New actions upon old objects: A new ontological perspective on functions

We argue in this paper that the systematic use of special software in instruction has a profound impact on the notion of function as an abstract entity to be constructed. We argue that through the medium of the computer, the objects in the graphical, tabular and algebraic settings can change their essence and thus become objects of a new kind we call representatives. Actions on representatives which naturally arise in this framework induce an ontological shift. A taxonomy of the skills involved in the learning of the concept of function through these new ontological lenses is presented, as well as software, and problem solving tasks that embody the same ontological perspective. Within the framework of a teaching experiment, students' acquisition of many of the identified skills was investigated by means of a questionnaire and interviews during computer supported problem solving sessions. The most salient results of the study indicate that a majority of students were able (1) to cope with partial data about functions (e.g., problems of interpolation and arbitrariness), (2) to recognize invariants (i. e., properties of functions) while coordinating actions among representatives from different settings, and (3) to recognize invariants while creating and comparing different representatives from the same setting.