IntroductionProfessional development of teachers can take highly diverse forms: institutionally organised or more personal and spontaneous; individual or in collaboration with peers or others; continuous or for a period. We well acknowledge that individual and continuous professional development is relevant and important. However, our interest focuses on collective forms of professional development. We are studying the ways of how shared knowledge is constructed in a community of experienced teachers, pre-service teachers and two teacher educators (UG and GK), and the positions and perspectives the participants of our study take during the process of our joint work.

1We start from two basic assumptions:(A) Every practice of mathematics teaching and learning is a locally emerging process with open ends. The course and the results of this process depend on the students’ and the teacher(s)’ capacities to interpret and influence the interaction in their classroom. How students and teacher(s) understand each moment of the lesson is crucial for their scope and margin to shape a lesson’s course. A mathematics lesson is exactly what those involved see in it. As a consequence of this assumption we can propose the following: If students and teachers were able to interpret the locally emerging processes of teaching and learning differently, then a different practice of mathematics education would be possible. The broadening of a teacher’s interpretative resources is crucial for her professional development.(B) Interaction in everyday mathematics classes is a complex issue.

Although interaction in the classroom is situated, and its course is contingent upon the perception and realisation of those involved, we nevertheless reconstructed four dimensions that provide a structure for analyses of what happens in mathematics classes (cf. Krummheuer 2004):* The mathematical concepts, theorems, procedures, and models, which students and teachers talk about.* The arguments and argumentation patterns which students and teachers produce.* The patterns of interaction.* The forms of participation of active and silent students.We take these four dimensions as a minimum model for understanding processes of mathematics teaching and learning. Particularly, these dimensions facilitate differentiation between two opposite forms of interaction in the mathematics classroom, interactionally steady flow vs. thickened interaction.

The first being characterised by fragmental argumentation, interaction patterns with inflexible role distribution, and less productive participation of all students; the second, in contrast, shows rather complete collectively produced arguments, flexible roles of students and scope for their involvement in the educational process. These two forms provide different favourable opportunities for student learning. Teacher development may be seen as a path towards better opportunities for students’ learning of mathematics, i.e. to facilitate thick interactions that interrupt the interactionally steady flow of everyday mathematics lessons.

Towards a heterogeneous community of interpretationBased on these two assumptions we offered a 14-weeks mathematics education course, in which 5 teachers (from two primary schools, teaching 3rd and 4th grade mathematics) and 13 university students studying for a career as primary teacher took part. Participants were divided into stable subgroups of one teacher and two or three students each. The teacher and the two or three students met one day of the week in the school of the teacher. There, students observed the interaction between the teacher and the pupils and among pupils, videotaped parts of the lessons, prepared themselves (supported by the teacher) for teaching the class and taught the class (observed by the teacher).The whole group met one day of the week at university for, what we call, collaborative interpretation of classroom interaction. For each of these meetings, one subgroup selected about 15 minutes of videotaped (and transcribed) classroom interaction from the mathematics lessons in their school. The task of the whole group then was to reconstruct the interactional dimensions of the 15-minutes-scene. The goal was to analyse what happened in the episode, to find markers why things went as they went, and how the course of interaction could have developed differently — eventually with optimised learning opportunities for the pupils.

The analysis aimed at uncovering the contingencies of the supposed natural and seemingly inevitable course of a lesson.Interpretation of videotaped classroom interaction is not trivial a task. If approached on the basis of common sense the videotape scenes do not look radically unusual and there seems nothing to be discovered under the surface. It is not before starting to scrutinise the videotaped interaction systematically, that is to say using techniques for focussing on specific dimensions of the interaction, that one can see alternative paths through the possible ramifications of teacher(s)’ and students’ talk.

For instance, some pupils’ utterances that on the first view appear to show a lack of understanding of the mathematical problem to be tackled prove to be thoroughly rational, sense making and potentially helpful — they are just misplaced within the course of the arguments. In the first group meetings, we introduced three techniques for interpretation of classroom interaction: interaction analysis, argumentation analysis, and participation analysis. These originate in the academic field of qualitative classroom research in mathematics education (e.g. Cobb & Bauersfeld 1995, Krummheuer 2000, Steinbring 2000).This is not the place to discuss these techniques. What we want to focus on is the group dynamics initiated by gathering teachers, university students and teacher educators together to interpret videotaped lessons prepared by the group members.

We consider this group a “community of interpretation” alluding to the seminal work of Lave and Wenger (1991) who describe modes of development from peripheral to full participation in communities of practice. The practice, here, is collaborative videotape interpretation and the members’ aim is to become more competent in interpreting the ongoing interaction in the mathematics classroom.As members of our community of interpretation collaborate in the schools as well as in the interpretation meetings their social positions within the group are (at least) bipartite. In schools, teachers take up to act as students’ mentors, thus positioning themselves as classroom experts or, in Lave and Wenger’s terms, as masters of teaching.

On the other hand, the students do not exactly perceive themselves as apprentices since their presence in school is limited and partly directed towards the exigency of collecting potentially interesting classroom episodes. In the interpretation meetings, the situation is different. Both teachers and students are novices as long as systematic interpretation of classroom episodes is concerned.They learn to use the techniques of interpretation by observing how the teacher educators (UG and GK) start analysing the video scenes and by applying the techniques gradually to the videotaped episodes.

Trial application of their local insights and operative improvements lead, by means of constant exchange in the community, to a co-operative work style and, to some extent, to shared knowledge. This is a typical developmental path from peripheral to full participation in a community of practice. However, the collaborative work in the community of interpretation may be influenced by the positions that teachers and students take up in their schools.We consider the community of interpretation a promising approach for bridging the divide between formal knowledge (about the contingency of classroom interaction and how to make use of it) and the practice of classroom teaching.

The importance of interaction patterns and interaction mechanisms is likely to be overlooked from the perspective of concrete teaching in schools. To analyse accounts of interaction is thus a crucial practice of learning from practice. A heterogeneous community of interpretation provides support for teachers’ and pre-service teachers’ learning from and for practice. This heterogeneity of the members is a rather unknown quantity within research on mathematics teacher education. We may suspect that the heterogeneity in our group leads to some specific socially distributed action patterns and to a particular structure of shared knowledge.Shared knowledge and socially distributed action patternsAs Lave and Wenger (1991, p.

98) say, “a community of practice is a set of relations among persons, activity, and world”. The social structure of this set of relations defines possibilities for learning and development. In order to describe the social structure of our community of interpretation we accommodate and elaborate some conceptual tools provided by Raeithel’s (1996) comparative study of ethnographical research on co-operative work.2A distinction can be made between the perspectives of somebody who is teaching (or has just taught) mathematics and of an observer of this teaching practice. This distinction applies to situations in schools, where students watch the classroom processes initiated by experienced teachers or where teachers observe the students having a try in their classrooms. From the perspective of the observer a detached analysis of what happens is possible, whereas the centred stance of practice requires a more intuitive grasping of situations in the classroom.Accordingly, the centred stance of teaching practitioners results, partly, in a shared generalisation of experience, whereas outside observers rather argue on the grounds of symbolic objectives and general rules.

These opposite perspectives, centred and de-centred, if not mediated, tend to generate a reserved relationship that is not productive for the learning from practice. The community of interpretation intends to overcome such opposite positioning by re-centring the analysis of classroom episodes. For that, it is, firstly, necessary to disturb the natural grasping of classroom interaction and to fathom the subtleties of the apparently straightforward course of action on the videotapes.Secondly, analysis should be directed away from assessment and judgement of what can be seen on the tapes towards a contentious or consensual dispute of central interactional distinctions and effects. The shared knowledge of the community of interpretation is fundamentally the result of legitimate participation. Disputes for convincing interpretations and for metaphors that guide future lesson design develop a specific interpretative sense-making capacity.

This gradually developing capacity is expressed through a specific language (oriented at the four interactional dimensions) as a characteristic feature of our community of interpretation. Briefly, the re-centring stance aims at legitimate semiotic self-regulation in the group (see table 1 for a summary).