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re: QuickThoughtAboutFractionManipulatives Wow! Thank you! This is why Life Changed when I met Carolyn. She's not just a mathematician herself; she spent years teaching math, and she is actively engaged in acquiring pedagogical content knowledge. Pedagogical content knowledge is a fancy way of saying that the things really good math teachers know are somewhat different from the things really good mathematicians know, and that the difference is important. (This is why neither Carolyn nor I feel that simply requiring math teachers to major in math is going to do the trick when it comes to raising math achievement. But that is a subject for another post.) While I was writing about rectangles being better than circles, I was visualizing circle manipulatives, and I was thinking:
But then I was thinking,
Now, here is Carolyn pointing out that it's going to be 'visually' impossible to tell a child that 3/2 represented as 1 and 1/2 circle is ONE THING, whereas it's going to be (reasonably) easy to tell a child that 3/2 represented as 1 and 1/2 of a bar is ONE THING. This observation has opened a window for me: I see that I hadn't progressed to the point of realizing that 3/2 should or even could be considered ONE THING. I have a ways to go. Still, this makes me hopeful that I'm beginning to develop some intuitive knowledge of math content and math pedagogy or teaching . . . because I could tell there was a reason why I'd grown more attached to rectangular fraction manipulatives over the year, not less. I just couldn't put my finger on it. Veering off on a tangent here, one of my very favorite books on the cognitive unconscious (tacit knowledge, or, sometimes, intuition) is Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious.
I remember Reber writing that one of the reasons the field of implicit learning got going in the first place was the question of how to make sure experts in one generation passed their knowledge on to the next generation. As I recall, the first thought everyone had was simply to ask experts, such as surgeons, how they did what they did. They figured the experts could tell them. It turned out the experts couldn't tell them. They were experts, not teachers. That raised the question of what we know that we don't know we know. I hope I'm developing some intuition about teaching math, and about the content of mathematics itself. But while intuition about how to teach math may be good enough, intuition about math itself probably is not. To be a good math teacher, it seems, you have to be able to put what you know about math into words and images.
Table of Contents, Implicit Learning
New Study on Manipulatives Part 2
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