Geometric proof-writing is a widely known cause of stress for secondary school students and teachers alike. As the textbook is the primary curricular tool utilized by novice teachers, a two-part qualitative study was conducted to determine (a) the types of proofs presented in a typical high school geometry textbook and (b) teachers’ preparedness and confidence to teach proof and proving. I conducted a qualitative analysis of the selected textbook based on its presentation of proofs and proof tasks, and then used said analysis to inform the creation of a five-question content assessment on proof, which was administered to 29 preservice and in-service secondary mathematics teacher participants. During the administration of the assessment, I interviewed each participant regarding their thought processes, as well as their knowledge, beliefs, and preparedness to teach proof and proving. The data were analyzed using a qualitative coding system to categorize participants’ responses to the interview questions according to their beliefs and attitudes, as well as issues with mathematical language and content that they encountered. The qualitative analyses indicated that the selected textbook largely underemphasizes the role of proof in the secondary school geometry curriculum, and that most participants are largely underprepared to teach proof at the secondary level. Participants expressed sentiments about the nature of proof and proving, verifying trends from the literature and providing the impetus for future study. The findings support that more studies are needed to analyze the intersection between curricular knowledge and content knowledge for secondary mathematics teachers.

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在國際評鑑計劃中，東亞地區學生的數理能力往往較西方國家為佳。現有學術文獻已就這個現象作深入探討，並提出東亞學生的文化特徵和模型。然而，鮮有研究以縱向的歷史角度，討論東亞教育制度下對於「學習者」的文化概念如何因應全球化的影響而演變。本文嘗試以香港的數學教育為研究個案，探討本地數學課程在教學重心及組織上的轉變。研究發現，該學科在教學上的轉變緣於現代個人主義的制度化過程 ── 社會整體愈益視學習者作為充權和自省的行動者。本研究致用縱向文本分析的研究方法，綜析四十二本於戰後出版的香港高小數學教科書的教學內容。研究結果顯示，歷年來課本及課程的教學愈加著重參與式教學、興趣啟發以及與扣連學生生活等概念。另外，政府官員、老師、本地壓力團體及學術專家等亦援用「以學生為本」的話語，推動課程改革。綜合而言，本研究認為世界政體理論相比歷史主義式的理論視野更有力地解釋數學課程內容的轉變。 / East Asian learners usually outperform their Western counterparts in international mathematics assessment programs. Although the cultural traits of East Asian learners are well documented and analyzed, the cultural dynamics of the changing conception of “learner“ in East Asia, especially under the influence of globalization, is under-explored. Taking Hong Kong as a case study, this paper explores the changes within the emphasis of instruction and organization of local mathematics curriculum across time. I contend that the shift of instruction within the school subject is due to the institutionalization of modern individualism, the process in which the conception of “learner as an empowered and self-reflexive actor becomes culturally legitimate in the society. I employ a longitudinal content analysis to track the changes of the instructional emphasis within 42 senior primary mathematics textbooks published in Hong Kong since World War II. It is revealed that despite the unique cultural and historical trajectory of educational development in Hong Kong, the curriculum embraces the ideas of participatory pedagogical approach, interest stimulation, and social relevance to students. Furthermore, it is found that governmental officials, teachers, local pressure groups and academic professionals draw scripts and models from the discourse of student centrism to drive curricular reforms. Results suggest that world polity theory is more applicable than the historicist perspective in accounting for the changes of the curriculum content. / Detailed summary in vernacular field only. / Yip, Chun Ting Patrick. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 60-66). / Abstracts also in Chinese. / Abstract --- p.iii / Abstract (Chinese) --- p.iv / Acknowledgement --- p.v / Table of contents --- p.vi / List of tables, figures and charts --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature review --- p.7 / Chapter 2.1 --- Endogenous explanation I: cultural essentialism --- p.7 / Chapter 2.2 --- Endogenous explanation II: historicist perspective --- p.8 / Chapter 2.3 --- Exogenous explanation: world polity theory --- p.9 / Chapter 3 --- Framework and hypotheses --- p.12 / Chapter 4 --- Method --- p.14 / Chapter 4.1 --- Longitudinal content analysis of textbook --- p.14 / Chapter 4.2 --- Textbook data --- p.15 / Chapter 4.3 --- Measures for trend analysis --- p.16 / Chapter 4.4 --- Textual analysis --- p.18 / Chapter 4.5 --- Secondary analysis of the process of curriculum development --- p.18 / Chapter 5 --- Result --- p.21 / Chapter 5.1 --- Trend analysis --- p.21 / Chapter 5.2 --- Textual analysis --- p.28 / Chapter 5.3 --- Explaining the rise of student centrism in Hong Kong --- p.38 / Chapter 6 --- Conclusion --- p.52 / Chapter Appendix A. --- Textbooks sampled --- p.56 / Bibliography --- p.60

published_or_final_version / Education / Master / Master of Education

This study compares formative assessment practices in primary mathematics classrooms in Guangzhou, Hong Kong and Melbourne. Nine schools (three in each research location) were studied to examine the similarities in and differences between formative assessment practices for mathematics, and to identity underlying factors that may account for these similarities and differences. Videotaped classroom observations and face-to-face semi-structured teacher interview were the main data collection methods employed. The study identifies several similarities and differences in formative assessment practices by reviewing a total of 1140minutes of videotaped classroom observations (380 minutes from each city). Four similarities were noted: (a)a common structure of formative assessment practice; (b) providing feedback on what students had done well and what they needed to improve; (c) encouraging students to engage actively in the feedback process; and, (d) infrequent use of practical assessments. In addition, six differences were found: (a) interpreting, judging and suggesting on students’ work; (b) forms of assessment (including self and peer assessment); (c) assessment items; (d) effective feedback;(e) specific format for presenting mathematics; and, (f) ways of receiving feedback. Three key factors have been suggested to account for the similarities in and differences between formative assessment practices in primary mathematics classrooms in the three research locations: (a) cultural influences on mathematics learning and examinations; (b) assessment reform initiatives; and, (c) teachers’ conceptions about formative assessment. This study contributes to the understanding of formative assessment practices in the classrooms by proposing a theoretical framework for comparing formative assessment practices that takes into account cultural, social, school and classroom factors. Potential directions for future research are suggested, including further comparisons of mathematics formative assessment practices at other schools in Guangzhou, Hong Kong and Melbourne, and in other countries with similar cultural backgrounds. / published_or_final_version / Education / Doctoral / Doctor of Philosophy

The purpose of this study was to identify preferred learning styles and strategies of secondary school students and to examine the prevailing problems that restrict them to use their own preferences. The study was intended to highlight a number of issues that need to be revealed and addressed in the learning of mathematics. The types of preferred learning styles and strategies students need to employ in learning mathematics, the assistance students require from their teachers, the conduciveness of the design of mathematics curriculum and the challenges they might face to use their own preferred learning styles and strategies in the learning of mathematics were addressed as basic research questions. The study dealt with various elements that were related to environmental, emotional, sociological, physiological and psychological categories of learning in the identification of the types of learning styles and strategies. This study is believed to contribute a lot in addressing the problems of learning styles and strategies, provide feedback to the concerned government bodies to help them improve the teaching learning processes in secondary schools. It is also to reduce the bias or prejudice on mathematics by assisting students to use their own preferred learning styles and strategies, and contribute to further investigations to make the learning of mathematics more enjoyable, participatory and lifelong career. This study was conducted in four secondary schools in West Shoa Zone. A qualitative method that was descriptive in nature was employed in the study while the instruments of the study were questionnaires and an interview. The sample comprised of 249 (128 male and 121 female) secondary school students and 30 (25 male and 5 female) secondary school mathematics teachers selected randomly. The result has shown that students were not learning mathematics on the basis of their preferred learning styles and strategies and the teachers were practicing autocratic teaching styles. Most of the students did not prefer learning mathematics through plasma television; they required brief outlines and concrete presentations, and indicated that there is not enough time to check and recheck the answers they found for the problems. These imply that the organization of secondary school mathematics curriculum requires reform to accommodate the preferred learning styles and strategies of students. / Further Teacher Education / M. Ed. (Mathematics Education)

Piaget and Freudenthal advocated thinking levels. In the 1950's the van Hieles developed a five level model of geometric thought. Judith Land adapted the model in 1990, utilising four levels to teach the concept of functions. These four levels have been considered here in the formation of concepts of sequences and series. The origin and relevance of sequences and series have been studied and the importance of visualisation, patterning and generalisation in the instructional process investigated. A series of lessons on these topics was taught to a group of six higher grade matriculation students of mixed ability and gender. Questionnaires related to student progress through the various levels were answered, categorised, graphed and analysed. Despite the small number of students, results seem to indicate that emphasising visualisation, exploring patterns and generalisation and teaching the topics as a reinvention had made a positive contribution towards progress through the various thought levels. / Mathematics Education / M.A. (Mathematics Education)