A thesis submitted to the School of Education, Faculty of Humanities, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy January 2017. / This research study focused on exploring prospective teachers’ knowledge of geometric reasoning in teacher preparation. Premised on the claims that learning mathematics is profoundly influenced by the tasks, by the learning context and by the tools that are used in mathematics instruction, mathematics prospective teachers’ technological pedagogical content knowledge was examined. The technological pedagogical content knowledge (TPACK) framework was employed to study the prospective teacher’s knowledge of circle geometry as proposed by Mishra & Koehler (2006). The main focus of the research was on investigating the empirical and theoretical questions of what characterizes aspects of prospective teachers’ technological pedagogical content knowledge. These aspects were geometry content knowledge (CK), geometry pedagogical content knowledge (PCK) and geometry technological content knowledge (TCK). This exploratory multiple case study explores the TPACK of six mathematics prospective teachers enrolled in a second-year undergraduate mandatory mathematics methodology course in an urban South African university. Data was collected through prospective teachers’ (PTs) responses to circle geometry tasks, interviews and screen cast recordings. Rubrics were employed as analytical tools. Duval’s (1995) cognitive apprehensions and processes were engaged as interpretative tools to understand how the PTs responded to the CK, TCK and PCK tasks. The results suggest that prospective teachers’ circle geometry technological pedagogical content knowledge constructed in a GeoGebra-based environment is characterized as weak emanating from weak geometry content knowledge (CK), weak technological content knowledge (TCK) and weak pedagogical content knowledge (PCK). The study has shown that a weak geometry CK was evidenced from the participating PTs’ weak display of cognitive apprehensions and geometry reasoning processes. This study contributes to the current debates on teacher professional knowledge and on an understanding of frameworks for which teacher knowledge can be premised in South Africa. A model was developed for classifying and describing forms of mathematics connections in geometry knowledge at teacher preparation level / LG2018

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg in partial fulfilment of the requirements for the degree of Master of Science, 2017 / This research explores the importance of Self- Directed Metacognitive questioning in the solving of Euclidean Geometry problems by grade 11 learners. A quasi-experiment was carried out at an urban school with fifty eleventh grade learners. Most researches in Mathematics Education aimed at unveiling socio-economic factors that hamper mathematics learning. In this research, I suggested that strategies that target the learners' metacognitive development can assist in addressing poor mathematics achievement. Metacognition (thinking about thinking) makes learners drivers of their own cognitive processes so that they can become better doers of the subject The research answered the question: Does teaching of metacognition to Further Education and Training (FET) learners help them to become better learners of Euclidian Geometry? This question was broken down into the following sub-questions: What is the effect of the use of Self -Directed Metacognitive (SDM) questions on the confidence level and preparedness of learners in the learning of Euclidean Geometry? To what extent does purposeful teaching and learning of metacognitive skills yield positive results in answering Euclidean Geometry questions? Metacognitive skills helped learners to perform better in problem-solving. This result agrees with previous researchers and is also consistent with the results of earlier investigations showing that achievement in mathematics can be raised through instruction enriched with metacognitive activity. Where previous research dealt with metacognitive training in an implicit manner this research on metacognitive training was done explicitly and it resulted in improved mathematics performance by learners. / XL2017

"Although there are some subjective opinions supporting the hypothesis that geometrical and topological topics are teachable at the elementary school level there is the need for definitive research which can answer the following questions: 1. What factors relate to a student's achievement with geometrical and topological topics? 2. At what grade levels are certain topics learned with a high degree of efficiency in terms of time and expended effort? 3. Which geometrical and topological topics are appropriate at various grade levels in terms of these topics serving to clarify and simplify other types of mathematical concepts? The primary purpose of this study was to investigate some factors which might serve as predictors of pre-determined levels of success with respect to some selected topics in geometry and point set topology, when taught to fifth-, sixth- and seventh-grade pupils via a programmed text"--Introduction. / Typescript. / "August, 1963." / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

Geometry is an essential branch in mathematics that helps students learn to grasp their environment and leverage that grasp into abstract understanding and reasoning. There has been an observable decrease in geometrical content in secondary education curricula, and particularly a “puzzling scarcity” in three-dimensional geometry, which has led to a decline in students’ geometrical abilities, spatial thinking and deductive reasoning abilities. This study addresses this issue by scrutinizing the enacted curriculum standards and the most influential textbooks related to three-dimensional geometry in two prominent countries, the US and China, both of which embrace the interplay of both conventional and innovative practices. This qualitative study used both content analysis and cross-cultural comparison methods to inquire about and to understand the current situation of three-dimensional geometry in high school. I focused on probing the communication types, objects, concepts, and spatial thinking abilities related to three-dimensional geometry in the standards and texts. To understand spatial abilities, I synthesized a spatial thinking abilities framework with six attributes and used this framework to exam the affordance of these abilities in the texts and requirements in the standards. The result and analysis reveal the details of each text and standards individually and offer an examination of the alignment between the standards and texts. The comparison of the two countries’ different approaches also sharpens the understanding of the issue. I also worked to unveil students’ multiple ways of making sense of geometry concepts by two geometry learning models, Piaget’s model and van Hiele’s model, as well as spatial thinking abilities.

Prospective teachers enter teacher education programs with previously formed conceptions of geometry and its teaching. These conceptions help them make sense of new information about teaching, their roles as teachers, and their translation of mathematics into learning activities. The purpose of this study was to investigate the relationships among preservice teachers' conceptions of geometry, conceptions of teaching geometry and classroom practices. Ten preservice mathematics teachers completed a card sort task with an interview. They also participated in a videotape task which consisted of viewing three experienced geometry teachers on videotape. Four of these preservice teachers were observed eight times each during their professional internship experience. All interviews and observations were videotaped and transcribed for data analysis. Results of this study indicated a complex relationship between the preservice teachers' conceptions of geometry and conceptions of teaching geometry. The preservice teachers could not discuss their conceptions of geometry without discussing the teaching of geometry. Their conceptions about geometry and their belief that geometry was linear, in nature were so strong that these views became connected with their views of teaching geometry. Clearly, the preservice teachers' conceptions of geometry influenced their conceptions of teaching geometry and the teaching of subject matter influenced the preservice teachers' conceptions of geometry as well. The relationship between the preservice teachers' conceptions of geometry and their classroom practices was directly influenced by the textbooks used. They believed geometry was ordered according to the textbook and their classroom practices also followed the textbook. The relationship of the preservice teachers' conceptions of geometry teaching to classroom practices indicated that what the preservice teachers said they believed and what they did in the classroom were not always consistent. Their beliefs about teaching geometry rarely emerged in their classroom practices. Finally, these preservice teachers had an overwhelming concern with classroom management. This concern governed their thinking about teaching. / Graduation date: 1996

published_or_final_version / Education / Master / Master of Education

Students nowadays are relatively confident in directly applying geometrical theorems and theories. Nevertheless, it has been a common phenomenon that students are not confident in constructing geometric proofs. They lack the confidence and sufficient experience and knowledge in conducting deductive geometrical proofs. To some students, they treat proofs simply as another type of examination questions which they can tackle by repeated drillings. Students make use of straightedges and compasses to construct different geometry figures in geometric constructions. Through geometric constructions, we can train our prediction and logical thinking skills when investigating the properties of geometric figures. Geometric constructions provide students with hands-on experience to geometry learning which requires students to have more in-depth thinking. This is an empirical study on the implementation of geometric construction workshops among junior secondary students in Hong Kong. Results have shown that students enjoyed the construction tasks during the workshops. Analysis has implied that geometric constructions help improve students’ ability in constructing geometric proofs and to raise their interests in geometry learning. / published_or_final_version / Education / Master / Master of Education

Geometry learning at the junior secondary level should focus on the connection between students’ intuitive, spatial thinking and formal, deductive reasoning. It is crucial for students at this stage to develop abstract concepts based on knowledge of and reasoning with geometric properties. Dynamic geometry tools are promising resources in classroom teaching for enriching students’ experience in geometry. Studies about students’ learning with dynamic geometry tools usually focus on the long-term development of tool use in the contexts of problem solving and explorations. Data are mostly obtained from interviews and observations with students working in small groups or individually. This study begins with an original approach. It aims at understanding students’ perceptions of geometric properties in simple dynamic figures without assuming any prior experience of students with dynamic geometry tools. A special web-based platform was developed to capture students’ results of dragging in dynamic figures. Quantitative and qualitative data were obtained for analysis from this platform and task-based interviews respectively. In a set of 8 tasks designed in this study, students were asked to drag free points in pre-constructed figures to create examples of geometric configurations with parallel lines. Results of the tasks were collected through the web-based platform from 1,589 secondary 1 to 4 Hong Kong students in 11 schools of different backgrounds. In the next stage, 24 secondary 2 students from another 6 schools took part in task-based interviews, enabling detailed observations and analysis of their reasoning. The basic assumption of this design is that students working on the tasks will generate examples of geometric figures that reflect their understanding of relevant geometric properties. Their results could then be analyzed in the form of collective and personal example spaces with critical dimensions of variation to be identified (Watson & Mason, 2005). The findings of the study indicate how participating students vary in their ways of discerning critical geometric properties in dynamic figures. In particular, the results reflect students’ general limited awareness of basic geometric properties, such as equal opposite sides or angles, while manipulating and interpreting pre-constructed figures. Graphical representations of collective example spaces, developed in this study, provide useful means for revealing dimensions of variation in examples generated. It is hoped that the findings and method of the study can inform classroom teaching with dynamic geometry tools that capitalizes on variation in students’ perceptions. / published_or_final_version / Education / Doctoral / Doctor of Philosophy

No description available.

The main focus of the research was to find out the causes of a poor performance in euclidean geometry especially in a grade eleven class. An easier way to find that information was to investigate the techniques that educators who are teaching grade eleven are following when they teach euclidean geometry. The necessary data was therefore collected from the educators as well as learners who were in grade eleven. This study is guided by the constructivist's VIew. The theoretical framework of this research is based on the ideas of theorists like Piaget, Vygotsky and other authors who conform to constructivism. Changes that affected the education system of South Africa due to the adoption of the new constitution were also visited. A shift from the traditional way of teaching and an Outcomes Based Education system, as a recommendation by the National Curriculum Statement was highlighted. The data was collected through both interviews and questionnaires. The semi-structured interviews of three educators from three participating schools were audio taped. In each school one educator was interviewed and six learners were given questionnaires to answer. The above gave a total of eighteen learners and three educators. Written responses from learners and audio taped responses from educators were kept and analyzed. The interview was focused on the techniques that educators employ in their teaching of euclidean geometry in grade eleven. The questionnaires administered to learners were aimed at confirming the responses from the educators. It is envisaged that the educators participated in the study can provide enough information which can assist in correcting the teaching approach in euc1idean geometry. The findings show that the conditions under which educators teach contribute to their methods of teaching euclidean geometry. The testing system and the focus on better results by the education department proved to be the main determining factors of the methods that educators resort to when they teach learners. It also came up from this study that some learners do not take mathematics out of their will. Their parents or the school forces them to take mathematics. Those who like to take mathematics are constantly discouraged by comments of educators who deem mathematics as a subject responsible for bringing down the pass rate of the school. The above diminishes the love of mathematics to learners and euclidean geometry becomes the section that suffers the most. Suggestions and recommendations aimed at improving the teaching and learning of the euclidean geometry have been made. / Thesis (M.Ed.)-University of KwaZulu-Natal, 2007.