The Complex Interplay between Theory and Practice: Reflections and Examples

This chapter addresses the complex issue of the relationship between theory, considered as the corpus of knowledge constituting mathematics education, and practice, seen as the act of teaching. Clearly this analysis is influenced by our model of teaching, which views the teacher as decision maker, whose decisions are conditioned by his/her knowledge, beliefs, emotions. This model has strong implications for research, leading to re-organize quality criteria such as reproducibility and relevance, but also for teacher training. In this framework we discuss as example some aspects of an Early Algebra Project (ArAl), which is also an in-service education process. In particular we sketch the main features of the Project, and describe the complex methodology of work with/for the teachers based on the sharing of reflections, made at different levels, about the processes enacted in different classes on the same didactical path. We conclude stressing the importance for teachers and researchers of a common reflection on teachers’ practice, that leads both to a greater awareness of the complexity of teaching, and suggests some issues for future research.