CALCULABILITY OF WRITTEN INSTRUMENTS FOR PROBLEM SOLVING

Résumé 0

In order to study spontaneous phenomena of semiotic instrument genesis in mathematical problem solving, we have designed an experimental situation in which students from 6 th grade to 11 th grade had to solve a series of problems commonly identified as "algebra problems with two unknowns". We examine how the students' writings allow to achieve the various cognitive functions involved in problem solving (i.e. their "calculability"), how the writings evolve into an autonomous status from the activity which produced them, as well as how they develop thanks to the repetition of isomorphic problems. This paper demonstrates that a significant minority of students makes use or invents written instruments. It also illustrates two categories of genesis phenomena : the adaptation of a curricular instrument to the problem constraints and the composition of a new instrument from previous isolated instruments in a "bricolage" perspective. Instrumental evolution from one problem to the next is rather rare and when it does occur it is mostly directed towards extending the use of previous instruments. In particular, the transformation of the pattern ax + by = d from a proof into a calculable representation, seems to be a privileged enough transition between arithmetic and algebra to be further studied.