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During
years 70 and 80, international congresses on
mathematics' teaching were speaking only about
"curricula", i.e., in some way : courses
planing and contents : Should such subject be
taught before or after such other subject ? Must or
must not such area be taught ? That is now known as
transition from savant knowledge to taught
knowledge. In this whole process the student did
not exist.
Then came a
didactic concerned by "epistemological
obstacles". That means people become aware that
in the history of mathematics some subjects were
more challenging for researchers than some others :
the focus was still on mathematics per se, the
students were always absent but history and time
were introduced, in other words mankind.
Then the students
appeared in didactics but only through their
performances in problem solving. This was
the time of statistical didactic; when, for the
same problem, one was looking for the frequency of
the different results, right or wrong.
The student was
look at as a black-box with the problem to be
solved as an input and the produced solution as an
output.
Hypothesis on the
internal performance of the black-box were then
made in studying not only the results but also in
trying to understand the various strategies
used to produce the results. To do so the
students' drafts were collected, if necessary, to
figure out how they worked out the solution. Those
drafts were then used to understand the various
search strategies leading to the solution of the
same problem. The student was still a mute
black-box.
Next didactical
researchers became conscious of the students'
speaking ability !
They studied again
the strategies but this time by asking the students
how they solved the problem. They got "after the
facts" explanations transcripts; very rationalized
explanations to justify their process, but
explanations allowing to become conscious of how
important the student words were.
A big step was
made by a female didactical researcher (Viennot)
who showed, by studying what the students were
saying in physical sciences, that they followed a
logic of their own, they were building
"spontaneous theorems" who, although
inaccurate, were useful in the problem solving
process.
In other words,
students were using "models" (some people say
concepts) of the various physical processes,
deductive mathematics logic was not the only one
used in student reasoning but another logic
existed.
The only missing
part was the student's unconscious in order
to take into account all the complexity of the
student's personality. That is what, following my
personal research, researchers like Claudine
Blanchard-Laville ; Benoît Mauret ; Jean
Claude Lafon ; Nathalie Kaltenmark-Charraud ;
Isabelle René etc..., are doing.
Others researchers
show also that the student is not alone but that
the group of students is important in the
learning process, in other words individual
psychism is influenced by group
phenomena.
This history of
mathematics' didactic is an example of the
complexification work we all must
do;
Complexification:
- of our student
reasoning process' model,
(See : general
scheme of mathematics
representations)
This very work
will enlighten us to find the answers to the
various questions we are asking ourselves about our
teacher work. This evolution show also how
important is the representation concept and to take
into account unconscious in teaching mathematics.
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