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Another case can be envisaged, that in which the established rules are from data corresponding
to observables on site but also from data perpetuated from these observables. The stability
of the latter therefore makes clear groups of “fixed” subjects (the students of a same sociocultural
milieu, “novice” vs. “experienced” teachers etc.) S.I.A. can then provide, either rules
linking actions, statements, the effects of these actions, of these words and these constituted
groups, or by the study of the contributions of subjects to rules, establish tendencies shared
by subjects from the same group, or on the contrary, of equally characteristic avoidances.
We will present several cases of studies in mathematical didactics exemplifying these
different usages.
2. Regulations of situated actions, rules established from observable
modalities.
2.1. Asymmetries of rules established and chronology of tasks
The example that we develop first is that of the study of responses of students of CM1-
CM2 (9 to 10 years) to an exercise that is composed of two successive tasks. Firstly they
are asked to put into order written decimals and fractions ½ ; 1,5 ; 2 ; 3,5 ; 4 ; 5,15 ; 5,6 ;
6,2 ; 9,5 ; 12 secondly to place them on a graduated line. In French the word numbers
associated with 5.15 and 5.6 are pronounced ‘five comma fifteen” and “five, comma six”.
This way of saying the numbers explains a frequent error at this school level, which consists
of placing 5.6 before 5.15, by only comparing the decimal parts of these numbers. However
the numbers have been chosen so that the reproduction of this classification error in the
second part of the task leads to a contradiction that the students – still of the school level
considered – can comprehend. In fact, to put a point corresponding to 5.6 on the graduated
line, then to put one that corresponds to 5.15, in moving back the first one by a space of “9”
( the space between 15 and 6) leads the student to place 5.15 erroneously on the point that
should correspond to 6.5 (5.6+0.9). This placement can seem contradictory with that which
corresponds to 6.2 and other points. We will say in this case that the information given to
the student by the erroneous placing of 5.15 is an element of the environment with which
the student interacts.
If consequently we expect certain students to commit errors in the classing of written numbers,
on the other hand, we wonder about the effects of the consequences of these errors during
the execution of the second task. Two types of common considerations in mathematical
didactics allow us to anticipate them. Firstly, the information that the erroneous placement
of the points on the line gives is not “naturally” interpreted in terms of a contradiction. In fact
the reading and the comprehension of this information necessitates the working of certain
knowledge : in fact it concerns considering the placement of 5.15 and 6.2 as “strange” and to
consider them as a consequence of the classification error of 5.6 and 5.15. Then the different
works done on the error or the problem in a scholarly situation lead us to differentiate the
recognition of a student of an error and the realization of that error. Or to put it simply
the perception of a contradiction in his results is often insufficient to lead a student in a
class to invalidate the latter because he still does not feel invested with the responsibility to
resolve the problem raised (Brousseau, Margolinas). The question of the study of students’
behaviour is therefore a legitimate question.
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