This study investigated pre-service primary mathematics teachers' teaching practice (TP) performance in the context of their subject content knowledge (SCK), pedagogical content knowledge (PCK) and attitude towards mathematics teaching (AMT). The interrelationships among these variables are analyzed too. Attitudes toward mathematics teaching and previous SCK achievement were elicited using twenty Likert-item statements and their previous highest public mathematics examination results respectively whilst information on student teachers' PCK achievement and TP performance were obtained through direct teaching observation. The sample consisted of 104 student teachers drawn from two different training programmes, 2-Year Certificate of Education (CE) Course and 4-Year Bachelor of Education (BEd) Programme, at the first stage. Hence 52 student teachers were invited to participate in the second stage, TP supervision. Differences across programmes, year groups and gender were investigated. This study found that AMT has a positive effect on TP performance and it is also positively correlated with PCK achievement. However, it is surprising that subject content knowledge (SCK) has no statistically significant relationships with AMT, PCK achievement and TP performance. There were no substantial gender-related differences on AMT and SCK achievement however, female student teachers performed better than male student teachers on tasks requiring patience on preparation of lesson plans and teaching aids, explanations of mathematical concepts and making discussion with pupils. Besides, and surprisingly, CE student teachers had better PCK than BEd student teachers and CE student teachers also improved more than BEd student teachers in AMT by comparing their Pre- and Pro-TP AMT scores. These findings have important implications for the design of teacher training programmes and teaching effectiveness in mathematics. It is recommended that similar research be replicated for secondary student teachers, full time kindergarten, primary and secondary teachers and with questionnaires written in Chinese if necessary.

This Action Research came about as a result of my concerns with the difficulties my young pupils experienced when applying their mathematical skills in other contexts. Initially my research questions focused on pupils learning and asked if problem based learning could improve thinking skills in mathematics, if collaborative learning was a more effective approach than working independently and what strategies were used by pupils when solving problems. My focus changed from examining my pupils' learning to examining my practice and as a result, four action-reflection cycles were carried out in order to bring about changes in my practice. Further key questions focused on the extent to which I used higher order questions to extend pupils' learning in mathematics and asked if increasing the number of higher order questions had an impact on their learning. I also examined the effect changing my practice had on the strategies pupils used to solve problems and the extent to which it enabled them to engage in discussions while doing so. My final research question focused in particular on generalising and the opportunities I provided for my pupils to specialise and generalise in mathematics lessons. My research found that when my very young pupils were given an appropriately challenging problem, they could engage in collaborative dialogue with the purpose of solving that problem. Evidence from this research also demonstrates how some very young pupils have the ability to make general statements about mathematics when given a rich mathematics task and challenging questions. However, there is a gap in the literature relating to how very young pupils generalise and specialise in mathematics and my research makes a contribution towards filling this gap as it describes how I enabled my pupils to get to this stage. It also contributes to the body of knowledge available on action research as it describes how my research journey travels alongside my mathematical journey. Although it may not be generalizable in the wider context it shows what can be achieved with action research .

Previous research has tended to focus on the development of separate number components (e.g. counting, addition, written numbers) and so, cannot comment on how development in one component affect development in others. The purpose of this thesis was to provide preliminary evidence towards a unifying view about the development of children's number competence, from early counting skills, at age four, to knowledge of place value, at age seven. To accomplish that aim 152 children from three different cohorts (Reception, Year 1 and Year 2) were given thirteen maths tasks, three times along one school year, assessing their understanding of four separate number components: counting and knowledge of the number-word sequence; generation of verbal number-words and the understanding of the structure of the numeration system; understanding of the arithmetical operations; and the ability to read and write numbers and understanding of the principles underlying place value. Beyond the assessment of these various number components, special emphasis was given to the separate role of each component and the developmental inter-relations amongst components in the child's development of progressively more complex ideas about number. Based on the children's performance on these tasks and the exploration of their relationships along time, it was possible to outline a preliminary proposal about children's number development. The evidence suggests that each number component plays a significant role at key times. For example, no children could develop the counting-on strategy or succeed in the arithmetical operation tasks without prior knowledge of continuation of counting. The data also showed that no development is possible without the inter-related development of several components, at other times. For example, no child could understand the structure of the decade numeration system without previous combined understanding of continuation of counting, addition and multiplication. Between 93% and 97% of the children fitted the model proposed in the various assessments. Although limited by the constraints of a correlational design, these findings suggest that the present inter-relational approach is relevant and worth further investigation through the introduction of intervention studies and the rigorous examination of causality.

The consensus in existing literature is that children's limited, and often appearance-based descriptions of shapes indicate that children view shapes as a whole and lack understanding of shape structure. This study approaches children's understandings of shapes from a different perspective, based on an alternative and more dynamic interpretation of the van Hiele model and with the acknowledgement that there might be multiple ways of knowing and expressing mathematical knowledge. This study examines the understandings young children have about the structure of shapes. It studies how this knowledge is expressed, and how it can be used in the process of constructing squares. Fifty-two children were engaged in three phase naturalistic task-based interviews. In Phase A (Description Task) the children were involved in classification and shape recognition activities. In Phase B (Construction Task) the children were asked to construct squares with the use of sticks and, in Phase C (Reflection Task) the children were asked to reflect on the construction process of Phase B. Even though during Phase A, the children, as supported by existing research, exhibited limited understanding about the structure of squares, through their involvement in Phase B, they exhibited much richer intuitive structural understandings. In Phase C, children tended to express structural understandings about squares. However the structural understanding that they exhibited was expressed in diverse and inventive ways. These findings challenge the view that children's limited verbal descriptions of shapes indicate lack of structural understanding. In the process of the interviews, the children articulated through the 'language' provided, existing intuitive structural knowledge of squares and at the same time they were able to situate their abstractions in the context of construction. Overall the findings indicate that, provided sufficiently sensitive techniques are employed, it is possible for children to express structural knowledge in diverse and often unconventional ways.

The nature of educational aims as criteria for worthwhile curriculum practice is explored and a cross-section of aims for mathematics education is discussed. An aim for mathematics education which emphasises the social aspect of the subject in its being, its conduct and its applications is identified and epistemological foundations for such a view of the nature of the subject are explored. It is argued that such an epistemological perspective of mathematics would be reflected in the social context of the mathematics classroom, arising from a methodology in which the subject would become more problematic and open to change, investigation and hypothesis. The aims of two major mathematics curriculum development projects (the Nuffield Mathematics Project and the School Mathematics Project) are examined to determine the extent to which their aims may take the 'social' nature of mathematics into account. The probable social context of mathematics classrooms using their materials is postulated in an attempt to characterise the nature of the subject as it is reflected in these materials. A view of the nature of mathematics held by practising teachers and by pupils is then established by drawing upon, and extrapolating from, evidence relating to the social context of mathematics classrooms at primary and secondary level. Conclusions follow, which suggest that fundamental change in mathematics education requires, as a first step, the adoption of a new epistemological perspective of the subject in order that the pursuit of the aim which emphasises the social nature of mathematics is achieved. It is suggested that this, in turn, ultimately could lead to the desired balance in the mathematics curriculum which hitherto has been lacking.

No description available.

This thesis examines the particular difficulties with multiplicative thinking experienced by students with very low attainment in school mathematics, and the representational strategies they use for multiplication and division-based tasks. Selected students in two mainstream secondary schools, all performing significantly below age-related expectations in mathematics, placed in ‘bottom sets’, and described by their teachers as having particularly weak numeracy, received a series of tuition sessions (individual or paired). These involved ongoing qualitative diagnosis of their arithmetical strengths and weaknesses, and personalised, flexible learning support, delivered by the author. Students engaged mainly in division-based scenario tasks designed to encourage their engagement in multiplicative thinking, and explored various visuospatial representational strategies tailored to their specific areas of conceptual and procedural difficulty. Multimodal audiovisual data collected from tuition sessions was analysed qualitatively across multiple analytic dimensions using a microgenetic approach. This led to the development of an adaptable framework for the analysis of nonstandard visuospatial representations of arithmetical structures and relationships. Analysis of changes in individual students’ strategies provided insight into some possible learning trajectories for multiplicative thinking. Parallel comparison of students’ varied representational strategies resulted in evidence for the psychological power of certain fundamental representation types, such as unit arrays and containers. The main findings of this thesis concern: the fundamentally componential nature of the concept and practice of division, the potential difficulties this causes in understanding, and the importance of modelling and manipulating unitary multiplicative structures; and the relationship between representational strategies, economy and efficiency in carrying out multiplication and division-based tasks. Conclusions are drawn on the relationship between the development of representational strategies and multiplicative thinking. Recommendations are given regarding learning and teaching practice for students with severe and milder difficulties in mathematics, and particularly the nature of 1:1 support provision for those considered to have Special Educational Needs.

The present study examines the possibilities of applying an individualised programme of Mathematics, namely SMP 7-13, in Greek primary schools. The research was prompted by the fact that the changes in the philosophy and practice of primary education as well as the evolution in the theory of Mathematics manifested at the outset of the 20th century hardly affected Greek educational matters, whilst they revitalised and transformed the British educational system. It was a programme which extended over three stages, the prepilot, the pilot and the main study. The latter was conducted for one year in three schools each one of which accepted children of different social strata. The application of the research exposed a number of difficulties innate either in the individualised programme itself or in the Greek school environment and function. Despite those difficulties, however, and the impediment of the limited sample the present research allowed for certain encouraging implications. An attempt to innovate the content, and moreover the teaching of Mathematics is not only imperative but it is also feasible provided, however, that certain conditions are taken into consideration. In relation to the applicability of SMP 7-13 to the current Greek educational system, the aspect adopted by the present study is that it could constitute a valuable supplement to the existing Mathematics programme.

This thesis reports on an investigation into the relevance of role play to the learning of mathematics. It is a case study of one Reception and one Year Four class during the academic year 2011-2012, where role play was used for children to explore mathematics that ‘made human sense’ (Donaldson 1978). This study is grounded in social constructivist theories of learning, seen as the product of interaction amongst members of communities. It starts from the theoretical perspective of play being a key medium for children’s learning, drawing on a body of literature positively linking play to young children’s development. It takes the view from accumulating educational research, that discussion and social interaction are significant in learning mathematics and that learners’ mathematical identities affect their engagement. The following questions were posed in this study: • What mathematics can be learned through role play? • What does mathematics learning look like in different role play contexts? • To what extent might role play affect the development of mathematical resilience, involvement and a positive attitude to the subject? • What particular classroom conditions positively affect mathematics learning through role play? • What adult intervention helps or hinders? Ontologically, this study considers participants’ interactions and behaviours and their interpretations of these, as central to a situation. In order to address these questions and reflect my views on knowledge, a qualitative, interpretative methodology was adopted for this study. The findings suggest that role play is useful for mathematical learning and that it is possible to engage in complex mathematics through role play. I argue that the potency of role play is its ability to suspend disbelief and engage children as participants in a community of learners. This study also concludes there is potential for developing children’s mathematical awareness and metacognition through reflecting on role play. Whilst the importance of reflection on learning is well established, how a child learns about themselves as a mathematician is under-researched. This study begins to consider this issue.

Improving the quality of classroom talk has been set as a target globally during the last four decades, considered as an indicator of improved teaching and learning; yet research globally indicates that this target still remains unresolved. The broad installation of Interactive Whiteboards (IWBs) in the UK was envisaged to support more interactive teaching and raise attainment. However, the initial waves of enthusiasm are now replaced by the realisation that synchronizing technological features to pedagogically informed methods which open up space for dialogic interactions is yet to be confirmed. This study investigates the impact of IWBs on standardised forms of assessment, on the quality of interactions during lessons and get an insight into pupils’ views of their own learning during IWB lessons. A mixed method methodology was applied, which employed a systematic review and a pupils’ questionnaire using targeted groups. Results of the systematic review indicate that there is no evidence indicating that the use of IWBs influence interaction and outcomes consistently and higher levels of interactivity are related to factors other than the installation of IWBs. Pupils’ responses from the questionnaire reinforce these results while, rather surprisingly, show that pupils consider techniques that are mainly attached to typical methods of teaching as learning ‘strengtheners’. This might be due to their experiences and expectations being narrowed in such teaching structures. All in all, investigating more rigorously the relation between the type of teaching and content of summative assessment might explain the durability of traditional teaching patterns.