Based on the findings above, it is recommended that the future study should explore further how teachers' problem schemas influence their teachings by means of classroom observation. The second recommendation is to study problem schema and other subject matter knowledge in one research to see a comprehensive view. It is also recommended that the development of teachers' problem schemas and subject matter knowledge should be strengthened in teacher education. / Teacher knowledge was an active research area in the last twenty years. It focuses on all kinds of knowledge that teacher knows related to teaching, The study explored subject matter knowledge in teacher knowledge. From the past researches on subject matter knowledge, we find that researchers often focus on teachers' understanding of mathematical concepts and procedures. This is the knowledge of "knowing that". The past researches on subject matter knowledge didn't touch on the knowledge of "knowing how". Based on the theory of problem schema in psychology, we studied teacher's problem schemas, which are related to problem solving. / The study shows that there are various types of problem schemas among secondary school mathematics teachers. They include: "schema organized by mathematics concept ", "schema organized by conclusion or unknown", "schema organized by basic figure ", "schema organized by mathematical structure", "schema organized by general mathematical problem solving strategies". Compared to non-expert teachers, expert teachers have a wider set of problem schemas. Expert teachers not only have the schemas non-expert teachers have, but also have the schemas non-expert teachers don't have. "Schema organized by mathematical structure", "schema organized by general mathematical problem solving strategies" are two of them. Generally speaking, expert teachers possess more profound problem schemas. Hence we see a relationship between problem schemas and teaching expertise. / To explore possible relationship between problem schema and teaching, we first identified the figure of schemas secondary school mathematics teachers have. Then we examined the differences between expert teachers and non-expert teachers to explore the relationship between teachers' problem schemas and their teaching context. In this study, secondary school mathematics teachers were chosen for case studies. Qualitative research methods, including questionnaire and interview were adopted to find out teacher's problem schemas. / 韓繼偉. / 論文(哲學博士)--香港中文大學, 2005. / 參考文獻(p. 154-169). / Adviser: Ngai-ying Wong. / Source: Dissertation Abstracts International, Volume: 67-01, Section: A, page: 0122. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in English. / School code: 1307. / Lun wen (Zhe xue bo shi)--Xianggang Zhong wen da xue, 2005. / Can kao wen xian (p. 154-169). / Han Jiwei.

李琼. / 论文(哲学博士)--香港中文大学, 2004. / 参考文献 (p. 218-233). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts also in English. / Li Qiong. / Lun wen (Zhe xue bo shi)--Xianggang Zhong wen da xue, 2004. / Can kao wen xian (p. 218-233).

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

This study investigated the experiences of four participating student teachers in using multiple representative approaches in the teaching of Grade 6 proportion word problems. The multiple representative approaches include the Between Comparison Method, the Within Comparison Method, the Diagrammatic Method, the Table Method, the Graph Method, the Cross-product Method, and the Oral Informal Method. An intervention programme was organised, using workshops where student teachers were prepared to teach Grade 6 proportion word problems using multiple representative approaches. The teaching practice lessons of the four participating student teachers in two primary schools were video recorded, and the focus group interview was conducted at the University Campus. With the exception of the Graph Method and the Cross-product Method, it was revealed that the multiple representative approaches were generally effective in the teaching of Grade 6 proportion word problems. The study further revealed that multiplicative relationships can be explored through using the different individual representative approaches. The study argues that the cross-product method is not the only way to teach Grade 6 proportion word problems. There are multiple representative approaches that should be used in conjunction with each other to enhance the teaching of proportion word problems. Furthermore, this study revealed that a number of challenges were encountered when using multiple representative approaches. The challenges include difficulties with the English language, different and unique abilities of the learners, lack of plotting skills and the lack of proficiency in the learners’ multiplication and division skills. This study made recommendations on the integration of multiple representative approaches in the mathematics education curriculum and textbooks. It further recommended that in-service workshops for teachers and student teachers on the integration of multiple representative approaches in the teaching of Grade 6 proportion word problems should be initiated.

Quality mathematics education relies on effective pedagogy which offers students appropriate and rich opportunities to develop their mathematical proficiency (MP) and intellectual autonomy in learning mathematics. This qualitative case study aimed to explore and analyse selected effective mathematics teachers' proficiency in the area of geometry in five secondary schools in five different Namibia educational regions. The sample was purposefully selected and comprised five mathematics teachers, identified locally as being effective practitioners by their peers, Education Ministry officials and the staff of the University of Namibia (UNAM). The schools where the selected teachers taught were all high performing Namibian schools in terms of students' mathematics performance in the annual national examinations. The general picture of students' poor performance in mathematics in Namibia is no different to other sub-Saharan countries and it is the teachers who unfortunately bear the brunt of the criticism. There are, however, beacons of excellence in Namibia and these often go unnoticed and are seldom written about. It is the purpose of this study to focus on these high achievers and analyse the practices of these teachers so that the rest of Namibia can learn from their practices and experience what is possible in the Namibian context. The mathematical content and context focus of this study was geometry. This qualitative study adopted a multiple case study approach and was framed within an interpretive paradigm. The data were collected through individual questionnaires, classroom lesson observations and in-depth open-ended and semi-structured interviews with the participating teachers. These interviews took the form of post lesson reflective and stimulated recall analysis sessions. An adapted framework based on the Kilpatrick, Swafford and Findell's (2001) five strands of teaching for MP was developed as a conceptual and analytical lens to analyse the selected teachers' practice. The developed coding and the descriptive narrative vignettes of their teaching enabled a qualitative analysis of what teachers said contributed to their effectiveness and how they developed MP in students. An enactivist theoretical lens was used to complement the Kilpatrick et al.'s (2001) analytical framework. This enabled a deeper analysis of teacher teaching practice in terms of their embodied mathematical knowledge, actions and interactions with students. procedural fluency (PF) and productive disposition (PD), were addressed regularly by all five participating teachers. Evidence of addressing either the development of students' strategic competence (SC) or adaptive reasoning (AR) appeared rarely. Of particular interest in this study was that the strand of PD was the glue that held the other four strands of MP together. PD was manifested in many different ways in varying degrees. PD was characterised by a high level of content knowledge, rich personal experience, sustained commitment, effective and careful preparation for lessons, high expectations of themselves and learners, collegiality, passion for mathematics and an excellent work ethic. In addition, the teachers' geometry teaching practices were characterised by making use of real-world connections, manipulatives and representations, encouraging a collaborative approach and working together to show that geometry constituted a bridge between the concrete and abstract. The findings of the study have led me, the author, to suggest a ten (10) principles framework and seven (7) key interrelated factors for effective teaching, as a practical guide for teachers. This study argues that the instructional practices enacted by the participating teachers, who were perceived to be effective, aligned well with practices informed by the five strands of the Kilpatrick et al.'s (2001) model and the four concepts of autopoesis, co-emergence, structural determinism and embodiment of the enactivist approach. The study concludes with recommendations for effective pedagogical practices in the teaching of geometry, and opportunities for further research.

The purpose of this project was to determine if there was a correlation between teachers' scores on fractions items on project assessments and the percentage of participation time in professional development activities.

Thesis (M.Ed. (Mathematics)) ---University of Limpopo, 2016 / This research explored the pedagogical content knowledge (PCK) of algebra as held by a group of senior phase teachers in Limpopo Province, South Africa. Sixty-one teachers from different districts in the province participated in this study. This qualitative study used a case study design. Data was collected using a test and content representation (CoRe) matrix. The results were analysed through analysis of narratives. The study revealed that most of the teachers could not identify the main concepts that are taught in algebra. It was discovered that some could not differentiate between algebra and arithmetic. The findings also point out that the teachers had inadequate knowledge of algebra subject matter. The participants were able to identify some of the concepts that learners were supposed to do in the next grades. The findings revealed that most of the participants did not know the importance of teaching algebra. It was revealed that most participants could not identify the learners‘ difficulties; they could not specify the procedures they followed when teaching the identified main concepts in algebra, or give reasons for using those procedures. Furthermore, they could not specify factors that influenced their teaching. However, they were able to indicate different methods they used to assess learners‘ understanding. It was concluded that these teachers had inadequate PCK of senior phase algebra. The study recommends development programmes for both subject matter knowledge and PCK for practising teachers. Keywords: pedagogical content knowledge, content representations (CoRes), algebra, senior phase mathematics teachers.

Dissertation submitted in fulfilment of the requirements for the Master's Degree of Technology: Information Technology, Durban University of Technology, 2010. / The main purpose of this study was to examine how e-learning can help resolve some of the most acute problems that are specific to the nature of the outcomes-based education (OBE) system in developing countries. This was accomplished by investigating the relevant literature on OBE and by designing an e-learning content prototype for the South African version of OBE, with the focus on the training of Mathematics and Mathematical Literacy educators. OBE is an education system centred on the theory of mastery of learning introduced by Bloom in the 1950s. It has been implemented worldwide in primary and secondary schools and also in tertiary institutions. Some studies have shown that OBE is problematic, and that both educators and learners are opposed to this system of education. Existing research also reports that the quality of education in OBE is very poor as compared to that of the traditional education system. This study was an attempt to identify the most acute problems experienced by educators and learners in the OBE context and to design a prototype for e-learning content that can be used in courses in an attempt to solve these problems. The research population of the perceptions survey consisted of Mathematics and Mathematical Literacy educators and learners from the 6 000 primary and secondary schools of the KwaZulu- Natal (KZN) province of South Africa. Eighteen schools formed part of the research sample, with an average of two schools per region (the KZN province is divided into nine regions). Both private and public schools were included in the study. Data for the study were collected from March 2007 to August 2007 in the form of a perceptions survey of 104 educators and 288 learners, yielding an average of six educators and sixteen learners for each randomly selected school. Results from the perceptions survey show that educators and learners do not understand OBE terminology. In addition, educators claim that they are not sufficiently trained for OBE. Research iii participants also report that schools’ basic infrastructure is unsatisfactory, and that their classrooms are overcrowded. Mathematics is perceived as the most difficult subject by both educators and learners. The survey also reveals that most public schools do not have computers and that neither educators nor learners have access to computers in their public libraries. A SCORM- (Sharable Content Object Reference Model) compliant e-learning course was developed in this study to address the most acute problems identified by the survey, based on the Software Engineering Unified Model. The designed e-course contains OBE terminology such as learning outcomes, OBE principles, assessment standards, assessment methods, national curriculum statement and learning fields. The e-learning course content also contains the Mathematics and Mathematical Literacy curriculum for grades 11 and 12. It was constructed using 16 documents extracted from the National Department of Education’s website: 7 documents under Further Education and Training, 4 under Teacher Guide, and 5 under General Education and Training. The evaluation of the e-learning content prototype was conducted through a survey among 36 educators from different primary and secondary schools of the Mnquma Municipality of the Eastern Cape province of South Africa. They were trained in the SCORM-compliant e-learning course content at the Walter Sisulu University’s Butterworth campus. The training took place from 10 to 13 November 2009. After the training, educators filled out a questionnaire on their perceptions of the effectiveness of the proposed e-learning content prototype with regard to the practice of OBE. Results from the SCORM e-course evaluation survey showed that the proposed SCORM software artefacts allow educators to have a better understanding of OBE terminology. The proposed software artefact is user-friendly and educators recommended its use not only for Mathematics but for all subjects.