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[Xmca-l] units of mathematics education

Ed, I can't answer your question, so I'll to revert to the other subject line.

A few ill-informed observations on the unit of analysis.

Philip's analysis into turns is of course a Bakhtinian analysis of classroom activity. Whatever else it may show, it is not going to tell us anything about the formation of concepts, mathematical or other.

I agree with you, Ed., that the text-mediated teacher-student interaction is "too big" to tell us anything about the learning of mathematical ideas. It is a lens for shedding light on the relation between teacher and student, which is doubtless helpful but still does not help us understand how a child may develop mathematical insight, only how to emulate what the teacher does, and after all, sometimes the teacher is wrong.

Vygotsky did come to the brink of talking about this topic when he studied the development of what he called pre-concepts, didn't he? "Pre-Concepts," in Vygotsky's lexicon, form only in older children, typically those who are already attending school and being confronted with school-like tasks, or engaged in social activities including processes such as measuring, buying and selling, calculating time, and so on. Such activities oblige the child to use culturally transmitted symbols of some kind (not necessarily numbers, for example coins or measuring sticks) to carry out processes requiring the abstraction of features from a concrete situation. And famously, Vygotsky used the artefact-mediated action as his unit for these studies. The teacher and the wider culture only entered the picture in the selection of the artefact and (presumably) the setting of the task (Vygotsky glossed over this).

In a brief off-line exchange, Ed has made me accept that mathematics, as practised by mathematicians, is a formal science. It is actually concerned only with sequences of symbols, not any material process outside the text. But I still think that mathematics is a science. Even symbols are material things. Mathematics is not merely a social convention, and I don't believe it can be learnt if it is taught as a formal discipline. Of course, if you believe that mathematics is just a social convention, then decoding texts under teacher direction is the essence of doing mathematics.

I accept that decoding word problems may be a typical way that mathematics is taught in school, but unless we are content to simply describe existing practices I am not satisfied that this unit, based on decoding texts, captures how students learn mathematics. And indeed it may obscure that.

*Andy Blunden*

Ed Wall wrote:

      I have never really figured out what reform-oriented oriented or, I guess the other, traditional, meant. If I ask many who name themselves as traditionalists to describe a reform-oriented classroom, they paint an inaccurate picture and, if I ask many who name themselves reform-oriented to describe a traditionalist classroom, they paint a picture just as inaccurate. Even worse if I go back and look at comments from the past, I find that the golden years of traditionalism were as contested (i.e. if transported back today's traditionalists would find themselves labeled reform-oriented). So I guess I more or less agree with Lave as regards consistency. Insofar as Piaget and Vygotsky are concerned, they both add things I think are useful to the mix and I discuss them both with my students (unfortunately my colleagues know little about Vygotsky so my students come a little less knowledgeable than I might wish).
      I really wouldn't know what it means for teaching to be consistent. I do have commitments as a mathematics teacher which I try to honor (I think you were part of the discussion when I laid these out on the list), but teaching is isn't a static profession (which I really like!) and I am always learning from my students and my colleagues. I agree with what you say about individuals and groups and grumble at my colleagues and my students (teachers-to-be) about "simply engaging students in group discussion with no real sense of how that activity is supposed to lead to learning or development in any sense." Even worse is the way quiet students are viewed and mentored.
Insofar as my original story was concerned, I can easily imagine some reform-oriented people thinking she was an excellent teacher and I can easily imagine some traditionalist teachers thinking the same (and vice-versa). As far as the story I asked Phillip about (and I was assuming others might answer), I know some mathematicians would see this class as a good idea and others would not. However, my objection was, in a sense, that it wasn't consistent (although perhaps not in the way you mean).
    So, directing the question at you (and others if they wish), why did I tell him nicely it was among one of the worst taught classes I had ever seen? I have given you a hint (smile) and I didn't have, I think, a theoretical reason (smile).