A number of issues have risen from reviewing the relevant literature for my study. These include the analytic resources and questions to consider and reflect on as the study unfolds. Concepts drawn include teacher education as intelligible and/or sensible (a practical accomplishment) in Davis et al. (2007) terms. These concepts enable the explanation of whether the discourse of engaging with LMT dealt with in mathematics teacher education is in the realm of the sensible and/or intelligible. Categorization of what entails the discourse of engaging with LMT following Even & Tirosh (2002) has also been illuminated to include developing in learners both instrumental and relational understanding, focusing on learner errors, and creating a classroom environment where teacher can listen to learners. This enables the establishment of ways in which teacher-educators’ and student-teachers’ talk resonate with these three categories.
In dealing with learner errors, a three-stage process for carrying out error analysis, in Peng and Luo’s (2009) terms and Jacobs et al.’s (2010) include identify, explain, and remediate. Analytic resources for explaining sources of learner error in Ryan & Williams’ (2007) terms include modelling error, prototypical error, error as a result of overgeneralization, and error as a result of inadequate process-object conception [proceptual thinking in Watson’s (2009) terms or process-object duality in Sfard and Linchevski’s (1994) terms]. Error as a result of overgeneralization is re-described, in Lima & Tall (2008) terms, as an issue of met-before or met-after. Other concepts for explaining sources of learner error in Hart et al. (1981) terms relate to learners’ interpretation of letters, and for my study these include letter evaluated, letter not used, letter used as an object, and letter used as a specific unknown. Having pulled the conceptual map from the literature review, one has to gaze on this through an
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understanding of how the relay produces some of these, and I now turn to Bernstein (2000). He provides the theoretical gaze that informs this study, hence the theories together with the literature reviewed make my theoretical field (Dowling & Brown, 2010).
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CHAPTER 4
4 A THEORETICAL PERSPECTIVE AND CONCEPTUAL
FRAMEWORK
4.1 Introduction
My study is informed by Bernstein’s (1982, 1996, 2000) theory of the pedagogic device (PD). Given my research questions, the relevant literature reviewed, and that I want to figure out what student-teachers know in relation to the discourse of engaging with LMT, I have to establish how they deal with this knowledge. Therefore, Bernstein’s (op cit) theory of the PD provides me with a special language to describe how educational knowledge is produced and reproduced in context, hence necessary to use in exploring and explaining the problem of engaging with LMT at teacher education level. Bernstein (op cit) argues that learning is about acquiring recognition and realization rules based on classification and framing values that are guided by contextual rules. In working towards establishing student-teachers’ preparedness for the tasks of teaching, focus is mainly on what is entailed in the discourse of engaging with LMT, how it is ‘taught’ and ‘learned’, and positionings, in particular, its use when dealing with school algebraic thinking.
From the aforesaid, I am in a way being responsive to Bernstein’s (2000, p. 27) invitation to not only focus on ‘the carried’ (what is relayed) but also on ‘the carrier’ (or relay). Brown and Dowling (1999) argue that a conceptual frame for research (its language of description), develops through the interaction between the theoretical and empirical fields of the study. As will become evident, my conceptual framework has just this interactive base. I thus need to briefly state what constitutes the empirical for the study, as the theoretical elaborations following are a function of my empirical foci. Details of the methodology follow in Chapter 5.
Teacher-educators’ discourses were elicited from in-depth interviews with four relevant mathematics teacher-educators, where they spoke about what and how they taught for LMT, and how they thought it was learned. These interviews were followed by focus group interviews with student-teachers, who similarly were asked to speak about what, how and
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where they learned LMT in their formal programme. Following this, student-teachers, in pairs were then interviewed on a range of algebraic tasks related to common learner errors from the literature and generated from tests that were administered to learners in selected local schools. In the discussion below, teacher-educators’ discourses thus refers to their interviews. Student-teachers’ discourses refer to the two distinct contexts in which this was elicited – interviews on LMT per se, and interviews where they engaged with learner errors.
As I progressed through the study, engaging iteratively with the empirical and further reading of literature reviewed in Chapter 3, the resonance between threads in the teacher-educators’ discourses of LMT and some of the literature became apparent, and so more explicit in the conceptual framing of the study. Specifically, to describe what engaging with LMT entails, I explored how teacher-educators’ and their student-teachers’ talk resonated with Even and Tirosh’s (2002) categories and described in Chapter 3. Moreover, student-teachers’ talk was mapped onto a framework drawn from Peng and Luo (2009), and Jacobs et al. (2010) to explore how engaging with LMT was used when focus was on scenarios designed around common algebraic learner errors. As for positionings, Morgan, Tsatsaroni and Lerman’s (2002) elaboration of Bernstein’s (2000) notions of classification and framing were used. These discourses and positionings then made it possible to establish and explain how student- teachers’ discourses of LMT relate to their teacher-educators’ discourses. As I elaborate my theoretical and conceptual framing below, the interaction between the theoretical and empirical will become apparent.
Finally, by way of introduction to this chapter, it is important to note that while Bernstein’s (1982, 1996, 2000) work is concerned with understanding the production or reproduction of social inequalities in terms of who gets what between the working class and the middle class learners through schooling, my work does not have social class as its focus. I want to understand the what and how of student-teachers’ acquisition of LMT. I am concerned with the opportunities for LMT the mathematics education teacher-educators make available for their student-teachers, and then whether and how what student-teachers acquire is similar or different from what is made available. In explaining this, Bernstein’s pedagogic device (the PD) is pertinent. It can be used to analyse “the processes by which discipline-specific or domain-specific expert knowledge is converted or pedagogised to constitute school knowledge (classroom curricula, teacher-student talk, online learning)” (Singh, 2002, p. 572).
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It is important to note that this process applies to any institutional knowledge such as a University, and not just the school. In line with this understanding the question I ask for my study is: What is constituted as the discourse of engaging with LMT in the mathematics education curriculum in formalized institutions of higher learning? In the sections that follow, I describe the PD and its intrinsic rules which regulate what counts as pedagogic communication, and how this provides a lens for my study.