Based on the classification from quantitative studies, total six teachers, two in each category, were interviewed to construct six individual qualitative case studies. The results suggested that teacher professional knowledge about function and beliefs about mathematics had impacts on their teaching approaches. The teaching approaches can be classified into three categories, i.e, content-centered with performance focus, contentcentered with understanding focus and student-centered with cognitive focus, which constituted traditional content-centered and student-centered perspectives on teaching approaches. Emphasizing process or product is still dichotomy for teachers. / In conclusion, this study showed the significance of subject matter knowledge, pedagogical content knowledge and beliefs about mathematics to teaching approaches of mathematics teachers, which pave the way for future education researches in this field. This study has also provided some valuable experiences in applying hypothetical situations to education researches. In the future, further explorations can be conducted in finding the effect of contextual factors to teaching approaches and how teaching approaches influenced student learning. / In past 30 years, under the global trend of curriculum reform, new requirements on teacher professional knowledge and belief were put on teachers. Accordingly, the focus of research on teaching was shifted from finding effective teaching behavior to in search for teacher professional knowledge, beliefs and cognitions behinds their behaviors. Before, the literatures on teacher professional knowledge were mainly focused on what professional knowledge that teachers have and/or should have. Only few literatures were interested in the effect of teacher professional knowledge and beliefs to teaching approaches. / The main objective of this study is to fill in the missing pieces and investigate the effect of teacher professional knowledge and mathematics beliefs on teaching for inservices mathematics teachers. Mixed method was applied in this study in order to build up some comprehensive understandings to the effects. In quantitative part, this study has surveyed ninety one high school mathematics teachers in M city to find out the deepness of professional knowledge and mathematical beliefs that teachers held. For professional knowledge, the results showed that teachers had different levels of subject matter knowledge and pedagogical content knowledge. For beliefs about mathematics, the results echoed with Ernest's (1989) classification of teacher mathematical beliefs, i.e. instrumentalist, Platonist, and problem-solver. / 張侨平. / Adviser: Ngai-Ying Wong. / Source: Dissertation Abstracts International, Volume: 73-03, Section: A, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 184-209). / 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, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in Chinese and English. / Zhang Qiaoping.

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.