time on the ramp to see if its true that the larger acceleration was paired with less time to accelerate, etc. As multiple groups finish, I have them check with each other on their answer and check with people across the room.

Once we are done, I do a quick summary of what we found, highlighting that its odd that both ramps end up giving the block the same speed. I restate the arguments we heard, and I emphasize that the argument for the right answer made it seem plausible that it could balance out, but why it exactly balances out seems like a puzzle to me. It didn’t just balance somewhat, it balanced out exactly.

I tell them that I want to consider another problem where we compare final speeds, but this time not with ramps. In this problem a baseball player throws a ball with same speed. In one case the ball is angled upward, and in the other case the ball is angled downward. The question is about the speed’s of the two ball’s just before impact, and how will they compare. Students vote. This time there is a split between two answers. Most students vote they will be the same, but don’t have good arguments. They are banking on it being similar to the last problem. Intuitively, it makes sense that the one thrown down will have more speed, and I support this argument a bit. If you are throwing it down, in the same direction of gravity, and its got a real direct path to the ground, isn’t it going to be a lot faster when it hits. There are some other really awesome arguments for why it should be the same, about why it must balance out, including consideration of what the one that goes up is like once its on the way back down. The best argument came from a student who had never spoke up in large discussion, so I spent some time re-voicing that argument and giving it space for consideration.

Once again, I turn the conceptual question into two problems to solve, adding angles and heights and an initial speed. We talk and collect strategy at the board. They solve the problems. I monitor progress, give extension questions, ask them to check with each other. Finally, I summarize and make connections at the end. I still try to keep the puzzle open: Why is it that when the two blocks fell through same distance, and ball’s fell through same distance that their final speeds were the same? Our current skills help us to calculate that this is the result that should happen, but it doesn’t help to explain why.

An interesting outcome of asking students questions to compare, and then asking them to compare pairs of questions, is that they start doing more and more comparing. Several groups started re-thinking the shooter-dropper experiments. Looking for connections across phenomena is something I want to promote and this kind of activity seems to promote more of it.

Anyway. So later this week, we’ll revisit these same two problems from an energy perspective, but I’ll also introduce puzzles for us to resolve that further our understanding of energy and kinematics. Namely, this time we’ll do a problem where two balls rolls down the same ramp, one with an initial speed and one from rest. In this case, they will neither end with same speed nor gain the same amount of speed. Rather they will gain the same amount of kinetic energy…

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* This used to bother me, having someone else test my students. But I now love it. My relationship with students is not of evaluator or judge. I am a learning coach. Sure, some of the evaluation is not meaningful. Sure, my students are learning things that aren’t evaluated. But my students do well on the evaluations for the most part, and students are constantly getting feedback from me on a broad range of their learning.

As usual, I ask students for feedback after they have taken the first test, but before they have gotten it back. Below is what they have to say. Lots of interesting themes. I have to say that the “mindset” curriculum has huge pay offs in terms of student learning, student attitudes, and student self-awareness.

The quizzes are good. They make me focus more on material now while its presented in class.

Making sense of the equations so they are easy to understand. It helps me to understand why I’m doing what I’m doing.

Example problems reinforce the methods. Quizzes allow me to learn the right way approach problems and then test it.

Hands on problems help me understand. The way you explain everything throughly helps me learn fully.

Whiteboarding problems with group, rather than figuring it out on my own. It helps me sort through my confusions.

I love working in groups and being able to draw out the problem on the whiteboard. I tis helpful because I normally don’t ask a lot of questions.

I like talking through problems with my group.

Instead of just learning equations, you give us reasoning behind them. That makes sense to me and helps me remember them better.

This class is more than just a physics class. This class helps me to find interest in learning, even with things I don’t want to learn.

Practicing problems in class helps a lot. Practice makes perfect.

Working problems as a group, help me see if I’m making a mistake-both where and why.

I love that we over multiple problems to understand a concept. What also is helpful is when you have break into our groups and have us write out every step at the whiteboard.

The clicker questions. Practice problems with w/ walk by assessments. This increases understanding. Example plus group work are great, too.

Your enthusiasm helps me keep focused on the material. The biggest thing that helps is working through problems with others in class. This is helpful because it helps me work through difficult things with other and see how they work it as well.

Showing examples of how to work problems without just picking a formula and plugging in numbers is helpful, because it shows what the problem is really asking and what we are trying to do and why.

Group work is helpful because we learn from others.It is helpfuly because you can explain to others reasoning in finding solutions.

The labs that point of the dynamics behind most everyday objects like hover pucks.

Group discussion provokes thinking and gives us a change to develop our own answers.

The hands-on instruction and conceptual learning.

Practice problems and quizzes that are akin to test questions.

Group work is very helpful. Talking to other people helps solidify my understanding.

Working and learning how to problems on the whiteboards, and discussing during class.

This is a good lab and the info is clearly communicated.

Working in groups with whiteboards really helps me learn. Being able to discuss problems with other people helps me understand better.

I get more from working out equations in my group and being able to hash out ideas I’m struggling with.

Hands-on activities; it gives my brain time to digest what we just went over (e.g., the hover pucks)

You discuss topics outloud in class that changes my thought process, and then we work thru problems, both visually and verbally to solve and bring understanding to topics.

I like the hands-on projects.

What do we do in class that unhelpful for you learning? Why is it unhelpful?

The labs.

The quizzes hurt me when it comes to learning, but it’s not a big deal.

Nothing really.

Labs. I know we have to do them. I just don’t feel like we are applying what we’ve learned when we do them.

I could benefit from would be some one-on-one tutoring, but I know that’s offered outside of class.

Explain more how to use the equations.

Sometimes we get off subject for too long-questions that have nothing to do with the assignment.

Quiz being taken in class. If we took them early, we would have an idea where we are before and after the lab.

I wish we went over the homework.

I wish the assessments were more difficult. I understand why they are what they are.

The “lab” lab part. Sometimes I feel that they are a waste of time.

The lab activities. I don’t get anything from them.

Sometimes we go really fast.

When you ask us to diagram a problem, annotate it, but don’t actually solve it. I feel like you stop the learning process for me half-way.

Annotating problems without solving them can get repetitive-more annoying them helpful

Sometimes I feel like I didn’t learn a thing, and have to wait for closure until the next class.

Is there anything else you’d like to tell me? Is there something on your mind? Or something we don’t do in class that you wish we did?

I sought outside help to determine exactly what force is. We still haven’t really figured it out.

We should have class outside more and do more hands on examples.

Before it gets cold, we should have class outside.

I don’t do well on quizzes. I know I need to do the homework and have a chance to digest the info to do better on quizzes.

You’ve been the most influential teacher I’ve ever had.

More group problems if possible.

I really do like this class, now that I have a better understanding of it. I just wish we didn’t have a quiz every class.

I’m batman

I recommend continuing talking about and teaching us about how to learn.

I wish we could do more group work.

More experiments and applications

I felt extremely prepared for the exam-very comfortable with the material. I am very glad to be in this class.

I like learning and doing a lot of problems and talking about strategies before the quiz. It’s hard for me to learn straight from the lecture notes.

Could we get some optional difficult problems to help understand and process more in depth. I always like a challenge.

We could use a little more time to complete the labs.

In our physics department, every physics major has to serve as an undergraduate TA. Most of them get assignments in our algebra-based introductory physics course.   Because of the manner in which most of these students were taught (i.e., find an equation and substitute numbers), they can easily find themselves feeling a bit lost in my class, especially if they think they are supposed to be an expert of the content.

For example, here’s a question discussed in class. A bowling ball is dropped from a height of 45m, taking 3 seconds to hit the ground. How fast is it moving the very moment before it hits the ground? The problem is intended to draw out the following answers and arguments, which we hash out.

10 m/s, because all objects fall at the same rate

15 m/s because you can calculate the velocity as 45m/3s = 15 m/s

30 m/s because it gained 10 m/s in each of the 3 seconds

Other more idiosyncratic answers come up as well, but not with high frequency.

The first answer points to the ways in which students haven’t yet teased apart clearly the meaning of acceleration and velocity. The second answer points to the ways in which students haven’t yet teased apart clearly the meaning of average and instantaneous velocity. The third answers is consistent with the idea of constant acceleration. We hear arguments, and counter-arguments, and at some point I help clarify the right reasoning, and what’s both so tempting and subtly wrong about the other answers.

So, here is the way the TA solved it, before class began.

xf = (vf + vi)/2 * t + xi

0 = (vf + 0)/ 2 * 3 + 45

0 = 3/2 v + 45

-45 = 3/2 v

v = – 30 m/s

While the TA could solve this problem, they didn’t have a rich set of ideas for thinking about. It didn’t seem obvious that 30 m/s makes sense, because of the idea that its 10 m/s/s, or because final velocity sould be twice the average velocity (since it accelerated from rest). For other questions without numbers that we discussed, the TA seemed just likely as students to give answers inconsistent with the concept of acceleration. I’m perfectly OK with that, but my suspicion is that the TAs aren’t prepared for this. They aren’t prepared to be wrong about so many things or confused about so many things. I wonder how I can better position them as learners in the class–learners who just know somethings that the first-time students don’t, but not everything.

Of other interesting note is this. In my physics content course for future physics teachers, the students that have had me for a semester or two are pretty rock solid on having a repertoire of ways of think about kinematics problems, and also for avoiding common pitfalls. The others are pretty much falling for all the pitfalls. The difference is pretty striking. The thing that I like is that the range of expertise we have allows for peer-coaching, but also some, “Hey, it’s OK. We were making those exact same mistakes 4 months ago,” and, “Yeah, get used to it. Brian isn’t too into solving problems by putting numbers into equations.”

One way I’ve gotten some decent leverage in getting students to really initiate with goal-less problems in early kinematics problems is to ask, “If you had taken this trip with a GPS device, what summary trip information could it give you?”

As a class we generate a list of things the GPS device would or should be able to calculate like:

Time of Arrival

Duration of Trip

Final Location

Total Distance Traveled

Average speed (while moving)

Average speed (during entire trip)

I usually tack on a few things, like it could tell us how far we are from where we started (and in what direction). I also say that a good one should also be able to make a graph. I may or may not introduce average velocity.

My job at some point is to connect each of these to formal language and algebraic symbols used in our text. Unfortunately, our text is sloppy with clock readings vs. time intervals. It’s also sloppy with displacement and position. So it’s a little difficult.

I also think it’s cool to have “average speed while moving” be something that the physics text book doesn’t have, and that we’ll have to completely invent our own way of calculating it.

Students practiced a problem today where a child goes down a slide that is 4m high. Students are asked to first calculate what the speed of the child at the bottom should be if there were no friction. Then they are given the actual speed data and asked to determine how much energy was “lost” due to friction. Everyone gets the first part right, so I want to talk about solution paths to #2

Solution Path #1:

Calculate the theoretical kinetic energy at the bottom and subtract from that the actual kinetic energy at bottom based on actual data.

Solution Path #2a:

Construct the Equation PEi + W = KEf (often based on pie charts), and solve for the work done by friction.

Solution Path #2b:

Construct the equation PE + W = KEf, and actually try to solve for the symbol f, by using W= f Δx, and often (mistakenly) plugging in 2m (which is height not distance along which friction acted). Some students go so far as to try to calculate μ, using f =  μ N.  I try to refrain from saying that these students are trying to solve for the force of friction or the coefficient of friction, because I think they are just solving for variables, not trying to determine any quantities in a physical sense.

Solution Path #3:

Calculate the Initial Energy (all PE), Calculate the Final Energy (All KE), and look at difference.

Solution Path #4:

Subtract the theoretical speed from the actual speed, and use that difference in speed to calculate a kinetic energy (essentially doing KE = 1/2 m (Δv)²

Solution Path#5:

Ignore the actual data. Calculate potential energy and then the theoretical final energy (based on speed answer to part one), and then examine the difference, actually finding a very small one due to rounding.

Solutions #1, #2a, and #3 all work. I find that Solution #1 and #3 are more thoughtful. Solution #2a can be thoughtful for some, but for many its just a routine. Solution #2b sends signal to me that student is in “algorithm of an energy problem mode”. They aren’t thinking; they are just doing. They probably also don’t understand what the difference between Work due to friction, force of friction, and coefficient of friction. Solution #4 is incorrect, but I still like it. It’s a plausible idea, and shows me they are thinking. There’s also something to build off, to learn from, etc. Solution #5 is odd. I suppose it’s good that they are trying to look at a difference, but they act of not including anything about the actual speed of the child sends a signal to me that they are also not thinking, they are just doing.

What do you all think?

I’ve been teaching using schema system diagrams, which I have just been calling interaction diagrams in my physics class. It’s the first time I’ve ever taught using them. I’m sold on them after one week.

Here is the biggest reason why I’m sold.

The diagrams provide a productive outlet for really good student ideas, which previously would have been considered misconceptions. An example:

Today, we started doing circular motion. We had a constant velocity buggy going around in a circle by means of a string. Just before we took some data for the time to get around and the radius of the circle, students were drawing interactions diagrams and free-body diagrams for the situation.

Three of eight groups included me in the interaction diagram, interacting with the string and the string interacting with the buggy. It’s a wonderful idea to think about that the motion we are observing hinges on the fact that I have pinned down the other end of the string. It’s insightful and correct—with out that interaction, there would be not constraint to move in a circle.  Now here’s the important thing: Previously, with out interaction diagrams to provide a place for that idea to go, that idea would have made its way to the free-body diagram. You could think that the reason I like the diagrams is because they prevented a mistake, but I really like the diagram because they provide a productive placeholder for valuable insights and ideas.

Three other groups included the motor in their interaction diagram. Each of those groups placed the bubble of the motor inside the bubble of the buggy. The really wonderful idea here is that none of the motion we are observing would not be happening without the motor. The buggy would screech to a halt.  Previously,when teaching without the interaction diagrams, that wonderful idea would not have had a productive outlet, so many students would have included a motor force on the free-body diagram.

So sure, one cool thing is that no group got the free-body diagram wrong. One reason to like the diagrams is that it leads to correct force diagrams. But the really cool thing is that students were thinking about the roles that both the motor and Brian were playing, which I hadn’t even thought about. It’s not merely preventing mistakes, it is generating insight and ideas about the different roles that interactions play inside, outside, or many degrees removed from a system.

Even if you showed me evidence that teaching system schemas doesn’t improve student learning, I’d still teach using them, because of how generative they are. It helps to create classroom environment in which student insights can be celebrated for what they are, rather than constrained to being misconceptions. By the way, the diagrams do seem to help student learning.

A question I’ve gotten a lot of leverage out the past two semesters is the following one:

You toss your keys straight up to a friend, who is 30m above you leaning out over a balcony. They keys leave your hand with a speed of 25 m/s. Will it get to your friend?

Sure this is a standard boring question. What makes it work is how the show is run. We start off by listing our best guesses about whether it makes it up and the top height they think it gets to: Their answers this semester ranged between 19m and 40m.

In my class, I actually work out this first answer for them (because I’m supposed to model a sample problem), but I ask for their help along the way.

First, I draw a motion map showing how the speed changes at 1s intervals, and we talk about the speed going from 25m/s to 15m/s to 5 m/s, etc, and how the time to the top is when v = 0 m/s. We talk about how much time it takes to get to 0 m/s if you are losing 10 m/s each second: it takes 2.5s to lose 25 m/s. We also talk about the average speed during the trip (12.5 m/s, half way in between 0 m/s and 25 m/s). This, of course, all builds on ideas we built up last week when talking about 1D acceleration problems.

The answer is immediately given as 12.5 m/s * 2.5s = 31.25m

The best guess this time was 32m, and kudos were given to that group.

Lot’s of students then want to talk about why it’s not 40 m (25m + 15m + 10m), and we get to talk about what constantly changing velocity means.

Because of class constraints, I typically re-derive the 31.25m in a way that is more typical of how they are expected to do it: Write down your knowns and unknowns and pick an equation or two to plug away with.

I then send them off to work on the next question. How fast are the keys moving by the time they reach your friend’s hand? Our guesses range between 1.25 m/s and 2.5 m/s.

The right answer is 5 m/s. And students are pretty surprised to find out that we all underestimated the speed. Every group got the right answer. Most students solved the problem by plugging away into equations. One group did so, but didn’t believe that 5 m/s was right, and so they took another approach, using two equations instead of one.

One group took this approach:

In the first second, the ball slows from 25 to 15, with an average velocity of 20 m/s. Thus in the first second, the ball covers 20 m. In the second second, the balls slows from 15 to 5, with an average velocity of 10 m/s. Thus in the second second, the ball covers 10m. That’s 30m covered, with a final speed of 5 m/s. That same group realized that for the first 2 seconds, the average speed was 15 m/s for 2 seconds, also giving 30m of travel.

Last semester, I had a group solve the problem by finding the speed of a ball dropped 1.25 m/s, arguing on the ground of symmetry that it had to be the same.

We ended the problem this semester by talking about the last 1/2 second, where the ball has an average speed of 2.5 m/s for 1/2 second, thus covering the final 1.25m, and why our guesses for the speed were so off.

Simple problem, but lots of places for intuition, lots of places for multiple approaches, and lots of opportunities to talk about velocity, distance, average velocity, and acceleration.