What is CGI?
Carpenter: I sort of think about CGI in a couple of ways. I think there’s sort of a small letter cgi and capital C, capital G, capital I. There are sort of first principles of CGI, and I think the underlying principle is that #1, we think of teaching in terms of affording learning of students and providing affordances for students to learn. So, we really think about teaching in terms of student learning rather then in terms of teacher activity. So, it’s not just that we don’t tell teachers what to do; [it’s] that that really becomes the way in which we think about the teaching. So, all of our conversations are around student thinking, not around what teachers are doing. Eliz is right, and we agree on this point: What teachers do is very, very important, but we think of what they’re doing in terms of affording learning of students. And the key underlying thing—I know this has sort of come up, but I’ll repeat it—is that [in] developing understanding, that children come to any learning situation, at any level, with knowledge, and a lot of that knowledge really provides a basis for learning more mathematics. And to develop understanding, you try to build on—you try and understand and then build on—the kind of knowledge that children bring. That’s sort of, I think, the fundamental key underlying idea to CGI.
The sort of capital CGI is in the details, that we’ve had some frames for thinking about students’ thinking: the kinds of problems, a trajectory of strategies, and so forth, that we hope that you can use to develop your own frames for thinking about it, but there are these sort of particular things; this is sort of the capital CGI. I think the devil is in the details, and I think those details really matter—at least that’s my perception of it. So, that’s sort of the capital CGI. Some people look at these details and think that’s the whole thing—the problem types and strategies—but the real principle is this idea of building on children’s thinking, starting where they are and going from there. But I think the details are important too.