Derivative Classes in Secondary Education

The aim of this study was to identify how an experienced teacher introduces the concept of derivative in secondary education in Lisbon (Portugal) and to verify whether commognition, as proposed by Anna Sfard, is applicable in this context. The qualitative case study approach was adopted, data were collected in a public secondary school classroom, using observation techniques. The analysis focused on identifying the teacher’s methodological approach, based on Sfard’s four categories: Word Use; Visual mediators; Endorsed Narrative and Routine. However, routines were further subdivided into two subcategories: classroom routines and mathematical routines. Six episodes illustrating mathematical routines were analyzed in detail. The findings showed that symbolic, graphic and gestural visual mediators were consistently present in all lessons. The endorsed narrative was constructed through stated definitions and theorems that are demonstrated, and are consistently present in all lessons. The concept of derivative was constructed from the concept of average rate of change, followed by the notions of approximation, limit and finally derivative at a point. The approach to the concept of derivative was formalized, with some appeal to intuition. The study concludes that traces of commognition, as proposed by Sfard, are observed classes. Given the limited research on the teaching of derivatives at both the secondary and higher education levels, this study contributes valuable insights into how this fundamental concept of Differential Calculus is taught in secondary education.