Torquetum

 

There are a lot of similarities between this chapter and the 2009 paper previously discussed. The main difference is the discussion here on classroom design experiments – the things that need to be central and the pitfalls you need to watch out for. I haven’t read much on design experiments before (although I probably should have) and I found that part very interesting.

 

An important feature, for instance, of a classroom design experiment is to “develop, test and refine theories, not merely to empirically tune ‘what works’ ” (p. 224). Also, “instructional design and research are interdependent” (p. 224). The design serves as a context for research and the analyses inform ongoing design. There are typically phases of a classroom design experiment: preparing, experimenting, conducting retrospective analyses.

 

In preparing, determining the focus of the experiment is of vital importance and can be a challenge. You need to know what aspects of students’ learning you want to focus on and what is background or of secondary importance. Often such experiments have as focus students’ development of domain-specific skills and reasoning and students’ identification with classroom activity is secondary. Cobb et al. support making students’ identification with classroom mathematical activity the focus. In the second phase, the objective is not to test whether the instructional design works (although that will necessarily be assessed), but to improve the design “by testing and revising conjectures inherent in the design about both the process of students’ learning and the specific means of supporting it” (p. 225). These ongoing analyses are highly selective and often involve “implicit, unarticulated assumptions” (p. 225). Cobb et al. argue that it is critical to “explicat[e] the key constructs used when making these interpretations so that underlying suppositions and assumptions are open to public scrutiny and critique” (p. 226). They put forward their empirically grounded (developed “while conducting retrospective analysis of a classroom design experiment” (p. 226)) framework of the key identity constructs as a way of addressing this concern of explication.

 

I won’t go into detail here on the key constructs (personal and normative identities) as it has been discussed elsewhere, but there is a lot of detail present in the chapter.

 

Interestingly, they discuss Sfard and Prusak’s (2005) narrative framing of identity at some length and make a particular point of discussing their “designated identity” which is related to trajectories into the future. To my mind, this ties in directly with Cobb and Hodge’s (2005) “core identity” yet in this paper they regard it as an “unfortunate oversight” that they (Cobb, Gresalfi and Hodge) did not include a construct related to future trajectory in their interpretive scheme. Surely they did? In 2005? With “core identity”? I’m confused. Anyway, they tie Sfard and Prusak’s designated identity to D’Amato’s extrinsic and intrinsic value (structural or situational significance), which is something I’d like to read more about.

 

I like the fact that they regard Sfard and Prusak’s framework as complementary to their own. Of all the identity theories which I have read, Sfard and Prusak’s narrative framework and Cobb et al.’s key constructs are the ones which make the most sense to me. I’d just like to read more on this lost “core identity” though …

 

Potential further reading:

D’Amato, J. (1992). Resistance and compliance in minority classrooms. In E. Jacob & C. Jordan (Eds.) Minority Education: Anthropological perspectives (pp. 181-207). Norwood, NJ: Ablex.

Horn, I.S. (2006). Turnaround students in high school mathematics: Constructing identities of competence through mathematical worlds. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA.

 

 

I am interested in finding various theoretical perspectives on identity and identity development of (preferably university) students studying mathematics. After having read widely, I could potentially compare and contrast, but right now I’m just trying to get a sense of what is out there.

 

In this paper, Cobb and Hodge introduce three key constructs of identity, namely normative identity, personal identity and core identity. Here are some definitions, taken from the text:

Normative identity: “The normative classroom identity is concerned with the obligations that a student has to fulfil in order to be an effective and successful mathematics student in that classroom. These obligations involve general norms for classroom participation as well as specifically mathematical norms.” (p. 11); “the type of person the students would have to become to be mathematical people” (p. 16)

Personal identity: “The facets of personal identity that emerged from our analysis concern students’ understandings and valuations of their general and specifically mathematical classroom obligations together with their assessments of their own and others’ developing mathematical competences.” (p. 25); “who they actually become in the classroom” (p. 16)

Core identity: “envisioned life trajectories … , of who the students viewed themselves to be and who they wanted to become” (p. 2); the person’s “life story” is of central importance to core identity, but core identity is not reducible to the life story (p. 27); They link to Gee’s notion of core identity, also Gee’s use of “life stories” which has echoes of Sfard and Prusak’s work. Gee’s identity constructs will be discussed in a later blog post, as will Sfard and Prusak’s.

 

An important feature of the normative identity is that it is classroom-dependent and is co-constructed by teacher and students. They refer to how the teachers and students “contribute to regenerations of patterns” (p. 12 and elsewhere). “ ... students are seen to contribute to the initial constitution and ongoing regeneration of the normative identity, and to develop their personal classroom identities as they do so” (p. 16, emphasis added). The norms that they looked at in their classrooms of interest were social norms and specifically mathematical norms, for instance taking notes and asking questions were social norms, and what constituted a mathematical explanation and how information is presented are mathematical norms. Interestingly, they point out that norms are most obvious when they are transgressed. The authors argue against the notion that students are invited to adopt a normative identity in a specific classroom, independent of their participation. Aspects of the core identity need to be reconciled with the regeneration of the classroom normative identity in order to develop an affiliation with mathematical activities. Or is it the other way around?

 

With reference to Martin’s (2000) study, they consider successful students to have reconciled their core identities with the classroom normative identity through their development of personal identity as doers of mathematics (p. 2). So, let’s see, students enter any classroom with an existing core identity related to mathematics which involves their internal trajectory of where they are going with relation to mathematics. They presumably also have some vestige of a personal identity as a doer of mathematics as constructed in previous mathematics classes, but potentially quite changeable depending on the specific classroom context they find themselves in now. During the practise of classroom activities, the students and teachers co-construct the normative identity (surely the teacher plays the larger role, though?) and it is through this co-construction that personal identities are developed or changed. The greater the sense of affiliation with the mathematical activities, the greater the reconciliation between aspects of core identity and normative identity. Hmm. I’m pretty sure I’m mangling this.

 

Cobb and Hodge report three difficulties they have with prior research on identity. First, they find a contradiction in identity theories from a situated perspective. They use as example Boaler & Greeno (2000) who describe identity as contextual yet also refer to a stable transcontextual identity. Secondly they feel that limited guidance is provided to teachers by other identity formulations. Thirdly they feel the word identity is used in too many different ways. They hope their three constructs counter these three difficulties.

 

They discuss (citing Nasir, 2002) how the development of a sense of identity and affiliation with a community of practice can motivate new learning and the gaining of new skills. They recognise the instructional value of cultivating such affiliation, but also argue that a sense of affiliation with mathematical literacy in general is an important goal in its own right. Again drawing on Nasir (2002), they illustrate the following (linear?) sequence: 

Development of more engaged personal identity leads to

reasoning about data in increasingly sophisticated ways, which leads to

participating in class in new ways, which leads to

affiliation with mathematical activity in class, as well as

increased mathematical competence.

They wield the hefty word equity in reference to the cultivation of interest in mathematics in and out of school and hence development of a personal identity as a doer of mathematics.

 

In closing, some lines from the Discussion (p. 31):

The authors argued “that an analysis of the personal identities that students develop in a particular classroom provides a way of accounting for students’ persistence, interest in, and motivation to engage in mathematical activity as it is constituted in that classroom. In addition, we argued that students’ development of personal identities that involve a sense of affiliation with mathematical activity should be an important instructional goal in its own right in that it related directly to issues of equity in mathematics education”

 

Boaler, J., Greeno, J. (2000) Identity, agency and knowingin mathematical worlds, in J. Boaler (Ed.) Multiple Perspectives on Mathematics Teaching and Learning (pp. 45 – 82). Stamford, CT: Ablex.

Gee, J.P. (2001) Identity as an analytic lens for research in education, Review of Research in Education, 25, 99 – 125. Note: check to see if there are other Gee (2001) refs, this C&H paper is unclear.

Gee, J.P. (2003) What Video Games Have to Teach Us about Learning Literacy. New York: Palgrave/MacMillan.

Martin, D.B.  (2000) Mathematics Success and Failure among African-American Youth. Mahwah, NJ: Erlbaum.

Nasir, N.S. (2002) Identity, goals and learning: Mathematics in cultural practice, Mathematical Thinking and Learning, 4, 213 – 248.