physics

Geometric Optics – hitting complexity first

I started what may end up being the last unit in physics with the idea that I would do things differently compared to my usual approach. I taught optics as part of Physics B for a few years, and as many things end to be in that rushed curriculum, it was fairly traditional. Plane mirrors, ray diagrams, equations. Snell’s law, lenses, ray tracing, equations. This was followed by a summary lesson shamefully titled “Mirrors and lenses are both similar and different” , a tribute to the unfortunate starter sentence for many students’ answers to compare and contrast questions that always got my blood boiling.

This time, given the absence of any time pressure, there has been plenty more space to play. We played with the question of how big a plane mirror must be to see one’s whole body with diagrams and debate. We messed with a quick reflection diagram of a circular mirror I threw together in Geogebra to show that light seems to be brought to a point under certain conditions. Granted, I did make suggestions on the three rays that could be used in a ray diagram to locate an image – that was a bit of direct instruction – but today when the warm up involved just drawing some diagrams, they had an entry point to start from.

After drawing diagrams for some convex and concave mirrors, I put a set of mirrors in front of them and asked them to set up the situation described by their diagrams. They made the connection to the terms convex and concave by the labels printed on the flimsy paper envelopes they were shipped in – no big introduction of the vocabulary first was needed, and it would have broken the natural flow of their work. They observed images getting magnified and minefied, and forming inverted or upright. They gasped when I told them to hold a blank sheet of paper above a concave mirror pointed at one of the overhead lights and saw the clear edges of the fluorescent tubes projected on the paper surface. They poked and stared, mystified, while moving their faces forward and backward at the focal point to find the exact location where their face shifted upside down.

After a while with this, I took out some lenses. Each got two to play with. They instantly started holding them up to their eyes and moving them away and noticing the connections to their observations with the mirrors. One immediately noticed that one lens flipped the room when held at arms length but didn’t when it was close, and that another always made everything smaller like the convex mirror did. I asked them to use the terms virtual and real, and they were right on. They were again amazed when the view outside was clearly projected through the convex lens was held in front of a student’s notebook.

I hope I never take for granted how great this small group of students is – I appreciate their willingness to explore and humor me when I am clearly not telling them everything that they need to know to analyze a situation. That said, there is really something to the backwards model of presenting complexity up front, and using that complexity to motivate students to want to understand the basics that will help them explain what they observe. Now that my students see that the lenses are somehow acting like mirrors, it is so much easier to call upon their curiosity to motivate exploring why that is. Now there is a reason for Snell’s law to be in our classroom.

Without planting a hint of why anyone aside from over excited physics teachers would give a flying fish about normals and indices of refraction, it becomes yet one more fact to remember. There’s no mystery. To demand that students go through the entire process of developing physics from basic principles betrays the reality that reverse engineering a finished product can be just as enlightening. I would wager that few people read an instruction manual anymore. Even the design of help in software has changed from a linear list of features in one menu after another to a web of wiki-style tidbits of information on how to do things. Our students are used to managing complexity to do things that are not school related, things that are a lot more real world to them. There is no reason school world has to be different from real world in how we explore and approach learning new things.

Building meaning for momentum from discussions, definitions, and data.

Today we started our next unit in physics with a ‘next time question’ from Paul Hewitt:

My reason for giving this was specifically because of the fact that we haven’t learned anything about it. I wanted the students to speak purely from their intuition. I asked them the following:

We aren’t quite ready to answer this by calculation, but I do want you to make a guess.

Will they move together faster than, slower than, or with the same speed as the ball?

Would your answer change if the ball bounced off Jocko instead of him catching it?

Student responses included:

We need to know if he bends backwards when he catches it, because that will affect it.
No matter how he does catch it, he will move slower. The larger mass will result in a smaller acceleration.
The clown has a non-conservative force, so the kinetic energy will decrease.

Interesting responses. We talked a bit about collisions and throws and catches of objects and what they ‘felt’ doing this with different objects. I introduced the idea that it might be nice to have a physics quantity that contains the direction and rate information of velocity, as well as the mass. I told them that physicists did, in fact, have such a quantity called momentum. They responded with a few non-physics related ways they had heard the term and described what it meant.

To figure things out about how momentum relates to collisions, I then had them analyze the three air track collision videos from the Doane Physics video library using Tracker. Their tasks were as follows:

Find the momentum of each cart before and after the collision for the video you are assigned. Calibration information is contained in the first frame of each video.
Find the total momentum of the system before and after the collision.
Find the total kinetic energy of the system before and after the collision.
What is thechange of the momentum of the system during the collision?
What is the change of the kinetic energy of the system during the collision?
Talk to your classmates and compare your answers for the three different videos.

It was pretty cool to see them jump in with Tracker and know how to analyze things without too much trouble. Fairly soon afterwards, we had some initial velocities and final velocities, and changes in momentum to compare.

I was, of course, leading them toward something with the change calculations.

We calculated the changes in momentum, which were non-zero. Were the magnitudes significant? A student suggested looking at the percent change compared to the initial momentum. For the first two videos, the loss was less than 1%, though for the third it was around 20%.

A student proposed the possibility that the change should be zero if no momentum is lost during the transfer. Comments were made about how that made sense in the context of our previous unit on energy – things feeling right when all of a quantity can be accounted for.

I then did a little pushing (since we were almost out of time) about what this might mean about total initial momentum and total final momentum. I also gave them definitions for elastic and inelastic collisions. I then assigned them a couple simple questions that I wanted them to figure out if we can say that the change in total momentum before and after is zero:

Then it was time for Calculus.

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I don’t usually like giving students information. I don’t like giving it away without some sense of where it comes from. I also like when students can discover quantities without equation definitions. Sometimes though, the simplicity of an idea like momentum and its power can come from taking the calculation itself as a tool that can be used to analyze a situation.

In previous classes, I have given the definition, shown situations in which momentum is conserved, and then asked students to use this idea of momentum conservation with their math skills to find unknown quantities. I really liked this alternate approach today of using momentum itself to analyze a situation and then have the idea of conservation come out of discussion. I think its potential for ‘stickiness’ in the minds of students is much greater this way.

Relating modeling & abstraction on two wheels.

Over the course of my vacation in New Zealand, I found myself rethinking many things about the subjects I teach. This wasn’t really because I was actively seeing the course concepts in my interactions on a daily basis, but rather because the sensory overload of the new environment just seemed to shock me into doing so.

One of these ideas is the balance between abstraction and concrete ideas. Being able to physically interact with the world is probably the best way to learn. I’ve seen it myself over and over again in my own classes and in my own experience. There are many situations in which the easiest way to figure something out is to just go out and do it. I tried to do this the first time I wanted to learn to ride a bicycle – I knew there was one in the garage, so I decided one afternoon to go and try it out. I didn’t need the theory first to ride a bicycle – the best way is just to go out and try it.

Of course, my method of trying it was pretty far off – as I understood the problem , riding a bicycle first required that you get the balancing down. So I sat for nearly an hour rocking from side to side trying to balance.

My dad sneaked into the garage to see what I was up to, and pretty quickly figured it out and started laughing. He applauded my initiative in wanting to learn how to do it, but told me there is a better way to learn. In other words, having just initiative is not enough – a reliable source of feedback is also necessary for solving a problem by brute force. That said, with both of these in hand, this method will often beat out a more theoretical approach.

This also came to mind when I read a comment from a Calculus student’s portfolio. I adjusted how I presented the applications of derivatives a bit this year to account for this issue, but it clearly wasn’t good enough. This is what the student said:

Something I didn’t like was optimisation. This might be because I wasn’t there for most of
the chapter that dealt with it, so I didn’t really understand optimisation. I realise that optimisation applies most to real life, but some of the examples made me think that, in real life, I would have just made the box big enough to fit whatever needed to fit inside or by the time I’d be done calculating where I had to swim to and where to walk to I could already be halfway there.

Why sing the praises of a mathematical idea when, in the real world, no logical person would choose to use it to solve a problem?

This idea appeared again when reading The Mathematical Experience by Philip J. Davis and Reuben Hersh during the vacation. On page 302, they make the distinction between analytical mathematics and analog mathematics. Analog math is what my Calculus student is talking about, using none of “the abstract symbolic structures of ‘school’ mathematics.” The shortest distance between two points is a straight line – there is no need to prove this, it is obvious! Any mathematical rules you apply to this make the overall concept more complex. On the other hand, analytic mathematics is “hard to do…time consuming…fatiguing…[and] performed only by very few people” but often provides insight and efficiency in some cases where there is no intuition or easy answer by brute force. The tension between these two approaches is what I’m always battling in my mind as a swing wildly from exploration to direct instruction to peer instruction to completely constructivist activities in my classroom.

Before I get too theoretical and edu-babbly, let’s return to the big idea that inspired this post.

I went mountain biking for the first time. My wife and I love biking on the road, and we wanted to give it a shot, figuring that the unparalleled landscapes and natural beauty would be a great place to learn. It did result in some nasty scars (on me, not her, and mostly on account of the devilish effects of combining gravity, overconfidence, and a whole lot of jagged New Zealand mountainside) but it was an incredible experience. As our instructors told us, the best way to figure out how to ride a mountain bike down rocky trails is to try it, trust intuition, and to listen to advice whenever we could. There wasn’t any way to really explain a lot of the details – we just had to feel it and figure it out.

As I was riding, I could feel the wind flowing past me and could almost visualize the energy I carried by virtue of my movement. I could look down and see the depth of the trail sinking below me, and could intuitively feel how the potential energy stored by the distance between me and the center of the Earth was decreasing as I descended. I had the upcoming unit on work and energy in physics in the back of my mind, and I knew there had to be some way to bring together what I was feeling on the trail to the topic we would be studying when we returned.

When I sat down to plan exactly how to do this, I turned to the great sources of modeling material for which I have incredible appreciation of being able to access , namely from Kelly O’Shea and the Modeling center at Arizona State University. In looking at this material I have found ways this year to adapt what I have done in the past to make the most of the power of thinking and students learning with models. I admittedly don’t have it right, but I have really enjoyed thinking about how to go through this process with my students. I sat and stared at everything in front of me, however – there was conflict with the way that I previously used the abstract mathematical models of work, kinetic energy, and potential energy in my lessons and the way I wanted students to intuitively feel and discover what the interaction of these ideas meant. How much of the sense of the energy changes I felt as I was riding was because of the mathematical model I have absorbed over the years of being exposed to it?

The primary issue that I struggle with at times is the relationship between the idea of the conceptual model as being distinctly different from mathematics itself, especially given the fact that one of the most fundamental ideas I teach in math is how it can be used to model the world. The philosophy of avoiding equations because they are abstractions of the real physics going on presumes that there is no physics in formulating or applying the equations. Mathematics is just one type of abstraction.

A system schema is another abstraction of the real world. It also happens to be a really effective one for getting students to successfully analyze scenarios and predict what will subsequently happen to the objects. Students can see the objects interacting and can put together a schema to represent what they see in front of them. Energy, however, is an abstract concept. It’s something you know is present when observing explosions, objects glowing due to high temperature, baseballs whizzing by, or a rock loaded in a slingshot. You can’t, however, observe or measure energy in the same way you can measure a tension force. It’s hard to really explain what it is. Can a strong reliance on mathematics to bring sense to this concept work well enough to give students an intuition for what it means?

I do find that the way I have always presented energy is pretty consistent with what is described in some of the information on the modeling website – namely thinking about energy storage in different ways. Kinetic energy is “stored” in the movement of an object, and can be measured by measuring its speed. Potential energy is “stored” by the interaction of objects through a conservative force. Work is a way for one to object transfer energy to another through a force interaction, and is something that can be indicated from a system schema. I haven’t used energy pie diagrams or bar charts or energy flow diagrams, but have used things like them in my more traditional approach.

The main difference in how I have typically taught this, however, is that mathematics is the model that I (and physicists) often use to make sense of what is going on with this abstract concept of energy. I presented the equation definition of work at the beginning of the unit as a tool. As the unit progressed, we explored how that tool can be used to describe the various interactions of objects through different types of forces, the movement of the objects, and the transfer of energy stored in movement or these interactions. I have never made students memorize equations – the bulk of what we do is talk about how observations lead to concepts, concepts lead to mathematical models, and then models can then be tested against what is observed. Equations are mathematical models. They approximate the real world the same way a schema does. This is the opposite of the modeling instruction method, and admittedly takes away a lot of the potential for students to do the investigating and experimentation themselves. I have not given this opportunity to students in the past primarily because I didn’t know about modeling instruction until this past summer.

I have really enjoyed reading the discussions between teachers about the best ways to transition to a modeling approach, particularly in the face of the knowledge (or misinformation) they might already have . I was especially struck by a comment I read in one of the listserv articles by Clark Vangilder (25 Mar 2004) on this topic of the relationship between mathematical models and physics:

It is our duty to expose the boundaries between meaning, model, concept and representation. The Modeling Method is certainly rich enough to afford this expense, but the road is long, difficult and magnificent. The three basic modeling questions of “what do you see…what can you measure…and what can you change?” do not address “what do you mean?” when you write this equation or that equation…The basic question to ask is “what do you mean by that?,” whatever “that” is.

Our job as teachers is to get students to learn to construct mental models for the world around them, help them test their ideas, and help them understand how these models do or do not work. Pushing our students to actively participate in this process is often difficult (both for them and for us), but is inevitably more successful in getting them to create meaning for themselves on the content of what we teach. Whether we are talking about equations, schema, energy flow diagrams, or discussing video of objects interacting with each other, we must always be reinforcing the relationship between any abstractions we use and what they represent. The abstraction we choose should be simple enough to correctly describe what we observe, but not so simple as to lead to misconception. There should be a reason to choose this abstraction or model over a simpler one. This reason should be plainly evident, or thoroughly and rigorously explored until the reason is well understood by our students.

Rubrics & skill standards – a rollercoaster case study.

I gave a quiz not long ago with the following question adapted from the homework:

The value of 5 points for the problem came from the following rubric I had in my head while grading it:

+1 point for a correct free body diagram
+1 for writing the sum of forces in the y-direction and setting it equal to ma_y
+2 for recognizing that gravity was the only force acting at the minimum speed
+1 for the correct final answer with units

Since learning to grade Regents exams back in New York, I have always needed to have some sort of rubric like this to grade anything. Taking off random quantities of points without being able to consistently justify a reason for a 1 vs. 2 point deduction just doesn’t seem fair or helpful in the long run for students trying to learn how to solve problems.

As I move ever more closely toward implementing a standards based grading system, using a clearly defined rubric in this way makes even more sense since, ideally, questions like this allow me to test student progress relative to standards. Each check-mark on this rubric is really a binary statement about a student relative to the following standards related questions:

Does the student know how to properly draw a free body diagram for a given problem?
Can a student properly apply Newton’s 2nd law algebraically to solve for unknown quantities?
Can a student recognize conditions for minimum or maximum speeds for an object traveling in a circle?
Does a student provide answers to the question that are numerically consistent with the rest of the problem and including units?

It makes it easy to have the conversation with the student about what he/she does or does not understand about a problem. It becomes less of a conversation about ‘not getting the problem’ and more about not knowing how to draw a free body diagram in a particular situation.

The other thing I realize about doing things this way is that it changes the actual process of students taking quizzes when they are able to retake. Normally during a quiz, I answer no questions at all – it is supposed to be time for a student to answer a question completely on their own to give them a test-like situation. In the context of a formative assessment situation though, I can see how this philosophy can change. Today I had a student that had done the first two parts correctly but was stuck.

Him: I don’t know how to find the normal force. There’s not enough information.

Me: All the information you need is on the paper. [Clearly this was before I flip-flopped a bit.]

Him: I can’t figure it out.

I decided, with this rubric in my head, that if I was really using this question to assess the student on these five things, that I could give the student what was missing, and still assess on the remaining 3 points. After telling the student about the normal force being zero, the student proceeded to finish the rest of the problem correctly. The student therefore received a score of 3/5 on this question. That seems to be a good representation about what the student knew in this particular case.

Why this seems slippery and slopey:

In the long term, he doesn’t get this sort of help. On a real test in college, he isn’t getting this help. Am I hurting him in the long run by doing this now?
Other students don’t need this help. To what extent am I lowering my standards by giving him information that others don’t need to ask for?
I always talk about the real problem of students not truly seeing material on their own until the test. This is why there are so many students that say they get it during homework, but not during the test – in reality, these students usually have friends, the teacher, example problems, recently going over the concept in class on their side in the case of ‘getting it’ when they worked on homework.

Why this seems warm and fuzzy, and most importantly, a good idea in the battle to helping students learn:

Since the quizzes are formative assessments anyway, it’s a chance to see where he needs help. This quiz question gave me that information and I know what sort of thing we need to go over. He doesn’t need help with FBDs. He needs help knowing what happens in situations where an object is on the verge of leaving uniform circular motion. This is not a summative assessment, and there is still time for him to learn how to do problems like this on his own.

This is a perfect example of how a student can learn from his/her mistakes. It’s also a perfect example of how targeted feedback helps a student improve.

For a student stressed about assessments anyway (as many tend to be) this is an example of how we might work to change that view. Assessments can be additional sources of feedback if they are carefully and deliberately designed. If we are to ever change attitudes about getting points, showing students how assessments are designed to help them learn instead of being a one-shot deal is a really important part of this process.

To be clear, my students are given one-shot tests at the end of units. It’s how I test retention and the ability to apply the individual skills when everything is on the table, which I think is a distinctly different animal than the small scale skills quizzes I give and that students can retake. I think those are important because I want students to be able to both apply the skills I give them and decide which skills are necessary for solving a particular problem.

That said, it seems like a move in the right direction to have tried this today. It is yet one more way to start a conversation with students to help them understand rather than to get them points. The more I think about it, the more I feel that this is how learning feels when you are an adult. You try things, get feedback and refine your understanding of the problem, and then use that information to improve. There’s no reason learning has to be different for our students.

From projectile motion to orbits using Geogebra

I was inspired last night while watching the launch of the Mars Science Laboratory that instead of doing banked curve problems (which are cool, but take a considerable investment of algebra to get into) we would move on to investigating gravity.

The thing that took me a long time to wrap my head around when I first studied physics in high school was how a projectile really could end up orbiting the Earth. The famous Newton drawing of the cannon with successively higher launch velocities made sense. I just couldn’t picture what the transition looked like. Parabolas and circles (and ellipses for that matter) are fundamentally different shapes, and at the time the fact that they were all conic sections was too abstract of a concept for me. Eventually I just accepted that if you shoot a projectile fast enough tangentially to the surface of the Earth, it would never land, but I wanted to see it.

Fast forward to this afternoon and my old friend Geogebra. There had to be a way to give my physics students a chance to play with this and perhaps discover the concept of orbits without my telling them about it first.

You can download the sketch I put together here.

The images below are the sorts of things I am hoping my students will figure out tomorrow. From projectile motion:

…to the idea that it is still projectile motion when viewed along with the curvature of the planet:

Continuing to adjust the values yields interesting results that suggest the possibility of how an object might orbit the Earth.

If you open the file, you can look at the spreadsheet view to see how this was put together. This uses Newton’s Law of Gravitation and Euler’s method to calculate the trajectory.You can also change values of the variable deltat to predict movement of the projectile over longer time intervals. There is no meaning to the values of m, v0, or height – thankfully the laws of nature don’t care about units.

As is always the case, feel free to use and adjust this, as well as make it better. My only request – let me know what you do with it!

A smattering of updates – the good with the bad.

I want to record a few things about the last couple of days of class here – cool stuff, some successes, some not as good, but all useful in terms of moving forward.

Geometry:

I have been working incredibly hard to get this class talking about their work. I have stood on chairs. I’ve given pep talks, and gotten merely nods of agreement from students, but there is this amazing resistance to sharing their work or answering questions when it is a teacher-centric moment. There are a couple students that are very willing to present, but I almost think that their willingness overshadows many others who need to get feedback from peers but don’t know how to go about it. What do I do?

We turn it into a workshop. If a student is done, great. I grab the notebook and throw it under the document camera, and we talk about it. (In my opinion, the number one reason to have a document camera in the classroom, aside from demonstrating lab procedures in science, is to make it easy and quick for students get feedback from many people at once. Want to make this even better and less confrontational? Throw up student work and use Today’s Meet to collect comments from everyone.

The most crucial thing that seems to loosen everyone up for this conversation is that we start out with a compliment. Not “you got the right answer”. Usually I tolerate a couple “the handwriting is really neat” and “I like that you can draw a straight line” comments before I say let’s have some comments that focus on the mathematics here. I also give effusive and public thanks to the person whose work is up there (often not fully with their permission, but this is because I am trying to break them of the habit of only wanting to share work that is perfect.) This praise often includes how Student X (who may be not on task but is refocused by being called out) is appreciative that he/she is seeing how a peer was thinking, whether it was incorrect or not. I also noticed that after starting to do this, all students are now doing a better job of writing out their work rather than saying “I’ll do it right on the test, right now I just want to get a quick answer.”

Algebra 2

We had a few students absent yesterday (which, based on our class size, knocks out a significant portion of the group) so I decided to bite the bullet and do some Python programming with them. We used the Introduction to Python activity made by Google. We are a 1:1 Mac school, and I had everyone install the Python 3 package for OS 10.6 and above. This worked well in the activities up through exercise 8. After this, students were then supposed to write programs using a new window in IDLE. I did not do my research well enough, unfortunately, as I read shortly afterward that IDLE is a bit unstable on Macs due to issues with the GUI module. At this point, however, we were at the end of the period, so it wasn’t the end of the world. I will be able to do more with them now that they have at least seen it.

How would I gauge the student response? Much less resistance than I thought. They seemed to really enjoy figuring out what they were doing, especially with the % operator. That took a long time. Then one student asked if the word was ‘remainder’ in English, and the rest slapped their heads as they simultaneously figured it out. Everyone enjoyed the change of pace.

For homework, in addition to doing some review problems for the unit exam this week, I had them look at the programs here at the class wiki page.

I had great success giving students immediate feedback on the physics test they took last week by giving them the solutions to look at before handing it in. I had them write feedback for themselves in colored pencils to distinguish their feedback from their original writing. In most cases, students caught their own mistakes and saw the errors in their reasoning right away. I liked many of the notes that students left for themselves.

This was after reading about Frank Noschese’s experience doing this with his students after a quiz. I realize that this is something powerful that should be done during the learning cycle rather than with a summative assessment – but it also satisfied a lot of their needs to know when they left how they did. Even getting a test back a couple days later, the sense of urgency is lost. I had them walking out of the room talking about the physics rather than talking about how great it was not to be taking a test anymore.

Today we started figuring out circular motion. We played broom ball in the hallway with a simple task – get good at making the medicine ball go around in a circle using only the broom as the source of force.

We then came in and tried to figure out what was going on. I took pictures of all of their diagrams showing velocity and the applied force to the ball.

It was really interesting to see how they talked to each other about their diagrams. I think they were pretty close to reality too, particularly since the 4 kilogram medicine ball really didn’t have enough momentum to make it very far (even on a smooth marble floor) without needing a bit of a tangential force to keep its speed constant. They were pretty much agreed on the fact that velocity was tangent and net force was at least pointed into the circle. To what extent it was pointed in, there wasn’t a consensus. So Weinberg thinks he’s all smart, and throws up the Geogebra sketch he put together for this very purpose:

All I did was put together the same diagram that is generally in textbooks for deriving the characteristics of centripetal acceleration. We weren’t going to go through the steps – I just wanted them to see a quick little demo of how as point C was brought closer to B, that the change in velocity approached the radial direction. Just to see it. Suddenly the students were all messed up. Direction of change of velocity? Why is there a direction for change in velocity? We eventually settled on doing some vector diagrams to show why this is, but it certainly took me down a notch. If these students had trouble with this diagram, what were the students who I showed this diagram and did the full derivation in previous years thinking?

Patience and trust – I appreciate that they didn’t jump out the windows to escape the madness.

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All in all, some good things happening in the math tower. Definitely enjoying the experimentation and movement AWAY from lecturing and using the I do, we do, you do model, but there are going to be days when you try something and it bombs. Pick up the pieces, remind the students you appreciate their patience, and be ready to try again the next day.

Testing physics models using videos & Tracker

I’ve gotten really jealous reading about how some really great teachers have stepped up and used programming as learning tools in their classes. John Burk’s work on using vPython to do computational modeling with his students is a great way to put together a virtual lab for students to test their theories and understand the balanced force model. I also like Shawn Cornally’s progression of tasks using programming in Calculus to ultimately enable his students to really understand concepts and algorithms once they get the basic mechanics.

I’ve been looking for ways to integrate simple programming tasks into my Algebra 2 class, and I think I’m sold on Python. Many of my students run Chrome on their laptops, and the Python Shell app is easily installed on their computers through the app store. It would be easy enough to ask them to enter code I post on the wiki and then modify it as a challenge at the end of beginning of class.. It’s not a formal programming course at all, but the only way I really got interested in programming was when I was using it to do something with a clear application. I’m just learning Python now myself, so I’m going to need a bit more work on my own before I’ll feel comfortable troubleshooting student programs. I want to do it, but I also need some more time to figure out exactly how I want to do it.

In short, I am not ready to make programming more than just a snack in my classes so far. I have, however, been a Tracker fan for a really long time since I first saw it being used in a lab at the NASA Glenn Research Center ten years ago. Back then, it was a simple program that allowed you to import a video, click frame by frame on the location of objects, and export a table of the position values together with numerically differentiated velocity and acceleration. The built-in features have grown considerably since then, but numerical differentiation being what it is, it’s really hard to get excellent velocity or acceleration data from position data. I had my students create their own investigations a month ago and was quite pleased with how the students ran with it and made it their own. They came to this same conclusion though – noisy data does not a happy physics student make.

I wanted to take the virtual laboratory concept of John’s vPython work (such as the activities described here) for my students, but not have to invest the time in developing my students’ Python ability because, as I mentioned, I barely qualify myself as a Python novice. My students spent a fair amount of time with Tracker on the previous assignment and were comfortable with the interface. It was at this point that I really decided to look into one of the most powerful capabilities of the current version of Tracker: the dynamic particle model.

My students have been working with Newton’s laws for the past month. After discovering the power of the dynamic model in Tracker, I thought about whether it could be something that would make sense to introduce earlier in the development of forces, but I now don’t think it makes sense to do so. It does nothing for the notion of balanced forces. Additionally, some level of intuition about how a net force affects an object is important for adjusting a model to fit observations. I’m not saying you couldn’t design an inquiry lab that would develop these ideas, but I think hands-on and actual “let me feel the physics happening in front of me” style investigation is important in developing the models – this is the whole point of modeling instruction. Once students have developed their own model for how unbalanced forces work, then handing them this powerful tool to apply their understanding might be more meaningful.

The idea behind using the dynamic particle model in Tracker is this: any object being analyzed in video can be reduced to analyzing the movement of a particle in response to forces. The free body diagram is the fundamental tool used to analyze these forces and relate them to Newton’s laws. The dynamic particle model is just a mathematical way to combine the forces acting on the particle with Newton’s second law. Numerical integration of acceleration then produces velocity and positions of the particle as functions of time. Tracker superimposes these calculated positions of the particle onto the video frames so the model and reality can be compared.

This is such a powerful way for students to see if their understanding of the physics of a situation is correct. Instead of asking students to check order of magnitude or ask about the vague question “is it reasonable”, you instead ask them whether the model stops in the same point in the video as the object being modeled. Today, I actually didn’t even need to ask this question – the students knew not only that they had to change something, but they figured out which aspect of the model (initial velocity or force magnitude) they needed to change.

It’s actually a pretty interesting progression of things to do and discuss with students.

Draw a system schema for the objects shown in the video.

Identify the object(s) that you want to model from the video. Draw a free body diagram.

Decide which forces from the diagram you CAN model. Forces you know are constant (even if you don’t know the magnitude) are easy to model. If there are other forces, you don’t have to say “ignore them” arbitrarily as the teacher because you know they aren’t important. Instead, you encourage students start with a simple model and adjust the parameters to match the video.

If the model cannot be made to match the video, no matter what the parameter values, then they understand why the model might need to be adjusted. If the simple model is a close enough match, the discussion is over. This way we can stop having our students say “my data is wrong because…” and instead have them really think about whether the fault is with the data collection or with the model they have constructed!

Repeat this process of comparing and adjusting the model to match the observations until the two agree within a reasonable amount.

Isn’t the habit of comparing our mental models to reality the sort of thing we want our students to develop and possess long after they have left our gradebook?

It’s so exciting to be able to hand students this new tool, give them a quick demo on how to make it work, and then set them off to model what they observe. The feedback is immediate. There’s some frustration, but it’s the kind of frustration that builds intuition for other situations. I was glad to be there to witness so we could troubleshoot together rather than over-plan and structure the activity too much.

Here is the lab I gave my students: Tracker Lab – Construction of Numerical models If you are interested in an editable version, let me know. I have also posted the other files at the wiki page. Feel free to use anything if you want to use it with your students.

I am curious about the falling tissue video and what students find – I purposely did not do that part myself. Took a lot of will-power to not even try. How often do we ask students to answer questions we don’t know the answer to? Aren’t those the most interesting ones?

I promise I won’t break down and analyze it myself. I’ve got some Python to learn.

Dare to be silent.

I made a promise to myself today – I was going to force the physics class to speak. It isn’t that they don’t answer questions and participate, it’s that usually they seem to do that to please me. Sometimes they will explain ideas to each other and compare answers, but it never works as beautifully as I want it to.

So today I told them I wasn’t going to talk about a problem I gave them. They were. And then I sat on an empty table and waited. It was really difficult for me. Eventually someone asked someone else for an answer. I stayed quiet. Then another person nodded and agreed and then said nothing. I stayed quiet. Then someone disagreed.

Full disclosure – at this point I gestured wildly, but still stayed quiet.

After about five minutes of awkward silence punctuated with half explanations that trailed off, something happened – I don’t know what the trigger was because if I did I would bottle it and sell it at educational conferences – a full discussion was suddenly underway. I was so amazed that I almost didn’t think to capture it – thankfully I did get the following part:
[youtube http://www.youtube.com/watch?v=YVSSjQVpB70&w=420&h=315]

Especially cool to see this knowing that English is not the first language of the students speaking.

I’m going to try to do this more often, though I again must point out that it was incredibly difficult working through the silence. The students in the end decided they had something to say, so they shared their thoughts with each other. I did nothing but wait for it to happen.

Physics #wcydwt – Indirect Measurement

While cleaning up after robotics class today, I noticed a statics problem involving an object hanging from a couple wires that was poking out from under one of my many piles of papers. We had looked at this question earlier during the week in class. A couple students were out for a volleyball tournament in Beijing, so I wanted to do something hands on and multimedia-esque that the missing students wouldn’t feel too upset about missing, but could somehow still be involved and connected with the class work from today.

I realized that we hadn’t yet used the spring scales during our discussion of forces. My obsession with #wcydwt lately has been on using the novelty of a minimum amount of information to get students to see a problem jump off the page/screen. I also wanted the students in class to get the joy of holding back information from their classmates to see if they could figure out the missing info. Lastly, I wanted there to be a simple physics problem that would serve to assess whether all of the students understood how to solve a 2D equilibrium problem.

So I grabbed the spring scales, some string, slotted weights, and told the students to put together a few pictures using these materials. We briefly discussed what information could be given, and what they wanted to leave out for the athletes to figure out on their own. I admit – I pushed them along, and given more time I would have given them more choice, but I don’t think my selfishness and excitement in doing this was too much. The other factor – the vice principal had given us an extra pizza to share – they were also really pushing for efficiency. It wasn’t all me.

And thus the spring scale picture project was born, thanks to one student’s iPhone and Geogebra:

The complete link of the assignment is at http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/e495b/Unit_2__Spring_Scale_Challenge.html.

I’m sure I am not the first to do this, but it was so simple to execute that I had to give it a shot, and I am sharing it because I’m trying to share everything I can these days. We will see what happens when the results come in next week.

Scheming with Schema…

When teaching physics before, I found the process of building free body diagrams with students to be a fairly smooth process. It took a lot of feedback to get there, but they way I introduced the topic was along the lines of the chart below:

This chart was based on one I had from my own physics notes taken during class with Mr. Bob Shurtz who influenced me both as a student (helping me explore the love of physics and engineering I didn’t know I had beforehand) and then as a colleague while designing my own AP Physics course in the Bronx.

I held students to the requirement in the beginning that every time they constructed a FBD they must make one of these charts because my feeling was it would help both in identifying the important forces acting on a single object and in discussions of Newton’s 3rd law. The students grumbled as they tend to do when we expect them to use organization scaffolds like this that they feel they don’t need. As time went on and FBDs were drawn correctly, I would loosen that requirement to the point that students were drawing diagrams and, minimally, felt guilty if they weren’t at least thinking to make sure all of those forces could be identified. Those charts were admittedly annoying, but I felt they at least got students in the right mindset for drawing free body diagrams, so it was a good thing to require.

When my fans on carts exploration with the students went long last week, I decided to push the introduction of FBDs to this past Monday. We did have time last week to talk out different types of forces (normal, gravity) so they at least had some ideas of what different forces could be included in the chart. This extra time gave me the weekend to take a closer look at Modeling Instruction, and more specifically, at the concept of drawing system schema. I had never heard this term, but it appeared all over the modeling literature, so I decided to take a closer look at the Arizona State University site on modeling where I found an excellent paper that details using them as part of the FBD development process. It seemed harmless enough. Worst case, it would be a scaffold like the chart I mentioned earlier, used in the beginning and then taken away over time.

It was especially lucky that shortly after reading this, Kelly O’Shea had posted an excellent guide on how she introduces the Balanced Force Particle model to her class. It seemed like such a natural way to analyze problems, so I introduced it to the class as part of drawing free body diagrams for the first time on Monday.

Some really interesting things happened during that class and during Wednesday’s class that deserve to be shared here. First, I was impressed how naturally students took to the idea of drawing the schemata. Not a complaint in the room.
They shared with each other, pointed things out, and quickly came to an agreement of what they should look like.
It was incredibly natural for them to then draw a dotted circle around the object they were analyzing and see the free body diagram nearly jump out at them. The discussions about directions and what should be in the diagram were matter of fact and clear with virtually no input from me. Score one for the schema.

The second thing that came up during class on Wednesday was in discussing a homework problem about a bicycle moving down a hill at constant speed due to a drag force of magnitude cv. The schema that one student had put together looked something like this:

The students were wondering how they would combine the friction from the ground and the air drag force into one to use the given information.

I was floored – after giving this problem for four years in a row, this was the first time the students even thought to think of anything about the friction on the ground. They decided to neglect this force after we thought about whether drag force had anything to do with the ground, but the fact that we even had this discussion was amazing and really shows the power of the schema to get students to think about what they are doing.

The final thing the class pointed out was an inconsistency that had again never even occurred to me. On Wednesday, we were looking at the following sketch as part of a kinetic friction problem:

The block was moving at constant velocity across a surface with coefficient of friction of 0.7. I asked the students to draw a schema, FBD, and figure out what the magnitude of the force F must be. They started working on their schemata, but then had these uncomfortable looks on their faces shortly afterwards.

Looking at the diagram, they had no problem identifying the effects of the entire earth and the ground, and they were fairly sure based on the situation that drag was not an important part of it. The thing they really didn’t know how to handle was that disembodied force F.

What object was causing it? Where was it coming from? How in the world could they include it in the diagram if they didn’t know what interaction was governing its presence?

At this point in previous years, students didn’t generally mind that random forces were being applied to blocks, spheres, or other random shapes – they just knew that they had to do a sum of the net force in x and y and solve for unknowns in the problem. In the context of the schema, however, the students were clearly thinking about the situation in exactly the way I had taught them to do and were genuinely concerned that there was no clear source of this force. This goes back to the fact that they were seeing the system schema as a representation of real objects, which is really what we want students to be doing! I had never thought about this before, but it was so amazing to know that they were having these thoughts on the second day of meeting the free body diagram.

We agreed on the spot, given my omnipresent power as a physics teacher, that any time a force appeared in a problem diagram that had no clear source, that it had to be because of an interaction with me, and they could include me in the schema to indicate that interaction. For the purposes of satisfying their newly found need for a source for every force (a possible catch phrase for schemata?) they now have permission to do this in their schemata.

I admit that my students in the past have gotten away with abstracting the process of equilibrium problems into barely more than a math problem. That capability has still gotten them to analyze some interesting situations and pushed them to explain phenomena that they observe in their own lives. Still, the way using the schema changed our conversations over the past couple days is an impressive piece of evidence in favor of using them.

In short? I’m sold. I’ll take twenty.

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