Over at Lani’s place, Max talks about expanding CGI beyond CGI.
“Another grand challenge could be to extend the kind of work that went into CGI [Cognitively Guided Instruction] to high school Algebra, Geometry, Probability, Statistics, and Calculus concepts, so that teachers teaching, say, right-triangle trig, or similarity proofs, or solving quadratic equations, or standard deviation, had a map of learning progressions in that space and ideas about kids’ informal intuitions and misconceptions, multiple representations that connected to various aspects of a fully operational mathematical understanding, and then linkages among those multiple representations.”
Max talks about detailing how kids learn topics. But I’m not sure that this is exactly how I read CGI, though I can see it through that lens. The question is, do we seek learning progressions mapping how students think about problems or how students think about concepts? Meaning, do we want knowledge about how kids think about addition or about how kids think about addition problems?
I don’t think that this distinction really causes issues at the elementary level — to know addition is to know how to solve addition problems, and we all agree on what addition problems are — I wonder if it’ll be harder to nail down on the secondary level.
- Comparing steepness visually, or by measuring the angles of the ramps
- Comparing steepness by comparing the sides of the triangles
- Comparing steepness by looking at both the height and width of the triangles, but not distinguishing between ratio of sides, difference of sides, etc.
- Comparing steepness using the height/width ratio of each triangle
- Comparing steepness using the ratio of any two sides of the right triangles
Here are some samples of student thinking to prove that I’m not totally pulling this out of my elbow.
I think it would be wrong to say that this is a way that my students thought about trigonometry. It’s certainly how my students thought about problems that are deeply related to trigonometry. It’s definitely how my students thought about “steepness comparison” problems.
Where does this leave us, if we’re interested in providing something of value to other trigonometry teachers?