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

In IJAIED 18 (1)

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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. Keywords. Problem solving, combinatorics, modeling of activities, the constructive method, incremental error detection, error schemas