Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Low achievement in Mathematics is a problem that faces schools, colleges and universities, not only in South Africa, but worldwide. Many efforts have been made to explore different ways in which teaching and learning can be more effective. ICT provides endless possibili–ties to enhance the teaching and learning of Mathematics. The purpose of this study is to determine teacher–students’ attitude towards ICTs in order to plan for effective, efficient and appropriate methods of Mathematics education. It further aims to determine to what extent these students will be willing to use ICTs in their own studies as well as in the teaching pro–cess in their classrooms, and aims to compare the students’ attitudes across three countries i.e. South Africa, Tanzania and Finland. The population consisted of Mathematics education teacher–students enrolled at the following institutions: School for Continuing Teacher Educa–tion (SCTE) at the North–West University in South Africa (317 students), the Iringa University College, University of Tumaini in Tanzania (111 students), and the University of Joensuu in Finland (56 students). This study followed a quantitative cross–sectional survey design com–prising a single mode research questionnaire to three groups of students in three countries. The instrument that was used was based on the Loyd and Gressard Computer Attitude Sur–vey (1984) (CAS), with a few additional questions to add to the relevance of this study. Questionnaires were completed by each group of students. The questionnaires were posted to each Mathematics teacher–student at the SCTE with a return envelope included. The re–searcher visited Tanzania and administered the questionnaire to the students at their univer–sity and some students in Finland completed the survey online while others completed a pa–per–based copy, which was returned to the researcher. Descriptive statistical techniques, reliability and validity of the instrument scale, inferential statistics (ANOVA), and cross–tabulations were used, and where appropriate, effect sizes were calculated. Findings indi–cated that the teacher–students have a positive attitude to computers and that they are willing to use ICTs in their own studies and in their teaching at school. The comparison of their atti–tudes revealed that the South African and Tanzanian teacher–students have a more positive attitude towards ICTs than the Finnish students. The teacher–students in the two African countries were also more willing to use ICTs than the students in Finland, despite their in–creased exposure and access to technology. To create an environment of effective Mathe–matics education, a number of factors play an important role. Mathematics education sup–ported by good pedagogical content knowledge can be transformed into effective Mathemat–ics education with the aid of a positive computer attitude and willingness to use technology. Together these aspects are supported by the technological pedagogical content knowledge (TPCK) framework. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2011.

Low achievement in Mathematics is a problem that faces schools, colleges and universities, not only in South Africa, but worldwide. Many efforts have been made to explore different ways in which teaching and learning can be more effective. ICT provides endless possibili–ties to enhance the teaching and learning of Mathematics. The purpose of this study is to determine teacher–students’ attitude towards ICTs in order to plan for effective, efficient and appropriate methods of Mathematics education. It further aims to determine to what extent these students will be willing to use ICTs in their own studies as well as in the teaching pro–cess in their classrooms, and aims to compare the students’ attitudes across three countries i.e. South Africa, Tanzania and Finland. The population consisted of Mathematics education teacher–students enrolled at the following institutions: School for Continuing Teacher Educa–tion (SCTE) at the North–West University in South Africa (317 students), the Iringa University College, University of Tumaini in Tanzania (111 students), and the University of Joensuu in Finland (56 students). This study followed a quantitative cross–sectional survey design com–prising a single mode research questionnaire to three groups of students in three countries. The instrument that was used was based on the Loyd and Gressard Computer Attitude Sur–vey (1984) (CAS), with a few additional questions to add to the relevance of this study. Questionnaires were completed by each group of students. The questionnaires were posted to each Mathematics teacher–student at the SCTE with a return envelope included. The re–searcher visited Tanzania and administered the questionnaire to the students at their univer–sity and some students in Finland completed the survey online while others completed a pa–per–based copy, which was returned to the researcher. Descriptive statistical techniques, reliability and validity of the instrument scale, inferential statistics (ANOVA), and cross–tabulations were used, and where appropriate, effect sizes were calculated. Findings indi–cated that the teacher–students have a positive attitude to computers and that they are willing to use ICTs in their own studies and in their teaching at school. The comparison of their atti–tudes revealed that the South African and Tanzanian teacher–students have a more positive attitude towards ICTs than the Finnish students. The teacher–students in the two African countries were also more willing to use ICTs than the students in Finland, despite their in–creased exposure and access to technology. To create an environment of effective Mathe–matics education, a number of factors play an important role. Mathematics education sup–ported by good pedagogical content knowledge can be transformed into effective Mathemat–ics education with the aid of a positive computer attitude and willingness to use technology. Together these aspects are supported by the technological pedagogical content knowledge (TPCK) framework. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2011.