Search

This study takes as its starting point Bernstein’s proposition that evaluation is central to pedagogy. Specifically, along with many researchers who draw on his work, Bernstein claims that explicit evaluative criteria are critical to the academic success of learners from working-class families and low economic status communities. The research problem stems from a hypothesis, derived from the literature, that social class differences in learner performances in school mathematics suggest differences in the functioning of pedagogic evaluation, and therefore differences in what is constituted as mathematics, and how, in pedagogic situations differentiated by social class (e.g. Dowling). The contention of this study is that insufficient finegrained analyses have been undertaken to surface the computational specificity of what it is that constitutes evaluative criteria in mathematics education studies of pedagogy. The study examines the functioning of pedagogic evaluation in what comes to be constituted as mathematics by teachers and their learners, and in the specialisations of mathematical thought in pedagogic situations. The study set out to investigate the functioning of pedagogic evaluation in two schools differentiated with respect to the social class membership of learners. Two Grade 10 teachers and their learners in each school served as research participants. Methodological resources for describing the functioning of pedagogic evaluation in terms of the computational activity of teachers and learners derive from the work of Davis, which draws on a computational theory of mind (e.g., Chomsky; Gallistel & King; Spelke). Bernstein’s theory of the pedagogic device, with its focus on who gets what knowledge and how, serves as a general descriptive frame structuring the study. The analysis reveals the following: (1) the commonly used descriptions of evaluative criteria as explicit/implicit are analytically blunt and consequently mask the complexity of criteria operative in pedagogic contexts; (2) differences as well as strong similarities in the functioning of evaluation and, therefore, differences and similarities in what is constituted as mathematics are evident in pedagogic situations differentiated with respect to social class; (3) an orientation to mathematics that constitutes mathematics as computations on the typographical elements of mathematical expressions is common to pedagogic situations involving learners from both upper-middle-class/elite families and working-class families; and (4) greater variation and inter-connectedness in computational resources is realised in pedagogic situations involving learners from upper-middle-class/elite families than in those involving learners from working-class families, where computational resources are relatively restricted and weakly connected. The differences between the two types of situations appear to be enabling of greater flexibility in mathematical thought and action for upper-middle-class/elite learners, on the one hand, and restricting for working-class learners, on the other. The contribution of the thesis is four-fold. The study: (1) provides a methodology for exploring the complexity of pedagogic evaluation by describing the computations performed by learners and teachers in mathematical terms, thus contributing to Bernstein’s account of pedagogic discourse as it applies to the teaching and learning of mathematics; (2) contributes to our understanding of the structuring effect of evaluation on learners’ mathematical thought; (3) contributes to the methodological resources developed by Davis for describing the constitution of mathematics in pedagogic situations; and (4) extends analyses of the constitution of mathematics in pedagogic situations to those populated by learners from upper middleclass/elite families in the South African context, albeit in a limited way.

There is longstanding and widespread concern that students find the transition from school to university mathematics difficult. There have been various practical responses to supporting students in this transition. Research conducted on these responses tends to focus on student perceptions and the impact on academic performance. However, research which explores the pedagogy implemented in support courses is lacking. Yet such research is needed if we are to understand what knowledge is valued and how it is communicated in support courses, which is an important first step in establishing whether these courses are replicable and whether they might indeed provide access to the knowledge valued in mainstream mathematics courses. My study investigates the implemented pedagogy of one particular mathematics support course for first-year engineering students. The pedagogy intended for the course is similar to the problem-centred approach (PCA), which is a competence pedagogy popular in selected white primary schools in South Africa in the 1990s. Critiques of school-level PCA - such as that it affords students insufficient "guidance" and that it is difficult to replicate – highlight the importance of understanding this support course's pedagogy. I made video records of one activity of the course in order to explore what knowledge the course values and how that knowledge is communicated to students. My theoretical framework is founded on Bernstein's (1996) theory of the pedagogic device, since it affords a language for speaking about the transformation of knowledge into pedagogic communication. I adopted theoretical tools from Davis's (2001; 2005) investigation of PCA at the primary school level. My study demonstrates the generalisability of these theoretical tools. Regarding what knowledge is valued in the course, I found that the central notion is problem solving. Problem solving serves as a vehicle for developing "sense-making". However, the notion of problem solving remains implicit since it is not discussed with students and students do not have an opportunity to solve the given problem independently. Regarding how knowledge is communicated, I found the implemented pedagogy to be a hybrid of Bernstein's competence and performance models. The former emerges in that much of the privileged knowledge remains implicit and the hierarchy between teacher and student is apparently flattened. The performance model is seen in teachers guiding students, both explicitly and implicitly. For example, they explicitly tell students to draw a diagram and how to check their answers. They implicitly guide students by modelling the problem-solving process and subtly positioning the students in complex ways. My results raise questions about whether students acquire the notion of problem solving in the course. Furthermore, the pedagogy identified may mitigate against students acquiring the sense-making disposition that the course intends to develop. My results bring into question the replicability of the course and how it may support students in their transition to university mathematics.

This dissertation is an investigation into the constitution of absolute value inequalities in Grade 12 students' solutions to the problem 2x − 3 < 4 in a baseline Mathematics test conducted in seven schools populated by students from working-class families in the Western Cape of South Africa. This study is located within the general problematic of the constitution of school Mathematics in pedagogic settings and the methodology draws from a study by Davis (2013a) who examined Grade 11 students' treatment of linear inequalities. This study uses a grounded theory approach (Glaser & Strauss 1967; 1999) as well as Weber's (1964) theory of ideal type categories to organize students' solutions into ideal type categories through comparative analysis. The methodology also draws from Bernstein's (1996) notion of the pedagogic device and in particular his evaluative rule and recontextualising rule as well as methodology by Davis (2010a; 2010b; 2010c; 2011a; 2011b; 2013b) for describing students' mathematical activity in terms of operations, domains and codomains. The production and analysis of the data is in two parts: part 1 analyses the recontextualisation of the topic of absolute value inequalities in the field of Mathematics in the relevant curriculum documents and a selection of relevant textbooks. Part 1 of the analysis informs part 2 which analysed the recontextualisation of the topic in students' solutions using ideal-type categories. From the analysis of students' solutions to the test item, three different levels of categories using three different sets of criteria emerged. The first level categorized attempted solutions in terms of how the notion of absolute value was maintained or disrupted, the second level categorized attempted solutions in terms of how the notion of order with regards to the logical connectives was maintained or disrupted and the third level categorized attempted solutions in terms of how the notion of order with regards to the order relations was maintained or disrupted. The results of this study confirm the general findings in the literature which show that students' treatment of inequalities is heavily influenced by their experiences of solving equations- as evidenced by students who inserted an equality symbol into their solution of an absolute value inequality problem. Another finding in the literature confirmed in this study is that one of the most common errors in students' solutions to absolute value inequalities is related to their inappropriate use or non-use of logical connectives. One of the most striking findings of this study is that the majority of students immediately treated the absolute value inequality as a linear expression or as two separate linear expressions, suggesting that for most students, the notion of absolute value is absent in their conceptions of absolute value inequalities. This study also found that the majority of students' computational activity consisted of operation-like manipulations such as "switching" which reverses the spatial orientation of the inequality symbol under certain conditions, thus constituting the topic, absolute value inequalities, as a combination of basic arithmetic and "operations" on symbols.