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.

Nice work when you can get it….

I graded my first set of physics tests today. With the group deciding not to take the class at the Advanced Placement level, we’ve been able to slow down and spend time experimenting and really engaging with kinematics and projectile motion. I assigned them problems and helped them learn, but I was more impressed with the experiments they worked on and their engagement level during those activities. I was concerned about what would happen when we returned to solving problems, but I was very pleasantly surprised.

I’m interested in sharing student work, so this is the first time I will be doing it. When I asked the student if it would be OK to share, the student agreed and was really excited that I would want to show the work to other teachers.

This student started out solving problems in a very scattered way: calculations here, sketches there, units nowhere to be found. When I showed the structure I wanted students to use to solve problems, it was initially a burden. The student didn’t like doing it. Upon grading, I was very happy to see this:

The degrees vs. radians issue is one that I always battle, made especially difficult this year because students have me for physics (when I insist on degree mode almost exclusively) and then calculus (when I change my insistence to radian mode) right in a row. Yes, the student should have noticed that multiplying by the ratio should not have resulted in a negative sign for initial y-velocity. Yes, it should have again become obvious that something was up when he found he needed to ‘add’ a negative sign in the answer at the end to make the sign of the answer make sense according to his own sign convention.

The fact that the student can notice these things (and that I can see where the errors are) would not even be something I could discuss if it wasn’t for the structure I put in place. By learning to use the structure to organize thoughts, this student became able to solve problems in a logical manner rather than with calculations all over the page. I don’t like teaching procedures, but this is an example of where it pays off.

We like students exploring and experimenting and constructing their own knowledge. These are really good ways for them to spend time in our classroom. I include using correct mathematical notation, showing steps, labeling axes, learning terminology, and other things of that nature as part of my class expectations – at times a battle that seems unimportant in the context of what I really want students to know how to do in five or ten years. There is room for procedural knowledge, however, and this student’s success is evidence of why we do it.

The main point (and the thing I’ve been working to change compared with how I did things for a while) is that these procedures should not be the meat of a lesson or the main focus of instruction all the time. These things CAN be taught by computers or videos and don’t necessarily need a human in the room. It is important for students to have skills and have access to resources that help them develop those skills.

But getting answers is not the point – this is the tricky part that we have to do a better job of selling to students, at least I do. Clear communication of reasoning and sharing the logic of our ideas are some of those “21st century skills” that students should have when they leave our classrooms. If a student needs to learn a structure to help them with this process, it is worth the time needed to help them learn it.

Modeling anyone? Fans on carts edition.

After reading a lot about the success that others have had with teaching physics using the modeling method, I’m giving it a shot as I start Newton’s laws with my physics class. When I taught this with my AP physics previously, I did a traditional development of Newton’s laws describing (I admit it – lecturing) about Newton’s understanding of what caused acceleration. We talked about acceleration being proportional to net force and inversely proportional to mass, and then went from there exploring what it meant for net force to be zero through a series of problems involving net forces, components, etc.

What I did seemed to work in so far as students were able to solve the problems I gave them. The undying assumption of course is that what I did was efficient and made me feel that I had got across the material to students, but along the way I wasted an opportunity for students to SEE the principles in action and try to figure things out on their own. Since my students this year are not taking the course at the AP level, I see no reason not to try this and see how it compares in the long run to student understanding and enjoyment of the exploration of physics concepts. It is the sort of thing that I can see doing even in the Physics B curriculum, as dense as it is, given the fact that students really need a chance to play to connect the mathematics of the equations to the fact that physics describes the real world, not just idealized situations.

Here’s where I’d love to get some input though – I am giving my students a test in the first half of the 85 minute period tomorrow, and then my plan is to let them spend the rest of the time watching some videos that I took this afternoon of toy fans attached to cars on an air track. The students will get to play with the actual air track, but I want to introduce to the way I want them to play by seeing these videos that I created.

I have posted the series of videos here at my wiki site. The general instructions for what I want them to do are there, but I might as well run through them here as well.

First, I want them just to watch all the videos. No physics, just observation. After they have done this, I’ve posted a number of questions I want them to use to classify, analyze, and predict based on constant velocity and non-uniform velocity cases. I plan to have them sketch what effect a single fan would have on the motion of the cart. My plan in the end is to have them construct a situation with the fans that results in a given scenario. For example: arrange the fans on the cart so that The cart has zero initial velocity and an acceleration to the left. Draw position, velocity, and acceleration graphs, and then use Tracker to confirm/refute what their models suggest will happen.

Let me know your thoughts either here or through Twitter (@emwdx) – I am excited to try this, and excited to give the students a chance to get some first hand experience testing their own ideas. I had a blast playing with it this afternoon, and while I do have a different standard for what is ‘fun’ at times, I don’t think this is one of those times.

Wiki site: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/52698/Unit_2__Carts_with_Fans.html

How China Keeps Me Learning: Part I

Ever since moving to Hangzhou, China in August of 2010, I’ve been amazed at the number of ways it has forced me to use my own problem solving and critical thinking skills. I’ve remarked inwards that talking about these experiences would help greatly in describing the sorts of experiences I want my own students to have, as well as the factors that have helped me be successful as I’ve explored. Now that I am taking the time to write about my experiences, I think this theme is a good one to return to from time to time to describe how these experiences I have relate to my classroom.

Hangzhou has a number of truly incredible places within its city limits. Some are incredibly beautiful. A few of them, however, are incredible for how they address my geeky-tinkerer side.

This building is one of two that sit on opposite sides of the road in the North-east section of Hangzhou. Inside are rows and rows of little booths that each sell electronic parts. Some specialize in motors or solar cells. Others have all different electronic components from resistors to circuit boards to jumper wires, all on display.

I’ve been to this place several times to get parts, other times just to wander around and gawk at the amazing quantity of raw materials there for projects not yet materialized. This week I returned for a different reason. My parents decided to take a big step and visit my wife (Josie) and I here in China, so they have been on numerous adventures with us for the past week. Another post on that is imminent, so stay tuned.

My dad is an engineer and was the first person I thought of when I walked into the building for the first time and saw what was there, so I knew I had to take my dad there for a visit. I also had a vague goal for what I wanted to get while I was there: sensors. Whether for robots or for upcoming units in physics, I knew it would be good to see what was available there so I had more available for experimentation in the classroom and to think ahead.

One other thing to be aware of: I don’t speak Mandarin. I know some basic greetings and scattered vocabulary, but don’t know ‘sensor’, ‘resistor’, or even ‘electric’ either in symbolic or spoken Mandarin. On every visit to the market, I have always had to resort to sketches and diagrams to communicate. This, however, is the most entertaining and enriching part of these trips to the market – figuring out how to say what I am looking for. This was my first visit to the market since my summer acquisition of an iPad, which together with Google Translate, tended to improve the quality of my communication with the dealers to an extent this time. It was, however, still a challenge.

After some wandering around and some awkward interactions with parts dealers that weren’t sure why we were there, my dad and I ended up in a booth with a pair of women intrigued by the site of us in their store. I get the impression on every visit that foreigners don’t enter the building with any regularity, so I’m used to it. I pulled out the iPad and entered ‘gas sensors’ , showing the translation to the women. They pointed to a column of plastic containers beneath a glass counter, gesturing and pointing while saying (in Mandarin) what each one was. Eventually with Translate’s help, they ended up identifying the various gases that they had sensors for, and I came to the conclusion that I needed to do more research before making any purchases. Bottom line – they had some great stuff, much of it exactly what I was looking for.

I went through a similar process in getting some platinum temperature sensors and aluminum blocks with strain gauges for measuring a cantilevered force.

Needless to say, the whole experience was a good one. We all left happy and having had a good time. Here’s just a start of what’s bouncing around in my head for how this experience connects to set up learning opportunities for my students:


I felt free to experiment and play in my learning environment.

I loosely defined goals for my time at the market, but there was no pressure for me to buy anything if I didn’t want to. If my attempts to communicate and find what I was looking for were unsuccessful, I would have other chances to figure it out later on. I wasn’t being evaluated on my time at the market – I was instead free to have fun and try my best to achieve the goals I set for myself.

How much time do we give our students to experiment and play with the material we want to teach them? How are we making the most of the tools we have available to let them do this?


I had the tools I needed to make up for my weaknesses.

The iPad translating capability really made it possible for me to communicate in the way I needed to communicate to achieve my goals. I do want to learn more Mandarin, but I don’t see it necessary that I learn Mandarin completely before I visit the market for my other learning goals. Since my goal had nothing to do with learning the language, but instead to use the tools I had (iPad, electronics market, seemingly amused dad looking on) to reach a desired outcome, I felt free to be creative in how I used the tools to have success.

I speak enough Spanish to be able to have been able to joke and shoot the breeze with cab drivers, store clerks, etc. in the Latin American countries that Josie and I have visited. I have really missed that ability here in China, though I am getting better. The technology lets me be comfortable and interact in a way that makes the entire process enjoyable rather than frustrating. Some frustration is to be expected when trying something new, but not so much to be uncomfortable throughout the process.

How much do the learning goals we set for our students require students have acquired previous skills? How do we address deficiencies in these skills when they arise? Do we give them the tools so they can reach the goals we set for them, or do we modify the goals themselves for these students?


I accepted that I was going to make mistakes, and felt comfortable changing my approach in response to these mistakes.

There were many times when even Google Translate failed to communicate exactly what I was saying (or what the parts dealers were saying) not to mention the challenges that arose in figuring out what I wanted to ask. There were times when I used the Mandarin I did have to confirm that I understood what they were saying, and many times they showed me that I did not. In either case, the dealers were incredibly patient and supportive in figuring out how to help me. It was clear that they were enjoying the process as much as I was, which made me appreciate the time they were willing to take to get me what I wanted. I knew instantly from their reactions to my translated questions whether I had communicated clearly to them, and we were both gesturing and checking that we understood each other as often as possible.

How do we encourage and acknowledge mistake-making as part of the learning process? How do our students feel about making mistakes? How do we develop an environment in which students feel comfortable experimenting and getting things wrong along the way to getting them right?

I love these trips to the market because the feeling of exhilaration and achievement I get when I succeed is worth every moment of frustration. The worst thing that can happen is I walk away empty handed. What usually happens is a scene like the one below:

Somewhere along the line in my classroom, however, students get the feeling that there’s a lot more at stake, that others (unfortunately including me) must be judging their abilities when they don’t get a question right the first time. Students get the feeling that they shouldn’t need to use the tools they have in front of them (graphing calculator, laptop, Geogebra, etc) to learn if they are smart enough. How do I show them that it isn’t about being smart, it is about working hard to get it right in the end? Is it enough to value the mistakes they make? Do I need to share my own mistakes in doing things? (This is part of my plan, at the moment, and is partly why I made the decision to commit time to blogging about what I do in the classroom.)

If I can turn my lessons into explorations and activities in which students feel safe experimenting with concepts, sharing their ideas and helping each other learn, it would make every other goal I have for what I want my students to achieve possible. I’m all ears if you have ideas on how to make this happen!

Lens Ray Tracing in Geogebra

One of my students came to me today to ask about ray tracing in preparation for his SAT II tomorrow in Physics. What happened is a good example of what tends to be my thought process in using technology to do something different.

Step 1 – I looked through some of my old worksheets, which I haven’t used in a while since I haven’t taught physics since 2009. The material I was happy with back then suddenly didn’t work for me. Given the fact he was standing there (and that time was of the essence) I wasn’t about to make a whole new worksheet.

Step 2 – I started drawing things on the board. This started working out fine, but I realized that every drawing I made would have to be erased or redone or saved in some other format. The student, after all, was most interested in learning how to do it and getting some practice. We did a couple sketches for mirrors, but when we got to lenses, I realized there had to be a better way. The sign for me for technology to step in is when I find myself doing the same thing over and over again, so the next step was pretty obvious.

Step 3 – Geogebra to the rescue. This is a particularly sharp student, so I was pretty happy with just talking him through what I was doing and asking him questions as I put together a quick demo of how to do this. He was pretty impressed with how logical the concept of ray tracing was, and had read the basic procedure in the textbook, but actually seeing it happen made a big difference. As he was standing there, his questions pushed me to make the applet (to steal Darren Kuropatwa‘s term) “a little more awesome.”

He asked what happens when the object is inside the focus of the lens. This led to throwing in some simple logic to selectively display the rays to show the location of the image when it is virtual and real. He asked what the difference is for a diverging lens. I told the student that I didn’t know what would happen if I switched the primary and secondary foci in Geogebra, but we talked about why that would relate to a diverging lens. Sure enough, the image appeared virtual and upright in the applet.

Step 4 – I then adjusted it a bit to show a diverging lens when the primary focus was on the left side of the lens, cleaned up some things, added colors, and now I have this cool applet to use when I get to working on lenses in the spring.


I like when I can think on my toes and use a tool like Geogebra to make something that will really make a difference. When I do this activity in the spring, it would be cool to put this side by side with an actual lens and an object and have students compare what is happening in both cases.

Check out the applet here: http://www.geogebra.org/en/upload/files/weinbergmath/Lens_Ray_Tracing.html

You can direct download the Geogebra file from here but be aware that I made the mistake of creating it in the beta version of 4.2. At some point, I’ll do it in the stable version.

You can drag the head of the arrow around, as well as move the primary focus F_p around to change it into a diverging lens. Clearly there are limitations to this – drag the object to the right side of the lens, for example, but I think it’s pretty cool that Geogebra can show something like this after an hour or so of playing around.

Have fun!