Torquetum

 

Previously, I read Cobb and Hodge (2005) which was quite a long report, published online. In it they outline three “key constructs” to describe and analyse identity as doers of mathematics. These key constructs are personal identity (who the student actually becomes in the classroom, including issues of affiliation or alienation), normative identity (obligations, expectations and norms co-constructed by teacher and students) and core identity (the trajectory the student is on, where they see themselves to be and where they see themselves to be going). I found core identity to be the most interesting, for my purposes, of the constructs. It is different from the others in that it reaches outside the classroom, whereas the others are by definition internal to the classroom.

 

In Cobb, Gresalfi and Hodge (2009) (a paper with exactly the same name as the prior report) only personal and normative identities are covered. They are explained clearly and in a lot of detail. The authors describe how these construct can be made empirically viable. I admit that I was disappointed not to see any mention of core identity. Was it omitted for lack of space? Certainly the journal paper had to be a lot shorter than the report. Have the authors abandoned that construct for some reason? Surely not. While the word “core” could be considered a problematic word (Sfard and Prusak () for instance, from their viewpoint of identity as narrative, refuse to consider an identity external to the discourse) the idea of identity linked to trajectory cannot be abandoned. I shall have to look around and see if these authors have published elsewhere on core identity.

 

The authors are particularly interested in three types of situations encountered in the classroom and reported in the mathematics education literature: “those in which students identify with classroom mathematical activity, those in which they merely cooperate with the teacher , and those in which they resist engaging in classroom activities and thus develop oppositional identities” (p. 41). They argue that their identity constructs provide an interpretive scheme which can “attend to the nature of mathematical activity as it is realized in the classroom; to what students come to think it means to know and do mathematics in the classroom; and to whether and why they come to identify with, merely comply with, or resist engaging in classroom mathematical activity.” (p. 41)

 

I can’t really define the constructs in better words than their’s, which are:

“building on Boaler and Greeno’s work, normative identity as we define it comprises both the general and the specifically mathematical opbligations that delineate the role of an effective student in a particular classroom. A student would have to identify with these obligations in order to develop an affiliation with classroom mathematical activity and thus with the role of an effective doer of mathematics, as they are constituted in the classroom. Normative identity is a collective or communal notion rather than an individualistic notion. In contrast, personal identity concerns the extent to which individual students identify with, merely comply with, or resist their classroom obligations, and thus with what it means to know and do mathematics in their classroom.” (pp. 43-44)

 

In the 2005 report, Cobb and Hodge argue that core identity and normative identity need to be reconciled for affiliation to occur, which is not a point argued in this 2009 paper at all. I really do need to read more widely…

 

So, normative identity subsumes general classroom obligations and specifically mathematical obligations. The general obligations are underpinned by the distribution of authority in the classroom and the opportunities for the students to exercise agency, and then agency itself, directly related to the distribution of authority, manifests as two kinds: conceptual agency “choosing methods and developing meaning and relations between concepts and principles” and disciplinary agency “using established solution methods” (p. 45).

 

I can see how a traditional classroom, with authority distributed only to the teacher allows students access to disciplinary agency only. I struggle to see easy ways of allowing access to conceptual agency in my classroom, though. Hmmm. Indeed, the authors mention “The algebra teacher had to accommodate typical concerns of content coverage and accountability within the school” and hence authority was distributed primarily to the teacher only and the students had opportunity only to practise disciplinary agency. This ties in with my situation, with a huge amount of content and a huge amount of accountability. I don’t have freedom to play around with all sorts of imaginative stuff, much as I’d like to. My responsibility and challenge comes in trying to find ways of teaching creatively under that weight of content and accountability.

 

Interestingly, the authors say that, if authority is distributed to teacher and students, then the students need opportunity to exercise conceptual agency otherwise those classrooms are going to be ineffective in supporting mathematical learning since the students “are not practiced at understanding whether or when particular kinds of disciplinary tools might be useful in solving problems” (p. 45). In general (although not always) a classroom where authority resides entirely with the teacher only allows the students to exercise disciplinary agency.

 

Ok, so the types of agency + to whom the students are accountable make up the general classroom obligations. The specifically mathematical obligations consist of the norms for mathematical argumentation + the norms of reasoning with tools and written symbols. These two types of obligations together make up the normative identity. Both types of obligation “are constituted in the course of the ongoing classroom interactions” (p. 46).

In the process of contributing to the “ongoing regeneration of the normative identity as a doer of mathematics as it is realized in the classroom” (p. 47) each student develop personal identities involved with affiliation with or alienation from classroom mathematical activities.

 

I’m struggling to understand how they make personal identity “tractable” for empirical analysis. They refer to the “moral dimension” of the classroom, referring to, I think, the fact that classroom norms and obligations are value-laden. I don’t really get that. How is that a “moral dimension”? Anyway, to investigate personal identity what is important is “documenting students’ understandings of their general classroom obligations, their valuations or appraisals of these obligations, and the grounds for their valuations” (p. 47).

 

The last word:

“Students would have to identify with the role of an effective student as delineated by these obligations in order to develop a sense of affiliation with mathematical activity as it is realized in their classroom. In this process, obligations for others would become obligations for oneself” (p. 63).