Design of a Learning Environment in Combinatorics based on Problem Solving: Modeling Activities,Problems and Errors

In this paper we present the design of a learning environment in a mathematical domain (elementary combinatorics) where problem solving is based more on modeling than on deduction or calculation. In this approach, we want to provide students with a presentation which is close to the natural language formulations that they tend to give spontaneously, while ensuring a rigorous mathematical reasoning. To do so, we have introduced three modeling levels: first, a mathematical formalization of the students' intuitive process, that we called the constructive method, then a conceptual and computational model that allows mathematical reasoning as well as communication with the student, and finally a presentation consisting of several"machines". We show that, in such a system, error detection is specific. We present an incremental mechanism of error detection. Specific knowledge necessary to detect and explain the errors is organized into a database of error schemas. The system Combien? founded on this research, has been used by university students since 2002.