In my teaching of Physics Class, we read chapter 2 of Aron’s Teaching Introductory Physics. As part of a reflection on that reading, I asked students to give a verbal interpretation of instantaneous velocity, much in the same way Aron’s gave a verbal interpretation for Average Velocity.

Here is what I got:

I think that “a velocity an object has at a given instant in time” works fine. Perhaps there’s more that I’m missing?

Instantaneous velocity is looking at a specific position that we have matched with our clock reading, and if we look at this instant, it’s the velocity that we would need to have for this object to be in this position from our arbitrary starting point.

Instantaneous velocity is the velocity of an object per unit time, such as speedometer reads.

Instantaneous velocity is the velocity of an object at a given clock reading. It is the rate of an object’s change in position at that time interval, and it represents that the velocity the object would have if the acceleration stopped at that point.

Instantaneous velocity is simply the velocity of an object at an exact time–the amount of distance covered in a given time segment if the speed were constant and acceleration zero.

Instantaneous velocity is how far an object would go in one second assuming that it didn’t change velocity.

It’s a mixed bag here, as to be expected… But I’m getting the creeping feeling this that is a sign of bigger things that really transcend the particular question and the particular responses. And I know I’ve felt this way before. See, last spring, when I was teaching an Learning Assistants course, I came to the conclusion that I couldn’t teach a “science pedagogy” course well without doing some science with the LAs. So, last spring I changed things up at some point, and we began spending half our time doing science, and then half our time talking about the science we were doing in light of the readings we were doing. In the LA class, I might have gone too far toward the side of doing science (in feedback, several students told me the pedagogy course was the best science course they’ve ever taken), but I don’t necessarily think that was a bad thing.

So, I think am reaching the same conclusion about this physics teaching course. I question, How I can teach a teaching physics course when students haven’t experienced (what I would see as) meaningful and transformative engagement with their own physics learning? I also think, that even if they had come in with such learning experiences, it would still be worthwhile to do some physics together, so that that common experience can become a thing we can point to in our immediate and shared lives. But how, and how much, and when? Maybe these are thoughts for another semester… or maybe they’re not.

Here is a sample problem I’m supposed to do for my class:

You are standing on the edge of the roof of a building. 8.0 m from the base of the building is the wall of a 5.0-m tall garage with a flat roof. You throw a ball with an initial speed of 8.3 m/s at a direction that is 42o above the horizontal. You release the ball 11 m above the base of the building.

Here is the problem I’ve decided to do instead.

You are standing on the edge of the roof of a building. 26 m from the base of the building is the wall of a 5.0-m tall garage with a flat roof. You throw a ball with an initial speed of 16.5 m/s at a direction that is 37° above the horizontal. You release the ball 20m above the base of the building.

Why these changes?

(1) So, The 16.5 m/s at 37° gets me an initial vertical speed of ~10 m/s, which is convenient for discussing change in velocity when the acceleration due to gravity is ~10 m/s/s. It also gets me an integer horizontal speed of ~13 m/s, which makes it convenient for talking about how far the ball will go each second.

(2) The 26 meters is used because the question of whether the ball get far enough to land on the adjacent roof is easier to think through with resorting to equations. The questions becomes “Does the ball stay in the air for at least 2 seconds (before it drops 15 meters)?”  Well, the ball lose it’s 10 m/s up in the first second, and gains back that 10 m/s (going down now) in the next second. So horizontally, the ball has gone the 26m by the time it’s at the same vertical height it started (now going down) leaving time to land further into the adjacent roof.

Now that’s we’ve made sense of what’s happening without equations, we can figure the rest out the way I’m supposed to model  the situation (even though reasoning it through is still really easy at this point). We really just need to figure out how much more time it spends in the air (1 seconds here), and then add 13m for ever additional second it gets to go (13 extra meters here)

What do people think? Is it worth it? Am I doing my students any disservice by trying to give them workable numbers?

So Many Class Ideas to Explain Box Theatre…

Bouncing, then Projecting Idea

Light goes from outside the box to inside the box through the hole. It then bounces off the paper and into the eye. Then our eyes project the image (that our Brain interprets) back onto the screen.

Extracting, then Stamping Idea

Light shines from the sun onto whatever object is outside (like a tree), and the sun light extracts color from the object. The light with extracted color goes into the box and “stamps” the inside of box with the image. But the light stops there, and doesn’t go toward the eye. Our eyes just see the light that is stopped on the box.

Hole Dependency Idea

The image is upside-down only because the hole we’ve made is at the top of the box, which forces everything to go down and invert on the screen. If the hole were at the bottom, maybe something different would happen.

Brain Flipping Ideas

We’ve heard it’s true that our eyes and our brains flip images: So, maybe the image is actually right-side up and our Brain is being tricked into not flipping it, or flipping it when it shouldn’t. OR Maybe the image is really upside-down but our eye/brain only flips thing with direct light, not reflected light, so our Brain isn’t flipping it.

Like a Mirror Idea

The box seems be similar in some ways to a mirror reflection. However, when you look into a mirror, things only get left-right reversed (not also upside down reversed). Also against this idea is the question, how can paper act like a mirror? It’s just paper, not a mirror. Out in open space, the paper just looks white. If paper can act like a mirror, would it be possible to use a piece of paper as a rear-view mirror if you created a hole and a dark space around it? Also, is there something special about white paper, or will this work just as well with the cardboard box and no paper.

Upside Down is Already Right-Left Backwards

If you take anything and turn upside down by turning it (clock-wise or counter clock-wise), left-and-right get mixed up as well? So upside down can already imply left-right backwards… it’s doesn’t have to be “flipped twice”. One flip can cause both kinds of backwards. … Question: Could the pin hole really be “turning” the light around?

Bouncing vs. Inverted Bouncing Discussion

If a Wilson ball bounced off a wall nothing would get inverted? It would just bounce, and be the same ball. So, how can bouncing cause things to flip? There must be an “inverting kind of bounce”–This led some to wonder if this is what the word refraction means, a bounce that also inverts?

But, If that wilson ball had paint painted on words, then it would leave an imprint on the wall of the word Wilson that was imprinted left-right backwards. Is that light an inverting bounce? Is that how the image gets stamped onto the back wall… However, this wouldn’t explain upside down part, only the left right

However, if with the wilson ball imprint on the board, you were standing on the opposite side of the wall, like looking from behind the wall, then it would look correct to you.

This led us to want to construct a double sided box (maybe with a translucent paper), to see if we could see image from different perspectives.

Like a Curved Spoon idea

Being upside down reminds us of looking in the inside of spoons, where you can appear upside down. Maybe this would explain why distance seems to matter, because it seems to matter with the spoon.

Crossing Light Idea

Things are upside down because light crosses through the hole. Light from top of object goes down through the hole. Light from bottom of objects goes up through the hole. At the hole, they cross, causing the image to be upside down. This group added that they like this idea because it didn’t appeal to how our Brain works, and seemed simple.

Snippet from Student reflection about Pendulum Video and Case Study:

“…Also, should the teacher have stopped the discussion earlier? Maybe about 10 minutes after they had gotten the right answer instead of continuing for an entire 45 minutes. My only thinking for this is because they could be thinking too hard and I feel like they continued to steer away from the correct answer more and more. She either should have stopped it earlier or helped them steer in the right direction.

I hear two great questions here: How long is too long to keep on discussing the same thing? And when should a teacher steer the class toward the right answer? So, I don’t have any answers for you. But I do have questions that I think are worth thinking about:

First, what in the video did you see the children doing that made you to think or feel the discussion needed to end earlier? In other words, can you support your claim with evidence from the video?

Second, I wonder if the decision to steer the children toward the correct answer or not depends on the “goal” of the discussion. If the goal of the discussion is only to get to the right answer, than I think makes sense to make sure that we all get “there”–to the right answer.  But, now I’m curious: What do you think the goals of the lesson might have been, and what would it mean to make sure all the children got “there”?

Yesterday, in my physics class, we discussed this question

A ball is tossed vertically downward from the same height that an identical ball is also dropped. Which ball will gain more speed before hitting the ground? [Ignore friction, air resistance, etc]

Here were our ideas:

“Almost to Top Speed Already”: The answer may depend on the exact height and the speed. However, since the thrown ball might already be close to the maximum speed of falling, it might not change all that much, while the dropped ball will have a lot of speed to gain to get up to that speed. The dropped ball gains more speed”

“The Rich get Richer”: If you start with more speed, your going to accumulate more speed. So the thrown ball will gain more speed.

“Gravity is the Same”: While the thrown ball will have a greater final speed when it lands, the changes in speed will be the same because the acceleration due to gravity is the still the same constant acceleration. The two gain the same amount of speed.

“It takes Time to Speed Up”: Acceleration happens over time. The thrown ball spend very little time in the air because its moving fast, and the dropped ball takes more time because it started slow. The dropped ball having more time, gains more speed.

What do I think of these ideas?

I’m not going to lie. In the “speed limit” idea, I love their close attention to discriminating change of speed, initial speed, and final speed. The students are wrong about the final speed (and about how objects freely fall), but that shouldn’t detract us from seeing some underlying value in the reasoning they are doing. Students easily confuse velocity and change in velocity, but these students are keeping that distinction clear here. Their reasoning would be correct, if the world worked a different way.

I’m not going to lie. I also love the “rich get richer idea”, because lots of things in the world work like this. It is the conceptual underpinning of exponential growth. The thinking applied here certainly contradicts the idea that constant acceleration is a linear accumulation model. I noticed as an instructor that we’ve seen thinking consistent with this idea before. As an instructor, I was also thinking, “I’m going to want this kind of reasoning later”, which is why I helped students give it a name: “The rich get richer.” It happens to be wrong in this situation, but the thinking is so useful, I couldn’t let is slip by unattended to.

I’m not going to lie, I’m torn about the gravity is the same idea. For some students, I think it might have been just an unthoughtful knee-jerk reaction that “gravity is always the same.” However, there is good reasoning here, especially for those students who were actually reasoning through this, thereby making a connection between acceleration and change in velocity, and also distinguishing velocity and change in velocity. Earlier in the year, most students though “more velocity means more acceleration”. So, the fact that they were seeing acceleration as connected to change in velocity (not just having velocity) is progress. This idea, would be true, if acceleration accumulated over distance.

The last idea is correct (and subtle). In this case, I love it, because of the way it happened in class. A student, who had initially thought that two would gain the same, was listening to the another student explain why the faster one would probably gain more, when she suddenly bursted out loud with excitement to share her idea that it’s not “the faster you start, the more speed you’ll gain”, but it’s  “the more time you have to speed up, the more speed you can gain”. That idea came spontaneously, and you could hear in her voice the certainty and joy she was experiencing about this new idea.

The question itself

I’ve been asking this question for a few years now. I actually don’t know where I got it from. It’s a great discussion question, which I suggest you save in your pocket until after students have spent a fair amount of time discussing acceleration and free-fall. It’s subtle and fun, and easily trips up members of the human race, even if they be physics majors, physics graduate students, or physics professors.

The question itself also seems to spontaneously generates new questions that students begin to discuss. For example, a group of students came to realize that the answer would be the same if the question had asked, “After 3 seconds of falling, how does the gain in speed compare?” because the two would gain the same amount of speed in the same time. In working through this, students were making contact with rabbit hole of ideas that Galileo struggled with.  In fact, on this blog, I’ve talked about what the world would be like if constant acceleration meant dv/ds rather than dv/dt, that is, if velocity accumulated in equal amounts over equal distances.

We are all struggling with the same ideas: Galileo, my students, and yours truly.

I asked students to write about what forces must be balanced when a box is pushed across a floor at constant speed. Here is what one student wrote.

At a constant speed, the force of gravity pulling the object down must balanced the force of the floor pushing the object up.

Do balanced forces have to be equal or “canceling out” forces? In the first question, if I’m constantly pushing the object at the same speed then the my force on the object is higher then the force of friction on the object. but my force stays constant and the lesser force of friction stays constant. I wanted to add this to my anwser on the first question but was a bit confused.

This is so important. The first question invoked in this student a response like, “Get down a correct thing you read from the text, and don’t risk getting it wrong by putting down what you actually think.” The second question invoked a different response, more like “Tell the instructor what you are really thinking and ask a question about how that thinking relates to the reading.

As instructors, these are some question we should always be thinking about: How are our students framing tasks of responding to the questions we pose? Are they looking to get correct stuff down and avoid getting incorrect stuff down, or are they honestly engaging in a dialogue between their ideas and the ideas they are encountering. How do we know whether its one or the other?  How do support a culture of the second?

So, this time in my inquiry class, I began the class by spending more time framing the class in terms of my students’ future careers as elementary school teachers. On the first day, among other things, we discussed what we thought would happen when a pendulum is let go just at the top of its swing,  then we watched and discussed a video of a 4th grade class discussing the same question, and finally we read for homework a case study written by the teacher of that class.

Here is what one student wrote in their response:

“I admit, as we talked about this in class, I thought the discussion was a little juvenile. I knew what was going to happen when the pendulum was finally dropped, and my reasoning was sound, but people kept bringing up different hypotheses that were wrong! It irritated me that this was the case. However, after watching the video and reading the case study I realize that I missed the whole point–which is, I am sure, what Brian was trying to teach me; it’s all about knowing how and why your students think, and nothing about what is exactly the right answer…The process itself must not be overlooked, and that is where I faltered. It was my own preconceived notion of what to expect in this class and this particular lesson that skewed me from, at first, seeing the value of what we were doing… [The way the lesson was taught] allowed the opportunity to think and allow arguments to be constructed and torn down”

So, we’ve been spending time in my teaching physics course examining artifacts and phenomena of student thinking. We’re going to continue doing that, but I want to shift toward, “OK, so we know some things about how student think. Now what?”

My plan is to introduce three broad ways of thinking about how student thinking fits into the classroom, which are not necessarily distinct.

Curriculum or instructional sequences that are based on knowledge of how student tend to think about some specific content or concepts. We might think of Tutorials in Introductory Physics (as curricular materials) or even Clement’s bridging analogy (as instructional sequence) as good examples.

In this case, as an instructor you might be largely relying on the knowledgeability of some third party to have built curriculum and instructional sequences based on student thinking.

Instructional practices that make students’ thinking visible to the instructor (and often students as well), which can help instructors make informed decisions about instruction. We might think of Peer Instruction (to instructor and students) or JiTT (just to the instructor) as good examples.

In this case, as an instructor you are hoping to create opportunities to learn about your students’ thinking in your class, not rely solely on patterns of how student thinking generally. As an instructor, you may of course be relying on a third party to have provide good “questions” or “demonstrations,” that are based on student thinking and that will be likely to help you learn about your students’ thinking.

A classroom structure in which student thinking and their ideas become part of the substance of the course, such that students are authors (not just consumers) of content. We might think of classrooms similar to the ones depicted by David Hammer in “Discovery Learning and Discovery Teaching“,  those being pursued by Leslie Atkins through Student-Generated Scientific Inquiry, or over by Mylene at Shifting Phases.

Of course, these aren’t distinct in practice. I find that I draw from each of these, and the extent to which I draw on these varies across the different classrooms I teach. I’m also not saying that each of the examples above are defined by the category. There are other features of the curriculum and instruction that are important in making them successful, which make them similar or different from another, etc. I’m just trying to introduce an initial framework for these future teachers to think about how student thinking can interface with classroom instruction.

Anyway, Today, I’m going to introduce these three ways of thinking about the roles that student thinking can play in the classroom, but then we’re going to dig into different examples of the first one.

On the docket are three flavors of the first: “Elicit, Confront, Resolve”, “Bridging Analogies”, and “Refining Intuitions”

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.

Here are some more student responses to this question, which I posted before. Each of these has interesting nugget to ponder over.

If it only took 3 seconds for the ball to drop 45 meters, it would make sense for the ball to move 15 meters every second. However, an object cannot fall faster than 9.8m/s if it has simply been dropped.

Because 15m/s is the average speed. The truth is that while this object is being dropped it accelerates 9.8m/s*s every time so the object speed probably reached a little more than 15 m/s at some point.

It would be assumed this is correct because 45m divided by 3s gives you 15m/s. This is incorrect because acceleration is calculated in m/s/s and therefore every second the ball falls its it accelerates 9.8 m/s so by the time the ball hits the ground it is traveling 58.8 m/s, traveling 9.8m/s after the first second, 19.6m/s after the second second, and 58.8m/s at the end of the third second.

A person would think that because they simply would divided 3 seconds by 45 meters. The reason this wrong because the question is asking for average speed. The average speed is the total time distance travel divided the total time.

One might this that the speed is 15 m/s because velocity is equal to the distance divided by times and 45 divided by 3 is 15. I’m not sure as to why this isn’t true but my guess would be that since the ball is being dropped it is in free fall which would means the acceleration of the ball is constant. Also, another thought is that since the object is falling in a downward position the acceleration would be negative so this could also depict a negative speed. But, honestly, I’m not sure. Please explain more in class.