*Y. M. Visetti, P. Dague*

This paper adresses the problem of building user models within the framework of Computer Assisted Instruction (ICAI), and more particularly for systems teaching elementary arithmetic or algebra. By "model building" we mean the understanding of the student’s performances, as well as a global description and evaluation of his/her ability (competence), including a representation of some errors. As an application domain we have here retained the learning of "calculus" in the field of rational numbers, as an intermediate area between arithmetic and algebra. The aim of our system is to control the way in which the pupil solves exercises. In the light of the particular nature of the chosen application, the main points to be stressed are the following : - calculations are described as plan generation and execution ; consequently the student’s modelling consists primarily in plan inferencing - the system takes into account the non deterministic nature of the task, and recognizes valid variants of expert calculation plans - numerous errors are detected and categorized - the system accepts that the student write the calculations in a more or less elliptic manner ; whenever ambiguities occur, the student is precisely asked about implicit steps of his calculations, and the system uses the answers given to reduce the uncertainties - a global model of the student is generated, which incorporates observations and appreciations ; this model, in turn, determines the subsequent interpretations. All these questions are discussed both at the fundamental and the methodological levels.

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