Thesis (M.A. (Mathematics Education)) --University of Limpopo, 2006 / Traditionally, mathematics has been perceived as objective, abstract, absolute and universal subject that is devoid of social and cultural influences. However, the new perspective has led to the perceptions that mathematics is a human endeavour, and therefore it is culture-bound and context-bound. Mathematics is viewed as a human activity and therefore fallible. This research was set out to explore geometrical concepts involved in the traditional circular buildings in Mopani district of Limpopo Province and relate them to the classroom learning in grade 11 classes. The study was conducted in a very remote place and a sample of two traditional circular houses from Xitsonga and Sepedi cultures was chosen for comparison purposes because of their cultural diversity. The questions that guided my exploration were: • Which geometrical concepts are involved in the design of the traditional circular buildings and mural decorations in Mopani district of the Limpopo Province?. How do the geometrical concepts in the traditional circular buildings relate to the learning of circle geometry in grade 11 class?. The data were gathered through my observations and the learners’ observations, my interviews with the builders and with the learners, and the grade 11 learners’ interaction with their parents or builders about the construction and decorations of the traditional circular houses. I used narrative configurations to analyse the collected data. Inductive analysis, discovery and interim analysis in the field were employed during data analysis. From my own analysis and interpretations, I found that there are many geometrical concepts such as circle, diameter, semi-circle, radius, centre of the circle etc. that are involved in the design of the traditional circular buildings. In the construction of these houses, these concepts are involved from the foundation of the building to the roof level. All these geometrical concepts can be used by both educators and learners to enhance the teaching and learning of circle geometry. Further evidence emerged that teaching with meaning and by relating abstract world to the real world makes mathematics more relevant and more useful.

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 / abstract / toc / Education / Master / Master of Science in Information Technology in Education

Over the past years geometry has posed a challenge to most learners in South African schools. The Government, in particular the Department of Basic Education (DBE), have tried and are still trying to implement new innovations and strategies for teaching mathematics more effectively. South Africa has experienced many changes in mathematics curriculum with an aim of placing the country on an equal footing with countries globally. This study was conducted while there was the implementation of the new Curriculum and Assessment Policy Statement (CAPS), which reinstated the geometry section within the curriculum. Geometry was relegated to an optional paper in mathematics in 2006, 2007 and 2008 in Grades 10, 11 and 12 respectively. This study is framed within the theoretical framework lens of social constructivism and situated learning, and is located within the qualitative research paradigm. It takes the form of survey research in one of the universities in KwaZulu-Natal, South Africa. This university is referred to as the University of Hope (UOH) in this study to protect its identity. The main aim of this study was to explore the pre-service teachers' (PSTs) knowledge of proof in geometry. The study used qualitative analysis of data generated through a survey questionnaire, task-based worksheets and semi-structured interviews for both the focus group and individual interviews. In total 180 PSTs completed task-based worksheets. Within this group of 180 students, 47 were 4th year students, 93 were 3rd year and 40 were 2nd year students. After the analysis of a task-based worksheet, a total of 20 participants from the 3rd and 4th year were invited to participate in focus group interviews. The findings of the study exhibit that the PSTs have very little knowledge of proof in geometry. The study revealed that this lack of the knowledge stems from the knowledge proof in geometry the PSTs are exposed to at school level. / Thesis (M. Ed.)-University of KwaZulu-Natal, Durban, 2012.

This qualitative study actively engaged a group of eight pre-service mathematics teachers (PMTs) in an evolutionary process of generalizing and justifying. It was conducted in a developmental context and underpinned by a strong constructivist framework. Through using a set of task based activities embedded in a dynamic geometric context, this study firstly investigated how the PMTs experienced the reconstruction of Viviani’s theorem via the processes of experimentation, conjecturing, generalizing and justifying. Secondly, it was investigated how they generalized Viviani’s result for equilateral triangles, further across to a sequence of higher order equilateral (convex) polygons such as the rhombus, pentagon, and eventually to ‘any’ convex equi-sided polygon, with appropriate forms of justifications. This study also inquired how PMTs experienced counter-examples from a conceptual change perspective, and how they modified their conjecture generalizations and/or justifications, as a result of such experiences, particularly in instances where such modifications took place. Apart from constructivsm and conceptual change, the design of the activities and the analysis of students’ justifications was underpinned by the distinction of the so-called ‘explanatory’ and ‘discovery’ functions of proof. Analysis of data was grounded in an analytical–inductive method governed by an interpretive paradigm. Results of the study showed that in order for students to reconstruct Viviani’s generalization for equilateral triangles, the following was required for all students: *experimental exploration in a dynamic geometry context; *experiencing cognitive conflict to their initial conjecture; *further experimental exploration and a reformulation of their initial conjecture to finally achieve cognitive equilibrium. Although most students still required the aforementioned experiences again as they extended the Viviani generalization for equilateral triangles to equilateral convex polygons of 4 sides (rhombi) and five sides (pentagons), the need for experimental exploration gradually subsided. All PMTs expressed a need for an explanation as to why their equilateral triangle conjecture generalization was always true, and were only able to construct a logical explanation through scaffolded guidance with the means of a worksheet. The majority of the PMTs (i.e. six out of eight) extended the Viviani generalization to the rhombus on empirical grounds using Sketchpad while two did so on analogical grounds but superficially. However, as the PMTs progressed to the equilateral pentagon (convex) problem, the majority generalized on either inductive grounds or analogical grounds without the use of Sketchpad. Finally all of them generalized to any convex equi-sided polygon on logical grounds. In so doing it seems that all the PMTs finally cut off their ontological bonds with their earlier forms or processes of making generalizations. This conceptual growth pattern was also exhibited in the ways the PMTs justified each of their further generalizations, as they were progressively able to see the general proof through particular proofs, and hence justify their deductive generalization of Viviani’s theorem. This study has also shown that the phenomenon of looking back (folding back) at their prior explanations assisted the PMTs to extend their logical explanations to the general equi-sided polygon. This development of a logical explanation (proof) for the general case after looking back and carefully analysing the statements and reasons that make up the proof argument for the prior particular cases (i.e. specific equilateral convex polygons), namely pentagon, rhombus and equilateral triangle, emulates the ‘discovery’ function of proof. This suggests that the ‘explanatory’ function of proof compliments and feeds into the ‘discovery’ function of proof. Experimental exploration in a dynamic geometry context provided students with a heuristic counterexample to their initial conjectures that caused internal cognitive conflict and surprise to the extent that their cognitive equilibrium became disturbed. This paved the way for conceptual change to occur through the modification of their postulated conjecture generalizations. Furthermore, this study has shown that there exists a close link between generalization and justification. In particular, justifications in the form of logical explanations seemed to have helped the students to understand and make sense as to why their generalizations were always true, but through considering their justifications for their earlier generalizations (equilateral triangle, rhombus and pentagon) students were able to make their generalization to any convex equi-sided polygon on deductive grounds. Thus, with ‘deductive’ generalization as shown by the students, especially in the final stage, justification was woven into the generalization itself. In conclusion, from a practitioner perspective, this study has provided a descriptive analysis of a ‘guided approach’ to both the further constructions and justifications of generalizations via an evolutionary process, which mathematics teachers could use as models for their own attempts in their mathematics classrooms. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.

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.

Orientador: Clotilzio Moreira dos Santos / Banca: Tatiana Miguel Rodrigues / Banca: Tatiana Bertoldi Carlos / Resumo: Na Grécia Antiga, os sábios buscaram a resolução de problemas que se baseavam na construção geométrica utilizando exclusivamente dois instrumentos: a régua não graduada e o compasso. Alguns desses problemas se tornaram clássicos por exigirem, dentro do desenvolvimento da Matemática, grandes esforços para se chegar a uma solução. São eles: a duplicação de um cubo, determinando o lado de um cubo, cujo volume é o dobro do volume de um outro cubo dado, a trisseção de um ângulo, que é dividir um ângulo em três partes iguais ou três ângulos de medidas exatamente iguais e a quadratura de um círculo, que consiste em construir um quadrado com área igual à de um círculo dado. Neste trabalho apresentaremos algumas construções geométricas com régua não graduada e compasso, algumas soluções encontradas que não estavam de acordo com as regras estabelecidas e desenvolveremos a fundamentação algébrica que demonstra a insolubilidade dos três problemas clássicos citados / Abstract: In ancient Greece, the sages sought to solve problems that were based on geometric construction using only two instruments: non-graduated ruler and compass. Some of these problems have become classics because they require within the development of Mathematics, great efforts to reach a solution. They are: the duplication of the cube, the side of a cube whose volume is twice the volume of a given cube; the trisseção of an angle, which is to divide an angle into three equal parts or three measures angles exactly equal and the squaring of a circle, which consists of constructing a square with the same area as a given circle. In this work we present some geometric constructions with non-graded ruler and compass, some solutions that were not in accordance with the rules laid down and develop the algebraic reasoning which demonstrates the insolubility of the three classic problems cited / Mestre

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.