People use metaphors to describe or understand one thing in terms of another. The central idea of this thesis is that metaphors can be used to teach mathematics, particularly abstract topics such as directed numbers. Using directed numbers as a context, this study develops a framework and a coding scheme that can be used as a tool for analysing the use of metaphors in the teaching of mathematics. The part of the theoretical framework of the coding scheme is based on the work of Lakoff and Johnson (1980) and Lakoff and Nunez (2000). In those studies, the authors classify metaphors used for teaching mathematics into one of three categories: ontological, orientational and structural metaphors. By considering the source domain of metaphors, they can be classified into either grounding or linking metaphors. Similarly, the target domain of the metaphors can be categorized by the intended learning outcomes and by the functions of the metaphors. One of the primary contributions of this thesis is the development of a coding scheme that is specifically designed to analyze the use of metaphors in mathematics lessons. The scheme was then used and validated through the analysis of mathematics lessons taught by two teachers with contrasting academic backgrounds and teaching experiences. Three lessons taught by each teacher on the topic of directed numbers at Secondary One level were recorded and analysed. The metaphors used by each teacher were identified, coded and analyzed in order to determine how metaphors can be extended and transformed into other metaphors.. Finally, this thesis compared how the two teachers differed in their use of metaphors, particularly in terms of the selection, sequencing and organization of the metaphors used. This can be indicative the level of conceptual learning that is made available for students in their classes. The research questions: 1. What kinds of metaphors did the teachers use to introduce and explain the concepts and computational processes of directed numbers? 2. What functions did these metaphors serve? 3. What is the developmental path of these metaphors within and across the lessons? 4. What were the differences in the selecting, sequencing, and organization of the metaphor used by the two teachers? Findings This thesis designed and tested an original coding scheme. The findings revealed that the two teachers had used many kinds of metaphors in their lessons. They were used for classifying different kinds of numbers, constructing concepts, and explaining the properties and computational processes of directed numbers. Most of the metaphors found in this study were used to provide a cognitive function that facilitates the introduction of new mathematical concepts and helps the students make sense of the operational processes; only a few metaphors served a memorable function. When comparing the use of metaphors by the two teachers, we can analyze their teaching philosophies. Teacher 1’s use of metaphors demonstrated a linear development path from a simple to a more advance perspective, whereas Teacher 2’s use of metaphors revealed more comprehensive, sophisticated and multi-layered perspective. Significance of the study This study provides insights into the meaning and implication of using metaphors in teaching mathematical concepts. At research level, this study extends the existing work of Lakoff and develops an analytical tool specifically designed to understand the pedagogical values of using metaphors to teach abstract mathematical concepts such as directed numbers. At pedagogical level, the metaphor coding scheme can act as an initial foray into how metaphors can be used in and for teaching. Moreover, the Metaphor-Concept Development Chart developed in this study is a practical tool that can help teachers to analyze and improve their own use metaphors, thereby furthering their professional development and teaching effectivenss. / published_or_final_version / Education / Doctoral / Doctor of Education

The purpose of this study was to gain insight into observed discrepancies between continuous assessment and final examination average marks in Grade 10 Mathematics in the Oshikoto region of Namibia. The study is framed as a case study and is grounded within the interpretive paradigm. A mixed methods approach was applied, eliciting both quantitative as well as qualitative data. The study took place in two phases. In Phase 1, continuous assessment and Grade 10 final examination average marks for 62 Junior Secondary Schools for the period 2008-2010 were gathered and analyzed. Schools were characterized in terms of the relationship between their continuous assessment and final examination average marks for each of the three years. Phase 2, which was informed by Phase 1, took the form of structured interviews with a sample of three Mathematics teachers and three principals along with a focus-group interview of twelve teachers in order to investigate more deeply the perceptions of teachers and principals toward assessment policy and practice. The study shows that Grade 10 assessment practice in Namibian schools is far from ideal. Many teachers are not fully conversant with the various continuous assessment components as outlined by policy, and teachers are not confident about setting appropriate continuous assessment tasks. There is a strong perception that continuous assessment marks can easily be inflated and those teachers who gave high continuous assessment marks to their learners were generally perceived as being either incompetent or dishonest. While continuous assessment was seen as an important component of teaching and learning, it is evident that teachers and principals would welcome greater clarity, along with standardization and moderation, with respect to continuous assessment practice.

M.Ed. (Subject Didactics) / Each child has his own personality and individuality. Children are learning in different ways, at different tempo's and achieve different heights of success with.their efforts. The degree to which the learner is able to master new concepts, is closely related to the reference framework and given pre-knowledge. However, the learning process is not always successful. Various reasons for this phenomena can be identified. This study focuses on the role which misconceptions play in this regard. In general, misconceptions can be defined as a distortion or misinterpretation of the learned concepts. synonyms used to describe this phenomena includes words. like "previous knowledge", "preconceptions" and "alternative frameworks" Misconceptions in Mathematics are numerous. In various studies conducted, misconceptions were identified in almost all areas of Mathematics. Likewise a great deal of misconceptions were found existing among students concerning the handling of fractions. It is an impossible task to research all misconceptions in Mathematics in one study. For this reason it was decided to do research on only one aspect, namely fractions where possible misconceptions can occur. With the empirical research which was conducted, certain misconceptions in the area of fractions were identified. These misconceptions include, amongst other, the following: 1. The sum of and difference between two fractions. There is very little or no notion of the smallest denominator. 2. Multiplying and division of fractions. The student is uncertain about the role which the numerator and the denominator play in the solution. As fractions play such an important role in Mathematical success, it is suggested that a plan of action will be set as soon as possible in order to prevent misconceptions influencing the student learning process.

The results reported here are the product of the research titled: Using history in the teaching of mathematics. The subjects are students in two classes of algebra II course at Florida State University High School-- 36 students-- makes and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra II is a course that is usually taken by high school seniors in 12th grade and a few 11th or 10th grade students which explains why the ages of the students are mostly 18 and a few 17 and 16 years old. In this investigation, both quantitative study and qualitative study were employed. The quantitative study was the main study-- a teaching experiment using quasi-experimental methodology that involves two groups-- group 1 and group 2. Group 1 is the control group, where various algebraic/mathematical concepts, or topics were taught or explained to students with the necessary formulas. Group 2 was the experimental group in which the accounts of the historical origin of algebraic/mathematical concepts and the mathematicians (Lewis Carroll, Archimedes, Pythagoras, and Sophie Germain) who brought forward or created the concepts were used to augment pedagogical lessons and exercises used for this study as the main feature of pedagogy. The qualitative study augmented the main quantitative study; it was a follow-up interview for students to probe further an in-depth rationale for the research theme, using history in the teaching of mathematics. The statistical analysis results indicated that there is a significant difference in the mean of score for the control group students and the mean of scores of the experimental group is greater than the mean on scores of student's performance in the control group; and the interview questions responses indeed corroborate the fact that the use of history in teaching mathematics does improve learning and understanding of algebraic/mathematical concepts. / Typescript. / "Submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisors: Elizabeth Jakubowski, Herbert Wills III, Professors Co-Directing Dissertation. / Includes bibliographical references (leaves 202-205).

published_or_final_version / Education / Master / Master of Education

The purpose of this study was to identify preferred learning styles and strategies of secondary school students and to examine the prevailing problems that restrict them to use their own preferences. The study was intended to highlight a number of issues that need to be revealed and addressed in the learning of mathematics. The types of preferred learning styles and strategies students need to employ in learning mathematics, the assistance students require from their teachers, the conduciveness of the design of mathematics curriculum and the challenges they might face to use their own preferred learning styles and strategies in the learning of mathematics were addressed as basic research questions. The study dealt with various elements that were related to environmental, emotional, sociological, physiological and psychological categories of learning in the identification of the types of learning styles and strategies. This study is believed to contribute a lot in addressing the problems of learning styles and strategies, provide feedback to the concerned government bodies to help them improve the teaching learning processes in secondary schools. It is also to reduce the bias or prejudice on mathematics by assisting students to use their own preferred learning styles and strategies, and contribute to further investigations to make the learning of mathematics more enjoyable, participatory and lifelong career. This study was conducted in four secondary schools in West Shoa Zone. A qualitative method that was descriptive in nature was employed in the study while the instruments of the study were questionnaires and an interview. The sample comprised of 249 (128 male and 121 female) secondary school students and 30 (25 male and 5 female) secondary school mathematics teachers selected randomly. The result has shown that students were not learning mathematics on the basis of their preferred learning styles and strategies and the teachers were practicing autocratic teaching styles. Most of the students did not prefer learning mathematics through plasma television; they required brief outlines and concrete presentations, and indicated that there is not enough time to check and recheck the answers they found for the problems. These imply that the organization of secondary school mathematics curriculum requires reform to accommodate the preferred learning styles and strategies of students. / Further Teacher Education / M. Ed. (Mathematics Education)

This study explores junior secondary students’ schemata on a line reflection construction task, the research of which was conducted in a secondary school in Hong Kong. The theories drawn on in this study come from the literature on theories of schemata and the corresponding knowledge embedded within, namely conceptual knowledge, manipulation and procedural knowledge. The research built on existing theories on schemata and attempted to categorize the different kinds of schemata as well as investigating the relationship between them among four junior secondary students in the construction of a line reflection task. The study also tried to find out how and why students manipulated in a line reflection construction task and the extent to which manipulation could lead learners to successfully tackle the task. This study researched on four junior secondary students, drawing mainly on qualitative data used in the analysis, including task-based interview with the employment of think aloud method in a designed line reflection construction task, as well as study of students’ drawings. The data analysis mainly focused on three areas. First, the analysis of each of the four cases was conducted by looking into the different kinds of schemata possessed by the student informants. Second, analysis of the different knowledge (conceptual knowledge, manipulation and procedural knowledge) embedded in the schema possessed by the student informants was done. Third, synthesis was drawn upon the analysis made in an attempt to answer the research questions posed in this study. Findings from the study confirmed the core role conceptual knowledge plays in the establishment of a learner’s schemata. Findings also revealed that different learners may possess different schemata towards the same concept such as the concept of same distance. When investigating the manipulative actions employed by student informants, it was found that there is a reciprocal relationship between a learner’s conceptual knowledge and his manipulation. This is also apparent in cases where there was a misconception in the learner’s schemata. The research also found that students exercised manipulation very differently and these manipulative actions were largely informed by their corresponding conceptual knowledge. With regard to why they manipulated, the research revealed reasons including manipulation for exploration, manipulation for representation and manipulation for verification. Based on the observation and analysis done in the four cases, it was found that manipulation helped students in the completion of the task to different extents. Learners with weaker conceptual knowledge in line reflection benefited more from the manipulation done in the construction task. These findings have implications for the teaching and learning of line reflection. Teachers are suggested to consider introducing using manipulative tools when approaching the teaching of line reflection, especially when they are dealing with students without rich conceptual knowledge in the area. The effectiveness of having hands-on experience implies that simply teaching definition and inviting learners to rote-learn does not necessarily lead to effective acquisition of knowledge in the Mathematics topic of line reflection. / published_or_final_version / Education / Doctoral / Doctor of Education

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. Johannesburg, 2016. / The purpose of this study, conducted at a mathematics clinic, was to investigate the misconceptions that learners display through errors they make when solving algebraic equations involving fractions. A teaching intervention to address those errors and misconceptions was done at a mathematics clinic. A mathematics clinic is a remedial facility where low-attaining students attend sessions, by choice or by referrals. In this study teaching intervention was used to address learners’ errors and misconceptions. The assumption of the study was that learners are knowledge constructors that use previously-learned knowledge as the basis of new knowledge. Since their previous knowledge contains errors and misconceptions, the construction of new knowledge results in errors. This research was mainly qualitative. Data were collected, using a sample of 17 grade 10 learners, though the work of only 13 of them was analysed. Two participants wrote the pre-test, but did not participate in the subsequent data collection, and the other two did not solve some of the equations in the pre- and post-tests. There were three stages of data collection; pre-test, teaching intervention and post-test. Pre- and post-tests were analysed for errors committed by learners, and the teaching intervention sessions were analysed for opportunities of learning provided. Transcripts were produced from the teaching intervention sessions. They were also analysed to check how students participated in constructing mathematical meanings, and also how effectively their attention was focused on the object of learning. The errors found in learners’ equation-solving were like-term errors, lowest common denominator errors, careless errors, sign errors and restriction errors. The comparison of the number of learners who committed these errors in the pre- and the post-test was insightful. Of 13 learners, 4 committed like-term errors in the pre-test and just 1 in the post-test; 4 committed LCD errors both in the pre- and post-tests; 9 committed careless errors (other errors) in the pre-test, and 6 learners in the post-test; 7 committed sign errors in the pre-test and 1 in the post-test; and 12 committed restriction errors in the pre-test, and 9 in the post-test. These findings suggest that teaching intervention is a necessary pedagogical technique, and needs to be employed when addressing learners’ errors and misconceptions in mathematics. Reduction in learners’ errors and misconceptions was evident after the teaching intervention suggesting that the mathematics clinic provided learning opportunities for participants. / LG2017

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This was an evaluative study undertaken to unpack some of the factors which could explain Transkei matriculation students' apparent poor conceptual understanding of Mathematics and to throw some light on possible solutions to their problems. In addition the study attempted to examine how Mathematics as well as the learning and teaching of Mathematics, were viewed by Transkei teachers and students at the high school level. The theory of quadratic equations, inequalities and their graphs constituted the mathematical content research area of this study. This topic was chosen because of the key role that it plays in the matriculation Mathematics syllabus. There were 8 research questions which led to 8 hypotheses. The research sample comprised 311 matriculation students taking higher grade Mathematics and their 10 Mathematics teachers from 10 schools in the Umtata education circuit. Four researcher-designed instruments, namely: a diagnostic test (students'), a student interview schedule, a teachers' questionnaire, and a teacher interview schedule were used. The diagnostic test consisted of 38 items aimed at addressing the first 7 research questions. Students' mean scores for each group of items of the test addressing a particular research question were computed and compared against a criterion score of 60%, using the "Z” statistic. In addition, an analysis of students' scripts was carried out and clinical interviews on a sample of the subjects (students) were conducted to find out their conceptual difficulties/misconceptions. The teachers' questionnaire and interview schedule were used to ascertain the teachers' disposition towards Mathematics teaching. Accordingly, teachers were divided into two groups A and B on the basis of their scores in relation to the median for the whole group. This enabled the testing of hypothesis 8. In this regard, means for the students taught by the two respective groups of teachers were comared by using "Z" statistic to establish if they were statistically different from each other. Teachers' reasons for their responses to some of the items in the questionnaire were analyzed and discussed with a view to finding out their favourite teaching styles and some of the difficulties they faced in order to be as effective as they wished to be. Analysis of data for research questions 1-7 showed that students did not have sufficient pre-requisite knowledge, and did not display a satisfactory level of mastery in solving quadratic equations and inequalities, and interpretation of graphs for quadratic equations and inequalities. Students' difficulties identified from the findings of this study were classified into 7 categories, namely: mathematical terms, mathematical symbolic language, mathematical skills, form in mathematics, over generalisations, translation and conceptual difficulties. The "Z" test for hypothesis 8 showed that students taught by teachers whose teaching strategies were more student-centred performed better than those who were taught by teachers whose teaching was inclined towards teacher-centredness. Finally, recommendations for teachers, curriculum planners, education authorities and other researchers are also made. / Thesis (M.Ed.)-University of Durban-Westville, 1994.