(The image above is part of The Modeling Comic that one of my students created last year)
I’ve been struggling for some time with the idea of a Model (within the construct of Modeling Instruction). Back up with me for a bit. In the summer of 2011 a colleague who teaches at another school in Minnesota introduced me briefly to Modeling Instruction. Being who I am, that is, someone who loves to learn, I promptly found a community of modelers on Twitter and learned quite a bit from blogs by Kelly O’shea, Frank Noschese, and Scott Thomas. I found out the hard way, however, that I had a fairly limited view of what a Model actually is.
In my excitement to get started Modeling I got stuck in a rut of thinking that 1) equations were physics (a statement I would never say to students but found myself thinking), and 2) that a model was an equation. For example, the CVPM model, to me, was x=xi+vt. Through the process of attempting modeling throughout the year, failing some, succeeding some, reading blogs, and finally taking the Modeling Instruction training this summer, I think I have a much better picture of what a Model actually is. The training was instrumental in my formation of a definition of a model, as was Sam Evan’s post on what it means to model. So let’s get to the good stuff.
I believe that a Model could be defined as a particular phenomenon that is described using a set of representations (diagrams, equations, descriptions, charts, graphs, and more). These representations should produce accurate and reproducible predictions for and explanations of the phenomenon, within the limitations of the model. This makes it very hard to use a representation to describe the model, because the model is about the phenomena, not the representation. The Constant Velocity Particle Model (CVPM) is not linear position vs. time graphs nor equally spaced motion diagrams. CVPM is a type of motion that occurs when neither speed nor direction are of an object are changing.
To demonstrate what I think a model is in terms of MI, I am going to use a model that is well known to the science community; the Bohr model.
The Bohr model is a the idea (concept? model?) that the nucleus of an atom is surrounded by electrons whipping around in fixed orbits.
The Bohr Model was proposed by Niels Bohr in an attempt to explain emission spectra, which it did fairly well for Hydrogen. Representations for The Bohr Model include the planetary-style diagram shown above and energy level diagrams (below).
Two notable equations are used to represent The Bohr Model; the allowed Bohr radii formula
and the allowed energy levels known as the Rydberg energies
The Rydberg energies also have a more general form for atoms with more than one proton in the nucleus;
We have a number of representations above that attempt to describe the model. The cool thing, I think, in using The Bohr Model as an analogy for how Modeling Instruction is structured is that the model can be broken. That is, the model is useful under certain conditions, but must be modified when extending beyond those conditions. It turns out that the phenomena of atoms is more complex than the relatively simple Bohr Model had suggested.
This is actually a good analogy to CVPM leading into the Constant Accleration Particle Model (CAPM), as CVPM is really a subset of CAPM for acceleration=0. The Bohr model works well for hydrogen because there are no other electrons interacting with the one that is ‘orbiting,’ thus Bohr’s assumptions work well; as soon as you add more electrons, the predicted emission spectra differs from the actual spectra and the model is broken. Using constant velocity to try to solve more complex motion problems where acceleration takes place is a bad idea, because the assumptions for CVPM no longer hold.
Still, because students need scaffolding and baby steps, it is pedagogically appropriate to teach CV before CA as a stepping stone; one could just teach CA, but it would be a bigger step to expect students to take. Similarly, the Bohr Model is the first step toward understanding quantum mechanics; in fact, it was a giant conceptual leap that allowed those who followed after Bohr to expand the model into something more complete.
I hope I have given some credence to what a Model is within the framework of Modeling Instruction. I very much appreciate any feedback you can give me!