The function concept at the transition to upper secondary school level: tasks for a situation of change

This paper is about the development of a task sequence to help overcome the fragmented understanding of the ‘function’ concept that students often bring with them into the initial stage of upper secondary school level. Our aim is to make the students’ use of functions more flexible in certain respects, for example when functions are required to be used as tools for modelling. The core idea of our task design is to interpret formulas as functional relations. The tasks are developed along the lines of a Design-Based Research approach in which several theoretical approaches are employed in a complementary way. The paper will show how this complementarity frames and informs the design as well as the analysis of data about how a student solves the tasks. Mediated by the task sequence, students’ development in terms of how their use of functions becomes more flexible is reconstructed. In the analysis, a key constraint for developing a flexible use of functions is identified: their poor understanding of the coordinate system and its scaling and utilization as a reference space for graphical representations of functions.