vygotsky's everyday concepts/scientific concepts dialectics in school ...

Students had to discover the crucial elements of the situation and organise their
hierarchical relations to back a procedure of calculation. This reflects the
development of an intentional and explicit use of additive relations and a mastery of
important systemic links. But they also dealt with the central notion of increase. This
everyday use of increase was crucial to interpret the problem situation in terms of
additive relations; at the same time the possibility of interpreting it with the scientific
concepts of the additive field made it necessary, and possible as well, to develop the
conceptualisation of increase (and specially to make explicit its dependence on time).
Teacher’s stimulation and direction of argumentation made this dialectics alive and
productive. As a result of setting this dialectic, the everyday concept of increase was
made more “scientific” (though remaining in an everyday use).
Evolution of the concept of increase
Development of systemic links of increase:
- distinction between height and increase: It was possible thanks to the implicit
relation to the number line, either by drawing or indirectly by mental reference
(because of student’s mastery of such elements of the additive conceptual field),
which contributed also to determine their relation to time. Increase was determined as
a variation of length through time;
- linking increase to reference situations: The real situation made it possible through
its dynamical character, thanks to the interpretation needed to solve the problem;
indeed increase it is a variation related to time that is not really visible, but has to be
elaborated. Also referring height to increase since birth as different from increase
between two dates had a positive effect: students had another reference situation and
another explanation of the role of time at their disposal.
Concerning students’ awareness:
students syntheses revealed that they assimilated this new understanding of the
concept (all of them referred to time) and were able to use it in a personal way to
answer the question put in general terms (it was not possible to just repeat the newly
established knowledge). The ability to determine and distinguish the different
components of height and increase was related to the possibility to master and
combine intentionally that set of additive situations (variation of length, reference and
dependence to time, and comparison of variations).
Extension of the scientific conceptualisation of the additive field: comparison of
additive transformations
Comparing transformations in the context of plants' increase (familiar because of the
long term work in the experience field of plant cultivation) was meaningful and
efficient: the growing of the plants confers to the additive situation a dynamical
character and a possibility of mental spatial representation. This allowed to master
and organise the various phases of calculus.
Development of systemic links of the additive conceptual field: