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

Basic mathematical competency seems to be lacking for engineering students starting their studies in this field. Students generally find the cognitive transition from secondary to tertiary mathematics challenging which in turn negatively influences their academic achievement in mathematics. The cognitive challenge is the transition from the application of mathematics to familiar questions to applying mathematical principles to varying practical application and problem solving. Mathematics provides the foundation for the cognitive toolset required for the development of skills required for analysing engineering systems and processes. It is therefore important to assess mathematical and cognitive competency and ability at the time of admission to a tertiary institution in order to identify and address gaps. This research demonstrates that first-year engineering students need to have a specific level of mathematical competency and cognitive ability to use mathematics within the context of engineering studies. This research attempts to connect the mathematic competency of first year engineering students to their academic results for subjects in the first year curriculum that rely heavily on mathematical competency. To satisfy the research question, the study firstly looks at relevant literature to identify the mathematical competency levels as well as the operational specification. Secondly, development theories and taxonomies were analysed to gain insight into the development processes associated with learning, cognitive development and the gap between cognitive competencies in transition from secondary to tertiary education. Further, cognitive competencies were identified that are essential for successful completion of first year engineering modules. Through synthesis of the different theories and taxonomies a framework was identified. This framework was used to analyse secondary data in order to measure mathematical and cognitive levels. Thirdly, the theoretical investigation was followed by a three-phase empirical study. A mixed quantative-qualitative (QUAN-qual) approached was followed. Phase 1 uses the assessment framework to measure first year students‟ mathematical competency at the inception of their studies as well as at the completion of their first semester. The mathematical competency at inception was measured with their Grade 12 mathematics marks and with relevant analysis of their initial bridging assessments, on a question by question basis. In addition, their first semester exams questions were analysed using the same approach as above. Phase 2 comprises the measurement of the relationship between the mathematical competency of first year enigineering students at admission and their achievement levels in selected first year subjects that required mathematical competency. Phase 3 includes the guidelines derived from the gaps and shortcomings identified. These gaps were identified in order to inform appropriate study support to first year students and to assists academic personnel with setting appropriate and dependable admission standards. The analysis of mathematical competency creates quality data that gives a clearer picture than a simple comparison of admission scores and first semester marks. The empirical study contributes to a better understanding of the problems associated with the transition from secondary to tertiary learning environments. From the study it was derived that study inception information of the students correlated only with their academic results on questions that tested mathematical and programming application. The inception information was not a predictor of mathematical achievement and results for both the lowest and highest mathematical competency levels. Futher study in this field is required to create frameworks for the measurements of both low and high levels of mathematical competency. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

The South African Department of Basic Education and Department of Higher Education and Training (2011a) made a call for the development of mathematics knowledge and practice standards in the foundation phase to serve as guidelines for the preparation of foundation phase teachers in the different higher education institutions. The purpose of the study in hand was to develop draft knowledge and practice standards in the number domain for the preparation of foundation phase teachers (referred to as mathematics knowledge and practice standards). These standards have to be refined and improved in further studies and should in the end serve as knowledge and practice standards for the preparation of foundation phase teachers in South Africa. Through a conceptual qualitative research methodology the researcher purposefully collected documents and analysed them through content analysis. The data-gathering process took place in three phases. During the first phase, policy documents with regard to general standards for teacher preparation, written school curriculum documents and mathematical standards for the preparation of foundation phase teachers in South Africa, the United States of America, Australia and the Netherlands were gathered. The second phase involved the purposeful gathering of articles, research reports and teacher preparation textbooks with regard to the preparation of foundation phase teachers to teach the number domain. During the first and second phases of data gathering, the documents were analysed according to mathematical knowledge for teaching as described by Ball, Thames and Phelps (2008) and the first draft of mathematical knowledge and practice standards was compiled. During the third data-gathering phase, critical evaluation reports were gathered from experts in the field of mathematics education (including researchers at universities and practising foundation phase teachers). The critical evaluation includes gaps/shortcomings in the draft mathematics knowledge and practice standards, as well as comments with regard to the clarity, applicability and functionality of the document. The draft mathematics knowledge and practice standards (MKPSs) for the preparation of foundation phase teachers include: Standard 1: Common content knowledge – The foundation phase teacher has a clear understanding of the common content knowledge of the number domain. Standard 2: Specialised content knowledge – The foundation phase teacher has a clear understanding of the specialised content of the number domain. Standard 3: Knowledge at the mathematical horizon – The foundation phase teacher understands how mathematical themes in the number domain relate to other themes in the different foundation phase year groups and in other phases. Standard 4: Knowledge of content and teaching – The foundation phase teacher is able to plan lessons and knows how to teach the number domain. Standard 5: Knowledge of content and learners – The foundation phase teacher knows the foundation phase learners and knows how they learn the number domain. Standard 6: Knowledge of content and the curriculum – The foundation phase teacher understands the South African school curriculum, as well as international trends in the school curriculum concerning the number domain. Those experts in the field of mathematics education in the foundation phase who participated in the study all indicated that the mathematics knowledge and practice standards in the number domain have the potential to boost the preparation of foundation phase teachers in South Africa. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Basic mathematical competency seems to be lacking for engineering students starting their studies in this field. Students generally find the cognitive transition from secondary to tertiary mathematics challenging which in turn negatively influences their academic achievement in mathematics. The cognitive challenge is the transition from the application of mathematics to familiar questions to applying mathematical principles to varying practical application and problem solving. Mathematics provides the foundation for the cognitive toolset required for the development of skills required for analysing engineering systems and processes. It is therefore important to assess mathematical and cognitive competency and ability at the time of admission to a tertiary institution in order to identify and address gaps. This research demonstrates that first-year engineering students need to have a specific level of mathematical competency and cognitive ability to use mathematics within the context of engineering studies. This research attempts to connect the mathematic competency of first year engineering students to their academic results for subjects in the first year curriculum that rely heavily on mathematical competency. To satisfy the research question, the study firstly looks at relevant literature to identify the mathematical competency levels as well as the operational specification. Secondly, development theories and taxonomies were analysed to gain insight into the development processes associated with learning, cognitive development and the gap between cognitive competencies in transition from secondary to tertiary education. Further, cognitive competencies were identified that are essential for successful completion of first year engineering modules. Through synthesis of the different theories and taxonomies a framework was identified. This framework was used to analyse secondary data in order to measure mathematical and cognitive levels. Thirdly, the theoretical investigation was followed by a three-phase empirical study. A mixed quantative-qualitative (QUAN-qual) approached was followed. Phase 1 uses the assessment framework to measure first year students‟ mathematical competency at the inception of their studies as well as at the completion of their first semester. The mathematical competency at inception was measured with their Grade 12 mathematics marks and with relevant analysis of their initial bridging assessments, on a question by question basis. In addition, their first semester exams questions were analysed using the same approach as above. Phase 2 comprises the measurement of the relationship between the mathematical competency of first year enigineering students at admission and their achievement levels in selected first year subjects that required mathematical competency. Phase 3 includes the guidelines derived from the gaps and shortcomings identified. These gaps were identified in order to inform appropriate study support to first year students and to assists academic personnel with setting appropriate and dependable admission standards. The analysis of mathematical competency creates quality data that gives a clearer picture than a simple comparison of admission scores and first semester marks. The empirical study contributes to a better understanding of the problems associated with the transition from secondary to tertiary learning environments. From the study it was derived that study inception information of the students correlated only with their academic results on questions that tested mathematical and programming application. The inception information was not a predictor of mathematical achievement and results for both the lowest and highest mathematical competency levels. Futher study in this field is required to create frameworks for the measurements of both low and high levels of mathematical competency. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

The South African Department of Basic Education and Department of Higher Education and Training (2011a) made a call for the development of mathematics knowledge and practice standards in the foundation phase to serve as guidelines for the preparation of foundation phase teachers in the different higher education institutions. The purpose of the study in hand was to develop draft knowledge and practice standards in the number domain for the preparation of foundation phase teachers (referred to as mathematics knowledge and practice standards). These standards have to be refined and improved in further studies and should in the end serve as knowledge and practice standards for the preparation of foundation phase teachers in South Africa. Through a conceptual qualitative research methodology the researcher purposefully collected documents and analysed them through content analysis. The data-gathering process took place in three phases. During the first phase, policy documents with regard to general standards for teacher preparation, written school curriculum documents and mathematical standards for the preparation of foundation phase teachers in South Africa, the United States of America, Australia and the Netherlands were gathered. The second phase involved the purposeful gathering of articles, research reports and teacher preparation textbooks with regard to the preparation of foundation phase teachers to teach the number domain. During the first and second phases of data gathering, the documents were analysed according to mathematical knowledge for teaching as described by Ball, Thames and Phelps (2008) and the first draft of mathematical knowledge and practice standards was compiled. During the third data-gathering phase, critical evaluation reports were gathered from experts in the field of mathematics education (including researchers at universities and practising foundation phase teachers). The critical evaluation includes gaps/shortcomings in the draft mathematics knowledge and practice standards, as well as comments with regard to the clarity, applicability and functionality of the document. The draft mathematics knowledge and practice standards (MKPSs) for the preparation of foundation phase teachers include: Standard 1: Common content knowledge – The foundation phase teacher has a clear understanding of the common content knowledge of the number domain. Standard 2: Specialised content knowledge – The foundation phase teacher has a clear understanding of the specialised content of the number domain. Standard 3: Knowledge at the mathematical horizon – The foundation phase teacher understands how mathematical themes in the number domain relate to other themes in the different foundation phase year groups and in other phases. Standard 4: Knowledge of content and teaching – The foundation phase teacher is able to plan lessons and knows how to teach the number domain. Standard 5: Knowledge of content and learners – The foundation phase teacher knows the foundation phase learners and knows how they learn the number domain. Standard 6: Knowledge of content and the curriculum – The foundation phase teacher understands the South African school curriculum, as well as international trends in the school curriculum concerning the number domain. Those experts in the field of mathematics education in the foundation phase who participated in the study all indicated that the mathematics knowledge and practice standards in the number domain have the potential to boost the preparation of foundation phase teachers in South Africa. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Abstracts in English, Afrikaans and Zulu / Assessment is considered as integral to the teaching and learning process of Mathematics where various types of assessment are used to develop feedback for several purposes. Research has highlighted the challenge of the dominance of summative assessment in classroom assessment practices. In recent years, world countries have been acknowledging the use of assessment for learning (AfL) to enhance the learning process and thus improve learner performance. This research explored how Mathematics teachers applied AfL in their classrooms. A theoretical framework to support AfL was presented through an overview of constructivism theory, social justice theory, connectivism theory, TPACK theory and Bloom’s Revised Taxonomy. A qualitative approach and a case study design were applied involving nine Mathematics teachers from nine primary schools. Data, collected through semi-structured interviews, non-participant observation and document analysis, were thematically analysed. The findings show a positive understanding of what AfL is and its importance to the teaching and learning of Mathematics. However, the application of AfL was found to be inconsistent with its purpose of creating an environment conducive to develop feedback that supports the learning process. Challenges that inhibit its application were found to outweigh successes experienced by teachers. Lack of theoretical understanding of the use of AfL, overcrowding, the language of learning and teaching and lack of resources emerged as some of the major challenges. Teachers pleaded for more in-service training opportunities to assist them with managing assessment for learning practices in Mathematics. / Assessering is onlosmaaklik met die onderrig en leer van wiskunde verbind. Wiskunde word op verskeie maniere geassesseer sodat terugvoering om allerlei redes verkry word. Volgens navorsing oorheers summatiewe assessering in klaskamers. In die laaste jare word assessering vir leer (AvL) wêreldwyd aangewend om die leerproses en leerders se prestasie te verbeter. In hierdie studie is nagevors hoe wiskundeonderwysers AvL in die klaskamer toepas. ʼn Teoretiese raamwerk vir AvL is opgestel uit ʼn oorsig van die konstruktivistiese teorie, die sosialegeregtigheidsteorie, die konnektivismeteorie en die TPACK-teorie en Bloom se Hersiene Taksonomie. ʼn Kwalitatiewe benadering en ʼn gevallestudie-ontwerp is gevolg in die verkenning van nege wiskundeonderwysers by nege primêre skole se assessering. Data is deur halfgestruktureerde onderhoude, waarneming sonder deelname en dokumentontledings versamel en tematies geanaliseer. Daar is bevind dat die onderwysers geweet het wat AvL is en die belang daarvan in die onderrig en leer van wiskunde besef het. Die toepassing het egter nie met die oogmerk van AvL gestrook nie. Die oogmerk is om ʼn omgewing tot stand te bring wat assessering bevorder om leer te ondersteun. Die toepassingsprobleme van AvL oorskadu die welslae wat daarmee behaal word. ʼn Gebrekkige teoretiese begrip van hoe AvL gebruik word, oorvol klaskamers, die taal van onderrig en leer, en ʼn gebrek aan hulpbronne is van die grootste uitdagings. Onderwysers bepleit indiensopleiding sodat hulle die assessering van leerpraktyke in wiskunde beter kan bestuur. / Ukuhlola kuthathwa njengento esemqoka ohlelweni lokufundisa nokufunda imethamethiksi lapho izinhlobo ezahlukahlukene zokuhlola zisetshenziswa ukwakha umbiko wakamuva ngesizathu sezinhloso ezimbalwa. Ucwaningo selukhombise inselelo yokuhamba phambili kwenhlobo yokuhlola i-summative assessment lapho kuqhutshwa umsebenzi wokuhlola emagunjini okufunda. Eminyakeni esandakwedlula, amazwe omhlaba kade amukela ukusetshenziswa kohlelo lokuhlola ukufunda (AFL) ukuqinisa uhlelo lokufunda kanti lokhu kuthuthukisa izinga lokufunda lomfundi. Lolu cwaningo beluhlola indlela uthisha wesifundo semethamethiksi ebesebenzisa uhlelo lwe-AFL emagunjini abo okufundisa. Isakhiwo sethiyori esiqonde ukuxhasa uhlelo lwe-AFL lwethuliwe ngamafuphi ngomqondo phecelezi we- -constructivism theory, social justice theory, connectivism theory, TPACK theory kanye ne-Bloom’s Revised Taxonomy. Indlela yocwaningo eyencike kwingxoxo (qualitative approach) kanye nedizayini yocwaningo lotho (case study design) zisetshenziswe kuxutshwa phakathi othisha bemethamethiksi abayisishiyagalolunye abavela ezikoleni zamabanga aphansi. Idatha iqoqwe ngokwenza inhlolovo eyakhiwe kancane, kanti kuye kwabhekisiswa imibono yalabo abangadlalindima kanye nokuhlaziywa kombhalo kuye kwahlaziywa ngokuthi kubhekwe indikimba. . Ulwazi olutholwe wucwaningo lukhombisa ukuthi uhlelo lwe-AFL kanye nokubaluleka kwalo kuzwisiseka kahle kakhulu ohlelweni lokufundisa nokufundwa kwemethamethiksi. . Yize kunjalo, ukusetshenziswa kohlelo lwe-AFL kuye kwatholakala ukuthi akuhambisani nenhloso yalo yokwakha isizinda esifanele sokwakha umbiko wakamuva oxhasa uhlelo lokufunda. Izinselelo eziqukethe ukusetshenziswa kwalo lolu hlelo ziye zatholakala ukuthi zedlula impumelelo eyenziwe ngothisha. Ukwentuleka kokuzwisisa umqondo wokusetshenziswa kohlelo lwe-AFL, inani eliphuphumayo labantwana, ulimi lokufunda nokufundisa kanye nokwentuleka kwemithombo yokufunda kuye kwavela njengezinye izinselelo. Othisha baye bacela ukunikezwa amathuba okuqeqeshwa basebenza ukuze lawo makhono abancede ukuqhuba izinhlelo zokuhlola imisebenzi yokufunda imethamethiksi. / Curriculum and Instructional Studies / D. Phil. (Education (Curriculum Studies))