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The competence of mathematics teachers and how to prepare competent future mathematics teachers have been hotly debated in recent years; pedagogical content knowledge (PCK) is a critical indictor of that competency (e.g., Ball & Bass, 2000; Ferrini-Mundy & Findell, 2010). This explorative study examines PCK and PCK change and the factors contributing to both among a group of secondary school mathematics student teachers in Hangzhou (the capital of and largest city in Zhejiang Province, China). Changes in PCK are investigated across the final two years of a pre-service secondary mathematics teacher education program. This program is traditional in nature, mainly consisting of mathematics teaching methods courses, teaching practica and advanced mathematics courses. Student teachers’ performance in three aspects of PCK — the substance of PCK, approaches to PCK and the structure of PCK — were assessed using a combination of quantitative and qualitative measures employed at two distinct stages of the program. At each stage, student teachers’ PCK was examined by a PCK questionnaire, a follow-up interview and three video-based interviews. The factors influencing PCK change were investigated using multiple phases and approaches of data collections. Specifically, rating schemes for each aspect of PCK were developed to evaluate student teachers’ responses and track the changes in their PCK. Interviews were conducted with student teachers at various stages of their professional growth to determine what they considered to be important factors affecting their PCK and changes to their PCK. In addition, observations of student teachers’ teaching practice during their teaching practica, together with interviews involving course instructors, mentor teachers and university teachers were employed to collect supplementary evidence on the impact of those factors. A quantitative analysis of the PCK questionnaire indicated that the participating student teachers generally did not perform well in PCK items in either stage. The follow-up interviews suggest that the different logic applied by the student teachers when responding to those items, their lack of sensitivity to contextual information, and their misunderstanding of terminology and incorrect assumptions all affected their performance. An additional qualitative analysis, based on three video-based interviews, indicated that student teachers’ overall performances in the three aspects of PCK improved in the second stage. Insights were gained into the major types of changes in PCK through paired responses. These changes were found to be influenced by changes in the student teachers’ knowledge of curriculum, of good examples/tasks/exercises, of clear lesson and teaching goals and of some affiliated affective factors. Other factors, including individual and social contextual factors, prior learning and tutoring experience, practicum experience and preparations for examinations and teaching competitions, are also examined for their direct or indirect impact on PCK. This study may contribute to current literature on the characteristics of Chinese student teachers’ PCK and PCK changes during the final two years of their pre-service teacher education. It provides a tentative explanation of how institutional and social contextual factors affect PCK and PCK change in different ways. Methodological and practical implications are also discussed. / published_or_final_version / Education / Doctoral / Doctor of Philosophy

There is no abstract available for this dissertation.

This research study has investigated the beliefs that prospective primary school teachers hold about the epistemology of mathematics, and the teaching and learning of mathematics. In particular, it considered the following questions: * What beliefs and attitudes about mathematics and mathematics education do first year primary school student teachers bring into their tertiary education? * Are any of the students' beliefs about mathematics and mathematics education similar to the beliefs of the teacher educators in mathematics education and how do students interact with first year mathematics education subjects in the teacher education course? * How do students' attitudes and beliefs influence their success in learning new mathematics at this stage of their lives? * How do students' beliefs and attitudes affect their ideas on good practice in the teaching of mathematics in the primary school? The research design was qualitative, using a case study investigation of 50 students in their first year of a teacher education course. The students' passage through the first year mathematics education subjects provided valuable insights into their beliefs, principally by means of interviews and open-ended questionnaires. The study was designed to have pedagogical outcomes for the students, by embedding the collection and interpretation of data in the teaching and learning of their course. My personal perspective throughout this research has been that mathematics is a socio¬cultural phenomenon, and that the learning of mathematics is achieved through the mediation of language, social interaction and culture. This perspective of mathematics and the learning of mathematics has influenced the choice of methodology and the research questions asked. Results indicated that students often held two or more philosophies of mathematics and moved between these philosophies, depending on context. Further, students generally considered that the characteristics of a good teacher included being supportive and enthusiastic. Good pedagogy was believed to incorporate practical activities demonstrating relevance, and providing 'fun' for pupils. However, an alarming result was that having higher order knowledge about mathematics was often seen by the students as being a disadvantage for a teacher, principally because students believed such teachers would be less empathetic to struggling pupils. These beliefs affected students' interactions with the first year university mathematics education subjects, as their beliefs about the importance of subject matter knowledge were at variance with the beliefs of the teacher educators. This dissonance led to devaluing of the mathematics education subjects by some of the students. The study has led to the conclusion that a number of the students' beliefs about mathematics, and the teaching and learning of mathematics, should not be left unchallenged. Those beliefs dealing with ideas on good pedagogy should be strengthened, while beliefs about the nature of mathematics and the value of subject matter knowledge should be made more transparent and addressed. On the other side of the coin, teacher educators need to acknowledge the differences in the beliefs that student teachers and teacher educators might hold, and to consider ways of making mathematics education courses more relevant and meaningful for students.

This research study has investigated the beliefs that prospective primary school teachers hold about the epistemology of mathematics, and the teaching and learning of mathematics. In particular, it considered the following questions: * What beliefs and attitudes about mathematics and mathematics education do first year primary school student teachers bring into their tertiary education? * Are any of the students' beliefs about mathematics and mathematics education similar to the beliefs of the teacher educators in mathematics education and how do students interact with first year mathematics education subjects in the teacher education course? * How do students' attitudes and beliefs influence their success in learning new mathematics at this stage of their lives? * How do students' beliefs and attitudes affect their ideas on good practice in the teaching of mathematics in the primary school? The research design was qualitative, using a case study investigation of 50 students in their first year of a teacher education course. The students' passage through the first year mathematics education subjects provided valuable insights into their beliefs, principally by means of interviews and open-ended questionnaires. The study was designed to have pedagogical outcomes for the students, by embedding the collection and interpretation of data in the teaching and learning of their course. My personal perspective throughout this research has been that mathematics is a socio¬cultural phenomenon, and that the learning of mathematics is achieved through the mediation of language, social interaction and culture. This perspective of mathematics and the learning of mathematics has influenced the choice of methodology and the research questions asked. Results indicated that students often held two or more philosophies of mathematics and moved between these philosophies, depending on context. Further, students generally considered that the characteristics of a good teacher included being supportive and enthusiastic. Good pedagogy was believed to incorporate practical activities demonstrating relevance, and providing 'fun' for pupils. However, an alarming result was that having higher order knowledge about mathematics was often seen by the students as being a disadvantage for a teacher, principally because students believed such teachers would be less empathetic to struggling pupils. These beliefs affected students' interactions with the first year university mathematics education subjects, as their beliefs about the importance of subject matter knowledge were at variance with the beliefs of the teacher educators. This dissonance led to devaluing of the mathematics education subjects by some of the students. The study has led to the conclusion that a number of the students' beliefs about mathematics, and the teaching and learning of mathematics, should not be left unchallenged. Those beliefs dealing with ideas on good pedagogy should be strengthened, while beliefs about the nature of mathematics and the value of subject matter knowledge should be made more transparent and addressed. On the other side of the coin, teacher educators need to acknowledge the differences in the beliefs that student teachers and teacher educators might hold, and to consider ways of making mathematics education courses more relevant and meaningful for students.

First Year College of Education students who have done and/or passed mathematics at matric level, often lack adequate understanding of basic mathematical concepts and principles. This is due to the fact that formal tests and examinations often fail to assess understanding at anything but a basic level. It is against this background that this study uses alternative and more direct means of assessing the level and nature of the understanding such students have of aspects of basic arithmetic and number theory. More specifically, the goals of the study are: 1. To determine the students' levels of understanding of the following number concepts: Rational numbers; Irrational numbers Real numbers and Imaginary numbers. 2. To determine whether the students understand the rules governing operations with negative numbers and with zero as principles rather than conventions. 3. To determine whether the students understand the rule governing the order of operations as a matter of convention rather than as a matter of principle. A survey of the literature concerning the nature of understanding as well as the nature of assessment is given. The students' understanding in the above areas was assessed by means of a written test followid by interviews. A sample of 50 students participated in the study while a sub-sample of 6 were interviewed. Some of the significant findings of the study were : 1. The students largely failed to draw clear distinctions between Real and Rational numbers as well as between Irrational and Imaginary numbers. 2. Very few of the students could explain the rationale behind the rules governing the. operations with negative numbers and zero. 3. Only half of the students had any knowledge of the rule governing the order of operations. Only one student demonstrated an understanding of the rule as a convention.

The problem of this study was to analyze doctoral programs in mathematics and education for the preparation of teachers of undergraduate mathematics. The purpose of the study was to determine (1) the need for such programs, (2) the attitude of college and university officials toward them, (3) the composition of present offerings and (4) recommendations to the future course their development should take.

This study explores the nature of pre-service Mathematical Literacy teachers' problem solving with a focus on intra-mathematics and extra-mathematics connections, across two years (2011-2012). The pre-service teachers were enrolled into a new three-year Bachelor of Education course, Concepts and literacy in mathematics (CLM), at a large urban University in South Africa. The CLM course aimed specifically at developing the teachers' fundamental mathematical knowledge as well as contextual knowledge, which were believed to be key components in ML teaching. The fact that the course offered a new approach to professional teacher development in ML (pre-service), contrasting the old model (in-service) reported in ML-related literature in South Africa, where qualified teachers from other subjects were reskilled, coupled with the need to grow the pool of qualified ML teachers, provided a rationale for conducting this study. Data relating to the pre-service teachers' responses to assessment tasks within the course, and their school practicum periods focusing on classroom mathematical working, combined with pedagogical orientations, was collected. PISA's (OECD, 2010, 2013) dimensions of the mathematisation process provided the theoretical framework while Graven and Venkat's (2007a) pedagogic agendas were used to make sense of the pedagogic orientations in practice. The results relating to both learning and practice suggest that the teachers' knowledge relating to model formulation, an aspect of extramathematics connections, was weak across the two years. Nevertheless, improvements in ways in which the dimensions ofthe mathematisation process occurred were noted across the two years, with localised errors. In terms of pedagogic agendas foregrounded by the teachers in ML classrooms, results indicate that agenda 2 (content and context driven) and agenda 3 (mainly content driven) featured more than agenda 1 (context driven) which supports the rhetoric in the ML curriculum. Two implications to teacher training have been noted; first the need for a focus on correctly translating quantities from problem situations into mathematical models, and secondly, the need for promotion of provision of solution procedures with pedagogic links. This study offers two key contributions namely; extending knowledge relating to pre-service ML teacher training, and extending theory for understanding steps in problem solving to incorporate aspects of pedagogy.

This study attempted to explore a possible relationship between diagrammatic reasoning and geometric knowledge of pre-service mathematics teachers. Diagrammatic reasoning skills, as a sequence of steps from visualization, to interpretation, to formalisms, are at the core of teachers' content knowledge for teaching. However, there is no course in the mathematics curriculum that systematically develops diagrammatic reasoning skills, except Geometry. In the course of this study, a group of volunteers in the last semester of their teacher preparation program were presented with "visual proofs" of certain theorems from high school mathematics curriculum and asked to prove/explain these theorems by reasoning from the diagrams. The results of the interviews were analyzed with respect to the participants' attained van Hiele levels. The study found that participants who attained higher van Hiele levels were more skilled at recognizing visual theorems and "proving" them. Moreover, the study found a correspondence between participants' diagrammatic reasoning skills and certain behaviors attributed to van Hiele levels. However, the van Hiele levels attained by the participants were consistently higher than their diagrammatic reasoning skills would indicate.

This study attempted to explore a possible relationship between diagrammatic reasoning and geometric knowledge of pre-service mathematics teachers. Diagrammatic reasoning skills, as a sequence of steps from visualization, to interpretation, to formalisms, are at the core of teachers"™ content knowledge for teaching. However, there is no course in the mathematics curriculum that systematically develops diagrammatic reasoning skills, except Geometry. In the course of this study, a group of volunteers in the last semester of their teacher preparation program were presented with "visual proofs" of certain theorems from high school mathematics curriculum and asked to prove/explain these theorems by reasoning from the diagrams. The results of the interviews were analyzed with respect to the participants"™ attained van Hiele levels. The study found that participants who attained higher van Hiele levels were more skilled at recognizing visual theorems and "proving" them. Moreover, the study found a correspondence between participants"™ diagrammatic reasoning skills and certain behaviors attributed to van Hiele levels. However, the van Hiele levels attained by the participants were consistently higher than their diagrammatic reasoning skills would indicate.