Adaptative Learning Environment for Geometry

Geometry with its formal, logical and spatial properties is well suited to be taught in an environment that includes dynamic geometry software (DGSs), automatic theorem provers (ATPs) and repositories of geometric problems (RGP). With the integration of those tools in a given learning management system (LMS), we can build an environment where the student is able to explore the built-in knowledge, but also to do new constructions, and new conjectures, allowing, in this way, a better understanding of the concepts presented in a given e-course. Dynamic geometry software are alreadywell know tools for their educational properties, (e.g., CINDERELLA, GEOMETER’S SKETCHPAD, CABRI, GCLC, EUKLEIDES), with them we can visualize geometric objects and link formal, axiomatic nature of geometry with its standard models and corresponding illustrations, e.g., Euclidean Geometry and the Cartesian model. The common experience is that dynamic geometry software significantly help students to acquire knowledge about geometric objects. The visualization and the possibility of dynamically modify some of the parameters of a given geometric construction, are a very important tool in the comprehension of the geometric problems. Automated theorem provers are less known as tools used in a learning environment, but geometry with its axiomatic nature is a “natural” field for a formal tool such as the ATPs. Automated theorem proving in geometry has two major lines of research: synthetic proof style and algebraic proof style (Matsuda & Vanlehn, 2004). Algebraic proof style methods are based on reducing geometric properties to algebraic properties expressed in terms of Cartesian coordinates. Synthetic methods attempt to automate traditional geometry proof methods. The synthetic methods (e.g. the Area Method (Chou et al., 1996a; Quaresma & Janičić, 2006a; Zhang et al., 1995)) provide traditional (not coordinate-based), human-readable proofs. In either cases (algebraic or synthetic) we claim that the ATPs can be used in the learning process (Janičić & Quaresma, 2007; Quaresma & Janičić, 2006b), and we will elaborate on this on the main body of this text. An important first integration of tools is given by the DGSs incorporating ATPs, e.g. the GCLC DGS/ATP (Janičić & Quaresma, 2006). With a DGS we can visualize a given geometric construction, we can also dynamically change some objects of the construction and test, in this way, if some property holds in all checked cases, but this still does not prove that the construction is sound. The integration of a given ATP in a dynamic geometry tool allow the