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.