Student Number : 0302922V - MSc research report - School of Education - Faculty of Science / The difficulty of teaching and learning mathematics in a language that is not the learners' home language (e.g. English) is well documented. It can be argued that underachievement by South African learners in most rural schools is due to a lack of opportunity to participate in meaningful and challenging learning experience (sometimes due to lack of proficiency in English) rather than to a lack of ability or potential. This study investigated how improvement of learners' English language proficiency enables or constrains the development of mathematical proficiency. English Computer software was used as intervention to improve the English Language proficiency of 45 learners. Statistical methods were used to analyse the pre- and post-tests in order to compare these learners with learners from another class of 48. The classroom interaction in the mathematics class before and after the intervention was analysed in order to ascertain whether or not the mathematics interaction has been enabled or constrained. The findings of this study were that proficiency in the language of instruction (English) is an important index in mathematics proficiency, but improvement of learners' language proficiency, even though important for achievement in mathematics, may not be sufficient to impact on classroom interaction. The teacher's ability to draw on learner's linguistic resources is also of critical importance.

Teaching mathematics towards mathematical proficiency cannot easily be described since mathematical proficiency is complex and no term alone capture all aspects. The purpose of the study was to examine how mathematical proficiency in conceptual understanding can be discriminated through the concept of subtraction in grade 1 and 3 primary school. Question raised by the study was 1. What different levels in quality can be discriminated in the ways students subtracts?   2. Can progression be discriminated within subtraction operations between grade 1 and 3? Concepts like mathematical artefacts, capability and ability have been useful tools when analyzing the material. Developing mathematical thinking in students is about developing conceptual understanding anchored in the process rather than focus on procedures and outcome. When subtracting with comprehension one knows that subtraction undoes addition since subtraction is the inverse operation of addition.      The methods used are qualitative studies using observations and interviews. The material consists of notes from observations, recordings of interviews and written solutions. The study developed an analysis tool in order to compare the different ways of thinking when doing subtraction activities. Developing the tool was critical in order to compare and discriminate different levels of mathematical thinking. The results indicate that the Swedish curricula in mathematics do not support mathematical teaching towards mathematical proficiencies because it’s lack of concrete measurable goals.

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Thesis (Ph.D. (Mathematics Education)) -- University of Limpopo, 2018 / The purpose of the study is to explore the effectiveness of Annual National Assessment (ANA) in monitoring the standard of mathematics education and to assess the mathematical proficiencies tested and exhibited by Grade 9 learners in South Africa. The research problem was premised on the dearth of data that justifies ANA as an evaluative assessment. As such, the study utilised five strands which were; procedural fluency, conceptual understanding, strategic competence, adaptive reasoning and productive dispositions as a theoretical framework to assess mathematics that was tested and exhibited by learners. To explore the research problem, the study used mixed methods in the context of exploratory sequential design. Document analysis was used first to capture mathematics content and cognitive levels examined by ANA. Second, learner responses were explored using four variables of achievement levels; no response, correctly answered, incorrectly answered and partially answered. First, the results from the analysis of ANA questions indicated that ANA mostly tested questions of low complexity. Second, the results from the learners’ responses revealed that the majority of learners were not proficient to ANA irrespective of low complexity testing. Third, the Porter’s alignment index for ANA and TIMSS was between moderate and perfect. Subsequently, content and cognitive levels were misaligned in the three consecutive years of ANA testing. It implies that learners were most likely to show a deficit of higher order problems solving skills which are a prerequisite of courses in advanced mathematics. Additionally, the results suggest that ANA had challenges of reliability and validity as an evaluative assessment due to inconsistency in the testing. As such, it is recommended that the complexity of ANA be addressed, the content areas where learners are not proficient be addressed and the alignment of ANA must be frequently calculated to monitor the standard of mathematics education in South Africa effectively. Keywords Alignment index, mean deviation, strands of mathematical proficiency, systemic assessment.

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

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The study attempted to determine the causes of poor performance among Grade 12 learners in tasks involving calculus, especially in cubic graphs and the application of differential calculus. The study was conducted in three schools of the Msukaligwa 1 Circuit in the Gert Sibande District, Mpumalanga Province in South Africa. Differential calculus is a branch of mathematics that is concerned mainly with the study of the rate of change of functions with respect to their variables especially through the use derivatives and differentials. Students have difficulties in learning and mastering this section of calculus as is revealed by examiners’ and moderators’ reports year after year. The purpose of this study was to investigate the possible reasons for the poor performance by Grade 12 learners in calculus-based tasks, especially in cubic graphs and the application in optimisation. The study sought to investigate the causes of the poor performance by Grade 12 learners in tasks based on these two subtopics of calculus. Three schools were selected by means of purposive sampling: one former model C, one Mathematics, Science and Technology Academy (MSTA) and one other school that does not fall in either of these two categories. This enabled the study to have participants from diverse backgrounds. A qualitative research design was used. Data was collected using learners’ scripts for the three formal tasks: May common test, June (midyear) and Trial (preparatory) examinations. Only the questions involving cubic graphs and the application of calculus were part of the study. Analysis was done in order to determine learners’ challenges, common mistakes, and misconceptions, but also of good responses given by learners. / Mathematics Education / M. Ed. (Mathematics Education)