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

Mathematics achievement has received much attention in recent years and results have been presented after examining results from different counties. This contribution deals with the implementation of new Mathematic promotion requirements in Namibia. The research was conducted in Shambyu circuit, Kavango region, within a selected combined public school situated fifteen kilometres from Rundu in the North-Eastern part of Kavango. Teachers and learners in Namibia have not performed well in the Southern and Eastern Africa Consortium for Monitoring Educational Quality (SACMEQ) projects, especially in Mathematics. The implementation of new promotion requirements was inevitable for improving achievement levels. The main aim of this study is to document the attitudes of teachers and learners towards the introduction of the new 2010 Mathematics promotion requirements. The complexity and the nature of attitudes are illustrated and some of the characteristics related to teaching and learning of Mathematics in the academic reform are presented. The researcher developed a conceptual framework to compare and contrast the theoretical positions on the topic. Attitude is defined from diverse perspectives, and relationships of attitudes pertaining to achievement to perform in Mathematics are argued. A qualitative case study was the preferred method of choice. The participants were sampled according to a non-probability purposive sampling strategy. Five teachers, six grade 7 and six grade 9 learners participated in the study. The participants were interviewed to gain insight into how they formulated their attitudes towards the implementation of the academic reform. Focus group interviews were captured though audio recordings. Patterns, themes and categories emerged from the data analysis, suggesting that teachers and learners demonstrate positive and negative attitudes which affect their stance towards the new promotion requirements. Research findings were compared with the relevant literature to identify strengths and weaknesses as extracted from the attitudes of the participating teachers and learners which confirm that attitudes of teachers and learners interrelate and affect teaching and learning of Mathematics. Strengths and weaknesses extracted from the attitudes of the teachers relate to teaching strategies, pedagogical content knowledge and practical application of the subject. A weakness of the policy change is that the Ministry of Education does not sustain involvement. Teachers need support through workshops to increase their pedagogical content knowledge and gain more information about the implementation of the new policy. Furthermore teachers expect educational support from the Ministry of Education through the provision of textbooks and teaching aids. Collaboration between teachers is crucial, as is the significance thereof for developing pedagogical content knowledge for the implementation of the new Mathematical policy. Strengths and weaknesses extracted from attitudes as viewed by learners in grade 9 are more related to their opinions about the teachers, their motivation and academic achievements. Learners’ natural Mathematics skills should be developed to instill feelings of accomplishment. Grade 9 learners experience fear and insecurity in Mathematics because learners experience teachers as too strict, owing to the absence of pedagogical content knowledge. The grade 9 learners distinguish the importance of ICT use in Mathematics as part of a process to prepare them towards greater goals and practical application as a strength. Both advantages and disadvantages of beliefs regarding Mathematics amongst the teachers and the learners guide grade 7 learner towards achievement. Further expectations drive the grade 7 learners towards achievement in order to increase career opportunities and level of schooling. In conclusion the in-depth qualitative exploration is summarized in order to investigate the phenomenon of attitudes towards Mathematics and academic reform. / Thesis (MEd (Mathematics Education))--North-West University, Potchefstroom Campus, 2013.

Mathematics achievement has received much attention in recent years and results have been presented after examining results from different counties. This contribution deals with the implementation of new Mathematic promotion requirements in Namibia. The research was conducted in Shambyu circuit, Kavango region, within a selected combined public school situated fifteen kilometres from Rundu in the North-Eastern part of Kavango. Teachers and learners in Namibia have not performed well in the Southern and Eastern Africa Consortium for Monitoring Educational Quality (SACMEQ) projects, especially in Mathematics. The implementation of new promotion requirements was inevitable for improving achievement levels. The main aim of this study is to document the attitudes of teachers and learners towards the introduction of the new 2010 Mathematics promotion requirements. The complexity and the nature of attitudes are illustrated and some of the characteristics related to teaching and learning of Mathematics in the academic reform are presented. The researcher developed a conceptual framework to compare and contrast the theoretical positions on the topic. Attitude is defined from diverse perspectives, and relationships of attitudes pertaining to achievement to perform in Mathematics are argued. A qualitative case study was the preferred method of choice. The participants were sampled according to a non-probability purposive sampling strategy. Five teachers, six grade 7 and six grade 9 learners participated in the study. The participants were interviewed to gain insight into how they formulated their attitudes towards the implementation of the academic reform. Focus group interviews were captured though audio recordings. Patterns, themes and categories emerged from the data analysis, suggesting that teachers and learners demonstrate positive and negative attitudes which affect their stance towards the new promotion requirements. Research findings were compared with the relevant literature to identify strengths and weaknesses as extracted from the attitudes of the participating teachers and learners which confirm that attitudes of teachers and learners interrelate and affect teaching and learning of Mathematics. Strengths and weaknesses extracted from the attitudes of the teachers relate to teaching strategies, pedagogical content knowledge and practical application of the subject. A weakness of the policy change is that the Ministry of Education does not sustain involvement. Teachers need support through workshops to increase their pedagogical content knowledge and gain more information about the implementation of the new policy. Furthermore teachers expect educational support from the Ministry of Education through the provision of textbooks and teaching aids. Collaboration between teachers is crucial, as is the significance thereof for developing pedagogical content knowledge for the implementation of the new Mathematical policy. Strengths and weaknesses extracted from attitudes as viewed by learners in grade 9 are more related to their opinions about the teachers, their motivation and academic achievements. Learners’ natural Mathematics skills should be developed to instill feelings of accomplishment. Grade 9 learners experience fear and insecurity in Mathematics because learners experience teachers as too strict, owing to the absence of pedagogical content knowledge. The grade 9 learners distinguish the importance of ICT use in Mathematics as part of a process to prepare them towards greater goals and practical application as a strength. Both advantages and disadvantages of beliefs regarding Mathematics amongst the teachers and the learners guide grade 7 learner towards achievement. Further expectations drive the grade 7 learners towards achievement in order to increase career opportunities and level of schooling. In conclusion the in-depth qualitative exploration is summarized in order to investigate the phenomenon of attitudes towards Mathematics and academic reform. / Thesis (MEd (Mathematics Education))--North-West University, Potchefstroom Campus, 2013.

Assessment, as an integral part of teaching and learning, gained unprecedented prominence in the curriculum in South Africa post 1994. When the new curriculum was introduced, it was assumed that teachers would effortlessly adapt their teaching and assessment practices, and swiftly implement the curriculum. Fourteen years after the inception of the new curriculum, majority of teachers are still grappling with issues of assessment. Previously, there was an exclusive bias towards summative assessment, which is mainly learning product–orientated and less or no focus on the other assessment typologies such as diagnostic and formative assessment, which are learning process–orientated. Of these typologies, diagnostic assessment is not being used maximally to enhance mathematics learning and inform the nature of the interventions to attend to learners’ needs. The study focused on diagnostic assessment by investigating the impact of a particular model, GIST model, on the learning barriers and learner achievement in mathematics among the grade 9 learners. The investigation of the impact of the GIST model was done through the experimental design in four schools with class sizes of d > 40. Data were collected quantitatively through Study Orientation Questionnaire (SOM) and Mathematics Achievement Test (MAT) as well as qualitatively through interviews, observations and document analysis. The t–test and the analysis of covariance (ANCOVA) revealed that the GIST model improved the learner achievement practically significantly (d = 0.79). However, the GIST model could not mitigate the learning barriers and improve correlations between SOM and MAT. The study, however, does find grounds to conclude that the latter findings can be attributed to teachers’ lacking understanding and implementation of diagnostic assessment, in particular the GIST components. Hence, certain recommendations are posed with regard to the applicable training of teachers in order to empower them to effectively utilize diagnostic assessment and to guide learners in overcoming learning barriers in mathematics. / Thesis (Ph.D. (Mathematics Education))--North-West University, Potchefstroom Campus, 2011.

Assessment, as an integral part of teaching and learning, gained unprecedented prominence in the curriculum in South Africa post 1994. When the new curriculum was introduced, it was assumed that teachers would effortlessly adapt their teaching and assessment practices, and swiftly implement the curriculum. Fourteen years after the inception of the new curriculum, majority of teachers are still grappling with issues of assessment. Previously, there was an exclusive bias towards summative assessment, which is mainly learning product–orientated and less or no focus on the other assessment typologies such as diagnostic and formative assessment, which are learning process–orientated. Of these typologies, diagnostic assessment is not being used maximally to enhance mathematics learning and inform the nature of the interventions to attend to learners’ needs. The study focused on diagnostic assessment by investigating the impact of a particular model, GIST model, on the learning barriers and learner achievement in mathematics among the grade 9 learners. The investigation of the impact of the GIST model was done through the experimental design in four schools with class sizes of d > 40. Data were collected quantitatively through Study Orientation Questionnaire (SOM) and Mathematics Achievement Test (MAT) as well as qualitatively through interviews, observations and document analysis. The t–test and the analysis of covariance (ANCOVA) revealed that the GIST model improved the learner achievement practically significantly (d = 0.79). However, the GIST model could not mitigate the learning barriers and improve correlations between SOM and MAT. The study, however, does find grounds to conclude that the latter findings can be attributed to teachers’ lacking understanding and implementation of diagnostic assessment, in particular the GIST components. Hence, certain recommendations are posed with regard to the applicable training of teachers in order to empower them to effectively utilize diagnostic assessment and to guide learners in overcoming learning barriers in mathematics. / Thesis (Ph.D. (Mathematics Education))--North-West University, Potchefstroom Campus, 2011.

Text in Afrikaans / Die doel van hierdie navorsing was om vas te stel of leerders in Graad R die nodige wiskundige vaardighede deur gestruktureerde spel bemeester het wat as grondslag vir verdere wiskunde-onderrig kan dien. In die literatuurstudie is die verskillende wiskunde konsepte in diepte ondersoek. Daar is ook gefokus op die oordra van wiskunde konsepte aan jong leerders. Verder is daar in die literatuurstudie ondersoek ingestel na die doel, waardes, tipes, ontwikkeling van spel asook spel in die moderne era. Vir die empiriese ondersoek is 103 Graad R-leerders van ses verskillende skole betrek. Daar is gebruik gemaak van ʼn ewekansige steekproef by die skole wat bereid was om aan die studie deel te neem. Die ondersoek het plaasgevind deur die gebruik van die positivistiese kwantitatiewe navorsingsmetode. Daar is gebruik gemaak van vraelyste om te bepaal wat die bemeestering van wiskundige vaardighede by Graad R-leerders is. Gestruktureerde spel is as hulpmiddel gebruik vir die insameling van die navorsingsmateriaal. Die resultate is bekom deur die vraelyste statisties te verwerk. Daar is ook vraelyste aan die onderwysers gegee om meer agtergrond oor hulle onderrigmetodes asook die onderwysers se eie gevoel teenoor en kennis van wiskunde te bekom. Die navorsingsresultate bevestig dat die meerderheid van die Graad R-leerders oor ʼn goeie wiskundige grondslag beskik. Dus is daar deur die studie gevind dat dit belangrik is dat jong kinders baie moet speel om nuwe vaardighede aan te leer, maar ook die geleentheid moet hê om die vaardighede te beoefen. / The purpose of this research was to determine whether learners in Grade R mastered the necessary math skills through structured play that could serve as a basis for further mathematics training. In the literature various math concepts were studied in depth. The main focus was on the transfer of math concepts to young learners. The literature study further focussed on the purpose, values, and types of play, development of play and play in the modern era. An empirical study using random sampling was done using 103 Grade R learners from six different schools who were willing to participate in the study. The study was done using the positivism quantitative research methodology. Questionnaires were used to determine the mastery of mathematical skills by Grade R learners. Structured play was used as a tool to collect the research material. The results were obtained by processing the questionnaires statistically. Questionnaires were also completed by the teachers to confirm their background knowledge, teaching methods as well as to determine the teachers‟ own feelings towards and their knowledge of mathematics. The research results confirm that the majority of Grade R learners have obtained a good math foundation. The study also confirmed that it is important that young children need to play to learn new skills and must have the opportunity to practise the new skills. / Psychology of Education / D. Ed. (Sielkundige Opvoedkunde)

Text in Afrikaans / Die doel van hierdie navorsing was om vas te stel of leerders in Graad R die nodige wiskundige vaardighede deur gestruktureerde spel bemeester het wat as grondslag vir verdere wiskunde-onderrig kan dien. In die literatuurstudie is die verskillende wiskunde konsepte in diepte ondersoek. Daar is ook gefokus op die oordra van wiskunde konsepte aan jong leerders. Verder is daar in die literatuurstudie ondersoek ingestel na die doel, waardes, tipes, ontwikkeling van spel asook spel in die moderne era. Vir die empiriese ondersoek is 103 Graad R-leerders van ses verskillende skole betrek. Daar is gebruik gemaak van ʼn ewekansige steekproef by die skole wat bereid was om aan die studie deel te neem. Die ondersoek het plaasgevind deur die gebruik van die positivistiese kwantitatiewe navorsingsmetode. Daar is gebruik gemaak van vraelyste om te bepaal wat die bemeestering van wiskundige vaardighede by Graad R-leerders is. Gestruktureerde spel is as hulpmiddel gebruik vir die insameling van die navorsingsmateriaal. Die resultate is bekom deur die vraelyste statisties te verwerk. Daar is ook vraelyste aan die onderwysers gegee om meer agtergrond oor hulle onderrigmetodes asook die onderwysers se eie gevoel teenoor en kennis van wiskunde te bekom. Die navorsingsresultate bevestig dat die meerderheid van die Graad R-leerders oor ʼn goeie wiskundige grondslag beskik. Dus is daar deur die studie gevind dat dit belangrik is dat jong kinders baie moet speel om nuwe vaardighede aan te leer, maar ook die geleentheid moet hê om die vaardighede te beoefen. / The purpose of this research was to determine whether learners in Grade R mastered the necessary math skills through structured play that could serve as a basis for further mathematics training. In the literature various math concepts were studied in depth. The main focus was on the transfer of math concepts to young learners. The literature study further focussed on the purpose, values, and types of play, development of play and play in the modern era. An empirical study using random sampling was done using 103 Grade R learners from six different schools who were willing to participate in the study. The study was done using the positivism quantitative research methodology. Questionnaires were used to determine the mastery of mathematical skills by Grade R learners. Structured play was used as a tool to collect the research material. The results were obtained by processing the questionnaires statistically. Questionnaires were also completed by the teachers to confirm their background knowledge, teaching methods as well as to determine the teachers‟ own feelings towards and their knowledge of mathematics. The research results confirm that the majority of Grade R learners have obtained a good math foundation. The study also confirmed that it is important that young children need to play to learn new skills and must have the opportunity to practise the new skills. / Psychology of Education / D. Ed. (Sielkundige Opvoedkunde)

The purpose of this study is to focus on constructing a modelling-based learning environment to improve learner performance in grade 6 mathematics classrooms. The purpose emanates from the continued poor performance of learners in mathematics at different school levels, especially grade 6. The teaching and learning of mathematics is explained from an ontological point of departure, focussing on constructivist paradigms. Different types of constructivism are discussed with special attention to the school mathematics domain. The learning, problem based learning, problem solving and learning environment are key components in the discussion. A theoretical perspective on the design of modelling as a powerful learning environment in primary schools mathematics classrooms is provided. Focus is placed on the applicability of the modelling-based learning environment on the South African mathematics curriculum and on study orientation as a key component to help develop an understanding of why learners perform or do not perform in mathematics. A mixed method research design, in which quantitative and qualitative are combined to achieve the outcomes of the research problem, is chosen for this research study project to provide a purposeful research framework. The findings of the research include not only learners’ improvement in dealing with non-routine, mathematical word problems but also in general-routine, mathematical word problems. A second finding shows that the overall SOM pre/post/retention showed good reliability, acceptable construct validity, good practical significance, and large effect but had low to medium effect in individual fields. The univariate analysis for the Crossover design used indicated that the problem solving field had statistical significance and practical significance, and the study milieu and mathematical confidence field might have statistical significance and practical significance. The third finding provided evidence concerning teacher administration, teacher and learner interaction, assessment and homework. The findings from the quantitative and qualitative data-analysis and interpretations, and literature review, guided the researcher in proposing a construct for a modelling-based learning environment as a means to improve learners’ mathematics performance in grade 6 mathematics classes in the John Toalo Gaetswe (JTG) District. The contribution that this study makes is to propose a construct for a modelling-based learning environment to improve learner performance in grade 6 mathematics. / 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

The purpose of this study is to focus on constructing a modelling-based learning environment to improve learner performance in grade 6 mathematics classrooms. The purpose emanates from the continued poor performance of learners in mathematics at different school levels, especially grade 6. The teaching and learning of mathematics is explained from an ontological point of departure, focussing on constructivist paradigms. Different types of constructivism are discussed with special attention to the school mathematics domain. The learning, problem based learning, problem solving and learning environment are key components in the discussion. A theoretical perspective on the design of modelling as a powerful learning environment in primary schools mathematics classrooms is provided. Focus is placed on the applicability of the modelling-based learning environment on the South African mathematics curriculum and on study orientation as a key component to help develop an understanding of why learners perform or do not perform in mathematics. A mixed method research design, in which quantitative and qualitative are combined to achieve the outcomes of the research problem, is chosen for this research study project to provide a purposeful research framework. The findings of the research include not only learners’ improvement in dealing with non-routine, mathematical word problems but also in general-routine, mathematical word problems. A second finding shows that the overall SOM pre/post/retention showed good reliability, acceptable construct validity, good practical significance, and large effect but had low to medium effect in individual fields. The univariate analysis for the Crossover design used indicated that the problem solving field had statistical significance and practical significance, and the study milieu and mathematical confidence field might have statistical significance and practical significance. The third finding provided evidence concerning teacher administration, teacher and learner interaction, assessment and homework. The findings from the quantitative and qualitative data-analysis and interpretations, and literature review, guided the researcher in proposing a construct for a modelling-based learning environment as a means to improve learners’ mathematics performance in grade 6 mathematics classes in the John Toalo Gaetswe (JTG) District. The contribution that this study makes is to propose a construct for a modelling-based learning environment to improve learner performance in grade 6 mathematics. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014