[Xmca-l] Re: Article on Positioning Theory

[Ed Wall]

Donna

       In a sense, I am talking about a 'definitional' eruption. Intriguingly and in a sense, it seems to be caused by a certain initial 'sloppiness' in the early grades and a certain learned 'rigidity' in the latter grades. For instance, during discussion of even and odd the case of six might come up. It behaves a bit like two - i.e. two groups of three - and a bit like three - i.e. three groups of two. Hence, if the idea of odd is still being negotiated, someone might claim six is both even and odd (by the way, Euclid, has the definitions even times eve; even times odd; and odd times odd). A bit latter there might be problems defining fractional portion of nonstandard geometric shapes (e.g. other than cookies - smile) as equal parts have been taken for granted. I could go on and be more specific, but perhaps this is sufficient.
        These 'blips' are, in a sense, developmental (I think of Vygotsky here) and, whether no one voices them sufficiently, usually lurking.  Thus it makes sense cayuse them surface. The problem is, of course, that can be uncomfortable for the teacher and the students. I can remember my own children saying to me, "Dad, don't explain! Just tell me the answer" (smile). I suspect teacher educators - such as myself - position our students - teachers and teachers-to-be -  to be comfortable (or to uncomfortable they are not comfortable).
     The term, by the way, came to be in a research group I frequented in which all we did, more or less, was watch classroom video of mathematics classrooms (especially those in which there was some sort of collaboration). After awhile it became clear that, unsurprisingly, ruptures were, to some extent, endemic. 

  I hope this is useful. 

Ed

On Mar 26, 2014, at  2:10 PM, Donna Kotsopoulos wrote:

> Ed,
> 
> I am intrigued by your reference to a "rupture." Can you talk more about this?
> 
> d.
> 
> 
> Donna Kotsopoulos, Ph.D.
> Associate Professor
> Faculty of Education & Faculty of Science, Department of Mathematics
> Wilfrid Laurier University
> 75 University Avenue West, BA313K
> Waterloo, Ontario, N2L 3C5
> (519) 884-0710 x 3953
> www.wlu.ca/education/dkotsopoulos
> www.wlu.ca/mathbrains
> 
> 
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>>>> On 3/26/2014 at 2:48 PM, in message <B0A6DB24-055F-488D-A87B-7960601D7E91@umich.edu>, Ed Wall <ewall@umich.edu> wrote:
> 
> 
> 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