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