Currently working on an ethnography of math instruction. Here’s where I’m at:
I’m realizing that when I started off planning to investigate the mathematics classroom as a discourse community, that was the linguist in me talking, not the math teacher. I’m defining discourse community in the John Swales sense of a group of people who share communicative norms and genres, so if math is a discourse community, then knowing and doing math is essentially a repertoire of ways of talking.
Rereading my fieldnotes, though, I don’t believe this is true, strictly speaking. There’s probably good work to be done on the linguistics of math that takes the discourse community as an operating framework, but for an ethnographic investigation, I’m trying to understand how people in the community understand it — and that’s not really in terms of discourse.
What math is, according to my best current understanding of my participants’ view, is not so much a way of talking as a way of knowing; less discourse community and more epistemic community, in Teun van Dijk’s terminology. The objects of mathematical study in the classes I’ve observed include functions, algorithms, and their properties; differentiation, for example, is an idealized abstraction of an emergent property of another idealized abstraction. To deal with all this abstractness, instructors constantly model the practice of articulating exactly how they know what they know:
There’s a property that says that we can do this.
What’s the important theorem that we know?
It’s zero over zero, we need L’Hopital’s rule.
… by the geometry of similar triangles, right?
In conversation, one teacher even went so far as to tell me that if a student was given the problem
3x2 × 4x3
it wouldn’t be enough for them to know that you multiply three times four, and then multiply the x terms; it wouldn’t even be enough for them to know that this is possible because you can multiply terms in any order. They would expect students to use the term commutative property of multiplication, or the answer would be imprecise.
So where does this leave me? At a micro-level, there seems to be some interesting work to be done around the mathematical concept of precision and the linguistic phenomenon of evidentiality: when and how do claims need to be backed up with appeals to authority or general principles? This feels like a manageable focus for a term paper project. In my interviews, I’m considering ways I could get teachers to comment on the nature of math as a community, and to what extent the epistemic community construct is accurate and useful. I’m also curious about the students’ perspective — what’s their view? What kind of community do they feel themselves to be peripheral members of?