[Xmca-l] Article on Positioning Theory

[Ed Wall]

I always enjoy reading about the dynamics of mathematics classrooms so thanks to Donna.

Some somewhat random thoughts (and as I am not entirely familiar with the terminology of positioning I may use it quite incorrectly) as I've been thinking about related issues.

Teachers are placed in classrooms (positioned?) with certain toolsets and among these is something that, in its various forms, is called collaborative learning. This is, in a sense, neither good or bad; collaborative leaning is simply a tool. When difficulties do arise, it is, in a sense, because it becomes a one-size-fits-all method (positioning?) for inducing dialogue. When it works it is very very good, when it is bad it is horrid (smile). The question, one might say, it gives a sort of answer to  becomes in mathematics classrooms, at least, how to give students opportunities to learn use publicly established ideas, methods, and language so as to make, validate, improve, and extend the mathematical knowledge of the class. Is this necessary or desirable? It depends on your point of view I guess.

Teachers are placed in some quandaries if they get up from their desk or relinquish their place at the blackboard. Collaborative learning of some sort (and the group could be two) forces this issue somewhat. However, it also surfaces the need for some careful grouping and the possible need to publicize appropriately in the collective class. That is, 'positioning' yourself as a teacher that supports some sort of collaborative work is usefully discomforting (smile). 

Along with this, if done thoughtfully, comes the ability to manufacture and juggle ruptures. Mitchell is a nice example of this although unfortunately his rupture does not seem to make it out of his group (I tend to see this, perhaps incorrectly, a misfire of the very idea of collaborative). What I find quite interesting in this regard is Donna's (I think I read this correctly) attempt to re-'position' Mitchell and the pronounced resistance from Mitchell's colleagues and, in a sense, from himself. Ruptures almost always arise with reasonable mathematics tasks and are to be cherished (all this is an opinion) for their potential. However, realizing that potential takes some serious teacherly skill and I'm not sure that re-positioning Mitchell is the solution (he may need to do this himself with, one might say, encouragement) although re-positioning his rupture may well be.

Finally, for some reason, I tend to read into the dynamics of Mitchell and his group Michel de Certeau's ideas of 'everyday' strategy and tactics. Mitchell (and I am, in part, reading myself into this) is engaged in tactical maneuvers (he says something to this regard) in the face of a somewhat strategic view of mathematics embodied by his colleagues (the omnipresent 'it'). He has also put something on the table that with a little teacherly push (although this needs some careful thought out) could usefully challenge that strategic view of mathematics. 

I have seen this activity done a number of times and when it 'succeeds' (my opinion) it usually is because a rupture surfaces for the entire class. What I don't know is how people position themselves, if they do, afterwards (including the teacher) in the light of the ensuing dialogue. Very interesting!!

Thanks

Ed Wall