The nature and experience of participation in school mathematics classrooms is considered through the analytical frames of community of practice theory (Lave and Wenger 1991; Wenger 1998) and the lifeworld by focussing on interactions generated by verbal teacher questioning of the whole class. The thesis reports on an explorative, personal inquiry that complements theoretical reflection with research material generated through interviews with learners of mathematics and by participant observation in school mathematics classes. The methodology is qualitative, hermeneneutical and engaged, and influenced by the principles of participative action research and co-operative inquiry in the context of post-modernist thought. The concept of usual school mathematics is developed to describe dominant social practices in mathematics in schools. Through an analysis of teacher questioning and the learners' experiences of it, the meaning of participation is problematised. The nature of marginal rather than legitimate peripheral participation in usual school mathematics classrooms shows that they are not communities of practice and are better described as regimes of practice. 'Ecologies of practices' is proposed as more flexible construct that allows the diversity of networks and practice in classroomsto be theorised. The nature of particular ecologies can be described(and researched) using similar dimensions to community of practice theory and this is illustrated by a case study which contrasts usual school mathematics practices with those that foster greater engagement by participants. The concept of the lifeworld supports understanding of the different experience of participants in ecologies of practice of mathematics and teacher questioning. The entities in mathematical lifeworld of learners of mathematics are not just simpler versions or a smaller subset of those in teachers' lifeworld but are existentially and ontologically different and this requires investigation. A case study of a mathematical lifeworld shows school mathematics can be deeply existentially alienating and marginalizing for learners. Students' views on ways of increasing engagement in questioning practices are presented and the implications of the research for engaged, transformative and democratic classroom practice with regard to teacher questioning is considered. Key conclusions of the thesis are: The analysis of the participation of participants in usual school mathematics practices as marginal demands that such practices be questioned; An ecological perspective allows the complexity of the variety of forms of participation, enterprises and engagement in classrooms to be analysed; The concept of the lifeworld supports an understanding of the diversity of meaning that arises for participants in the same ecologies; There are questioning practices that foster fuller participation by learners.

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The need for fostering students' higher-order thinking as one of the twenty-first century mathematics education goals was the motivation for this study. Additionally, mathematics (especially geometry) teaching has gone through continuous changes incorporating the use of technology since the early 80s. The arrival of dynamic geometry software (DOS) like Geometer's Sketchpad (GSP) provides an incredible dynamic tool for exploring geometry. Thus, this study sought to investigate whether and how students' higher-order thinking (cognitive skills of applying, analyzing, evaluating and creating) can be supported by using DGS in learning mathematics (for the topic of loci). The methodology of the research consisted of a pretest-posttest with nonequivalent groups (consisting of an experimental group and a control group) quasi-experimental design. Furthermore, both quantitative and qualitative approaches and measures were used in the research for answering two research questions. A class of Form Five students from a secondary girl school in Hong Kong was selected for data collection. Half the class formed the experimental group (N = 20), engaging in learning using DGS, while the other half of the class formed the control group (N = 20), engaging in learning in a more traditional teaching approach. Quantitative results indicated that students in the experimental group achieved numerically, but not significantly, higher than students in the control group. However, qualitative results showed that using DOS for learning geometry provides a learning environment that supported students to enhance their higher-order thinking skills. Furthermore, a common underlying pattern consisting of two different possible problem solving processes (Making conjectures-Validating conjectures-Proving and Exploring- Making and Validating conjectures-Proving process) was developed. Besides, which GSP-specific features that tapped higher-order thinking of the students during the processes was also identified. Finally, implications for professional practitioners and further research are discussed at the end of the thesis.

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Despite mathematicians valuing the ability to visualise a problem and psychologists finding positive correlations of visual-spatial ability with success in mathematics, many educationists remain unconvinced about the benefits of visualisation for mathematical understanding. This study compared a „visual‟ to a „verbal‟ teaching approach by teaching a range of early secondary school mathematics topics to two classes using one or other approach. The two classes were compared by considering their scores on a post-intervention test of mathematical competency, on which the verbally taught class scored significantly higher. A major interest of the research was individual differences in underlying abilities or preferred learning styles, seen as underpinned by visual-spatial and verbal cognitive processes. A test was developed to measure participants‟ general tendency to process information visually or verbally and the mathematics test results were also considered from the perspective of cognitive style. No interactions were found between teaching style and the learners‟ preferred styles. The pupils identified as „visualisers‟ did tend to perform more poorly on the mathematics test. However, further examination of the classroom performance and approaches taken to mathematics by these and other students led to doubt about the validity of the visualiser-verbaliser test used and indeed about the underlying constructs of visualiser and verbaliser cognitive styles.

It is widely recognised that generality is at the heart of the learning and teaching of mathematics. Motivated by a desire to understand what it is about generality which presents such an obstacle for so many students, this study examines the variety and complexity of ways in which generality is expressed in mathematics classrooms. Systematic reflection on my own experience of teaching over a year revealed a wide range of types of generalisation taking place in mathematics classrooms. The main study then analyses transcripts of fifty-two lessons taught by six teachers teaching at least four hundred students, sampled over a period of two months. The focus is on 'ordinary' lessons where expression of generality is not the main objective. Infonned by the literature, observation notes and student work, a framework is developed with five categories used to distinguish between types of generalisations, which emerge from the transcribed data . These categories are: the object of generalisation, its presumed longevity of relevance, its justification, its origin and the awareness being promoted. Having established the Ubiquitous richness and complexity of expression of generality in mathematics classrooms, the study looks in closer detail at the expression of generality pertinent to mathematical procedures and to mathematical concepts. The study uses the framework, and draws on second language education literature, to re-examine the fifty-two main study lessons. This analysis highlights the complexity of expressing generality through natural language, and suggests that natural language exhibits many of the pitfalls and ambiguities of algebraic expression. Further, it suggests that algebraic notation might offer a clearer means of expressing generality in many cases. The framework developed for considering characteristics of expressions of generality is then applied to the researcher's own classroom, demonstrating how awareness of ways in which generality is expressed can inform pedagogic choices as well as provide a structure for reflection on practice.

This study investigates 14 year-old students' development of graphical concepts using graphing calculators. Two learning models based on two broad interpretations of Vygotsky's "zone of proximal development" were implemented to gauge the role of graphing calculators in technology-based learning. Epistemological case studies were used to ascertain the extent to which the graphing calculator facilitated the learning of key graphical concepts. To this end, students of different levels of mathematical attainment were observed to determine the different kinds of understanding they derived from using the technology. The 24 students participating in the study were pre and post tested, and formed into two groups. One group was taught according to a structured, teacher-led learning model, and the other group was taught according to an open-ended, activity-led learning model. What emerges from the study is the complexity of the teaching and learning situation when technology is incorporated. A student's learning of graphical concepts with the graphing calculator was the result of an interplay between his/her knowledge of the functionality of the graphing calculator, existing mathematical knowledge and the nature of teacher intervention. The use of the graphing calculator raises the issues of the ordering or sequencing of learning of graphs from simple linear equations to those perceived as more complex polynomials. With the graphing calculator, students were able to learn much more than was thought possible. Changes in students' mathematical learning were accompanied by a change in the role of the calculator from a static display tool to a mediational tool. The study also highlights the issues of teachers' roles when technology is incorporated including teacher's content knowledge, and the ways in which teachers intervene with students, in particular how teachers deal with students' semantic and syntactic errors in using the calculator.

The primary aim of this researchis to investigate students' use of graphical calculators for high school mathematics. I see appropriation of the technology to be central to this and therefore I discuss the term appropriation and outline the definition of appropriation I will adhere to. In particular I followed six students through the academic year September 2003 to July 2004 with a view to establishing how, why and when they used their graphical calculators and what benefits they gained from its use. I selected the two schools from where the students came and the students volunteered to take part in my project. My research is broadly socio-cultural as I collected data not only from the students but also about the context in which the students learn. I used a case study approach, focussing on a small number of cases -a case being a student-with-a-GC-in-school. Overall I adopted a naturalistic paradigm for my study and collected qualitative data about the 'natural setting' – the classrooms and schools - and made every attempt to minimise the disruption to the students during their daily routines. The data was collected through a variety of methods- interviews, observations journals and key-stroke data from the students' graphical calculators. The key-stroke data are central to my work. The key-stroke capture software used provides an exact record of a student's use of the graphical calculator. This method of collecting data is not widely used or known and I have dedicated a chapter to outline its main features and make a critical analysis of it as a data collection tool. I see appropriation as a central issue to students using a graphical calculator and as such I reflect on the evidence with this at the forefront. I report on what are the signs that a student has appropriated their graphical calculator and what are the barriers to appropriation. I found that the six students appropriated their GC to varying degrees. The extent of their appropriation was influenced by a variety of factors including the tension between the old tool and the new tool, the teacher, the institution, the curriculum and personal aspirations. I examine these factors in detail and examine the stages of appropriation of each student.