So “Modeling” is, in my mind at least, this huge thing. Kids should take some big problem and, with as little scaffolding as possible, solve it. That often looks like using some mathematical abstraction to represent something in the world, using the abstraction to make a prediction about the world, and interpreting that prediction in a specific context.
That’s a lot of stuff. I’ve thrown a bunch of modeling tasks at my kids this year, with the “you can always add, but you can’t subtract” mentality. And I always add. Scaffolding. Lots of it. Modeling is hard. And it’s fine for kids to need scaffolding, and to come up short — the magic of Mathalicious and Robert Kaplinsky’s lessons and Three-Acts is that they’re engaging, and even if kids didn’t figure the whole thing out on their own, they usually enjoyed puzzling through it, making some predictions, and being scaffolded their way to the answer.
Anyway, here’s my favorite snapshot of the modeling cycle, from Bill McCallum of Illustrative Mathematics.
There’s a lot here, and much of it is absent from typical mathematics curricula. There are plenty of daily objectives for “identify the variables”, and a lot fewer for “validate conclusions by comparing them with the situation”. I think math teachers have a lot of room for growth here.
One specific piece I’m thinking about today is the idea that an equation, corresponding to a graph, can be used to represent something physical in the world. I spent some time playing around on Desmos, and came up with a few graphs for a very specific goal: That students understand that a graph can be used to model a relationship in the physical world. Nothing more. No predictions or calculations to suck away from working memory or frustrate them. These are meant to be pretty simple to play with, build some intuition about how functions work, but mostly just help students internalize that functions (or non-functions) can be used to represent physical phenomena. So when I say, which of these models is more consistent with the situation?”, it isn’t a big deal, and during a modeling lesson students can focus on what’s really important — the evaluation of the model, and its implications in the problem.
Anyway, I’m talking too much. Model 1:
Drag the sliders to see if the function fits the McDonald’s double arches. Does it? How do you know? How could it fit better? On what domain does it fit?
Drag the sliders to see which function fits the Gateway Arch. Which is it? How do you know? How could it fit better? On what range does it fit?
This is the Gateway Arch, but photographed from a different angle. Does the angle change the model? Why do you think this is?
This is a church in Iceland called the Hallgrimskirkja. Use the sliders in each folder to see which function fits better. Which is it? Why? What parts of the function fit best? What parts of the function don’t?
I’m not sure if this is useful. I think it builds a useful building block to help make modeling tasks — where the goal is for students to complete the task without teacher help from start to finish — more accessible. But does it cut out the usefulness of modeling, where we are making predictions about the world and using them to learn more about the situation?
I’m planning to teach this lesson not long after break. I think it’s a necessary precursor to my lessons on Ebola and population decay in the Americas. But I’m also nervous. Is there worthwhile learning bound up in thinking about these graphs, and is it necessary before my students tackle the modeling cycle successfully?
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