Specialised knowledge network activated in teacher education to answer to a mathematical why on fraction division

Background: one way to deal with the complexity inherent in teaching is to advance in understanding the relationships between the different types of knowledge used in teaching practice. Objective: to characterise specialised teaching knowledge and its connections established in a formative context of elaborating an answer to students on why to invert-and-multiply to divide fractions. Design: qualitative analytical-descriptive study, associating the mathematics teachers’ specialised knowledge (MTSK) and a teacher education course anchored in a teaching practice situation. Setting and participants: a teacher education workshop for in-service and preservice mathematics teachers, from which we carried out an in-depth analysis of the data collected from two participants (a preservice teacher and a teacher). Data collection and analysis: we obtained data during the workshop and an interview with the subjects and used content analysis, the MTSK analysis instrument, the MTSK cluster and the MTSK network diagram enabled to describe the connections between types of knowledge. Results: the activated network contains knowledge from all MTSK subdomains, exercising different levels of importance and roles (from the action itself to its support). The route of construction of the answer started from a mathematical layer towards didactic elements, reflecting the level of development (including gaps) of the subjects on justifications of algorithms and didactic aspects. Conclusion: a seemingly simple teaching action (answering a student question) mobilised all the MTSK subdomains and activated an intricate procurement network, requiring specialised, reasoned, and intentional teaching preparation