Thesis ((Ph.D. (Mathematics Education)) -- University of Limpopo, 2016 / The focus in this thesis was on the approaches that seek to compare two study arms in the absence of randomisation when the interclass correlation coefficient is greater than zero. Many reports on performance of learners in Grade 12 have used ordinary regression models (such as logistic regression model and linear regression models) which ignore clustering effect, and descriptive statistics (e.g., averages and standard deviations for continuous variables, and proportions expressed as percentages and frequencies). These models do not only bias point estimates but also give falsely narrow confidence intervals. The study was applied to two of the nine provinces of South Africa: Gauteng Province and Western Cape Province in 2008, 2009 and 2010 academic years. Causal models, and in particular, hierarchical models (or disaggregated approach), unlike descriptive analyses, are more powerful as they are able to adjust for individual covariates. For the analysis of continuous variables; Western Cape Province was expected to significantly score higher marks than Gauteng Province in 2008 (Crude estimate: 0.782) and 2009 (Crude estimate: 0.957 ) while Gauteng Province was expected to score higher marks than Western Cape Province in 2010 (Crude estimate: −0.302). Adjusted models indicate that Western Cape Province performed better than Gauteng Province in 2008 and 2009 but not in 2010 where Gauteng Province performed better than Western Cape Province after adjusting for gender. In case of binary outcome; the crude estimates favoured Western Cape Province than Gauteng Province in 2008 (Odds ratio = 1.16) and 2009 (Odds ratio = 1.19). Otherwise, the crude estimates favoured Gauteng Province in 2010 (Odds ratio = 0.11). The proportion of female learners in Gauteng Province ranged between 54.48% and 54.99%, while in Western Cape Province it ranged between 56.78% and 57.16%, in 2008 through 2010 academic years. Proportion of female learners in Western Cape Province were found to be higher than those in Gauteng during this period. At least 70.42% of learners in Gauteng and at least 73.96% of learners in Western Cape Province passed Grade 12 during the years 2008 to 2010. Through the application of causal model we have learned that although gender is not a significant predictor of the overall learner performance in Grade 12, the effect of gender gave the mixed findings depending on the nature of the outcome. The xi effect of gender on continuous endpoint (marks) suggests that a model of single-sex classrooms or single-sex schools may be adopted so as to mitigate the inherent perceptions and stereotype regarding learner-gender. However, the results based on binary endpoint (pass/not pass) suggest that coeducation system is the best bet. A school quintile is a significant predictor of the overall learner performance in the two provinces. The resourceful schools are more likely to produce learners with higher marks. Also, the resourced schools than the less or under resourced schools are more likely to produce the favourable results (higher marks (%) or/and pass) in the two provinces.

This study investigated the problem-solving skills of Grade 12 learners in probability. A total of 490 Grade 12 learners from seven schools, categorised under four quintiles (socioeconomic factors) were purposefully selected for the study. The mixed method research methodology was employed in the study. Bloom’s taxonomy and the aspects of probability enshrined in the Mathematics Curriculum and Assessment Policy Statement (CAPS) document of 2011 were used as a framework of analysis. A cognitive test developed by the researcher was used as an instrument to collect data from learners. The instrument used for data collection passed the test of validity and reliability. Quantitative data collected was analysed using descriptive and inferential statistics and qualitative data collected from learners was analysed by performing a content analysis of learners’ scripts. The study found that the learners in this study were more proficient in the use of Venn diagrams as an aid in solving probability problems than in using tree diagrams and contingency tables as aids in solving these problems. Results of the study also showed that with the exception of Bloom's taxonomy synthesis level, learners in Quintile 4 (fee-paying schools) had statistically significant (P-value < 0.05) higher achievement scores than learners in Quintiles 1 to 3, (i.e. non-fee-paying schools) at the levels of knowledge, comprehension, application, analysis and evaluation of Bloom’s taxonomy. Contrary to expectations, it was revealed that the achievement of the learners in probability in this study decreased from Quintile 1 to Quintile 3 in all but the synthesis level of Bloom's taxonomy. Based on these findings, the study argued that the quintile ranking of schools in South Africa may be a useful but not a perfect means of categorisation to help improve learner achievement. Furthermore, learners in the study demonstrated three main error types, namely computational error, procedural error and structural error. Based on the findings of the study it was recommended that regular content-specific professional development be given to all teachers, especially on newly introduced topics, to enhance effective teaching and learning. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)