Kyle Webb drops a WCYDWT video on circle area and perimeter:
First, let's pay respect to how fast the video moves, how it sets a scene and establishes a problem in just 14 slides and 57 seconds. Webb knows his audience and its attention span. Also, none of this is stock photography. Every photo selected is of high bandwidth and relates directly to the problem. After 12 seconds, we have three different views of the lawn. After 15 seconds, a panoramic shot. I'll begin my redesign 23 seconds in, when he mentions the lawn is 75 steps across.
This is really, really close to my textbook's own installation of the problem. The text would ask a question like "how far is it around?" or something with a real-world spin like "how large would the ice rink be?" (standing in for "what is the area?") and then it would explicitly define the only variable we need: 75 steps. My students would identify the formula and then solve.
This kind of instructional design puts students in a strong position to resolve problems the textbook draws from the real world but in no position to draw up those problems for themselves. This kind of instructional design also yields predictably lopsided conversation between a teacher and his students.
The fix is simple but difficult: be less helpful.
Let's start here: is circle area just something math teachers talk about to amuse themselves or do other people care? If they care, why do they care? How do we convey that care to our students? Maybe someone needs to fertilize the lawn. Maybe someone wants to spray paint the dead lawn green in the winter. Without this component, the answer to the question "how far is it around?" is little more than mathematical trivia to many students.
So put them in a position to make a choice, a tough choice that's true to the context of the problem, a choice that math will eventually simplify.
For instance: "how many bags of fertilizer should I buy to cover the entire lawn?"
Or, a little weirder: "how many cans of spray paint should I buy to cover the entire lawn?"
In both cases, we're putting every student on, more or less, a level playing field. They are guessing at discrete numbers (ie. "fifty bags — no — sixty bags.") and drawing on their intuition, which, from my experience, is a stronger base coat of for mathematical reasoning than the usual lacquer of calculations, figures, and formula.
This approach also forces students to reconcile the fact that the problem is impossible to solve as written. This is an essential moment. They need more information, but what? What defines a circle? Would it be easier to walk across the lawn's diameter or around the lawn's circumference? Which would be more accurate? Why is the radius difficult to measure? Did Kyle really walk through the center of the lawn or does he just think he did?
When you write "75 steps" on a photo, that conversation never happens.
My thanks to Kyle for jogging my thoughts here.