In recent times a positive relationship between mathematical achievement and the ability to distinguish between fingers has been observed and training to develop finger acuity has been shown to lead to gains in mathematical skills. An association between mathematical achievement and non-symbolic number magnitude comparison has also been established. This mixed methods two phase study therefore addressed whether an intervention programme designed to increase connections between the symbolic and non-symbolic representations of number using fingers as a tool could help students construct a deeper understanding of number and thereby increase their scores on mathematical tests. The phase one studies trialled the intervention programme with classes, groups and one individual who were experiencing difficulties in learning mathematics for a variety of different reasons in order to see if improvements in their mathematical skills could be observed irrespective of these reasons. Phase two was designed to evaluate the effectiveness of using the combination of fingers and games in the intervention activities in comparison to using either part separately or no intervention at all. Results demonstrated that the pupils involved in the intervention demonstrated gains in mathematical achievement greater than those who did not take prut regardless of the reasons for their initial difficulties with mathematics. Results also confirmed that the full intervention groups made significantly more increases in mathematics achievement tests than those who experienced part or no intervention. This suggests that the combined intervention of finger gnosis training and mathematical games with the visual representation of fingers and dot patterns acting as mediators could help children to make connections between symbolic and non-symbolic representations of number and thereby raise their mathematical achievements.

This study explores women secondary mathematics teachers' experiences of professional learning. Life histories were elicited through semi-structured interviews in the form of guided conversations, supplemented by time-lines of mathematics and of professional learning. Analysis focused on constructed personal experience narratives. Although research demonstrates the features of effective professional learning, teachers' experiences of learning throughout their careers remains under explored. A particular focus in this study is on the ways in which professional learning is supported, providing opportunities for reengagement with mathematics, a subject frequently viewed as inaccessible and masculine. The women's stories are peopled with significant others who provide both models and encouragement, frequently drawn from their own school days and early professional experience. Much professional learning is informal, arising from unstructured reflection on teaching, with teachers accorded neither agency nor consistent support for their learning. The women's narratives provide a perspective on lived experiences of professional learning. Frequently learning is unsupported and spaces to discuss mathematics learning and teaching limited. Teachers appear isolated in restrictive school environments which contribute to a perception of reduced agency. Where opportunities for collaborative professional learning exist, women participate actively in the wider mathematics education community. Analysis of the narratives suggests that teachers' agency over their professional learning needs to increase to create spaces for women to collaborate on mathematics focussed professional learning. The allocation of resources to teacher professional learning should be prioritised. These glimpses reveal the restricted landscape of women mathematics teachers'learning opportunities. Despite these restrictions, however, teachers push at the boundaries. The narratives will support teachers who may draw on the voices of others to help them to make sense of their own development. Further research is warranted to explore the way individuals might develop and utilise their own professional learning narratives.

This thesis describes the analysis of the implementation of a professional development programme for secondary mathematics teachers in England. The research used a mixed methods multiple case study design with three secondary schools. The aim of the study was to understand mathematics teachers' professional learning in the context of this professional development programme. However, through analytic generalisation, i.e. generalising to theory, these findings may have broader application to understanding teachers' professional learning. Social learning theory was used as a framework for explaining professional learning, within this are two components, observational learning and self-efficacy. Teachers learn to teach through observing behaviours and models of teaching; they implement the approaches that they are confident will be effective in their classroom - that they are self-efficacious about. I show how this explains the prevalence of traditional teacher-centred teaching in secondary mathematics and how, through observing models of alternate approaches in PD, and through developing self-efficacy in that approach, teachers can implement new approaches in their teaching. In this research, I show that the PD designed to support teachers in teaching to develop students' problem solving skills had an effect on teachers' practices: their teaching became more student-centred. It also had a positive effect on teacher self-efficacy in the suggested approach. In the qualitative analysis of multiple individual cases, I explore how observational learning processes work, in the context of the PD, and the mechanisms by which teacher self-efficacy is developed. However, a contextual analysis demonstrates that the extent to which the ideas in the PD are implemented and sustained are influenced by context at a national level and within the school. High-stakes accountability and lack of integration of PD initiatives into school strategic plans lead to PD efforts not being sustained. The main contribution of this thesis is in bringing a new theoretical approach to the field of mathematics teachers' professional development and professional learning, that of social learning theory: one that has the potential to improve the design and evaluation of professional development and teacher education in the future.

Number pattern generalisation is often regarded a difficult topic for students to learn. To explore this perception, the present study undertakes an empirical investigation with the main aim of providing a comprehensive description of how 14-year-old secondary school students in Singapore generalise figural patterns and justify their generalisations when varying the formats of pattern display and the types of function. Comprising two interrelated parts, the study first examines 515 students’ strategies and justifications and probes systematically the influence of the formats of pattern display and the types of function on their generalisations through a specially developed paper-and-pencil test. The other part, through a specially designed questionnaire, looks at their beliefs about which strategy would best help them to derive the rule for predicting any term of a figural pattern as well as their ability to construct the rule using their choice of strategy. The first part uses an independent-measures research design to examine whether different formats of pattern display have any effect on students’ rule construction and a repeatedmeasures research design to determine whether their rule construction is influenced by the different types of function. In the second part, a survey study is employed with all students asked to identify their choice of best-help generalising strategy. This is then followed by interviews with 16 of the 515 students to probe whether they are able to derive a correct functional rule using their chosen strategy. This study complements many previous studies mainly undertaken in the west in that its findings indicate that the more academic students are competent in developing a functional rule for linear patterns but falters when working with quadratic patterns. There is a widespread failure of the less academic students in both linear and quadratic patterns, confirming the oft-regarded view that expressing generality is elusive. Successful students perceive the patterns in several ways and generate wide-ranging functional rules, predominantly symbolic, to describe them. They employ a variety of generalising strategies, especially the figural type, and some of which are new in the literature. Both the test and the survey confirm that the figural strategy involving the breaking up of the whole configurations into non-overlapping parts is their clear favourite. For rule justification, verifying it using the numerical cues and drawing diagrams to explain its development are their favourite approaches. Task features such as the format of pattern display and the type of functions do contribute to student difficulties in generalisation. Based on these findings, some useful teaching strategies for teachers and teacher educators are then suggested to help them improve their teaching of pattern generalisation. The findings also point the direction for future research studies on pattern generalisation by suggesting some recommendations for researchers.

This action research project focussed on the key components of the construct mathematical resilience and how mathematical resilience can be developed in learners who are working towards their GCSE in mathematics. Split-screen lesson objectives, one related to a mathematical skill and the other related to a learning skill, were used to focus the learner’s attention onto each skill. These learning skills were chosen to encourage a particular group of learners to gain the confidence, persistence and perseverance to allow them to work inside the Growth Zone. The overall aim of this action research project was to improve the attainment of learners in their GCSE mathematics examination.

This research uses a case study approach to examine the learning experiences of mature non-specialist first year undergraduate university students studying mathematics as an ancillary subject. The challenges faced by such students taking mathematics as a subsidiary subject within their main degree have not been adequately addressed in the literature: this study seeks to address this gap. The research took place in a UK inner-city post-1992 university which has a very diverse student intake. A qualitative data set was generated from in-depth and focus group interviews of 22 mature students, the majority of whom were non-specialists taking mathematics as a required ancillary subject. An additional quantitative data set was derived from a questionnaire distributed to 250 students taking first year mathematics modules, either as an ancillary or as a specialism subject. A small number of mature students specialising in mathematics in both the interviews and the survey were included in order to compare the experiences and views of the both specialist and non-specialist groups. The Mixed Methods Research Design adopted combined results from the qualitative and quantitative analyses, and was accompanied by a post-structuralist theoretical framework which examines the discursive practices students were exposed to in relation to their construction of mathematics as a subject and their experiences of learning mathematics. The study shows that the major perceived factors that affect mature non-specialist students learning of mathematics include the pedagogical model that is used; the attitudes and beliefs of the learners; the support available to aid learning; and the prevalent discourses about the learning and perceptions of mathematics. These findings have a number of important implications for policy and practice for teaching mathematics to such students, for our understanding of student identities and for widening participation. The evidence from this study suggest that there should be a shift of government policy on access and financing for mature students; a review of mechanism of financial support for mature students; changes in the organisation and resourcing of small classes; a review of curriculum and pedagogy to fit the diverse background of learners; and the development of mathematics support provisions that are embedded in courses that require mathematical skills.

Following recent research linking executive functioning to children 's skills, this thesis explores the relationship between children's inhibition effciency and mathematical ability. This relationship was initially explored using six Stroop task variants containing verbal, numerical or pictorial stimuli. The results indicated that, in the numerical variants only, children of lower mathematical abilty possess less effcient inhibition mechanisms, compared to children of higher mathematical ability. Thus, it is proposed that low-abilty mathematicians may possess a domain-specifc problem with the inhibition of numerical information. The increased interference scores of the lowability mathematicians, however, were only evident under those conditions which also required a degree of switching between temporary strategies. A series of experiments also examined children's ability to inhibit prepotent responses and switch between strategies whilst performing mental arithmetic. The aim of these experiments was to provide a more naturalistic and appropriate exploration of the hypothesized relationship between mathematical abilty and inhibition effciency. These results also indicated that low-ability mathematicians possess fewer executive resources to cope with increased inhibition demands. A further systematic manipulation of switching and inhibition demands revealed that the low-abilty mathematicians experienced a particular difculty when both types of inhibitory demands (i.e. inhibiting a prepotent response and inhibiting an established strategy)were present. This suggests that their reduction in inhibition effciency stems from the amount of demands, rather than the type of demands placed on the executive system. Furthermore, the results indicated that inhibition effciency may be a specifc element of mathematical ability rather than an element of intellectual ability in general. The final study involved a group of low-abilty mathematicians and examined the disturbing impact of irrelevant information on their arithmetic word problem solving abilty. This study revealed that irrelevant numerical (IN) information has a more detrimental impact on performance than irrelevant verbal (IV) information. It is proposed that it is more difcult to inhibit IN information, as it appears more relevant to intentions, and thus, enters WM with a higher level of activations. In sum, the results indicate that low-abilty mathematicians have a reduced domainspecific working memory capacity, characterized by ineffcient inhibition mechanisms.

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Σκοπός της εργασίας αυτής είναι να μελετήσουμε την έννοια της γενίκευσης στη Μαθηματική Παιδεία. Αρχικά διερευνούμε την ετυμολογία και την ιστορικότητα του όρου και αναδεικνύουμε της σχέση μεταξύ γενίκευσης επαγωγής και αφαίρεσης. Αφού μελετήσουμε τη λέξη και την έννοια του φυσικού αριθμού, δύο γενετικά καθοριστικές για την ανθρώπινη νόηση γενικεύσεις, προσπαθούμε να αναδείξουμε, με βάση την υπάρχουσα βιβλιογραφία, το ρόλο της γενίκευσης στην απόκτηση γνώσης, αλλά και να εξετάσουμε την ικανότητα του ανθρώπου για γενίκευση. Έχοντας εντοπίσει το ρόλο της γενίκευσης στην ανάπτυξη επιστημονικών εννοιών και θεωριών, περνάμε στη περιοχή των Μαθηματικών και εξετάζουμε τις έννοιες του Ορισμού, του Θεωρήματος και της Ευθείας Απόδειξης, αναφορικά με την έννοια της γενίκευσης. Μέσα από τα ερευνητικά δεδομένα αναδεικνύονται τα προβλήματα που αντιμετωπίζουν οι μαθητές στην προσπάθεια τους να παράγουν κατάλληλες γενικεύσεις. Εστιάζουμε την προσοχή μας στις δυσκολίες των μαθητών αναφορικά με την κατανόηση της έννοιας της μεταβλητής αλλά και της Τέλειας Επαγωγής και παραθέτουμε διδακτικά αξιοποιήσιμα παραδείγματα. / --

This study aimed to explore teachers’ and students’ perceptions of metacognition in relation to mathematics teaching and learning in secondary schools in Saudi Arabia. This research adopted an interpretive paradigm. This meant that a socio-cultural perspective was central to examining perceptions of metacognition in relation to mathematics among secondary students and their teachers in Saudi Arabia. The use of case studies was a methodical means to achieve elaborate data and to shed light on issues facing the study. The instruments used for data collection were semi-structured interviews, group discussions and classroom observation. The participants consisted of two case study classes from secondary schools in Saudi Arabia. There were three stages of the study’s fieldwork: the pilot study and the two subsequent stages which comprise the main body of fieldwork. These last two stages were carried out in order to enable the formulation of a clearer and more complete picture of mathematics teaching and learning through metacognition in Saudi Arabia, before and after the implementation of the IMPROVE programme, regardless of improvements in specific strategy or any boost to students’ achievement. Several findings were drawn from the data, the first of these being that the traditional method can hinder mathematics teaching and learning through metacognition. Secondly, although metacognitive mathematics instruction should be planned, the strategy that is introduced should be directly targeted at improving the monitoring and regulation of students’ thought when dealing with mathematics problems. Thirdly, metacognition should be given priority to improve students’ consciousness of the learning processes. This is because conscious reflection enables students to develop an ability to choose the most appropriate strategies for learning concepts and solving mathematics problems. The findings underlined the importance of the student’s role in learning through metacognition. The study presented a perspective for dealing with metacognition along with a practice-based model of metacognitive mathematics teaching and learning. These are in the educational context of Saudi Arabia and are set out after the implementation of the IMPROVE programme. In addition, this study asserts that metacognition can be enhanced through the creation of a suitable socio-cultural context that encourages the social interaction represented through cooperative learning.