The phenomenon of prior knowledge is deep rooted in the rhetoric of education. There is much discourse within pedagogy about its value and pivotal role in the formulation of new learning. However teachers are not able to use prior knowledge effectively as they do not have a working sense of it, but are using it intuitively and colloquially. While researchers provide a multitude of definitions of prior knowledge, no one has examined its elemental structure in a way that provides a model for teachers to use and support learning. This deficit is surprising as prior knowledge is a universally accepted pedagogical notion. The aim of this thesis is to fill the deficit and establish a structure of prior knowledge. The research was situated within Year 1 primary mathematics classrooms following eight teachers across five schools over one academic year. Using naturalistic research methodology, the data were gathered through audio recordings of the interactions between teachers and children during mathematics lessons. These recordings were analysed using grounded theory and content analysis. The research explored and produced a partial model of prior knowledge emerging from the data which includes at least eight interconnected elements – abstraction, acculturation, cognition, context, individual motivation, metacognition, perception and social group. These can be seen as elements which can shape children’s memory – the central feature of the prior knowledge that they bring to each mathematical task. Children may manifest different degrees of these elements, and possibly of others which did not appear in these data, in different proportions and balances. Such a prior knowledge model, even though it remains partial, gives a deeper understanding to a common but widely misunderstood term. The implications of knowing and understanding more and in more depth about the structure of prior knowledge are potentially far-reaching for children, schools, teachers and curriculum development.

The current study examines the pupil, classroom and school level characteristics that influence the attainment and the progress outcomes of young Maltese pupils for mathematics. A sample of 1,628 Maltese pupils were tested at age 5 (Year 1) and at age 6 (Year 2) on the National Foundation for Educational Research Maths 5 and Maths 6 tests. Associated with the matched sample of pupils are 89 Year 2 teachers and 37 primary school head teachers. Various instruments were administered to collate data about the pupil, the classroom and the school level characteristics likely to explain differences in pupil attainment (age 6) and pupil progress. The administered instruments include: the Mathematics Enhancement Classroom Observation Record (MECORS), a parent/guardian questionnaire, a teacher questionnaire, a head teacher questionnaire and a field note sheet. Results from multilevel analyses reveal that the prior attainment of pupils (age 5), pupil ability, learning support, curriculum coverage, teacher beliefs, teacher behaviours and head teacher age are predictors of pupil attainment (age 6) and/or pupil progress. Residual scores from multilevel analyses also reveal that primary schools in Malta are differentially effective. Of the 37 participating schools, eight are effective, 22 are average and seven are ineffective for mathematics. Also, in eight schools, withinschool variations in teaching quality, amongst teachers in Year 2 classrooms, were also elicited. Illustrations of practice in six differentially effective schools compared and contrasted the strategies implemented by Maltese primary school head teachers and Year 2 teachers. A discussion of the main findings as well as recommendations for future studies and the development of local educational policy conclude the current study.

The many valuable aspects of Chinese education have been seen throughout the world. For example Chinese teachers have been invited into the UK to instruct UK teachers on how to improve the quality of Mathematics education there. However, from 2001 onwards the government of China has sought to import pedagogies from the West in order to tackle observed problems with their own education system. Many scholars feel the problems the government were trying to address – the lack of critical thinking, a tendency to rote learning and an exam focused mind-set – would lead to failure for these techniques. However, these studies have all been with older students in high school and beyond. Little or no work has focused on students in the primary grades. Casual observation of Chinese primary school students would imply that they have no trouble in coming up with ideas and discussing them. The initiative for the study in this thesis has therefore been to see if this age range of students would be open to discussion based classes. A study was undertaken to determine if the removal of the teacher from control of the discussion would facilitate this age group to partake in face to face discussion. Other aims were to see whether constructivist learning would result or would the face based, hierarchical Confucian background education system prevent this. The original study was encouraging and as a result a technology based intervention was developed to see if this could help to improve the discussion and would allow further opportunities for students to feel able to engage. To encourage this the new system was also made anonymous. A third study was introduced to see if this approach could prove beneficial to the teachers also as to get such approaches adopted in Chinese schools the teachers would also need to see the benefit of the approach. The resultant study has demonstrated that not only do Chinese primary school students engage in face to face study but they also can be further encouraged by use of an online system. Further developments also indicate that the system can be valuable to the teachers who can use it as an aid to find out the preconceptions of their students and thus help them in developing a more focused curriculum. The thesis ends by describing ways in which this study can continue to have a positive impact in developing students’ critical thinking skills.

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This thesis examines number sense in primary mathematics. I begin by presenting literature to demonstrate how a cognitive definition of number sense, dominates understandings of mathematical development. I argue that this has influenced fixed-ability practices in mathematics (e.g. Boaler, 1997; Marks, 2014) presenting number-sense as a natural ability. I outline the political landscape and explore data which demonstrates that mathematics education systematically disadvantages some people (Zevenbergen, 2001). After reviewing mathematics learning from a range of theoretical perspectives, I demonstrate a gap in the literature: a sociological exploration of number sense in primary school and illustrate the need to examine school structures and their implications for equitable outcomes for all children. To address this gap I have employed Bourdieusian tools of habitus, field and capital, to explore number sense development. Through ethnographic methods in Year 4 classrooms, I examine how number sense positions children in the field of primary mathematics. This research was undertaken during the first year of statutory implementation of the National Curriculum (DfE, 2013) allowing insight into the lived experiences of children at this time. My findings show that facts, fluency and flexibility are key ways children demonstrate their number sense. Through rapid recall of facts children are seen by their teachers, peers and themselves as ‘able’ at mathematics, leading to explicit reproduction of social class, as these facts are usually learned at home. Similarly, a demand for fluency has led to a focus on procedural accuracy with calculation. Based on this, children are sorted into ability groups magnifying infinitesimally small differences between them (Bourdieu, 1986). Finally, children demonstrate flexibility through different calculation strategies; however, lessons usually rehearse single methods, hiding this key mathematical practice. Each aspect of number sense differentiates children, advantaging those with middle-class habitus and therefore reproducing educational inequalities.

The research for this doctoral study focused on children’s learning in mathematics and its relationship with independent pupil-pupil talk. In particular the interest was in how younger lower attaining children (aged 6-7) exchanged meaning as they talked together within a mathematical task. The data for the doctoral study had been gathered as part of the Talking Counts Project which I directed with colleagues at the University of Exeter. The project developed an intervention to encourage exploratory talk in mathematics with younger lower attaining children. Video material and transcripts of the mathematics lessons from nine classrooms that were part of the TC Project were used as the data set for the doctoral study. The focus of the analysis was on the independent pupil-pupil talk from one pre intervention session and one post intervention session from these nine classrooms. In using an existing data base, analysis was carried out in more depth and from a new perspective. A Vygotskyan sociocultural approach was maintained but analysis of the learning in the doctoral study was refocused in line with theories of situated meaning in discourse and with theories of the emergence of mathematical objects. Hence my examination of the children’s learning for the doctoral study went beyond the original research carried out in the TC Project. Within an interpretivist paradigm the methods of analysis related to the functional use of the children’s language. Interpretations were made of the children’s speech acts and their use of functional grammar. This enabled a study of both social and emotional aspects of shared intentionality as well as personal, social and cultural constructs of mathematical objects. The findings suggested that, where the talk was productive, the children were using deixis in sharing intentions and that this use could be related to the exchange of meaning and objectifying deixis.

In the UK, as many as 20% of children in primary schools are more than two years behind their peers in mathematics. Research-based intervention for such disadvantaged children has been shown to be effective but not always sufficient, such that alternative approaches might sometimes be necessary. One alternative might involve digital games. This study used a design-based research approach to investigate a prototype digital game, that implements principles of an effective numeracy intervention and draws on insights from learning theory and the cognitive sciences, designed for children in primary schools who are low-attaining in mathematics. It comprised three cycles of design, intervention, analysis and reflection. The first research cycle involved the initial design of a prototype digital game, which was researched in one school. The second research cycle involved a second iteration of the game, designed in response to the feedback of teachers and children, which was researched in three schools. The third research cycle involved the design of a final iteration of the game, which to achieve theoretical saturation was researched online with twenty-four schools. The study has shown that a game that implements principles of an effective numeracy intervention and that draws on insights from learning theory and the cognitive sciences can be designed and can be useful in schools for children who are low attaining in mathematics. However, for it to be taken up by schools, the game has to be perceived by teachers to have achieved a quality threshold. In any case, such a game is of limited use in and of itself. Where the prototype game has been shown to be most useful is when it serves as a fulcrum for social interaction and educationally productive discussion between the children and teaching staff: when it becomes an artefact that both supports individual learning and stimulates, scaffolds and mediates dialogue-based collaborative learning.

L'objectif de cette recherche est de mettre en lumière les représentations sociales des compétences pour enseigner les mathématiques des professeurs brésiliens et français et d'étudier comment celles-ci se manifestent dans leurs pratiques professionnelles. Nous partons de l'hypothèse que la professionnalité, particulièrement dans l'enseignement des mathématiques, est influencé par les représentations sociales que les professeurs ont de leurs compétences professionnelles à l'oeuvre dans leurs pratiques dans des contextes sociaux spécifiques et qui fonctionnent comme des guides de conduite dans le développement de leur action en classe et en dehors la classe, dans des activités de préparation de l'enseignement par exemple. Ainsi, dans le développement méthodologique de la recherche, nous avons procédé à l'identification des représentations sociales des compétences pour enseigner les mathématiques et plus spécifiquement des compétences pour organiser les enseignements relatifs à un thème, pour préparer une séance de classe et pour faire la classe. Nous avons administré une association libre des mots, des entretiens et des observations de classe de certains professeurs d'établissements d'enseignement publiques au Brésil et en France. Les résultatsde recherche obtenus dans le champ empirique nous ont permis de mettre en évidence des ressemblances et des différences entre les représentations sociales des professeurs brésiliens et français, selon les domaines de la professionnalité (disciplinaire, pédagogique, didactique, déontologique entre autres) et le lien avec leurs pratiques d'enseignement des mathématiques / The purpose of this research is to study how social representations of Brazilian and French mathematics teachers concerning the competence to teach mathematics influence their teaching practice. We assume that the professionalism, particularly in the field of mathematics teaching, is influenced by the social representations held by the teachers concerning the professional competence that they mobilize in their practices in specific social contexts and that serve as behavior guidelines in the development of teachers’ action inside and outside classrooms (for example, considering the preparation of lessons). Thus, in the methodological development of the research we have identified social representations of competences to teach mathematics and particularly the competences to plan the teaching of a mathematics theme, to prepare a lesson and to give the lesson. We have administered a test of free association of words, interviews and class observations in Brazilian and French public schools. The outcomes of the research point out similarities and differences in the social representations of Brazilian and French teachers, according to the domains of the professionalism (disciplinary, pedagogical, didactic, deontological, among others), as well as a relationship with their mathematics teaching practices / O objetivo deste trabalho é analisar as representações sociais de competência para ensinar matemática de professores brasileiros e franceses e como elas interferem na prática docente. Partimos do pressuposto que a profissionalidade docente, em particular em matemática, é influenciada pelas representações sociais que os professores possuem de sua competência profissional engendradas nas práticas de seus contextos sociais específicos e elas funcionam como guias de conduta no desenvolvimento das ações dos professores em sala de aula e também fora dela (nas atividades de preparação do ensino). No decorrer de nosso estudo tecemos algumas considerações sobre a relação entre “profissionalidade, ofício e profissão” e “profissionalidade, qualificação e competência” a fim de melhor cotejarmos a identificação das competências profissionais docentes no campo empírico e no campo das prescrições institucionais. Especificamente, sobre a noção de competência ressaltamos o papel do contexto social e dos recursos mobilizáveis em suas diferentes dimensões (cognitiva, afetivomotivacional, ético-política e social). Para o desenvolvimento metodológico da pesquisa tomamos como referência a abordagem psicossocial da Teoria das Representações Sociais desenvolvida por Serge Moscovici e a Teoria do Núcleo Central de Jean-Claude Abric. Inicialmente, procedemos à identificação das representações sociais de competência para ensinar matemática através da aplicação de um Teste de Associação Livre de Palavras e de hierarquização de itens, com a participação de duzentos e cinquenta e seis professores de matemática do Brasil e da França. Na análise dos resultados do TALP, utilizamos o software Trideux na quantificação das palavras ou expressões mencionadas pelos professores e a técnica de análise de conteúdo de Bardin para sua categorização. Em seguida, com base nos resultados obtidos na hierarquização (2/6 das palavras consideradas como mais importantes do TALP), a fim de identificarmos a organização interna das representações, foi aplicado um teste de verificação e entrevistas com trinta professores dos dois países Na análise das entrevistas utilizamos o software Alceste como suporte à identificação dos sentidos atribuídos pelos professores aos referidos elementos. Finalmente, foram realizadas ao total vinte e nove horas-aula de observação da prática docente em cinco escolas da rede pública de ensino no Brasil e na França, com a colaboração de um professor em cada uma delas. Dentre os resultados obtidos na investigação do campo empírico, expomos as aproximações e distanciamentos entre as representações de competência para ensinar matemática de professores brasileiros e franceses e de sua recorrência em diferentes domínios da profissionalidade docente (disciplinar, deontológico etc.), de como se situaram em diferentes dimensões (cognitiva, afetivo-emocional etc.) e em que medida elas dialogam com as competências prescritas no campo da qualificação pelos Ministérios de Educação do Brasil e da França. Por fim, apresentamos o que foi identificado na prática docente com base no NC das representações de competência para ensinar matemática de professores brasileiros e franceses e destas correlacionadas às representações de competência para organizar o planejamento, preparar aula e dar aula. Por exemplo, destacamos a análise da prática docente a partir das representações do conhecimento matemático e da capacidade de dominá-lo. E, de modo particular, no caso dos professores brasileiros, a partir do compromisso com a função de ensinar e dos professores franceses, do conhecimento das dificuldades em matemática de seus estudantes

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This doctoral research was concerned with the role of language and its implications for the learning, teaching and assessment of mathematics for children aged 4-9 years. Earlier research by the author had established language and assessment as bridges enabling learning although they had the potential to increase the divide between teacher and learner. Reflection raised the question on how children achieved in mathematics despite potential difficulties with language and assessment. Review of the literature concluded that resources and sociocultural norms were also bridges between learner and teacher. A model was established of the relationships and processes between all perceived variables that provided an external, theoretical structure to be evaluated against structuralist, pragmatic and integrational linguistic approaches and empirical outcomes. The overarching approaches adopted were institutional ethnography and case study. An appropriate methodology was devised whereby sophisticated ICT equipment captured all visual and speech events during classroom interactions. Frequency analysis at word level, content analysis at utterance level and discourse analysis at total speech level triangulated with content analysis of interviews and evaluation of documentation completed the empirical research. Data analysis revealed five registers of children’s talk. Evidence suggested that the peer-peer ‘conditioned talk’ used in focused group work was the most effective for learning as it enabled them to discern the small steps in the inferential leaps in discourse made by their teachers, work out problems together, inform their peers, share findings and reinforce each others’ learning. Learners’ language showed aspects of structural, pragmatic and integrational linguistics, confirming a conclusion of the literature review that the various linguistic approaches discussed should be used to support and not exclude each other. The contribution made to knowledge is the ethnomethodology provided by the model, ICT resource and the five registers of talk revealed by the linguistic approach to discourse analysis. Teachers would be able to understand nuances of language used by their pupils and acquire essential skills and tools to put into effect the personalised learning agenda. Peer-peer observation of teachers would be an appropriate platform for the observation of the different registers used by learners, the resources that generate those registers, and their most effective use to close the gap between natural language and the subject specific language of mathematics.