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This dissertation traces the history of mathematics education in Palestine Hebrew secondary schools from the foundation of the first Hebrew secondary school in 1905 until the establishment of the State of Israel in 1948. The study draws on primary sources from archives in Israel and analyzes curricula, textbooks, student notebooks, and examinations from the first half of the 20th century as well as reviews in contemporary periodicals and secondary sources. Hebrew secondary mathematics education was developed as part of the establishment of a new nation with a new educational system and a new language. The Hebrew educational system was generated from scratch in the early 20th century; mathematical terms in Hebrew were invented at the time, the first Hebrew secondary schools were founded, and the first Hebrew mathematics textbooks were created. The newly created educational system encountered several dilemmas and obstacles: the struggle to maintain an independent yet acknowledged Hebrew educational system under the British Mandate; the difficulties of constructing the first Hebrew secondary school curriculum; the issue of graduation examinations; the fight to teach all subjects in the Hebrew language; and the struggle to teach without textbooks or sufficient Hebrew mathematical terms. This dissertation follows the path of the development of Hebrew mathematics education and the first Hebrew secondary schools in Palestine, providing insight into daily school life and the turbulent history of Hebrew mathematics education in Palestine.

The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal introduction to proof in geometry in the United States. The lens through which this study aimed to examine the intended curriculum was by identifying and reviewing the modeling of proof and deductive reasoning in the most popular and widely circulated algebra textbooks throughout the United States. Textbooks have a major impact on mathematics classrooms, playing a significant role in determining a teacher's classroom practices as well as student activities. A rubric was developed to analyze the presence of reasoning and proof in algebra textbooks, and an analysis of the coverage of various topics was performed. The findings indicate that, roughly speaking, students are only exposed to justification of mathematical claims and proof-type thinking in 38% of all sections analyzed. Furthermore, only 6% of coded sections contained an actual proof or justification that offered the same ideas or reasoning as a proof. It was found that when there was some justification or proof present, the most prevalent means of convincing the reader of the truth of a concept, theorem, or procedure was through the use of specific examples. Textbooks attempting to give a series of examples to justify or convince the reader of the truth of a concept, theorem, or procedure often fell short of offering a mathematical proof because they lacked generality and/or, in some cases, the inductive step. While many textbooks stated a general rule at some point, most only used deductive reasoning within a specific example if at all. Textbooks rarely expose students to the kinds of reasoning required by mathematical proof in that they rarely expose students to reasoning about mathematics with generality. This study found a lack of sufficient evidence of instruction or modeling of proof and reasoning in secondary school algebra textbooks. This could indicate that, overall, algebra textbooks may not fulfill the proof and reasoning guidelines set forth by the NCTM Principles and Standards and the Common Core State Standards. Thus, the enacted curriculum in mathematics classrooms may also fail to address the recommendations of these influential and policy defining organizations.

The purpose of this study was to determine how the study of number theory might affect high school students' metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. The study utilized questionnaire and/or interview responses of seven high school students from New York City and 33 high school students from Dalian, China. The questionnaire components served to measure and compare the students' metacognitive functioning, mathematical curiosity, and mathematics attitudes before and after they worked on a number theory problem set included with the questionnaire. Interviews with 13 of these students also helped to reveal any changes in their metacognitive tendencies and/or mathematics attitudes or curiosity levels after the students had worked on said number theory problems. The investigator sought to involve very motivated as well as less motivated mathematics students in the study. The participation of a large group of Chinese students enabled the investigator to obtain a diverse set of data elements, and also added an international flavor to the research. All but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics after the students had worked on the assigned number theory problems. The results of the study thus have important implications for the value of number theory coursework by high school students, with respect to the students' metacognitive processes as well as their feelings about mathematics as an academic discipline.

The purpose of this study was to investigate the knowledge base and instructional practices of two highly qualified experienced secondary mathematics teachers within the context of their classrooms during a unit in a geometry class. Data collected from interviews, classroom observations, pre and post-observation questionnaires, and detailed analyses of several lesson segments were used to create case studies for each teacher, which were compared to reveal any patterns in their instructional practices. The theoretical framework used for this study was Schoenfeld's (1998) model of teaching-in-context that included three factors that affected teachers' decisions during instruction: beliefs, goals, and their knowledge bases. The supporting questions that were investigated in this study dealt with teachers' conceptions of mathematics and teaching and learning mathematics, instructional goals, instructional strategies and curricular materials used during the unit, and any modifications made to instruction. Both teachers in this study used a more traditional lecture and discussion style of instruction that closely followed an explicit model of teaching instead of a more reform-based style of teaching. The teachers incorporated the processes of mathematics such as proof and reasoning and representation into their instruction through modeling instead of incorporating activities into instruction designed to engage students in the processes. Although both teachers were aware of and had used reform-based methods, they perceived that the traditional instructional methods were more efficient and effective. Contextual factors played a dominant role in the decisions the teachers made about their instruction. The contextual factor that had the greatest effect on instruction for these two teachers was the pressure to teach all of the topics in the required curriculum to prepare their students for the state standardized high stakes test. Other contextual factors were large class sizes, limited physical space, and limited access to technology. The results of this study indicated that although the teachers had strong content knowledge and knowledge of both traditional and reform-based pedagogy, they chose a more traditional instructional style and this decision was affected by contextual factors such as high stakes testing, a required curriculum, and the demands of their jobs. / Graduation date: 2006

published_or_final_version / Education / Master / Master of Education

The reformed Hong Kong mathematics curriculum for the 21st century consists of three components, namely generic skills, values and attitudes and, lastly, traditional cognitive development. The first two are newly emphasized and expanded. Theoretically, these components correspond closely with communication, socioculture and constructivism respectively, which are the central concepts of dialogic learning (DL). In DL, students are autonomously engaged in egalitarian dialogue, in which they share, reflect and form a learning community. Through DL, a student is expected to develop into an all-rounded and life-long learner. Contrary to the reform, dialogue is still deficient in mathematics classrooms. The role of this study is to present examples of students’ experiences in DL, found in the mathematics club of a secondary girls’ school. This study aims to explore and investigate: (1) the existence of DL in the club, (2) what the members learnt and (3) how they did it. This is an ethnographic research, which emphasizes first hand understanding, grounded theories and thorough intricacies. Therefore, I observed the students’ activities as a participant, interviewed them, and then described, analyzed and interpreted my findings accordingly. Based on my synthesis of relevant literature and the insight I gained from decades of teaching and otherwise interacting with students, I constructed a pentahedral framework to help investigate DL in a more comprehensive and intensive way. It involves the development of various generic skills and the cultivation of values and attitudes, which are usually unrecognized in examination syllabuses and the old curriculum. It consists of five facets, concerning cognitive knowledge, sharing and negotiation, learning skills, metacognition and values and attitudes. And here are the findings. All significant elements of DL from literature have been identified to exist in the club. As for what the students learnt, they recalled fruitful experiences in all five facets of the DL pentahedron. These findings were then combined with the learning histories of three subjects to yield four representative learning patterns, namely those of a ‘cognitive developer’, a ‘communicative daily life explorer’, a ‘eureka torchbearer’ and a ‘frustrated sharer-explorer’. These 4 learning patterns were further combined with (i) the purposes for mathematics study from pure examination results to ‘liberation’ and (ii) the understanding of mathematics learning from pure cognitive knowledge to inclusion of generic skills and values and attitude, to form a conceptual model of learning styles. The styles of the ‘eureka torchbearer’ and the ‘communicative daily life explorer’ were found to be exemplars of the ideals of people who favour the most liberal implementation of the curriculum reform. The ‘frustrated sharer-explorer’ was stuck with the style favoured by conservatives who are against hasty reforms. The ‘cognitive developer’ was somewhere in between. These findings may contribute to the framework of policy debate on mathematics education. In the school and classroom level, they may help teachers overcome learning disaffection and other difficulties, in both theory and practice. Organizers of extracurricular activities may also be inspired by the students’ rich experiences of dialogic learning. / published_or_final_version / Education / Doctoral / Doctor of Education

The examination-oriented assessment methods have been widely employed in Macau but the over-dependence on such methods may hinder students’ balanced development of mathematical proficiency (Morrison & Tang, 2002; Schoenfeld, 2007). Peer assessment may compensate the limitation of those methods by engaging students actively to assess. However, little research has focused on the implementation of peer assessment in Macau secondary school. This dissertation reports a study which implemented a five-step peer assessment in an eighth grade mathematics lesson in Macau based on Ploegh at al.’s (2009) and Tillema et al.’s (2011) frameworks, in which the quality criteria are taken into account for revising the procedures. 16 students participated in three action cycles and the action plan was modified to explore how the changes to the peer assessment may influence students’ learning and students’ views towards the implementation of peer assessment. The results show that it is effective to establish a formative peer assessment to promote students’ mathematical learning in Macau by adopting the frameworks. The students in general held positive attitude towards the implementation of the peer assessment. They regarded it as a fair assessment, appreciated the extra opportunity to discuss mathematics, and treated it as a way to collect more feedback on their strength and weakness. Peer assessment also served as a learning activity which helped them gain deeper understanding of mathematics. It was found that students’ involvement in the setting of the assessment criteria, making judgment and writing narrative feedback improved students’ use of mathematical language to express their ideas. Providing more opportunities to judge and discuss mathematical problems also fostered the development of their mathematical proficiency. This study also reveals that asking peers for feedback and discussion about the feedback is an efficient way to develop students’ adaptive reasoning. The students’ change of performance in the action cycles also suggests that peer assessment has the potential to help the students access higher level of development in their zone of proximal development (ZPD) and balance the role of authority in mathematics classroom. / published_or_final_version / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education