CGI teachers are concerned about the effects
of sending their students on to a non-CGI classroom.
Do you have any thoughts about that?
Fennema: I’m concerned about that too, and it’s something that has concerned teachers from the very first group we worked with. It is a problem. And frankly the way I respond to it, you know, “It’s better to have loved and lost than never to have loved at all.” That in some ways, that’s a trivial thing, because it doesn’t solve the problems. You know one of the other problems with CGI and the study of it—and I don’t know how we could have done it; the culture of the school is so influential about what happens, and if you could get the culture of the school to all be CGI dominated, there would be no problems. But there’s other things, and you have to work within that culture of the school, so you cannot go out and trivialize the idea. That is a problem. And the only thing I have to say about this are anecdotes that have come from teachers. One was that the children, when they go on, second-grade kids are pretty malleable. They’re pretty good kids; they’re not going to argue, but they will say to their teachers, “Well don’t you want to know how we solved it?” And if the teacher is halfway sensitive, the teacher will begin to try to figure out what’s going on and try to listen to them. So many times the children themselves take the issue ahead without you ever worrying about it. The other thing is I think communication within a school, as you all know, is so important, and I think the research data is very convincing. Bring in your workshop leaders and have them talk about what the studies show about what children learn. And those teachers [will see] that your children should be coming in about as well prepared if not better prepared. Those are the sorts of things; children can influence it, and somehow you can help influence the next teacher. I would never suggest that you quit teaching it. Now some teachers have quit and tried to do, you know, a 5-week unit or something on memorization, and kids probably won’t do that after you’ve had them think all year. But the only other thing I think is wonderful is one teacher in particular would send her children, you know, [she’d] pick out a child and be so excited about how the child had solved the problem, and send the child down to the principal to tell the principal how they solved that problem. Now, I think the first few times—I’m sure she sent a brave child the first time, ‘cause she didn’t prepare the principal, and the first few times that happened the principal was a little startled and then began coming into her classroom to see what was going and recognizing the quality of the thinking. So, your own unique ways are the only way I—we know nothing about it. I can tell you all about cultures in schools; I can’t tell you how to change them. But I can tell you, children are probably your best apostles on that. Let them go out and ....
Carpenter: One of the big things that I see happening in CGI classes is kids developing a sense of identity of themselves as mathematicians, and that’s something that endures. If you can get kids to have that identity, that there’s something there, and that that gives them a lot of power to go into a variety of kinds of situations and deal with that. Everything Eliz said—we need to try and transform the culture of the schools in a lot of ways so that they are more receptive. But kids are going to go on—those of you who are teaching in the upper grades—are going to go to middle school and things are going to be different; some of these things are going to happen. Some teachers have sort of addressed this issue directly, have essentially had conversations with their kids: “We do things this way.” [They do] not say, “Stop thinking; you’re going to be asked to stop thinking and just go along with it, go along with the show,” but to try and prepare them to continue to do what they’re doing as they move into situations that are maybe not quite as supportive of the kinds of thinking they’ve been doing.