# Contributing to develop contributions - a metaphor for teaching in the reform mathematics classroom

Sammanfattning: This thesis aims at contributing to the theoretical research discourse on teaching mathematics. More precise, to explore a teacher’s role and actions while negotiating meaning of mathematical objects in discursive transformative practices in mathematics. The focus is to highlight the teacher as an active contributor to the classroom mathematical discourse, having an important role in shaping the mathematics. At the same time, the teacher is acknowledged as an individual who learns and develops as a lesson and semester progress.Three research papers illustrate the state, at that time, of an inductive analysis of three teachers, teaching a series of lessons based on probability theory at two Swedish primary schools. The teachers worked together with the students to explore an unknown sample space, made up out of an opaque bottle with coloured marbles within that showed one marble at each turn of the bottle. They had to construct mathematical tools together to help them solve the mystery. The analysis focused on teacher–student interactions during this exploration, revealing complex connections in the process of teaching.The three papers presented the development of a theoretical framework named Contributing to Develop Contributions (CDC). The frameworks’ fundamental idea is that teachers learn as they teach, using the teaching metaphor learning to develop learning. That metaphor was developed, in light of the ongoing empirical analysis, into CDC by drawing on a theoretical idea that learning can be viewed as contributing to the collaborative meaning making in the classroom. Teaching and teacher learning are described and understood as reflexive processes in relation to in-the-moment teacher-student interaction.Contributing to develop contributions consists of three different ways of contributing. The analytical categories illustrate how students’ opportunities to contribute to the negotiation of mathematical meaning are closely linked to teachers’ different ways of contributing. The different ways are Contributing one’s own interpretations of mathematical objects, Contributing with others’ interpretations of mathematical objects, and Contributing by eliciting contributions. Each way of contributing was found to have the attributes Transparency, Role-taking and Authority. Together, these six categories show teacher– student interaction as a complex dynamical system where they draw on each other and together negotiate meaning of mathematical objects in the classroom.This thesis reveals how the teaching process can be viewed in terms of learning on different levels. Learning as thought of in terms of contributing to the negotiation of meaning in the moment-to-moment interaction in the classroom. By contributing you influence the collective’s understanding as well as your own. A teacher exercises and develops ways of contributing to the negotiation of meaning of mathematical objects, in order to develop students’ contributions. In a wider perspective, the analysis showed development over time in terms of transformation. The teachers were found to have transformed their understanding of classroom situations in light of the present interactions. Contributing to the negotiation of meaning in the classroom was understood as a process in such transformation, in the ever ongoing becoming of a mathematics teacher.

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