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.

The aim of the thesis is to describe the decision making process of students in the so-called simple categorization, i.e., decision whether a particular object is or is not an element of a category. This process is examined in the context of categories of mathematical objects. The theoretical part of the thesis presents arguments why the study of simple categorization of mathematical objects is important for mathematics education. These arguments are not only based on the available literature in mathematics education, but also partly draw on historical, mathematical and psychological literature. The practical chapters of the thesis describe the design and piloting of a research tool suitable for this research. The dominant elements of this tool are the measurement of the binary answers (yes / no) of the respondent and of his/her reaction time. This tool is then used in the Main study based on mixed, qualitative-quantitative methodology. It was found that with the help of the proposed tool, while adhering to appropriate methodological rules, it is possible to distinguish different approaches of respondents to categorization. In addition, the basic patterns in the decision-making process of the respondents were described. These are, for instance, differences in the categorization of examples and non-...

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Étudier l'émergence de savoirs lors de situations didactiques non finalisées par un savoir préfabriqué et pré-pensé nécessite un bouleversement des points de vue, aussi bien épistémologique que didactique. C'est pourquoi, pour l'étude de situations didactiques pour lesquelles le problème est l'essence, nous développons une nouvelle approche historique et repensons des outils pour les analyses didactiques. Nous proposons alors, pour un problème particulier, l'exploration des pavages archimédiens du plan, une enquête historique centrée sur l'activité du savant cherchant et sur l'influence de la relation aux objets dans la recherche. De ce point de vue, l'étude des travaux de Johannes Kepler à la recherche d'une harmonie du monde est particulièrement instructive. Nous proposons également, pour l'analyse des savoirs émergents en situation didactique, une utilisation d'outils liés à la sémiotique qui permet de mettre en évidence la dynamique de l'évolution des objets mathématiques. Nous pouvons finalement conclure quant à la possibilité de construire et mettre en œuvre des ≪ Situations Didactiques de Recherche de Problème ≫ assurant l'engagement du sujet dans la recherche, l'émergence et le développement d'objets mathématiques, la genèse de savoirs. L'étude nous conforte dans la nécessité d'une approche pragmatique des situations et la pertinence d'un regard différent sur les savoirs à l'école / The study of the emergence of knowledges in teaching situations not finalized by a prefabricated and pre-thought knowledge requires an upheaval of point of view, epistemological as well as didactic. For the study of learning situations in which the problem is the essence, we develop a new historical approach and we rethink the tools for didactic analyzes. We propose, then, for a particular problem, exploration of Archimedean tilings of the plane, a historical inquiry centered on the activity of the scientist in the process of research and on the influence of the relationship with objects. From this perspective, the study of Johannes Kepler’s work in search of a world harmony is particularly instructive. We also propose, for the analysis of the emerging knowledge in teaching situations, to use tools related to semiotics, which allows to highlight the dynamic of evolution of mathematical objects. We can finally conclude on the opportunity to build and implement “Didactic Situations of Problem Solving”, which ensure the commitment of the subject in the research, the emergence and development of mathematical objects, the genesis of knowledges. The study reinforces the necessity of a pragmatic approach of situations and the relevance of a different look at the knowledge at school

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