This thesis explores the publishing practices of the mathematics education research community in England with the aim of better understanding the ways in which education research is shaped by its social, institutional and political context. This is of vital importance in debates about its goals, nature and future, particularly at a time of rapid change in the higher education sector, with changing funding patterns, a drive for research 'impact' and the association of publication with accountability through the Research Assessment Exercise and the Research Excellence Framework. Mathematics education research is explored on three levels: as a field, following Bourdieu; through its external relations with other areas of research, with institutions, government and society; and through the sense-making of individuals who are part of it. The focus on publications cuts an analytical cross-section/seam across these three levels since publication is intimately bound up in both internal and external struggles. Interviews with academics and social network analysis of publication data are brought together through an analysis of existing literature which examines the autonomy, boundaries, entry conditions and doxa of mathematics education research as a field. Semi-structured interviews with nine academics at English universities were used to reconstruct some of the narrative resources drawn on in making sense of publishing practices. These suggest that positive narratives around the value of publication to the research field itself are lacking. This finding is linked to the nature of education research as a field of study connected to professional practice, as well as to the link between publication and accountability. Exploratory social network analysis of publication data from fourteen mathematics education research journals over a ten-year period allowed a structural examination of the patterns that the ties formed by collaboration. This analysis was then linked with interview data on individual positioning within the field, suggesting the varied ways in which similar patterns of collaboration arise. Implications are drawn for mathematics education research in the UK and for the role of publication in social sciences research, particularly in a field of study connected with professional practice.

This study aims to examine how teachers' mathematical and pedagogical knowledge develop as they learn to use a multi-representational technological tool, the TI-Nspire handheld device and computer software. It is conducted as an enquiry into the learning trajectories of a group of secondary mathematics teachers as they begin to use the device with a focus on their interpretations of mathematical variance and invariance. The research is situated within an English secondary school setting and it seeks to reveal how teachers' ideas shape, and are shaped by, their use of the technology through a scrutiny of the lesson artefacts, semi-structured interviews and lesson observations. Analysis of the data reveals the importance of the idea of the 'hiccup'; that is the perturbation experienced by teachers during lessons stimulated by their use of the technology, which illuminates discontinuities within teachers' knowledge. The study concludes that the use of such a multi-representational tool can substantially change the way in which both the teachers and their students perceive the notions of variance and invariance within dynamic mathematical environments. Furthermore, the study classifies the types of perturbations that underpin this conclusion. The study also contributes to the discourse on the design of mathematical problems and their associated instrumentation schemes in which linked multiple representations offer a new environment for developing mathematical meanings. This thesis makes an original contribution to understanding what and how teachers learn about the concept of mathematical variance and invariance within a technological environment.

Ten mathematics projects have been developed recently for the secondary school in the U.K. From among these projects the SMP syllabus was chosen as most suitable for this research. This syllabus has been taught for a relatively long period, and deductive reasoning ability is one of the most vital objectives of teaching mathematics. None of the earlier research has investigated the development of this ability as influenced by methods of teaching the S.M.P. syllabus . The present research attempts to fill this gap. It is hypothesized that "an amalgamation of some methods of teaching could be more effective in developing of students' deductive reasoning ability than a particular and different amalgamation", To test this hypothesis two samples (O-level and A-level) were drawn at random from four secondary schools within SO miles radius of London. The students received a deductive reasoning test in the first term of the session 1977- 78 (pre-test). During the second term the teaching behaviour of the mathematics teachers, while teaching each sample, was observed using an observation system. The methods of teaching were identified and grouped to form amalgamations. According to the latter the students in each sample classified in groups. At the end of the third term the post-test was completed. Students' scores were statistically analysed. The results revealed that the second amalgamation (synthetic method and problem solving method) was more effective than the first one (synthetic method, problem solving method and lecture method) in developing a-level students' deductive reasoning ability. The three amalgamations employed in A-level (synthetic method and problem solving method), (synthetic method, problem solving method, lecture method) and (synthetic method, problem solving method, inductive method and lecture method) did not affect A-level students' ability. Consequently, the hypothesis was accepted in a-level, and rejected in A-level.

Initially, the research stemmed from previous work which had clarified the objectives of C.S.E. mathematics courses. A number of objectives had been classified as relating to the affective domain and, in terms of C.S.E., were not being evaluated. This study set out to identify affective behaviour and, if possible, to rectify the absence of any affective measures. The work developed in three distinct phases. The first concerned itself with recognising traits of affective behaviour in C.S.E. children. The method used employed a questionnaire technique which was evaluated by factor analysis. The varimax and promax analyses resulted in three definable affective factors. These related to (i) an attitude identified as tending to see mathematics as an algorithmic, mechanical and stereotyped subject; (ii) an attitude recognising an intuitive, open-ended and heuristic approach; (iii) an attitude representing commitment, interest and application to mathematics. The second phase involved restructuring the initial instrument and narrowing it to relate only to the three defined factors. An improved questionnaire was then used in a pilot trial consisting of four schools with 358 candidates. The evaluation of this trial produced evidence on the affective behaviour of the children involved and also information on the acceptability of the instrument. Before proceeding further, a revalidation of the content of the test was undertaken. The third and final phase consisted of a field trial involving 2690 candidates in a wide variety of schools. The outcome of the results has contributed to two major areas; namely knowledge concerning individual children's affective behaviour and information concerning attitudes to fields of study and content of courses. In terms of the qualities involved, and the effect of courses upon them, the research provides some very challenging questions to mathematicians. The individual pupil profiles which were developed within the study and which combine affective and cognitive behaviour, should be useful both to teacher and employer if taken in conjunction with the other information normally available.

Teacher assessment and formative assessment have a crucial place in the teaching and learning process. Their value especially was increased due to the changing perspectives of educational assessment in the last decade. This study therefore, has been designed to identify, examine and compare secondary mathematics teachers' assessment practices and formative assessment in Turkey and England. For this study multiple case studies, approached under the qualitative paradigm were chosen as the main approach. Semi-structured interviewing and non-participant observation (being a complete observer) were adopted as the methods of data collection. The study was carried out with 12 mathematics teachers of 11-14 year olds in England and Turkey. The findings show that questioning played a crucial role in teachers' assessment processes during interactions between the teachers and the pupils. The teachers collected information about their pupils by observing them, while pupils are practicing their skills and using their knowledge, and examining pupils' products as a part of their teaching. Formative assessment appeared as generally correcting errors and an approving type of feedback. The results of the study also show that both English and Turkish teachers used almost the same assessment techniques. However, how they used these techniques and what they looked for was different between Turkish and English teachers. It seems that these differences came from the different teaching styles of teachers as well as different teaching contexts. Differences in assessment practices and formative assessment appeared not only between teachers from different countries, but also among teachers from the same country. According to findings, a model of teachers' assessment practices was devised. Some issues were found with Turkish teachers assessment practices because of the influence of assessment policy and lack of use of assessment methods.

When studying for a mathematics degree, it has been shown students have great difficulty working with proof (Moore, 1994). Yet, to date, there has been surprisingly little research into how we could improve the way students study mathematical proofs. Furthermore, there is relatively less research on students' proof comprehension skills when compared with that of their proof construction skills (Ramos and Inglis, 2009). The aim of this thesis was therefore to build upon the existing proof comprehension literature to determine methods of improving undergraduate proof comprehension. Previously, text based manipulations (e.g. Leron, 1985; Rowland, 2001; Alcock, 2009a) have been tested as a way of improving proof comprehension but these have often not been as successful as we would have liked. However, an alternative method, called self-explanation training, has been shown to be successful at improving comprehension of texts in other fields (Chi et al., 1989; Wong et al., 2002; Rittle-Johnson, 2006; Ainsworth and Burcham, 2007). This thesis reports three studies that investigate the effects of self-explanation training on proof comprehension. The first study confirmed the findings of previous self-explanation training research in other fields. Students in the study who received the self-explanation training showed a significantly greater understanding of the proof text compared to that of a control group. Study 2 used eye-tracking analysis to show that self-explanation training actually changed the way students in the study read proofs; they concentrated harder on the proof (as measured by mean fixation durations), and made more between-line transitions. The final study revealed that self-explanation training can be implemented into a genuine pedagogical setting with relative ease and also showed the positive effects on proof comprehension last for a longer term of three weeks. From the findings of the research reported in this thesis it can be concluded that many students who participated in these studies appeared to have the knowledge required to understand proofs, it is perhaps they just needed some guidance on how to apply their knowledge. Self-explanation training appears to do this as it significantly improved proof comprehension in the short-term as well as offering longer-term benefits. More research will be needed to confirm these findings, given that the studies here involved participants from only one UK university on what would be considered as typical mathematics degree courses for the UK. However, these findings are promising and provide the foundation for improvements in undergraduate proof comprehension.

This study explores mathematics teachers' classroom practices in Turkey and is centrally informed by socio-cultural theories. The research examines mathematics teachers,instructional practices in relation to the wider institutional context in which teaching practices are situated. The study takes a naturalistic approach, with minimum prior assumptions on the way in which teachers' classroom practices are examined. The structure of the examination of practices is grounded in the data itself. A multiple-case study methodology was used for this purpose. The main data included observation of four mathematics teachers' lessons from different institutional backgrounds (two state school and two private preparatory courses). I video recorded teachers' lessons while they were teaching the topic 'functions' over a period of one month, a total of 52 lessons. Other sources of data included semi-structured interviews and a questionnaire administered to 87 teachers. The findings from the analysis of interviews suggested that all of the teachers described their lessons in the same manner and I conclude that all four teachers' instructional practices contain two main elements: 'content', where the theory of the mathematical knowledge to be taught is presented; and 'example solving', where the theoretical knowledge presented was essentially put into practice. Analysis of the video data suggests different patterns of practices in the teachers of different institutions. My attempt to make sense of these differences revealed an emergent theme that I pursued: that the institutional context influences teachers' practices more than I expected and more than is reported in the mathematics education literature. The findings reveal associations between specific instructional materials, teaching practices and institutions. The analysis of data also shows that the institutional context influences teachers' practices to an extent that teachers subordinate their own views regarding their teaching practices, i. e. teachers adopt teaching practices the institution they are working in promotes, even though they believe that these are not the most appropriate teaching practices to facilitate student understanding of mathematics. On the basis of these findings: I argue that individual differences in teachers' practices may be reduced by the institutions, depending on the degree of influence of the institution concerned; I argue that institutions influence mathematics teachers' professional development; I introduce the construct 'contextual density' to describe the varying degrees of influence of institutions on teachers.

Understanding mathematics teacher knowledge is an international endeavour, seen by researchers as a key part of improving pupil learning. Within the last few decades, several conceptions of teacher knowledge have been proposed within the literature including Mathematical Knowledge for Teaching (Ball and colleagues) and the Knowledge Quartet (Rowland and colleagues). However, multiple criticisms of these conceptions exist, prompting the introduction of a new approach to considering teacher knowledge within this thesis. Rather than seeking to categorise a knowledge unique to teaching different than the mathematical knowledge required for other professions, this research aims to examine how knowledge changes within the context of trainee secondary teachers in England. The poor mathematics results of school leavers in the UK as well as a shortage of mathematics teachers, has influenced government policies on teacher training. Bursaries differentiated by degree class and the introduction of government-sponsored ‘subject knowledge enhancement’ (SKE) courses (to graduates from numerate disciplines) attempt to increase the quality and supply of teachers. By examining how knowledge changes over a teacher training course, with emphasis on the divide between SKE and non-SKE course participants, it is proposed that further insights into the knowledge useful for teaching and how this knowledge needs to be organised can be gleaned. This mixed methods study employs questionnaires, interviews and observations to track the knowledge change of a sample of Postgraduate Certificate in Education (PGCE) students over their year-long course. Results of the current study suggest that changes in the quality rather than quantity of knowledge take place over a PGCE course, in other words, a change in the organisation of knowledge. In addressing the research questions, this study also: raises questions about what the Mathematical Knowledge for Teaching items measure; suggests potential changes to the Knowledge Quartet codes; evaluates the proposed alternative approach to knowledge; and, discusses implications for teacher training policy.

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The aim of the research is to rethink how students of mathematics learn, by employing the ‘late’ Husserl’s theory of knowledge. To do this the mathematical investigations of three prospective teachers of mathematics are explored, focusing on ‘objectification’ and ‘intuition’, thinking about how students acquire new knowledge in order to resolve mathematical tasks; in particular how students come to grasp mathematical objects in order to conceptualise the tasks. For this purpose I employed Husserl’s phenomenological attitude and his methods of reduction and bracketing, informed by Merleau-Ponty’s extension of the Husserlian frame. The course from where the data were collected included thirteen students, and the teacher facilitated my phenomenological perspective, by deliberately suspending guidance and any other intervention apart from introducing the operational aspect of the tasks. By using the Husserlian theory and methodology and by following the teacher’s non-intervention strategy I managed to track the ‘moments’ of objectification and the critical role of intuitions—in the Husserlian sense—in the process of objectification. The embodied, founding powers of the living body (the body-subject) and the pre-reflective and reflective intentional forces manifested their significance for the students’ objectification processes. Most importantly, intuitions appeared as the critical acts that enabled each objectification to take place. In summary, the main findings of the research and the related contributions of the study to knowledge are the following: Intuitions are critical objectifying acts, preparing as well as constituting mathematical objects. Three genetic features of intuitions are identified, thus allowing their tracing as such. Empirical and abstract intuitions were shown as essentially interrelated, and the description of the transition from empirical to abstract knowledge through according objects was exemplified in a number of cases. The general structure of the (Husserlian) abstract intuitions was clarified, leading to suggestions for teachers to introduce abstract objects in accordance to the aforementioned structure. A novel phenomenological gaze on the mathematical learning experience is introduced, one that transforms ready-made mathematical objects to objectified lived experience. The contribution to knowledge suggested by this gaze is that it takes into account the complexity and richness of the learners’ lived worlds, that it has the potential to reorient teaching practice into a student-oriented inclusive praxis, and finally, to enable the researcher to cash in Husserl’s theoretical and methodological reflections as a “working philosophy”.