published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

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)