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discriminate between the variables studied. To study the relations between S.I.A. and the
didactics of mathematics consequently questions the theoretical status that mathematics
didacticians can grant to these models of rules as well as the nature or the status of data
from which the modalities of variables subject to the S.I.A. are constructed.
If one can define mathematical didactics as a plan of scientific study of the phenomena
tied to the transmission of disciplinary knowledge, this has the consequence that the
didacticians’ preferred field of observation is that of the mathematics class : the mathematics
class in the sense of the material space, certainly (as there one can explore the posters,
the students’ notebooks…) but also in the sense of symbolic space (the mathematics class
still exists when the teacher prepares his classes, when the student learns his lessons, at
home, at study…). This class exists consequently once the interactions between subjects
places them one as teacher, the other as student, in relation to an object of disciplinary
knowledge. Very schematically, one of the principle objects of study of didactics is that of the
manifestations of this relationship between these three interdependent elements, the master,
the student and the knowledge of the discipline, of its establishment and its maintenance
over time. Two consequences can be drawn from this modelling. Firstly the observables
are consequently constructed as interactions between master and student, master and
knowledge, student and knowledge. Then this study requires the analysis of the regulations
that are simultaneously going to be generated by this system of interactions and assure its
functioning. Thus for example, one of the most fruitful problems in mathematical didactics is
that of the regulations which affect the interactions between the student and the knowledge
at issue if the observables are in this case actions in which the student is engaged (linguistic
or not), the regulations that govern these actions the engagement of procedures, some
choices made…) or that are produced by these actions (the abandonment of certain ways
of acting, certain controls…) are essential to highlight.
Some of the rules made evident or revealed through I.S.A. are thus interpretable in
mathematical didactics in terms of regulation of interactions. Two positions can then be
adopted : either these rules have the status of hypotheses for the didactician, it being his
responsibility to invalidate them or confirm them through other methods of analysis (as for
example the undertaking of interviews) or they have the status of facts of experience (thus
allowing to contradict or not invalidate the analysis a priori).
The asymmetry that these rules present is also to be taken into account in the interpretation
that mathematical didactics can make of them. It poses the problem of asymmetries
explicable by the modelling in terms of a system of interactions. The didactic system that we
have summarily evoked (a triplet of interactions between student, teacher and knowledge)
is a system in which the dissymmetries of the characteristics tied by a rule can be explained
in a number of different ways.
Let us begin with the most classic case, in which the rules established are from data
corresponding to situated observables the actions or the effective declarations of students
or teachers gathered on site). A dissymmetry between observables must correspond to the
dissymmetry between modalities of variables linked by a rule which has resulted from S.I.A.
‘all the subjects having the characteristic A have the characteristic B’. This dissymmetry
leads to question didactically the fact that very few students have done B without having
done A, have succeeded in B without having succeeded in A, have answered B without
having answered A, that very few teachers have done B without having done A etc. These
regulations of doing, saying and of their effects can be the effects of temporality, of differences
between the tasks proposed, of organization of knowledge…
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