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)

M.Ed. (Mathematics Education) / This dissertation was on mathematical proficiencies that learners displayed in their response to final examination questions on sequences and series, differential calculus, analytical geometry and trigonometry. The main focus was on four mathematical proficiencies, procedural fluency, conceptual understanding, strategic competence and adaptive reasoning. The initial goal was to review examination questions to identify mathematical proficiencies that the examination questions demanded and then an analysis of learners’ scripts to identify the mathematical proficiencies that the learners displayed in their response to examination questions. Review of literature first focused on the theoretical framework of the study, constructivism, its guiding principles and how it contributes towards mathematics learning. The framework for mathematical proficiency was shown with an aim of viewing other underlying issues on mathematical proficiency. The four mathematical proficiencies; conceptual understanding and procedural fluency constitute mathematical knowledge and strategic competence and adaptive reasoning are mathematical skills. Mathematical tasks are reviewed as properties of well-structured mathematical tasks in which examination questions were also mathematical tasks. The taxonomy of learning was discussed with an aim of finding properties of the taxonomy that inform the structure of the examination questions. Examination questions on sequences and series, differential calculus, analytical geometry and trigonometry were reviewed, to find mathematical proficiencies that the examination questions demanded and those that learners display in their responses.

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In many cases topics for research are presented to a student in capsulated, clearly defined terms, either as the result of his own experience or as a request by some institution. In other cases the topic takes shape but gradually, very often as the result of a student slowly becoming aware of a field of research through repeated observation of related factors. In some cases the aim of research is to determine whether there is a relationship between various factors; or disprove such in others the main aim may be to prove relationship in unequivocal terms. A large body of research is, however, concerned mainly with the statement of a problem or the finding of facts. The work presented in the following pages can be regarded as falling in the last-mentioned category. A vague suspicion was gradually strengthened by observation and experience until it finally crystallised to form the basis of the research. Facts and figures were gathered and analysed and some conclusions drawn, conclusions that gave rise to more questions and problems than fall within the scope of this work. It was, in fact, found that this research raised more questions than were answered by it and served mainly to underline the magnitude of the problem rather than to offer a solution.

This study researched the nature of tasks used in Grade 10 mathematics tests and end-of-year national examinations. The study was carried out in three, purposively selected, Grade 10 schools in the Ohangwena region in Northern Namibia. For the purpose of this study, a mixed method approach was employed to analyse a combination of both quantitative and qualitative data. A sample of three tests per mathematics teacher from the three participating schools and national examinations question papers for the past three consecutive years (2011-2013) were analysed using Stein, Smith, Henningsen, & Silver’s (2000) framework of cognitive demand. The study was divided into two phases. Phase 1 was the analysis of teacher test items and national examination items in terms of their cognitive demand. Phase 2 involved semi-structured interviews with three selected teachers to probe their views and find out their basis for selecting test items. The findings of this study revealed that there was no substantial difference in the distribution of the levels of cognitive demand in both tests and national examinations items. The study, however, showed that mainly tasks requiring only procedures without connections dominated the tests and the examinations. The number of higher level tasks in both tests and examinations analysed was low. There was no single task coded at level 4 in any of the teachers’ tests. Only 2% of tasks could be classified at level 4 in the examination items. The study also revealed that since tests and examinations assess the same learning objectives from the syllabus, most of the test items set by teachers were extracted from the national examinations question papers. The paper recommends that more tasks at a higher level category need to be included in assessment tasks to promote critical thinking amongst learners.

The purpose of this study was to investigate the nature of mathematical connections used by selected teachers when teaching the topic of algebra and to investigate their perceptions of their use of connections. The participants were selected on the basis of teaching experience as well as their willingness to share their ideas. An interpretive paradigm was used to collect and analyse data. The data was collected from three participating teachers. These participants were selected from the three secondary schools in the town of Tsumeb in Namibia. I used video recordings of two lessons per teacher as well as semi-structured interviews as my tools to gather data. After the two lessons were video recorded, I conducted a workshop with the teachers to introduce them to the 5 types of mathematical connections pertinent to this study. We analysed the videos together using Businskas' framework as a basis for analysis. This then formed part of the stimulated recall interviews. It was found that, even though teachers were not aware of the concept of mathematical connections before our interactions, there was strong evidence of connections being made and used in their lessons. The two types of connections that were used most frequently (24.1 percent each) were procedural and instruction-oriented connections respectively. Part-whole relationships connections were used the least with a frequency of 12 percent. All three teachers agreed that they needed to make more connections when teaching and that they would think more about connections in future, particularly when preparing their lessons. The study makes recommendations to encourage the continuous use of connections in teaching mathematics.

D.Ed. / Each student undergoing tuition, is unique and one of a kind. Each student has his own personality and individuality. Students have different ways of learning; progress differently and reach different degrees of success with their methods of study. The success of the student's learning process is closely related to the student's existing pre-knowledge. The orientation student at the technical college finds himself in a unique situation. As this course is a bridging course, the student must overcome the backlog in his field of study and also be prepared for the studies that will follow. Learning of mathematics is a complex matter. No two answers will correspond if inquiring into the method in which mathematics is mastered. The same is also true if inquiring into the teaching strategies to be followed in order to acquire success in teaching this subject. In this study the theories of Piaget, Bruner, Ausubel among others, were scrutinised. Numerous teaching strategies can be followed to ensure success in the classroom. This study concentrates on co-operative learning, since the point of view is held that it provides the overall framework within which effective tuition can be achieved. Cooperative learning has been researched by applying it in the mathematics classroom. A very positive response was received from the students as well as the teachers concerned. However, certain problems were experienced. These include, inter alia, that some students found the classroom discipline lacking. Others complained that the lecturers did not do enough explaining. The following conclusions can be drawn from this study: Traditionally the classroom is where the lecturer has the sole right to teach. A change is necessary. New teaching strategies will have to be looked at. To achieve this change, co-operative learning is strongly recommended. It provides for active involvement of the student in the learning process; it provides the opportunity for the student to accept responsibility for his own learning success as well as that of his fellow students and it provides the opportunity for mastering social skills which are a necessity for our modern, complex and integrated society.

M.Ed. (Curriculum Studies) / Mathematics is felt to be one of the most important subjects in the school curriculum by educators, parents and society at large. As Bishop (1988: 1) puts it: "Anyone who cants to get on today, needs to study mathematics,. and preferably computing too." Diab (1987) also argues along these lines and says that mathematics has been a key subject in the school curriculum and it is still a basic ingredient in the educational make-up of a person who wishes to find his place in today's increasingly technological world. Mathematics promotes the development of the mental, social, emotional and occupational life of a person (Grove & Hauptfleisch, 1979: 228). ~ardner, et al. (1973: 18) outline the reasons why a person should learn mathematics. They argue that, inter alia, mathematics should be learned and taught at schools for the following reasons: ii) Living. Mathematics for living, refers to those aspects of mathematics which an individual must know in order to function adequately as a member of society. At primary level this clearly includes such topics as number, time, money, length and weight. More and more information is presented in statistical form and this trend will continue. An educated person must be able to evaluate and interpret such data effectively if he is to playful and useful part in society. Most people will, at some time or other, be involved with such complex activities as house purchase and insuarance...

M.Ed. / This study is an investigation into cognitive mapping as strategy in the constructivistic approach to mathematics education to learners with a visual disability with the view to describe the change in pupils' thoughts on mathematical concepts, as well as their experiences during the process of cognitive mapping. The rationale for the investigation was derived from the shift in South African Mathematics teaching from traditional teaching to constructivistic (or problem-centered) teaching. As this implies a shift in paradigm, teachers will be in need of relevant constructivistic training to equip them with effective teaching strategies. The objective of this study is to examine cognitive mapping as a possible supportive strategy to constructivism . The study commences with a theoretical framework in which constructivism is clearly explicated. The principles of radical and social constructivism, the roots of which can be traced back to the epistemological theories of Piaget and Vygotsky, are explored. The constructivistic view of knowledge, with the relationship between public knowledge and the forming of personal knowledge, is discussed and extended to include the formation of mathematical knowledge. The focus then shifts to the concepts of instruction and learning and the role they play in the constructivistic paradigm. In the constructivistic view, learning implies cognitive restructuring, which is facilitated by assimilation and accommodation. The implications of this view of learning for instruction, and in particular mathematics instruction, is then discussed. This chapter concludes with the working definition the researcher has used to conduct the remainder of the study. The theoretical framework is structured furthermore to give background regarding cognitive mapping. According to the constructivistic approach, learning implies conceptual change. Cognitive maps externalise conceptual change by means of visual representations, and therefore it was decided to investigate them as a teaching strategy. Some definitions, as found in the literature, are given, and three types of maps are illustrated as examples. The different uses of cognitive maps, among which study strategy, lesson planning and means of evaluation, are discussed. A discussion on the different methods of constructing a map follows, with specific focus on how to include the whole class in the activity. The advantage of social interaction while constructing knowledge, is highlighted. Lastly, the advantages and disadvantages connected to cognitive mapping as teaching strategy, are discussed. The theoretical framework is complemented by a chapter on the design of the research, substantiating the choice of format and methods of data collection and analysis. The data is reported in the succeeding chapter, and examples of raw data from transcriptions, journals of the pupils and cognitive maps are presented. Finally, the consolidated data is interpreted. In the concluding chapter the findings of the study are discussed. The most significant findings of this study are: cognitive mapping, as mathematical teaching strategy, improved the understanding of grade nine learners, with a visual disability, of real numbers; the learners experienced the teaching strategy of cognitive mapping positively; the number of group members involved in the construction of a cognitive map, influenced.

The questions guiding this research seek to discover the factors that affect high school math course completion and college readiness in a Texas suburban public school district. The first research question examines the relationship between 8th grade completion of Algebra I and high school mathematics course taking patterns and college readiness. The second question evaluates the relationship between race, gender, socioeconomic status and enrollment in the Advancement Via Individual Determination (AVID) program to college math readiness and high school mathematics course completion. Participants included 841 high school graduates of the class of 2006; 76% of the graduates were White, 15% Hispanic and 7% African American. Twenty-three percent of students were economically disadvantaged and 46% of students completed Algebra I in 8th grade. Chi-square, Cramer's V, and multiple regression were conducted to evaluate possible relationships between variables. The Chi-square and Cramer's V showed statistically significant (p<.05) relationships between 8th grade algebra completion and both college readiness and high school math course completion. A significant statistical relationship was also found between college readiness and each of the independent variables, ethnicity, economic status, completion of 8th grade algebra and enrollment in AVID. The number of math courses completed in high school was statistically related to ethnicity and economic status.. The findings of this study indicate that early access to Algebra I can positively affect the number of high school math courses a student completes and the likelihood that the student will be college ready after high school graduation.