"There are some parts of the common core standards that I would express as mathematicians’ thought experiments.” Confrey, Nov. 2, 2012
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- Does that mean we need a term like "imaginary trajectory" or "fantasy progression" to describe parts of the CCSS?Nov 2, 2012
- I'd describe it as progressions of the logic of mathematics as opposed to progressions of the psychology of the child.Nov 2, 2012
- I wish I remembered the paper in which I first read a statement about structuring the curriculum around the structure of mathematics, which isn't necessarily structured the same way we learn mathematics. Things like that have a way of making a person challenge their own assumptions.Nov 2, 2012
- Yeah, this is my big problem with Devlin -- although he takes great pains to caveat that he is not an educator and has no expertise in education, he simultaneously advocates for a model of mathematics education based on "the" (or rather, his perceived) structure of mathematics, and not on the way we invent mathematics.Nov 2, 2012
- Last year at NCTM there was an outstanding session with Steffe, Hackenberg, Norton and others that demonstrated the incompatibility with the CCSSM trajectory for fractions K-5 with the research evidence on students' conceptual structures that are in place at the various grade bands. When there is a mismatch, such as a trajectory being modeled on (some) structure of mathematics rather than on students' conceptual development, we're placing teachers in a real bind. They can either teach to the students they have, thus abandoning the standards and facing the test-related consequences, or they can follow the standards, thus abandoning goals of actually having students understand what's going on.
What is Devlin's argument for basing math ed on (his) structure of math? The drawbacks are clear. I'm not sure what the advantages would be.Nov 2, 2012
- Mostly, I think his argument is based on efficiency:
In particular, it may in principle be possible for a student, with guidance, to learn all of mathematics in the iterated-metaphor fashion described by Lakoff and Nunez, where each step is one of both understanding and competence (of performance). But in practice it would take far too long to reach most of contemporary mathematics. What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion.
And (my emphasis added),
Rather, a mathematician (at least me and others I've asked) learns new math the way people learn to play chess. We first learn the rules of chess. Those rules don't relate to anything in our everyday experience. They don't make sense. They are just the rules of chess. To play chess, you don't have to understand the rules or know where they came from or what they "mean". You simply have to follow them. In our first few attempts at playing chess, we follow the rules blindly, without any insight or understanding what we are doing. And, unless we are playing another beginner, we get beat. But then, after we've played a few games, the rules begin to make sense to us - we start to understand them. Not in terms of anything in the real world or in our prior experience, but in terms of the game itself. Eventually, after we have played many games, the rules are forgotten. We just play chess. And it really does make sense to us. The moves do have meaning (in terms of the game). But this is not a process of constructing a metaphor. Rather it is one of cognitive bootstrapping (my term), where we make use of the fact that, through conscious effort, the brain can learn to follow arbitrary and meaningless rules, and then, after our brain has sufficient experience working with those rules, it starts to make sense of them and they acquire meaning for us. (At least it does if those rules are formulated and put together in a way that has a structure that enables this.)
(http://www.maa.org/devlin/devlin_12_08.html)Nov 2, 2012
- Ah yes, the common stance of generalizing from one's own perspective/experience. The tricky part is that the vast majority of students are not budding mathematicians, and what worked for mathematicians when they were young 'uns tends not to work for the other 99%.Nov 2, 2012
- Not to mention that the approaches that emphasize understanding are better by virtually every measure, so it's not even true that you are giving up on the possibility of reaching advanced mathematics.Nov 2, 2012
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