This study is inspired by and utilises the van Hiele theory of geometric thought levels, currently acclaimed as one of the best frameworks for studying teaching and learning processes in geometry. The study aims both to explore and explicate the van Hiele levels of geometric thinking of a selected group of grade 10, 11 and 12 learners in Nigerian and South African schools. The study further aims to provide a rich and indepth description of the geometry instructional practices that possibly contributed to the levels of geometric conceptualisation exhibited by this cohort of high school learners. This collective case study, presented in two volumes, is oriented within an interpretive research paradigm and characterised by both qualitative and quantitative methods. The sample for the study comprised a total of 144 mathematics learners and 6 mathematics teachers from Nigeria and South Africa. They were selected using both purposive and stratified sampling techniques. In using the van Hiele model to interrogate both learners’ levels of geometric conceptualisation and teaching methods in geometry classrooms, the study employs a qualitative and qunatitative approach to the data-collection process, involving the use of questionnaires (in the form of various pen-and-paper tests, hands-on activity-based tests), interviews and classroom videos. Although the data analysis was done largely through descriptive statistics, the whole process inevitably incorporated elements of inferential statistics (e.g. ANOVA and Tukey HSD post-hoc test) in the quest for indepth analysis and deeper interpretation of the data. Learners were assigned to various van Hiele levels, mainly according to Usiskin’s (1982) forced van Hiele level determination scheme. The whole process of analysing the classroom videos involved a consultative panel of 4 observers and 3 critical readers, using the checklist of van Hiele phase descriptors to guide the analysis process. Concerning learners’ levels of geometric conceptualisation, the results from this study reveal that the most of the learners were not yet ready for the formal deductive study of school geometry, as only 2% and 3% of them were respectively at van Hiele levels 3 and 4, while 47%, 22% and 24% were at levels 0, 1 and 2, respectively. More learners from the Nigerian subsample (53%) were at van Hiele level 0 than learners from the South African subsample (41%) at this level. No learner from the Nigerian subsample was at van Hiele level 4, while 6% of the South African learners were at level 4. In general, learners from the Nigerian subsample had a poorer knowledge of school geometry than their peers from the South African subsample, as learners from the latter subsample obtained significantly higher mean scores in the van Hiele Geometry Test (VHGT) and each of the other tests used in this study. Results relating to gender differences in performance generally favour the male learners in this study. For each of the participating schools, learners’ van Hiele levels (as determined by their scores on the VHGT) strongly correlate with their performance in geometry content tests and mathematics generally. For each of the Nigerian and South African subsamples, for n ≤ 2, learners at van Hiele level n obtained higher means on nearly all the tests administered in this study than their peers at level n–1. This finding provides support for the hierarchical property of the van Hiele levels.

This study provides a qualitative examination of the role of visualization through an understanding of the thought processes that occur during visualization when Grade 9 learners engage in data handling and spatial tasks. Data were gathered in a problem-centred context from learners' written responses in order to determine the students' visuality. Visuality is defined as how often learners used visualization. In addition interviews were conducted with the learners who described the thought processes that they engaged in during visualization while involved in problem solving. The role of visualization was highlighted through the processes that learners described during the interviews. The tasks which provided manipulative materials helped learners create visual images which promoted the process of visualization. Certain recommendations were made. Knowledge of the role of visualization enables the educator to encourage the use of visualization during the teaching of mathematics. / Educational Studies / M.Ed. (Mathematical Education)

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

published_or_final_version / Education / Master / Master of Science in Information Technology in Education

published_or_final_version / Education / Master / Master of Science in Information Technology in Education

published_or_final_version / abstract / Education / Master / Master of Education

This is a study of secondary mathematics teachers' knowledge of the dynamics of learning and teaching mathematics in multilingual classrooms in South Africa. It probes teachers' articulated and tacit knowledge through a qualitative methodology that includes In-depth interviews, classroom observations, and reflective workshops. The sample is purposive and theoretical, comprising SIX teachers drawn from three different multilingual school contexts. Categories of description and analytic narrative vignettes enable a qualitative, layered analysis of what the teachers said and how they acted.

Applying mathematics to real world problems, mathematical modeling, has risen in priority with the adoption of the Common Core State Standards for Mathematics (National Governors Association and the Council of Chief State School Officers, 2010). Teachers are at the core of the implementation of the standards, but resources to help them teach modeling are relatively undeveloped. This multicase study explored the perspectives of teachers regarding mathematical modeling pedagogy (the modeling cycle), instructional materials, and professional collaboration, with the assumption that understanding teachers’ views will assist authors, publishers, teacher educators, and administrators to develop better support for modeling instruction. A purposeful sample of six high school mathematics teachers from a variety of school settings across the country was interviewed using a semi-structured protocol. A conceptual framework developed by applying the theories of Guy Brousseau (1997) to the modeling literature guided the analysis. Qualitative methods including elements of grounded theory were used to analyze the data and synthesize the study’s results. The research showed that teachers structure their instruction consistently with the modeling cycle framework, but it also uncovered the need for additional detail and structure, particularly in the initial steps when students make sense of the problem and formulate an approach. Presenting a modeling problem is particularly important and challenging, but there is inadequate guidance and support for this teaching responsibility. The study recommends the development of additional materials and training to help teachers with these steps of the modeling cycle. Furthermore, teachers find that modeling problems are engaging, and they help students make sense of mathematical concepts. Teachers would employ modeling problems more often if they were more available and convenient to use. The study recommends that features for an online depository of modeling materials be researched and developed, including a course-based, chronological organization, a diverse variety of materials and formats, and tapping teachers to contribute their lessons.

M. Tech. Education / Given the challenges of shortage of skills as informed by low performance in Grade 12 Mathematics, it seemed appropriate that a study on factors behind poor performance in Mathematics amongst grade 12 learners be undertaken. The study explored various factors that contribute to poor performance of learners in Mathematics in grade 12.

published_or_final_version / abstract / Education / Master / Master of Education