Computational Thinking – Why do we need to do this?

I’ve been pushing programming tasks on my Algebra 2 class. Pushing is the operative word.

Specifically, I’ve given them tasks that require them to use programs written in Python to do something related to what we are learning. Previously I showed them a program I wrote for a homework assignment as part of my work for Programming a Robotic Car , which went just fine for what it was: a short activity. I’ve also given them a few tasks during previous classes asking them to adjust a program I had written. IN one case this was to calculate a solution to a linear system; in another to evaluate the quadratic formula.

What I admit I did a bad job of, however, of making it clear why we would even want to do so. I think I erred on the side of them seeing how great it was that the computer could do this, which is silly considering I’m also pushing applications like Wolfram Alpha that do much more than my programs, and in a much more eye-catching way. I imagine it’s sort of like showing my students a typewriter that can post on twitter and insisting they find it cool because such a thing exists. The power of programming is in making the computer do work that makes sense for a computer to do. The power of showing programming to students is to both demonstrate how a computer can do this type of work for them, and then empower the students to apply this knowledge on their own.

After reading a couple posts by David Wees on programming and mathematical thinking,  I realized that I was doing things backwards. I needed to establish a reason for doing and teaching computational thinking. I was going to have some students out for athletics last week, and those tend to be the days when I experiment. I mentioned in the previous class with this group that I was planning to do a lesson with Python, and the reaction was instant and violently vocal. Those that were going to be out were thrilled. The others looked about as excited as if they learned that lunch for the following week would be nothing but spinach.

I designed the lesson knowing that if there was going to be any Python at all, it would need to be a task that justified itself quickly, clearly, and with as little trouble as possible.

The topic of the day was composition of functions. The warm up activity was this:

  • Suppose f(x) = x^2. What is f(f(2))?
  • This should also work for complex numbers. What is f(f(1 + i))?

What was neat was that a couple of the students were stuck on the first question playing around with it. I didn’t have to ask a student to perform the composition three, then four, then five times to its own value – they did it on their own to postpone having to deal with the complex numbers. Not a problem. They got the point that they could do this, but that it was tedious. That was phase 1.

This came next:


The program they used can be found at .

They quickly figured out how to use the program to run a whole bunch of times. Some found that they weren’t sure if an initial value escaped or not because the value was close to 1 or -1, so they figured out that they could change the number of iterations in the program to give them a better indication. Others realized they were looking at scientific notation and that they needed to review that that meant. 

We talked about why the computer was the way to go for this, and then related it to the second warm up problem. Could it be possible to use the computer to do this for the more tedious task of managing complex numbers? Enter my second program at
The students used the second program to determine whether these points diverged/converged (I occasionally slipped these words in with hand motions to link them to escaping or being trapped) and found the concept pretty straight forward, as it had been with the points on the number line. I continued to ask them how they could do this manually using pencil and paper, and was met with groans – the computer clearly was the more logical tool to use for this. (Yes!)
My final task for them was simple: plot the points that are trapped. I gave them this:
I said I wanted them to color in any of the squares with bottom left hand corners that represented values that converged. That’s a lot of grid squares, but that was kind of the point. There were nine in class that day, so they started dividing up the space. Some picked points randomly. Others were more methodical, with some starting to trace the border of the region, but generally there was only a scattering of filled points on the class copy where I had asked them to record their colored in grid squares.
Having predicted how this would likely end, I was ready with this modified version of a Processing sketch written by Daniel Shiffman that basically did the entire task on the computer. The students were staring at the clock after doing this for about ten minutes and only having filled in a minimal portion of the plane. The students understandably said that this process was stupid and that there most likely was a better way for them to be spending their time. That’s when I said they were right. 
You know when you design activities to carefully manipulate your students’ emotions so as to realize a particular point, and it totally works? My students have a pretty distinctive facial expression that they each make when they realize I’ve done this, and right about then, it spread like wildfire through the class.
I showed them this:
This didn’t contradict what they had come up with, but it was significantly more complete. I asked if there was any recognition, but there was none. So I decreased the pixel size more:
Still no recognition, but there was more recognition that the shape of the region was an odd one. One more iteration made it pretty clear:
We had looked at a video zooming into the Mandelbrot set earlier in the year, so they had seen it before. I wanted to push that the computer made it possible to investigate this sort of mathematics. This sort of thing could not have been done by hand at this level. Having the computer available to do repetitive calculations and construct graphs according to simple rules made it possible to investigate the mathematics of these areas in ways that were never known before. They were impressed that the mathematics of fractals was not investigated until 1980, which is recent enough for them to perhaps see that math is actually a dynamic field, in contrast to the way it is usually presented.
I liked this lesson, and plan to continue to push my students to use computation when necessary. Our first unit problem for the exponential and logarithmic functions unit was a twist on the penny problem (get one penny the first day,two the second, four the third, and continue on for a month, or just take a lump sum of $50,000) and I insisted they use some computational tool to answer this, or just straight out find it extremely difficult.
This is an important skill, and I believe in it. To make it happen, I am committing to deliberately committing time for students to learn how to use computers to do the computation work so we don’t have to. I hope my days of seeing students solve the bee/train problem through tedious methods of manually adding terms together will soon be over.

Stand up and wave!

I’ve had the skeleton of a lesson on standing waves in physics floating around in my brain for a long time. In the past, I’ve used just a white board and drawn some diagrams, and have occasionally shown some animations from the web to help students visualize the relationship between a vibrating string and the waves moving around on it.

Here was an attempt scanned from my first lesson back in 2006:

And more:

Nothing to write home about. My lesson had a bunch of information in it. Now in my defense, much of this I was eliciting from students. It was a very carefully designed progression of thoughts, and my students did end up understanding it well enough to then apply it to problems. (Better, in fact, than I was when I first tried to learn this in high school. This was one thing in physics that didn’t make sense to me, and I was committed to having my students get it.)

There wasn’t much for the students to grab onto or see – nothing dynamic or moving to engage them. The following year, I added some interactive applets, such as the one here. I didn’t have a class set of laptops, so it was still a demo at the front of the room, but here it was possible to show much more clearly what was actually going on and students themselves could play around with the applet and get some intuition for how it worked.

Fast forward to the present. I’ve been teaching a non-AP level physics class and have been able to write my own curriculum along the way. As a result, we’ve been able to slow down and play with models, do experiments and play with data. Waves are inherently hard to visualize given the frequency of the sound waves with which we typically interact (not to mention their invisibility) so I wanted to try something different. Kate Nowak’s recent post about using technology to do lessons that would be impossible also got me thinking how I could use the technology I have available to do this lesson in a fundamentally different way.

Here is what we did today:

First we watched this video on YouTube showing Tuvan throat singers in action. The kids had never seen this, and it shocked me the first time I heard it too back in high school during a class in electronic music. We also watched parts of the video of Daniel Palacios’ standing waves art piece (Thanks to John Burk, @occam98 for sharing this find yesterday!)

We talked a bit about how the singers might be doing this, but there was no consensus. So I brought them to the back of the room where we had a spring waiting for us on the floor.

I challenged two students to create a single wave with the spring bouncing up and down. This quickly devolved into attempts to make two, three, and eventually six loops in the spring. Hands-on discovery of the relationship between the number of loops and frequency? Check. I had students identify points of maximum amplitude and stationary points, but no vocabulary yet.

We then made our way to another station where I had set up a strobe light and some tuning forks. We played around with the tuning forks and the strobe frequency to be able to visualize the movement of the tines of the fork during vibration. I showed a similar demonstration with a student plucking a guitar string. Objects vibrating up and down in the same way as the spring create sound. Simple idea.

None of this is really revolutionary, I admit it. The good stuff is coming, I promise.

During a workshop I attended with Nick Jackiw on Geometer’s Sketchpad, I learned of a feature that I didn’t know about. With periodic functions, you have the option is to add an ‘action button’ to your sketch that allows you to ‘Play’ the function as a sound. In the previous class when we investigated superposition, we learned how waves added together, even those of different frequency.  I created a simple GSP sketch that added three waves together in which the amplitudes and frequencies could all be changed.

So…wave superposition – they could see it happening in front of them mathematically. Now they could hear the effects of superposition, changing amplitudes and frequencies, and understand how different frequency waves could add together to create the sounds that we hear. Standing waves are things bouncing up and down–> bouncing things up and down make sound waves –> we can also make sound waves by combining waves of different frequencies together.

The real key to making the final connection comes from applying some technology on loan from National Instruments. I have been working with their myDAQ student data collection device in finding ways to adapt it for use in math and science classes. The device has a number of analog and digital inputs and outputs, which are nice, but they pale in comparison with the Audio-In port in terms of ease of getting data into the computer. That data can then be analyzed very nicely by LabVIEW to create a frequency spectrum of the audio data in. I downloaded the software from an NI support site describing a beer bottle music project.

Now to lead up to the climax of the class  – I had a separate function generator create a 880 Hz sine wave which we played over the speaker. We had just seen a graph of this function, so students knew what it looked like graphically, and then what it sounded like. I had them whistle to imitate the frequency, and  now I showed them the graph of the amplitude vs. frequency plot in Labview, which looked like this:


I could change the pitch of my whistle and students could see how the peak moved left or right as the frequency went up or down. This was an easy introduction to the concept of frequency space and what it could be used for.

As with most of my classes, I’m the only one usually willing to sing in front of everybody, so I gave it a go:

A plastic bottle I had left over from the day before (though I kept this part to myself during class) was just asking to be played, and that gave some interesting results blowing lightly across it when it was empty…

…partly empty:

…and more full than empty: (Look at those odd harmonics shine!)

What I liked the most about this activity is that we could pretty much put any sound the students wanted to make, and the frequencies would pop out as they did above. They could see how singing a note and changing the shape of one’s mouth changed the amplitude of the various harmonics associated with the sound, and that was what we were really hearing as different sounds.

This all  breaks down to what I think is the most challenging concept for students to really understand about standing waves: If the sound waves we hear are made up of all of these different frequencies (as shown by these graphs we generated in real time), then it means the objects generating those sounds (bottles, voices, strings, tuning forks) are somehow able to vibrate at a whole bunch of frequencies at the same time.

At what various frequencies does it then vibrate?


Now we get to look at those diagrams relating length, wavelength, and frequency that used to show up at the beginning of my lesson before. Fifteen minutes at the end of class to find the frequencies of vibration for a string and a tube closed at one end – they figured it out pretty easily without much prodding.

This was NOT what I used to think a good physics lesson (or any lesson) needed to be: a series of presentations of content that made logical sense. Then we drill like crazy through problems to use the ideas.

Instead, we had a series of experiences that let the students interact in a visceral way with the material. Students could see waves and feel them. They could hear them. They could see what it sounded like when they were added together. They could observe how this new special graph took sounds in and spit back the frequencies of the individual waves that made them. I told them briefly about Fourier and showed them what happens when you play a triangle wave or sawtooth wave into the spectral analyzer – they easily saw the patterns of harmonics for each and (I think) had some intuition for what that meant. No need to have a bunch of theory first before seeing it in action.

I could not have done this with just a whiteboard. Or just a projector. The ease with which I was able to generate a frequency spectrum, create audible sine waves, model super position in a visible and tangible way – these were because of the convergence of technology in front of me. This is how technology is changing things as we speak – the tools to do these things are getting more and more easy to obtain by anyone. We have to get these tools in the hands of our students – they will run with it and get more out of it than we might even understand. We need to be there to guide them in the right direction – we need to help them decide from which parts of the fire-hose to drink.

This reminds me of two years ago when my dad was doing some data collection of sound levels near wind turbines in North-Eastern Ohio. He learned that there were already some applications available to do basic audio analysis on the iPhone that was similar to what the more expensive scientific equipment could do. That was two years ago. Now you can do this (from Faber Acoustical):

Fundamentals are still important. My students wouldn’t understand any of what we did today if they didn’t know what a sine wave is and what it means for a wave to have a frequency.

My argument is that we can bounce back and forth between theory and application. Let’s meet at a happy point in the middle where we can all do a big dance, capture it on video, and send it to YouTube. The whole “they need to understand the basics first” or “do their time” with the boring stuff argument is a hard one to maintain with these amazing tools available that used to only be accessible to universities and companies with deep pockets. Kids take to technology like ducks to water – using it to enable learning in an engaging and relevant way is one of the most powerful aspects of its presence in our classroom.

This lesson felt great today. I’ve always wanted to teach standing waves and show in depth what they are all about, but have never had the right tools at my disposal. This  is exactly the sort of style I want for all my classes – exploration and play that leads to a lesson that motivates itself. Ultimately I hope this lesson leads to a stronger connection and conceptual understanding of the material – they got it a whole lot faster than I did.

I hope that says more about them than it does about me….

Reflections on EARCOS Teachers Conference 2012 – Friday

I decided to use a few digital tools to record my thoughts at the EARCOS conference. At other workshops, I tend to take notes on paper, leave them in a folder, and possibly go back to them when inspiration hits, if I remember I have them. Since I am on my computer so much of the time (and NOT digging around in a filing cabinet to see what is in there) I think this will keep the ideas from this conference fresh and nearby.

I attended a few fantastic workshops Friday and tweeted extensively about each one as important ideas came up. The #earcos12 archive and search function will be really useful for going back and reminding myself of the ideas that came to mind during those workshops.

Workshop 1 – The Geometer’s Sketchpad Workshop: Beyond Geometry with Nicholas Jackiw

It was really a treat hearing the person that defined dynamic geometry talk about the philosophy of his software that implements the model. Having learned mathematics using GSP back in 9th grade, I’ve always seen the dynamic geometry as a natural lens through which geometric concepts can be viewed. Nick mentioned that mathematicians initially had a problem with the concept because two triangles with vertices A,B, and C that aren’t congruent are not the same. Since dynamic geometry defines triangles in terms of the relationships of vertices, two triangles with the same vertices connected in the same way represent the same geometric object. This means that any triangle ABC can be turned into any other triangle ABC just by dragging vertices around the screen.

We went through the basics of plotting points, lines, and measuring slope using the tools of Geometer’s Sketchpad. I hadn’t used it for a while, but it still remains a great program. Nick is a genuine guy with a love for mathematics and what his software can do for students learning concepts. He has a solid grasp and had some great activities that could be used for students to actively learn concepts through exploration rather than listening to a teacher go through a list of boring definitions.

I had the pleasure of sharing with Nick that I used Geometer’s Sketchpad to use geometry in ninth grade and that I still had print outs of the assignments I did using the software. Back then we printed out computer assignments and turned them in, much different from today when turning things in electronically is quite easy. I was a little star-struck talking to him, but as with most good teachers I meet, he was really friendly and appreciative of my comments.

Workshop 2 – The Harkness Method: The Best Class You Never Taught – Alexis Wiggins

One of the things I want help doing is improving the quality of classroom discussions. The shelf life of the discussions we have isn’t much longer than the class period itself. I have been able to extend that a bit having students create wiki pages, interact onor create videos describing their understanding of problems.

I think this workshop provided a real possibility for restructuring my class to do this far more effectively.

Alexis shared how the Harkness method (originated at Exeter Academy) has transformed her classroom and itneraction with students. Students spend class time discussing, arguing, and critiquing arguments. In the process, they learn extensively how to be good community members, be constructive in their criticism, and communicate their ideas. She shrewdly hooked us math/science teachers at the beginning (why are we always the cynics?) by sharing that Exeter does this in their math department. Alexis also shared that she does need to do direct instruction once in a while – her ratio is around 60% discussion, 40% other methods. She also does not do this for the entire class period, particularly for the younger (9th grade) students. Modeling the process and explicitly teaching students skills that make this successful in her room is a key part of her process. She made clear that it takes time to get them to be good at it.

Alexis posted her materials at

Workshop 3 – Rules of Engagement – Using Technologies to Motivate Rather than Distract – Doug Johnson

We are constantly having discussions at our school (which is 1:1 Macbooks) about how to maximize student time on task during class – I think this is something almost everyone in schools is currently battling. The presence of technology has so many potential positive applications for learning. It is easy, however, to fixate on the negative aspects almost entirely and stall the process of making these potential benefits available to students in the classroom.

Along with having one of the most useful handouts I’ve ever received at a workshop, Doug Johnson made a number of fantastically relevant points about how school communities can think about the issue. The question he posed at the beginning was “How do teachers compete w/ tablets, smart phones, netbooks, mp3 players, portable games, etc?” What I found most interesting throughout was that he showed how it didn’t need to be a competition. Instead teachers can capitalize on the opportunity

His emphasis on the distinction between entertainment and engagement really resonated with me, as I always wonder if the activities I do with my students are actually helping them learn or not. We then worked together to identify ways to make technology an active part of classroom activities, including a lot of modeling using and references to other similar sites such as

Doug’s presentations can all be found at

Workshop 4 – Digital Citizenship: The Forgotten Fundamental Kim Cofino

This workshop from the excellent Kim Cofino was a perfect pairing with Doug’s workshop and a good ending point for the day. She clearly described her process at the Yokohama International School of rolling out (all at once, which she said was the best idea ever) her 1:1 laptop program with students.

The most important takeaway was how much deliberate planning and community collaboration went into not only creating the acceptable use policy but actively sharing that philosophy with the students, teachers, and parents. The school year started with two days of 1:1 boot camp activities – students discussing and debating different aspects of the policy. She also mentioned that the students will soon repeat some elements of this training and discussion now that the community has been through several months of living out the policy.

An important element of this is that students are explicitly taught and engaged in activities that teach them digital citizenship. She made clear that this does not happen by accident, or by hoping that students will know how to act when they are suddenly given the power afforded them by technology. This is one of the key things I will be taking back with me to Hangzhou.

Her presentation and resources can be found at


This has been a really fantastic experience being at the conference this year – I am learning so much at the workshops and through meeting the incredible collection of teachers here. I appreciate that everyone has been so positive and open in sharing their work and ideas with me. I admit it – I’m addicted to this conference atmosphere. Thankfully, I’ll be able to keep in touch with the people I have met here, and continue learning from them well after I have left Bangkok.

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.
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.

Math is everywhere! – fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you’re like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:


The cooler thing than seeing this in the first place was discovering that it’s a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:








In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.

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.

Graphical Systems in Geogebra and crashing LEGO robots in Algebra 2

In the Algebra 2 class, we started our unit on solving systems of equations. From a teaching perspective, this provides all sorts of opportunities for students to conceptualize what solutions to systems mean from a graphical, algebraic, and numerical perspective. Some students seem to like the topic because it tends to be fairly straight forward, is algorithmic, and has many ways to check and confirm whether it has been done correctly.

I used this as my warm-up activity today:

a) Estimate the solution of the system.

b) Write an equation for each line in standard form.

c) In Geogebra, select CAS view and type the following using your two equations: Solve[{7x+3y=6,3x-4y=12},{x,y}]

d) Use your calculator and convert these values to decimals. How close are these to your estimate?

We had some great discussions about the positives and negatives of graphical solutions to equations. Weaker students got some much needed practice writing equations for lines. For all students, this led to some good conversations about choosing two points that the lines clearly pass through for writing equations (if possible) rather than guessing at the y-intercept. The students also got the idea of how Geogebra can solve a system of equations exactly as a quick check for their algebra, an improvement over substituting (which is at times more trouble than it’s worth for students with poor arithmetic) and slightly faster than solving for y on a graphing calculator and finding the intersection.

I also like the unit, though I don’t tend to like the word problems. It’s hard to convince students about the large scale importance of coin problems (especially in an international school with everyone used to different currency) or finding how many tickets were sold at the door or advance since anyone with a brain would just ask the person tallying the tickets.

I also found myself thinking about Dan Meyer’s post over the summer about how many word problems are made up for the purposes of math, rather than using mathematics to analyze cool situations and create problems out of the situations. Getting students to figure out how to use the math to do this is ultimately what we want them to learn to do anyway. Figuring out when trains pass each other is not exciting to students, but I realized this morning while brushing my teeth that doing this problem with real robots either crashing into each other or racing adds a neat dimension to this problem. The question of figuring out both when they will crash or catch up to each other, and also where they will do so is a clear motivation for finding a solution to a system of equations describing their positions as functions of time.

So I gave the students the two robots (videos of them posted at and . I told them I was going to set them apart a certain distance that was tentatively 80 centimeters, but said I wanted the ability to change that at any time. I wanted them to predict when and where they would collide.

The rules:

No, you can’t just run the experiment and see where they crash. That not only defeats the purpose of this exercise, but we will be doing this sort of activity in a couple different ways during the unit, so being able to do this analytically is important. You also can’t run both robots at the same time – that’s for those of you that are going to try to be lawyers and break that first rule.

You can measure anything you want using any units that you want using either robot individually.

At some point, you should be able to show me how you are modeling the position of each robot as a function of time.

And I set them off to figure things out. Despite the fact there were only two robots, the 12 kids naturally divided themselves up into a couple teams to characterize each robot, and there was some good sharing of data amidst some whining about how annoying it was to actually measure things. In the end, most students at least had some idea of how they were going to put together their models, and some had actually written out what they were. As one would hope for these types of activities, there were plenty of examples of students helping others to understand what they were doing. The engagement was clearly there, as confirmed by students visibly excited to run the robot and time how long it took for it to move around.

It was a fun exercise that I plan to return to in a few ways during this unit – perhaps some interrobo-species interaction (my iCreate robot is charging up as we speak). Fun times.

UPDATE: This is the video of the next day’s class when students solved their functions. I set the robots apart from each other and the students did the rest.

What do you do when they don’t need you?

I’ve tried an experiment over the last two days – my advanced algebra students and geometry students each had some challenging tasks that I sort of left to them to figure out. Last year, I taught them very explicitly how to do the tasks at hand, modeled some examples along side their own work, and then gave them time in class to practice. For homework, I gave them more problems that were similar to those we did in class, giving them more chances to practice what I had assigned them.

This year I turned it around. In geometry, we are starting proofs. I gave them a couple relatively simple ones, and asked them in groups of two to construct some sort of logical reasoning to go from a starting point to proving the statement I had given them. There was a lot of struggling, difficulty stating using facts why one logical statement led to another. Over time, they did start communicating with each other and sharing what they were thinking. I did occasionally poke one group in a certain direction, but didn’t lead the whole group in that way. Eventually they were all thinking along the lines that I envisioned at the beginning. I could have modeled for them what I did last year, but I saw a lot of really good conversations along the way. By the end, they were much closer to making their own proofs than they had in the beginning. By the end, they were clearly seeing the connections between thoughts. This was only the second class period during which we had talked about proofs. While I don’t think any of them would wager large sums of money over constructing geometric proofs, I think they at least see how the system can be used to make logical statements that are irrefutable.

I did something similar with the advanced algebra group which was to figure out graphing absolute value functions during our lesson last Friday. I gave them an exploration that was, in hindsight, confusing and didn’t do much aside from frustrate them with Geogebra commands. I told them that I wanted them to use Geogebra, the textbook, Wolfram Alpha, and any other resources available to learn how to graph any arbitrary absolute value function by hand. At the end of the class I broke down and apologized for giving a poorly designed exploration. I told them I would put together a video on graphing functions, and I did – posted it on the wiki.

When we had class on Tuesday, I found out that none of the students had actually taken advantage of the video. They had looked in the textbook. They had graphed functions over and over. When I asked students to share what they had figured out, one student used a table of values and a piecewise function based on the sign of the argument of the absolute value function. Another student had graphed both the argument of the absolute value function and its opposite since that was what this student had observed, and then erased either the top half or bottom half of the graph. Another student broke down and did what the book said to do. By the end, all of them were graphing absolute value functions using their own method. I wasn’t sure about understanding, but in the end, I admit that I didn’t quite mind. They all had their own models for what was going on, and they were confident that they could use the technology to confirm whether what they were doing was right or not.

I have always wondered about what I would do in the situation where it was clear a group of students didn’t need me to be there. In a way, this is part of my fear of tools like Khan academy – if there are others out there that are more engaging, better at explaining ideas, or better at coming up with really interesting questions that got students thinking about what they were doing, and these people happened to make videos: what would I do in my classroom if students got hold of these videos? Would I be mad? Or would I figure out a way to take advantage of the fact that students had figured something out on their own and use it to do something even more interesting or impressive?

I think I do a good job of engaging students – today we were talking about differentiability and we were joking like crazy about whether functions would be differentiable at a given point or not. Really? Were we really joking about this? It seemed like everyone was minimally entertained, but based on the questions I was bouncing around from person to person, it seemed like they also understood the concept. I know teachers that are more effective though at making kids understand and be entertained and do problem after problem until concepts are so painfully clear that they become automatic. These teachers could easily make a career in stand-up or television based on their comedic brilliance and presence – what if they decide to make videos?

What then? What do I do? If I get to that point, is it the end of my usefulness as a teacher?

Or is that just the beginning? Maybe that’s the point where I can assume my students have a certain level of basic knowledge and I can then build off of that level to do even cooler stuff. Maybe that’s where I can assume, for once, that my students have a base level of skills, and can then rise above to analyze bridges or patterns in nature or create a mathematical model that inspires a student to choose to be a doctor or engineer where he or she never would have if I hadn’t done the right project or group activity or lesson. Maybe the fact that I entrusted my students to try to figure out something on their own is enough that they feel empowered to try things even if failing again and again is a possibility. Make mistakes and come back asking questions about why their theories were incorrect. I remember worrying at one point in my career what horrible things would happen if I somehow introduced students to a concept in a way that it caused them get a question wrong and cause them just one more failure in a line of failures. If I could teach in a way that makes students feel OK coming in the next day saying “I didn’t get it, but this is what I tried” I’d feel pretty good about myself. That is, ultimately, the sort of resilience that a person needs to survive in this world.

If the students show us that they don’t need us to show them how to solve a specific problem, then we as teachers should honestly accept this fact. Our goals, if they can do the problem in a way that is mathematically correct, should shift to applying that ability to doing something more profound and relevant, be it communicating that solution or applying the solution to a new situation that is different but connected in a subtle way. Our job puts us in contact with some really amazing minds that are eager to do what we say in some circumstances. In the cases that they demonstrate that they don’t need us – that is when we must apply our professional judgment and teach them to expand their knowledge to something bigger than themselves.

Not sure why I’m waxing so philosophical today, but I’ve been really impressed with my students this week, and it’s only Wednesday! After these few days, it feels like what I’m doing with this group is like using a super computer to do word processing. I only hope they are enjoying the process as much as I am.

Climbing Mount Tai – #wcydwt edition

I am spending an amazing few days with students on this year’s class trip to Shandong province in China. We spent a couple days wandering around Qufu, the home of Confucius, and the location of the temple and mansion constructed for his relatives. There were some cool opportunities to think about mathematical thinking in Chinese architecture (more on that later) but nothing ready for prime time.

Today’s trek led us to the foot of Mount Tai, China’s #1 mountain for it’s cultural significance (not due to it’s height.) we decided as a group to trek up the mountain from the Heaven’s Gate which reduced the climb somewhat, but will descend the full height of the mountain in the morning after watching the sunrise.

From Wikipedia (to be replaced by my own pics when I get home, I promise.)


The realization that I might be able to do something really cool with this came after regretting that I had decided to leave two of my favorite data collection devices (heart rate monitor and hiker’s GPS) at home being unsure during packing if they would really be worth bringing. I had done this hike in March and had several conflicting reports of the exact height we climbed up and down. The students were asking me how many steps there were, and I vaguely recalled something around 7,000, but I wasn’t sure. This question actually popped out from a few different students as we passed the first set of steps. It got me thinking. Is it possible to take either one of these numbers (height or number of steps) and try to calculate or estimate the other? If the students were asking it standing at the bottom looking up, there might be a possibility they would be interested in answering it on their own if posed the right way.

I grabbed my camera and grabbed the best standard length measure I had on me: my iPhone.


(It probably isn’t necessary to say this, but this is just an example I took in the hotel.)

I took a number of pictures like the one above on way up the steps, trying to come up with a fairly random sampling of the size of stairs compared to the phone along the entire height. Through some combination of Geogebra, pencil & paper calculations, and some group discussion, I can see some height calculations for the climb coming out of this.

On the way up, there was also a perfect “answer” to this challenge posted in the form of a placard fixed to the wall that says both the vertical height and the number of steps – again, I will include a picture of this when I can transfer photos from the camera I used to take the good photos. I could see cropping this photo in a way that hides the answer, though I’m sure there is a more dramatic Act 3 to this challenge out there.

I think there is some potential here for some fun, as well as for good student discussion and writing about how close the number actually gets to the right answer. This is the second time in a week that I’ve been able to find something good that could work for a class activity, and I wanted to get the details out while still buzzed about its prospects.

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