This thesis explores one group of undergraduate mathematics students’ experiences throughout their three year degree course, to gain a better understanding of why some students’ attitudes to mathematics change during this period. Research by the "Students Experiences of Undergraduate Mathematics" (SEUM) project (Wiliam, 2005) explored some of the factors influencing undergraduates. This study extends that work by investigating the experiences of another cohort, looking specifically at their approaches to learning, conceptions of mathematics, the support they encountered during their degree, and how these impacted on their attitudes. These themes were investigated throughout the students’ degree course, by taking a mixed methods approach to the research design. Questionnaire data was used to compare the cohorts’ approaches to learning, and conceptions of, mathematics, at the beginning and end of their course, and to investigate whether these factors related to the students’ examination marks. No statistically significant changes over the period were found, and contrary to previous research, no relationship was found between these factors and examination attainment. Four student case studies, combining both questionnaire and interview data, are presented to help explain these results, illustrating how contextual factors of the teaching and learning environment affected outcomes. Analysis of interview data demonstrated that the type and degree of support experienced was an important influence on these students. Dividing the analysis between the social support from peers, and the academic support of peers and staff in their department, led to insights into ways students do, or do not, integrate into the university context. This work highlights the importance of the social aspects of being an undergraduate, and of academic support in developing the students’ sense of belonging. This sense of belonging, or lack of, was a salient factor.

Pupils in mathematics lessons in England rarely have an opportunity to comment on their experiences. When they do most researchers validate their comments through classroom observations or interviews with their teachers. This study was concerned only with the views of pupils in mathematics classrooms. Eighteen pupils in a middle school in England were interviewed to establish how they perceived their mathematics lessons. Data were collected when the pupils were in Year 6 and 7 and consisted of questionnaires and semi-structured interviews. A variety of original sorting tasks were used to prompt discussions and probe the issues raised by the pupils. The issues the pupils talked about were authority, identity and community. These manifested themselves as setting, assessment and classroom organisation. The pupils designed an Ideal Mathematics Classroom which contained features that they felt were more likely to support their learning of mathematics. They believed that in the Ideal Mathematics Classroom they would be more likely to be successful learners of mathematics. Based on the comments from the pupils, the thesis contains recommendations to teachers on how to create a classroom environment that is conducive to pupils developing a positional identity as successful learners of mathematics.

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The words that students use to paint a picture of mathematics are very different from those which they use to describe their experiences in art and music. In the views of students, mathematics is pointless and repetitive while the arts are creative, relaxing and an expression of themselves. This thesis reports on the findings of a two part research project designed to investigate the attitudes of high school students when learning mathematics, art and music. The focus of this study was a comparative look at their confidence and enjoyment in learning these subjects. A questionnaire was designed and developed for use in a study of students in the United States and England (n = 1226). The intent of this questionnaire, which contained seven Likert-type questions and one open-ended response, was an in-depth look at the existence of confidence and enjoyment in learning mathematics, art and music. The results indicated that, the highest frequency of students in the mathematics group were confident in their ability in mathematics but did not enjoy learning it. This study also found, however, that there were very low percentages of students that were confident in art and music but did not enjoy learning them. Additionally there was a high frequency of students who had no confidence in art but did enjoy learning it compared to a low frequency of students in mathematics who were not confident but enjoyed learning it. To further explore these findings, repertory grid interviews were conducted on a selection of questionnaire participants from the United States (n = 42). Honey's method of content analysis was used to analyse the data. Among the differences found between students' confidence and enjoyment in learning mathematics compared to the arts were their perceptions of the routine nature of their daily lessons in mathematics versus their active, creative, personally engaging experiences while learning art and music.

This is an investigative and exploratory study focussing on better understanding the landscape of disaffection with school mathematics. A case is made that disaffection is a disabling condition that harms the capability of young people to learn mathematics and has economic, social and individual consequences. At the same time, and despite its importance, a claim is made that disaffection with school mathematics is under-researched and poorly understood. Those studies that have either focussed on disaffection, or have encountered it, have done so often from the point of view of the quantitative study of attitude. The case is made that this, whilst valuable, does not give a comprehensive and insightful enough picture of the phenomenon. Thus in this study disaffection is characterised as a primarily motivational and emotional phenomenon, and a mixed methods approach, emphasising the value of qualitative interviews, has been adopted. The study is undertaken from a constructivist and interpretative perspective, and Reversal Theory has been adopted as the main theoretical framework, used to inform the design of the study, and the interpretation of the data. Innovative methods and instruments have been devised to arrive at a ‘thick’ description of the subjective experience of disaffection. The data has been presented as a series of case studies, together with analysis and discussion across cases. Inferences and generalisations have been drawn to add further to knowledge in the field.

E-assessment in higher education mathematics is explored via a systematic review of literature and a practitioner survey, and compared with other assessment approaches in common use in higher education mathematics in the UK. E-assessment offers certain advantages over other approaches, for example question randomisation allows individualisation of assessment, but it is restricted in the range of what can be assessed due to the limitations of automated marking. A partially-automated approach is proposed in which e-assessment techniques are used to set an individualised assessment which is taken and marked by hand. This approach is implemented in a higher education mathematics module. The module uses individual coursework assignments alongside group work to attempt to account for individual contribution to learning outcomes. The partially-automated approach is used as a method for reducing the risk of plagiarism in this coursework, rather than replacing it with a written examination or e-assessment. Evaluation via blind second-marking indicates that the approach was capable of setting a reliable and valid assessment. Evaluation of student views and analysis of assessment marks leads to the conclusion that plagiarism does take place among the undergraduate cohort, was a risk during this assessment, but was not in fact a particular problem. The partially-automated approach is recommended as an appropriate addition to the repertoire of higher education mathematics assessment methods, particularly in cases where an assessment carries a high risk of plagiarism but the need for open-ended or deeper questions make an examination or automated marking system sub-optimal.

The purpose of this study is to investigate how teachers cater for students’ individual differences in the context of a reform-based Mathematics curriculum, using the topic ‘Similar triangles’. A group of six Hong Kong secondary schools in different locations, and with different banding and setting policies, took part in the study. The students were in the age-group 12 to 13 years. A naturalistic research design, without any interference from the researcher, was chosen to examine teacher behaviour. The emphasis was on observing, describing, interpreting and exploring events in the complex setting of the classroom, via a case study approach. Data were collected from the six teachers through interviews, questionnaires, and video and audio recording of five to six lessons for each teacher. There was also one focus group interview with students from each teacher’s classes. This research reports on how the methods suggested in the Curriculum Guide for catering for individual differences were implemented in the classroom. In general, the teachers involved: (1) attempted to check students’ prior knowledge, but only a small number of students was involved; (2) asked questions at different levels, but did not know about the students’ learning progress; (3) chose content which was most likely to follow the textbook; (4) were unable to vary the focus to help students to learn; and (5) could not identify what was hindering students in working out problems during seatwork. This study indicates that teachers are using their own methods to try to solve the problem of catering for student diversity, but the approaches they employ are not of a high enough quality to help students. Also, the ways in which teachers catered for individual differences in students varied considerably. This was found to depend on: the learning atmosphere; the opportunities created for student responses; variations in the scaffolding used; and the level of students’ motivation for learning. It is strongly recommended that teachers open their minds to contacts outside the classroom to refresh their teaching repertoire, and try to use some new methods which are related to the theories discussed in this research. Also, it is suggested that policy makers could build on the teachers’ experiences to enhance their ability to handle student diversity.

The topic of the thesis is proof. At Year 9 Greek students encounter proof for the first time in Algebra and Geometry. Thus the principal research question of the thesis is: How do students’ perceive proof when they first encounter it? The analysis tool in order to obtain an image of students’ perception of proof, the Harel and Sowder’s taxonomy, is itself a research question in what concerns its applicability under Greek conditions. Its applicability, of which there is strong evidence, provides the space to shape an image of students’ proof fluency, proof appreciation, proof readiness etc. In order to collect data with regard to answering the research questions in collaboration principally with the class teacher I constructed the two tests on proof that are presented in this thesis. The first test was administered to the students of Year 9 at the beginning of the school year 2010-2011 before the teaching of proof. The second was administered after the teaching of proof of the same school year. Students’ answers were analyzed and provided strong evidence that the Harel and Sowder’s taxonomy is applicable on them. Thus every answer was characterized in terms of the taxonomy. As a result every individual student but also the whole sample is depicted by proof schemes. The major findings of the analysis are the two following: • Students’ proof fluency is higher in simple proof issues. Although they face difficulties when the issues are more demanding, they show high proof appreciation. • There is strong evidence of the applicability of the Harel and Sowder’s taxonomy in a completely different socio-cultural and educational environment in comparison to that of its original invention and application. In the same vein the research proposes the mixture of proof schemes within one proof as theoretical and methodological contribution. Finally from the findings emerge new research questions as e.g. • How teaching and curriculum affect students’ proof schemes? • What is the origin of mixed proof schemes?

Abstract for a thesis submitted for the DPhil degree, Hilary term 2011. Given the central functions of reasoning in mathematics learning, knowledge of students' reasoning is crucial for mathematics educators. Yet there is little research evidence about the reasoning of students with low prior attainment. Reasoning can be understood as argumentation, or as thinking used to draw conclusions from available evidence. I address theoretical and tactical difficulties in researching both senses of students' reasoning in classroom context: the fragmented nature of interactive mathematical talk, and the multiple possible connections between what is said and underlying inferential thought. Toulmin's model of argument is adapted using pragmatics, to enable constrained inference to plausible interpretations of students' reasoning as argumentation. Analysis of novelty in each teacher-mediated task, in the constantly developing context of the knowledge and problem-solving skills made available to students, is used to identify opportunities for reasoning as thinking. This eclectic framework was used to produce a description and evaluation of reasoning from students with low prior attainment in mathematics. In a six-week focused case study of two classes of 15-16 year old students, mathematical conversation was transcribed and analysed for argumentation, using multiple data sources to support analytical inferences. Reasoning as thinking was identified from successful argumentation on novel tasks. The core Toulmin argument abstracted from students' explicit argumentation was often only partly stated, but once minimal inferred content was taken into account, core arguments were usually complete. The students worked on geometry, but few provided perceptual warrants. Most students worked with appropriate deductive warrants when deriving information about figures from theorems. Students often struggled with the calculation elements of geometrical work, so some argumentation appeared procedural. Opportunities for students' reasoning as thinking were limited, with little opportunity to generalise. When tasks did involve novelty, many students showed facility in use of deductive derivation, and there was some use of visual transformation of mathematical structure. Supplementing Toulmin argument analysis with pragmatics provides a tool for further research on incomplete and procedural argumentation, and a framework for use in professional development with teachers. Findings on students' reasoning as thinking are used to raise questions about teacher and curricular assumptions about other groups of low attaining students.

The present study aims to explore the factors underlying student-teachers' perceptions concerning 3D-descriptive geometry education in Mozambique. Elements of a grounded theory mixed methods approach were employed to gather data from a series of six focus groups' discussions, ten interviews and a questionnaire with 120 participants. The initial findings of the study show six emergent categories, comprising the Practical 3D-descriptive geometry programme, Spatial learning process, Learning descriptive geometry through 3D-based tasks, Applied 3D-descriptive geometry teaching methods, Spatial geometry teacher educators' role, and Spatial geometry-related sional studies.