An investigation into the constitution of absolute value inequalities by Grade 12 students in a selection of Western Cape State schools as displayed in students' solutions to a baseline test problem - Ward, Julianne (School of Education, 2017)
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.