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The main objective of this study was to determine the progress made in the development of mathematical proficiency, in the domain of number skills, by learners in the two cohorts (namely: Mathematics and Mathematical Literacy) during Grade ten. The research was conducted at Temple Valley Secondary School, which is situated at Verulam, north of Durban in KwaZulu-Natal. Furthermore, this research study was questionnaire-based employing basic descriptive statistics as well as qualitative analyses. Data was collected through numeracy (number) skill tests, a questionnaire and focus group interview responses of grade ten mathematics learners. A total of two skill tests were administered and 3 focused group interviews were conducted with six grade ten learners each. The first test and first attitudinal questionnaire was administered in May 2006 when learners had branched off into the two paths of mathematics. Thereafter, a second test and second attitudinal questionnaire was administered, during October 2006; when learners had completed most of the school year. The October-test was followed by the focus group interviews. A convenient sample was used in selecting the learners for this research study and a purposeful sampling technique was used for the focus group interviews. All grade ten learners at Temple Valley Secondary School were selected as the sample. The results showed that the mathematics learners slightly outperformed the mathematical literacy learners in the development of mathematical proficiency for number skills. The findings from this research could inform: Teachers of grade ten learners with an interest in improving the mathematical proficiency, in number skills, of learners and; curriculum developers and materials development specialists who prepare mathematical material for grade ten classrooms. The results showed that none of the grade ten learners, from either cohort, were mathematically proficient in May or October. From the study, it was noted that the female grade ten learners generally outperformed their male counterparts. The focus group interviews revealed that learners had a positive attitude to the learning of the subject mathematics, despite the poor test results of this study. / Thesis (M.Ed.)-University of KwaZulu-Natal, 2008.

This study encompasses the use of concept maps to investigate learners' conceptual understanding of mathematics of a grade eight class in the predominantly Indian suburb of Shallcross. The grade eight learners that were the participants of this study were in the secondary school for the first time. They were in the middle of the Senior Phase within the General Education and Training Phase of their schooling career. This study is embedded in an action research methodology and is conducted from an interpretivist paradigm. Operating on a mixed methods theory, concept maps constructed by learners were analysed through the duration of the study. Towards the end of the study learners completed a questionnaire. Based on certain responses in the questionnaire and the analysis of the concept maps, six learners were then selected and interviewed to probe learners' conceptual understanding of concept maps. The results of the study reveal that most learners find concept maps enjoyable, fun and a creative way to learn mathematics. Further analysis of the questionnaire indicates that learners enjoy the social dynamics that group work provides and they found that working together also enhances their understanding. Further analysis of individual and group concept maps reveals that group work is effective in the use of concept mapping. The results of the study suggest that concept maps might be successfully used in assessing learners' declarative knowledge. An important implication of this study is that concept maps can be practically and effectively applied to a variety of outcomes within the constraints of the classroom. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2009.

This study examines the mathematics education of youth at-risk in South Africa. It explores how two learners at the margin understand and perform in mathematics in two radically different educational environments. It also examines what provisions, if any, are incorporated into the mathematics curriculum to accommodate these pupils. One of the research participants attended Thuthukani, a residential school for youth at-risk and the other was based at Sanville Secondary, a mainstream school. The differences between the two contexts were in the scarcity of resources, limited space and class sizes. The qualitative case study, which was the preferred method of choice, was carried out in two stages. Observation of learners at the residential school was stage one. Stage two was the observation of a learner at the margin in a mainstream school. Observations were captured through audio and visual recordings and photographs. Pupils' written reflections and workbooks, combined with the information acquired through interviews, informal discussions and a research diary, supplemented the instruments to produce a rich data for analysis. The analysis suggests that each of the components of this study, namely: the educational environment (context), the mathematics curriculum, the teacher and the learner at the margin influence the teaching and learning in the classroom. The study concludes with the researcher's recommendations on the mathematics education of learners at the margin. / Thesis (M.Ed.)-University of Durban-Westville, 2003.

This study intends to foreground visual literacy within the wider context of visualisation and visual thinking in mathematics teaching and learning. Visualisation in general has been receiving attention in mathematics education research. I distinguish visual literacy from visualisation by referring to visual literacy as the combination of visualisation and logical thought. Visual literacy emphasises construction of meaning through the process of decomposition, comprehension and analysis of visual representations. The section on functional relationships is located in the National Curriculum Statement (NCS) for mathematics in the Further Education and Training (FET) phase for Grades 10-12 (Department of Education, 2003). Graphical functional relationships, which form an integral part of functions and algebra in the FET phase for Grades 10-12, demand visual literacy, which includes graphical interpretation and comprehension skills. Therefore, the conceptual understanding associated with graphs is dependent on the way graphs are presented. This study examines learners‟ and educators‟ procedural and conceptual understanding of the graphs they sketch and interpret in the FET curriculum. The data analysis contributes towards the fast growing body of knowledge on visualisation in mathematics with the significant impact visual literacy has on the conceptual understanding of mathematical graphs. The analysis reveals that the overarching theoretical framework of constructivism embracing the Process-Object, Visualizer-Analyzer and Semiotic models are useful in illustrating and justifying the link between visual literacy and the conceptual understanding of learners. In examining the visual understanding of graphical representations of ten Grade 11 learners and the two mathematics educators that teach them, the data reveals that learners display a somewhat skewed understanding of the nature of the Cartesian plane, the characteristics of graphs, functional notation and graphical terminology. In fact their educators, in some instances, displayed similar understandings. Learners display procedural understanding of graphical representations to a large extent. The educators‟ visual understanding does suggest that learners‟ interpretation of graphs is in some way influenced by the way they teach. The overriding contribution of the research study is that visual literacy plays a significant role in the conceptual understanding of functional graphical relationships. The relationship between graphical representations and logical thought is central to visual literacy. Key concepts: visual literacy, conceptual understanding, graphical representations, visualisation, analytical thinking, constructivism, process-object, Visualizer / Analyzer, semiotics and vehicles of reasoning. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2009.

This study explores characteristics of students’ repertoires of representations in two mathematical contexts: whole number multiplication and the comparison of common fractions. A repertoire of representations refers to a set of representations which a student can reconstruct as needed. Of particular interest are (1) how multiplicative relationships among units were represented, and (2) whether continuous measurement was an underlying conceptual framework for their representations. In addition, the characteristics of students’ representations and interpretation of units of linear and area measurement were explored. Data were collected through a series of interviews with Grade 5 and Grade 7 students. Some results of the study were as follows. Each repertoire of representations was exemplified by a dominant form of units, either discrete or contiguous. Within a repertoire, all forms of units were related, first through a common system of measurement (either numerosity or area), and second through their two-dimensional characteristic. In the multiplication context, some repertoires were comprised only of representations with discrete units, but others also included some representations with contiguous units. Students sought characteristics in their representations which reflected those based on continuous measurement, however linear or area measurement was not used as a conceptual framework. Instead, all representations were based on the measurement of numerosity. Also, students exhibited different limits in their representation of multiplicative relationships among units. Some represented no multiplicative relationships, but most represented at least a multiplicative relationship between two units. Relationships among three units were seldom constructed and difficult to achieve. Common fraction repertoires were based on the measurement of either numerosity or area, but the physical characteristics of the units varied. Some repertoires had only contiguous representations of units, others also included representations with discrete units, and a few did not represent fractional units at all. Students’ representations reflected characteristics of area-based representations, however area measurement was not necessarily a conceptual framework. In addition, students’ beliefs about what constituted units of area measurement were variable. As a result, they either represented no multiplicative relationships among units, or fluctuated between representing two-unit and three-unit relationships. Linear measurement was notably absent as a basis for representations in both mathematical contexts. The one-dimensional characteristic of linear measurement did not fit students’ dominant framework for constructing mathematical representations. With respect to measurement, students represented linear units in terms of discrete points or line segments. Counting points and interpreting the count in terms of the numerosity of line segments was problematic for nearly all students. When partitioning regions into units of area, a few students also equated the number of lines with the number of parts. The direct relationship of action and result in counting discrete objects was generalized without consideration of other geometric characteristics. When comparing quantities having linear or area units, numerical reasoning was not always used. Alternatively, either quantities were transformed to facilitate a direct comparison, or only perceptual judgements were made. No students consistently used numerical reasoning to compare fractional units of area. In the latter situations, the part-whole relationship among units seldom was observed. In general, there was no direct relationship between the forms of representations used by students in the two mathematical contexts and the characteristics of their representations of units of the measurement contexts. The development of repertoires of representations appears to be context specific. The repertoires were strictly limited in terms of the forms of representations of which they were comprised.

In this thesis, I describe how interactive journal writing was used to improve the understanding of mathematics, and to foster communication with two groups of remedial grade ten students. Mathematics is a gatekeeper course in high school, and students who are not successful with this subject are at a distinct disadvantage, both in terms of their education and in their future careers. A persistent source of difficulty for these students is related to language; students often struggle both to understand what is being taught, and how to explain concepts or problem solutions in their own words. Interactive journal writing was initiated as a means of addressing this situation, and of meeting the objectives proposed by the Quebec Education Plan, which specifies three closely related competencies: (1) solve situational problems; (2) use mathematical reasoning; (3) and communicate by using mathematical language. There is ample proof in the research literature that communication plays an important role in supporting learners by helping them clarify, refine and consolidate their thinking. / This study demonstrates the importance of allowing and encouraging students to use writing as part of their learning processes. By writing about what they are being taught, students are forced to slow down, examine and reflect on the steps they use to solve problems. Sharing what they write promotes meaningful dialogue and personal engagement, essential ingredients of successful learning.

In recent years much has been said and written concerning the widening gap between the newer developments in mathematics and that which is traditionally taught in secondary schools. Not unnaturally, leading scholars in mathematics have looked at the school programmes and found them wanting in many respects. [...]