Beginning a Short Unit on the Seasons

So, we are ending the year in inquiry by studying the seasons. We started by talking about the following situations:

(1) You are at a concert. What could you do to increase or reduce the impact of the sound on your ears?

(2) You are by a fire. What could you do to get hotter or colder?

After they muddled with those situations, I introduced a third test case.

(3) A person in the room has a smelly perfume? What could you that would make your experience of the smell more or less intense?

 The goal was to generalize a set of general patterns on what affects the intensity of “emanating stuff”. Our initial list was the following:

Volume (how strong the actual source of smell, sound, or heat is)

Proximity (how near are far you are from the source of smell, sound, or heat)

Duration (how much time you spend around the smell, sound, or heat)

Protection (how many barriers, blockers, or filters are between you and the source of heat, sound, smell)

We went into the detail explaining how these might work in each case, but that’s the gist.

An Experiment to Foster Thinking about A New Mechanism

Two identical heat lamps were set 10 inches away from a sheet of paper. Under the sheet of paper was a thermometer. The identical lamps fixed the volume. The 10 inches set the proximity. We set the duration time to 1 minute. The identical paper fixed the level of protection.  One lamp was set to shined directly down on the paper, and the other was set to shine at a very shallow angle (being careful to keep the 10 inch separation from the thermometer).

Students were asked to discuss what would happen to the temperature when I turned on the lamps.

Most groups believed correctly that the lamp shining straight down would make it hotter. Here is how we eventually built pieces of an explanation for why angle matters:

  • You are more likely to be burned by the sun from in the middle of the day, than the morning or evening, the sun’s rays must in some way be stronger when overhead than we angled low in the sky.
  • Direction also matters for our previous example. With fire, you can turn your cheeks toward fire to give it more direct access to fire’s heat. With a sound you can turn your ears away. With sound, you can turn your nose away.
  • With angled light, the rays of light hit the paper at a shallow angled creating a “glancing blow“, like skipping a stone on water, or a car hitting a wall at angle (vs. throwing a pebble straight into the water or driving you car head-on into a wall). The shallow angle creates only a glancing blow, which has less impact than a “head-on” collision.
  • With angled lamp, the light rays end up hitting a large area on the paper; where as the angled down rays hit the paper in a small area. This changes the concentration of the heat. It’s like heating up a large room or a small room with the same space heater. The large room will take longer to warm up, and may not even get up to same heat, because the heat gets spread out more.

We did the experiment, and in one minute the overhead lamp heated the thermometer to 130 degrees, while the angled lamp only heated up the thermometer to 78 degrees. Huge difference. I even rigged the deck in opposite way so that the angled lamp was actually closer than 10 inches and it got the thermometer that read a little higher. It was no contest. We added a new factor to our list, so that we now have:  Volume, proximity, duration, protection, and now direction.

Our goal over the next 3 days will be to figure out which of our 5 factors are most significant for explaining the seasonal changes to temperature–that is to collect evidence and arguments for the relative importance of each and to refine our sense of mechanism about how they work in the case of earth.

Why I’m liking this approach?

#1 We are drawing on knowledge from everyday experience : sitting by a fire to keep warm, smelling something rotten, being around loud music, etc. Had I asked what causes the seasons, it would have been about orbits and tilts. That would lead us down a frustrated track of sterile and unproductive school knowledge.

#2 We were generalizing quickly from a set of particulars, and naming them to help support generalization. We were not not just swimming in a vast sea of specific situations, and hoping that abstraction and connections were made. I specifically asked them to connect case specific mechanisms and to come up with general names.

#3 We are making sense of contrived situations in terms of everyday mechanism, such as getting burned, car crashes, skipping stones, and heating rooms. While I suggested the situations early on, students quickly extended to and built on other everyday sources of knowledge. This suggests that I helped “frame” the conversation as building on everyday knowledge. Going to the contrived could have tipped us out of, but it didn’t.

#4 Keeping the initial conversation away from the learning target (i.e., the seasons) and toward other phenomena (i.e., fire, sound, smell), keeps my “misconceptions” ears from perking up. Instead, I’m listening for useful ideas, analogies, observations, mechanism, insights, etc.  My listening patterns in turn influence my interactions with students, which in turn influences the nature of the discourse that emerges. My commitment and ability to focus on the good students say rather than the wrong stuff depends on the context I set up. I’m setting up a context, not only in which students will hopefully draw on everyday ideas productively, but I’m setting up a context in which I will be more likely to hear and draw on their ideas productively.

#5 I hope this will get us to “tilt” last, which is the empty vacuous understanding that many students have. Instead, I hope we will initially focus our explanations on locally observable changes, such as changing amounts of daylight and changing altitude of the sun in the sky. Tilt will be, hopefully, for the purpose of explaining the changing daylight and changing altitude. Thus, changing daylight and changing altitude will be the explanation for the seasonal variation in temperature.

Ways of Knowing…

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.”

Student Solutions from Today

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?

A misconception is just an insight without a productive place to go?

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.

What is force like?

The student quotes below are in response to the following prompt:

Explain why someone might think that objects can “have” force, or that you can “give” force to an object. Then explain why force is not like something you can have or give. In your own words, what is force like?

I’m curious about which is your favorite and why.

“Because people may be confused with the definition of momentum. They feel that if I give this object this much velocity because it has this much has then I can make it have a huge force when it impacts against something. You can’t give or have force because force is always there, there are different forces acting on everything. Force to me is this like a bully hat is always around and very active but you don’t notice until a bigger force is around to put that force in its place. Sort of… ok I’m not really good at explaining this.”

“Someone might think that objects have force because the objects are the source or cause of the reaction to the force. But, force is just a result from the movement or actions of an object and isn’t anything that an object can ‘have’ or ‘give’.”

“Someone can think that an object has force because when two objects collide they react to each other and this is motion is what people see as force. you cant have force or an item cant have force because force is the energy that is expelled when the items collide with each other. the energy that is expelled on the the second object is force and is only present when the items collide with each other.”

“Someone might think an object would have force because it contains  mass and could therefore put force onto another object. Force is not something you can have or give because it is just the attraction between two objects; it does not contain mass. I would describe force as a push or pull on an object that could cause it to move or accelerate due to some type of attraction between the objects.”

“Someone might think think that you can give force to an object because an object moves when someone pushes on it. You can’t give an item your force. When you put a force on an object it will move.”

“Someone might think that objects have force due to their mass when it pushes an object. Force depends on acceleration and mass, so without acceleration there would be no force even if it had a mass. With a constant velocity, the acceleration will equal 0 and there will be no force.

“One cannot have force because when one object exerts a force onto a second object the second object exerts a force of equal strength and opposite direction onto the first object. Force is like the ability to move objects.

“Someone might think that objects can have force, or that you can give force to an object because you can obviously push or pull something if you wanted to as well a heavy object pushing against you. However, force is not like something you can have or give because force is the direct interaction between two objects when a push or pull is done. You can not have force until it is acted upon an object. Therefore, force is an interaction between objects. The force of an object to another object is equal in opposite directions.”

“You could think that you can give something force because, when you apply force you may transfer it to the object. On the other hand force may just be being applied to the object. Force is what happens when two objects interact.”

“Force is not something you can just have or give away between two objects. It may be common to think it is due to the fact that is what many people have heard throughout their elementary science classes. However force is a relationship between two objects causing motion to occur. The way I think about force is if there is a heavy box on the ground and I am trying to push it I cant simply walk up to the box and touch it and expect it to move. I have to push with my legs against the ground and apply a force through my arms to push the box and cause a movement.”

Objects don’t have a force, a force is exerted on an object. People might think that you can give force to an object because when they push it or pull it them they think that is what would be considered force.”

“An example of someone thinking an object might have force would be pool. When you hit the que ball, you are giving it a force.

“Someone might think that an object “has” force if it doesn’t break when touched. For example, someone might say a chair is applying force when someone sits on it, since it doesn’t break. Someone might also think that by touching or pushing an object this adds force to it. Force is more of a measure to describe how the movement of an object changes – as the result of a change in the object’s mass or acceleration. It doesn’t describe what someone is doing to the object, but what happens as a result of someone’s contact with the object.”

“I think that people get force and momentum mixed up. momentum is determined by the mass and velocity of an object. A car at a low speed crashing into a wall is not going to cause as much force as for instance a 18 wheeler going at a higher speed. I think that’s what people think is force or that fact that you can give something force when really it is momentum. Also, I could apply force by pushing down on a button, but i am applying it not giving the button itself force. Force is an application or influence. It is not transferred, given, or something something already has.

“To me force is something that doesn’t happen until you exert it. If you push something you give it enough momentum to get to where its going, but if it hits something it will exert a force on that object and make that object go while the first object either slows down considerably or stops all together.”

Getting the most of out standard kinematics questions

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.

Distinctions: velocity at an instant, average velocity, and acceleration

This week’s online pre-class question:

An object is dropped from a height of 45m and takes 3 seconds to hit the ground. Explain why someone might think the object’s speed just before hitting the ground is 15 m./s. Then explain why that can’t be correct.

Three responses representing very different places students can be:

“First of all, wow! That’s the exact answer I had in mind and that is because if it’s dropped from a height of 45 meters and it takes 3 seconds to hit the ground you would want to divide the 45 meters by the 3 s to speed per second (15 m/s), but that is if it was going at a constant speed. So you also have to keep in mind that it was dropped at rest/zero so the speed will increase slowly not constant. I’m still confused.”

“Someone might think it is that because they would divide distance (45m) by time (3 s) which would come out to be 15 m/s. But that would be the average speed.  To find the final speed you would take the initial speed (0 m/s) and add it to the acceleration (9.8 m/s^2) multiplied by the time (3 seconds). The final speed of the ball before it hits the ground would be 29.4 m/s.”

“Because most people would think just divided 45 into 3 to get 15m/s but we haven’t put in  our minds about the acceleration of gravity, which is 9.8m/s that can round up to 10m/s then if you was to times 10m/s by 3s you would get 30m not 45m.”

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