Exploiting the potential of dynamic asymmetry in dragging to foster students’ understanding of functions and their cartesian graphs

The notion of function is considered central in mathematical thinking and in mathematics curricula, especially at the high school level. In particular, being able to interpret a function’s behavior by looking at its Cartesian graph and to construct a graph starting from the function’s properties are processes that are widely considered essential in mathematical thinking. These processes are based on two key aspects: (1) understanding basic notions in functional thinking, such as variable and covariation and (2) reconstructing the relationship between variables from a Cartesian graph of a function, and, vice versa, constructing a Cartesian graph of a function based on the relationship between the variables. However, many studies have shown that mastering these concepts is complex. We designed and implemented a sequence of activities in a digital interactive environment, capitalizing on its dynamic asymmetry potential (DAP), the potential to let students experience both the variations and the asymmetry in the behavior of varying objects that can be dragged on the screen. Through examples from an implementation of these activities, we show how students can come to appreciate specific aspects of the notion of function, tightly related to the DAP and to recognize them in the traditional static Cartesian graph representation.