Affiliations: University Pierre et Marie Curie (Paris 6),
LIP6-MOCAH, 104 avenue du président Kennedy, 75016 Paris, France.
Abstract: 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