This study investigated how Grade 11 learners from a school in KwaNdengezi, near Pinetown, in Durban, understood the cosine function with software known as The Geometer’s Sketchpad. This was done on the basis of what they had learnt in Grade 10. The timing was just before they had covered the topic again in their current grade. The researcher hoped, by using The Geometer’s Sketchpad, to contribute in some small way to teaching and learning methods that are applicable to the subject. This may also, hopefully, assist and motivate both teachers and learners to attempt to recreate similar learning experiences in their schools with the same or similar content and concepts appropriate to them. In this research project, data came from learners through task-based interviews and questionnaires. The school was chosen because of the uniqueness of activities in most African schools and because it was easily accessible. Most learners do not have access to computers both in school and at home. This somehow alienates them from modern learning trends. They also, in many occasions, find it difficult to grasp the knowledge they receive in class since the medium of instruction is English, a second language to them. Another reason is the nature of the teaching and learning process that prevails in such schools. The Primary Education Upgrading Programme, according to Taylor and Vinjevold (1999), found out that African learners would mostly listen to their teacher through-out the lesson. Predominantly, the classroom interaction pattern consists of oral input by teachers where learners occasionally chant in response. This shows that questions are asked to check on their attentiveness and that tasks are oriented towards information acquisition rather than higher cognitive skills. They tend to resort to memorisation. Despite the fact that trigonometry is one of the topics learners find most challenging, it is nonetheless very important as it has a lot of applications. The technique of triangulation, which is used in astronomy to measure the distance to nearby stars, is one of the most important ones. In geography, distances between landmarks are measured using trigonometry. It is also used in satellite navigation systems. Trigonometry has proved to be valuable to global positioning systems. Besides astronomy, financial markets analysis, electronics, probability theory, and medical imaging (CAT scans and ultrasound), are other fields which make use of trigonometry. A study by Blackett and Tall (1991), states that when trigonometry is introduced, most learners find it difficult to make head or tail out of it. Typically, in trigonometry, pictures of triangles are aligned to numerical relationships. Learners are expected to understand ratios such as Cos A= adjacent/hypotenuse. A dynamic approach might have the potential to change this as it allows the learner to manipulate the diagram and see how its changing state is related to the corresponding numerical concepts. The learner is thus free to focus on relationships that are of prime importance, called the principle of selective construction (Blackett & Tall, 1991). It was along this thought pattern that the study was carried-out. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the cosine function from their Grade 10 work in all four quadrants to check on: * What understanding did learners develop of the Cosine function as a function of an angle in Grade 10? * What intuitions and misconceptions did learners acquire in Grade 10? * Do learners display a greater understanding of the Cosine function when using Sketchpad? In particular, * As a ratio of sides of a right-angled triangle? * As a functional relationship between input and output values and as depicted in graphs? The use of Sketchpad was not only a successful and useful activity for learners but also proved to be an appropriate tool for answering the above questions. It also served as a learning tool besides being time-saving in time-consuming activities like sketching graphs. At the end, there was great improvement in terms of marks in the final test as compared to the initial one which was the control yard stick. However, most importantly, the use of a computer in this research revealed some errors and misconceptions in learners’ mathematics. The learners had anticipated the ratios of sides to change when the radius of the unit circle did but they discovered otherwise. In any case, errors and misconceptions are can be understood as a spontaneous result of learner's efforts to come up with their own knowledge. According to Olivier (1989), these misconceptions are intelligent constructions based on correct or incomplete (but not wrong) previous knowledge. Olivier (1989) also argues that teachers should be able to predict the errors learners would typically make. They should explain how and why learners make these errors and help learners to correct such misconceptions. In the analysis of the learners' understanding, correct understandings, as well as misconceptions in their mathematics were exposed. There also arose some cognitive conflicts that helped learners to reconstruct their conceptions. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2010.

This qualitative investigation took the form of a case study and fell within the interpretive research paradigm. The Mathematics Chair at the Education Department of Rhodes University launched the Mathematics Teacher Enrichment Programme (MTEP) in 2010 in order to combat poor Mathematics performance of learners in the lower Albany district of the Eastern Cape. The challenge that the participating MTEP teachers faced was a lack of time available to implement new teaching ideas. This was because most of their time was spent catching up “lost” or untaught concepts in the classroom. To address this problem, the Catch-Up Project was launched, whereby selected Mathematics teachers in the area taught lost concepts to Grade Ten learners during afternoon classes in an attempt to improve their fundamental Mathematics knowledge. In order to establish which sections of Mathematics were more difficult for the learners in this programme, bench mark tests were administered biannually. Whilst these tests certainly identified deficient areas within their Mathematics knowledge, the poorest performance areas were the sections of the syllabus which were spatial in nature, such as Space and Shape and Geometry. However, a more in depth assessment tool was required to establish which specific spatial skills the learners were not able to employ when doing these Geometry tasks. To this end, the Spatial Skills Assessment Tasks (SSAT) was developed. It consisted of traditional text book type Geometry tasks and real-world context tasks, both of which were used to assess six spatial skills deemed crucial to successfully facilitate learning Geometry. The case study took place in two of the schools which participated in the Grade Ten Catch-Up project. The case was focused on Grade Ten learners and the unit of analysis was their responses to the SSAT instrument. The learners that participated all did so on a strictly voluntary basis and great care was taken to protect their wellbeing and anonymity at all times. The results of the SSAT instrument revealed that the real world context tasks were in general far more successfully answered than the traditional text book type questions. Important trends in learner responses were noted and highlighted. For example, geometric terminology remains a huge challenge for learners, especially as they study Mathematics in their second language. The ability of the learners to differentiate between such concepts as congruency and similarity is severely compromised, partly due to a lack of terminological understanding but also due to a perceived lack of exposure to the material. Concepts such as verticality and horizontality also remain a huge challenge, possibly for the same reasons. They are poorly understood and yet vital to achievement in Geometry. Recommendations for the development and strengthening of spatial skills support the constructivist approach to learning. Hands on activities and intensive sustained practice over a period of a few months, in which both teachers and learners are actively involved in the learning process, would be considered most beneficial to the long term enhancement of these vital spatial skills and to the learning of Geometry in general.

Visualisation in general and visualisation processes in particular have received much attention in the mathematics education research literature. Literature suggests that the appropriate use of visualisation helps learners to develop their conceptual understanding and skills of geometry as it allows them to visually interpret and understand fundamental mathematical and geometrical concepts. It is claimed that visual tools play an important role in communicating mathematical ideas through diagrams, gestures, images, sketches or drawings. Learning mathematics through visualisation can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. This interpretive case study focused on how selected teachers taught concepts in geometry through visualisation processes for conceptual understanding as a result of an intervention programme. The study was conducted at four high schools by four mathematics teachers in the Kavango East Region in Northern Namibia. The participants were involved in a three-week intervention programme and afterwards taught three lessons each on the topic of geometry. The data collection method of this research was: focus group and stimulus recall interviews, classroom observations and recorded videos. This research is located in constructivism. I used vertical and horizontal analysis strategies to analyse the data. My analytical instrument consisted of an observation schedule which I used in each lesson to identify how each of the visualisation processes was evident in each of the observed lessons. This study revealed that the participant teachers used visualisation processes in most of their lessons and these processes were used accurately in line with the requirements of the grade 8 mathematics syllabi. The visualisation processes were used through designed visual materials, posters and through the use of geometrical objects such as chalkboard ruler, protractor and compass. The results from this study also confirmed that visualisation processes can be a powerful instructional tool for enhancing learners’ conceptual understanding of geometry.

The purpose of this research was to investigate hypothesized relations of visuospatial and logical reasoning skills, and span of short-term memory to achievement in geometry. In addition, major subfactors of visuospatial ability (visualization, speeded rotations, spatial orientation, and disembedding) were assessed to determine which were significant predictors of geometry achievement. Vernon's (1965) model of intelligence and Baddeley's model of working memory provided the theoretical framework for these hypotheses. Subjects (N = 110) were students in seven sophomore level geometry classes in two schools in southwest Virginia. Cognitive measures of speeded rotations, visualization, spatial orientation, disembedding, Gestalt closure, logical reasoning, and short-term memory span were administered. Two measures of geometry achievement were used: The standardized New York Regents Geometry Exam, and z-transformations of the classroom final grade. A model of geometry achievement is proposed and major predictions of the model were supported. within this sample, regression analysis showed the measures of visualization, logical reasoning, and short-term memory predicted achievement on the New York Regents Geometry Exam. Separate regression analyses for each gender revealed visualization predicted geometry achievement for the girls, while logical reasoning and short-term memory span predicted geometry achievement for the boys. Gender differences favoring boys were found on measures of speeded rotations, spatial orientation, and Gestalt closure. Girls had significantly higher scores on the measure of short-term memory span and the classroom measure of geometry achievement. / Master of Science

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.

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

The study reported here utilised a theory of levels of geometric thinking. This theory was proposed and developed by two Dutch mathematics educators, Pierre van Hiele and his wife, Dina van Hiele-Geldof. The van Hiele theory enables investigations into why many students experience difficulties in learning geometry. In many nations, such as the UK, the USA, Netherlands, the USSR and to a certain extent, Nigeria and South Africa, research evidence has indicated that the overall students’ mathematical competencies are linked to their geometric thinking levels. This study is the first of its kind to apply the van Hiele theory of geometric thinking in the Namibian context to analyse geometrical conceptualisation in Grade 12 mathematics students. In all, 50 Grade 12 students (20 from School A and 30 from School B) were involved in this study. These students wrote a van Hiele Geometry Test adapted from the Cognitive Development and Achievement in Secondary School Geometry test items. Thereafter, a clinical interview with the aid of manipulatives was conducted. The results from this study indicated that many of the School A and School B students who participated in the research have a weak conceptual understanding of geometric concepts: 35% of the School A and 40% of the School B subsamples were at the prerecognition level. 25% and 30% of the School A, and 20% and 23.3% of the School B students were at van Hiele levels 1 and 2 respectively. An equal number of students but different in percentages, 2 (10%) in School A and 2 (6.7%) in School B, were at van Hiele level 3. Only one student from School B attained van Hiele level 4. These results were found to be consistent with those of previous similar studies in UK, USA, Nigeria and South Africa. The findings of this study also highlight issues of how the Namibian Grade 12 geometry syllabus should be aligned with the van Hiele levels of geometric thinking as well as the use of appropriate and correct language in geometrical thinking and problem solving.

This study reports on the extent to which selected mathematics teachers facilitate the teaching and learning of geometry at the van Hiele levels 1 and 2 at a Grade 10 level in selected schools in Namibia. It also addresses and explores the teaching strategies teachers employ in their classrooms. Kilpatrick et al.’s model for proficient teaching and the van Hiele model of geometric thinking were used to explore the type of teaching strategies employed by selected mathematics teachers. These two models served as guidelines from which interview and classroom observation protocols were developed. Given the continuing debate across the world about the learning and teaching of geometry, my thesis aims to contribute to a wider understanding of the teaching of geometry with regard to the van Hiele levels 1 and 2. There are no similar studies on the teaching of geometry in Namibia. My study concentrates on selected Grade 10 mathematics teachers and how they teach geometry using the van Hiele theory and the five Kilpatrick components of proficient teaching. As my research looks at teaching practice it was important to deconstruct teaching proficiency with a view to understanding what makes good teachers effective. The results from this study indicated that the selected Grade 10 mathematics teachers have a good conceptual understanding of geometry as all of them involved in this study were able to facilitate the learning and teaching of geometry that is consistent with the van Hiele levels 1 and 2.

The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The focus is on the use of van Hiele's theory to explore problems encountered in circle geometry by grade 11 learners and making some policy recommendations concerning the curriculum structure and teaching of the geometry at all grades. The interpretation is based to the learners' background in geometry i.e. their prior knowledge and experience of learning geometry. The study was carried out over a period of three years. The data collection process took a period of two months (April and May 2003) with a group of 21 grade 11 mathematics learners in a rural senior secondary school in the Eastern Cape. The researcher used document analysis, worksheets, participants' observation, van Hiele tests, a questionnaire and semi-structured interviews to collect data. The study showed that the structure of the South African geometry syllabus consists of a some what disorganized mixture of concepts. It is not sequential and hierarchical and it sequences concepts in a seemingly unrelated manner. The study revealed that the South African high school geometry curriculum is presented at a higher van Hiele level than what the learners can attain. The findings of the study showed that many of the grade 11 learners were under-prepared for the study of more sophisticated geometry concepts and proofs. Three categories of reasons could be ascribed to this: Firstly, there was insufficient preparation of learners during the primary and senior phases. Secondly the study indicated that there is overload of geometry at the high school level in the South African mathematics curriculum. Thirdly, the over-reliance on the traditional approach to teaching geometry, poor presentation of mathematical technical concepts and language problems, were identified as possible additional reasons for the poor learner understanding of geometry in general and circle geometry in particular. The study recommends that the structure of the South African geometry curriculum should be revisited and redesigned. Teachers should be empowered and developed to be more effective in teaching geometry through further studies in mathematics and in-service workshops. They should also be engaged in the process of implementing the van Hiele's theory in the teaching of geometry in their classrooms.