Instrumented Techniques and Reflective Thinking in Analytic Geometry

In a previous study that explored epistemological perspectives on solving problems with Computer Algebra Systems (CAS) we concluded that awareness of the special ways that the software utilizes symbols in algebraic manipulations and in implicit plotting should be encouraged (Zehavi, 2004). Such awareness is required for, and encouraged by treating geometry analytically with a symbolic-graphical system. In this paper we compare a traditional solution of a problem in analytic geometry with CAS-based solutions to the same problem. The discussion will focus on the role of reflective thinking, namely selection of techniques, monitoring of the solution process, insight, and conceptualization, play in the creation of instrumented techniques (Guin & Trouche, 1999). Teachers, who experienced learning activities from a resource e-book for teaching analytic geometry with CAS, contributed to the design of tasks and to the analysis of instrumented techniques.