1. Design and Redesign of an In-service Course: A case study of a course concerning mathematics communication Lisser Rye Ejersbo, Learning Lab Denmark, DPU

2. Two Conflicts What happened on the course The teacher’s goals What happened in the classrooms My goals Lisser Rye Ejersbo, Learning Lab Denmark, DPU

3. Classroom Problems Analytical Tools Reflection Skills Mathematical Communication Lisser Rye Ejersbo, Learning Lab Denmark, DPU

4. Meta-Didactical Transposition A ‘didactical transposition’ denotes the process of transferring mathematics as a scientific discipline into an educational subject at school. [Chevallard, 1985] Scientific Mathematics Teaching Tools Mathematics Education Research In-service Teaching Tools

5. Task-oriented and User-oriented Tools Lisser Rye Ejersbo, Learning Lab Denmark, DPU

6. Design Research Design research is a continuous process between instructional design and classroom based analysis, where the researcher, when preparing for the design experiment, formulates a hypothetical trajectory. Developing theory should be one of the primary goals of design research and theory must do real design work in generating, selecting and validating design alternatives at the level at which they are consequential for learning. (Cobb, diSessa, Gravemeijer) In this case, the classroom is an in-service classroom and the arrows symbolise the process from design and preparation to a realisation and redesigns. Classroom- based analyses Instructional design Lisser Rye Ejersbo, Learning Lab Denmark, DPU

7. The Guiding Principles On the in-service course, each teaching session is based on a particular theoretical concept from the research literature. The theoretical concept is transformed into practical activities so that –the theoretical concept is made into a tool; –the teachers can use the method in their own classes; –the teachers’ tacit knowledge becomes apparent; –room for reflection is shaped along with the activities. In the preparation phase, theory precedes activities; in the practical phase, activities precede theory. Lisser Rye Ejersbo, Learning Lab Denmark, DPU