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1. macrodidactic or global choices that guide the whole of the engineer-
ing;
2. microdidactic or local choices that guide the local organization of the
engineering, that is, the organization of a session or a phase.
In the present example, the main choices made on a global level are the
following:
1. Making explicit the contractual change in the status of the graphic set-
ting through the introduction in the teaching of a work module on functions
and their representations that breaks away from normal practice in sec-
ondary teaching. Here the didactical and cognitive constraints linked to the
status of the graphic framework in teaching have to be tackled, and, at the
same time, the students have to be prepared for the mobility between the
registers of symbols required by qualitative solving.
2. Use of computers. In these situations, computers initially seem to pro-
vide a way of breaking up the complexity of qualitative solving. Indeed,
they are used in order to embed qualitative solving into a structured set of
tasks of varying complexity (tasks of association between equations and
phase portraits, tasks of interpretation of phase portraits, tasks of more or
less assisted drawing of phase portraits or solutions with given conditions) –
a simplification that is more difficult to set up in a traditional environment.
Of course, they also appear to be a means of engaging in an approach to
numerical solving. Furthermore, as they allow nonelementary situations to
be controlled, they help to counter simplistic representations of the field.
3. The explicit teaching of methods for qualitative solving. Following the
ideas developed in Schoenfeld (1985) or Robert, Rogalski, and Samurcay
(1989), this means facilitating the construction of knowledge recognized as
being complex by introducing an explicitly metacognitive dimension into
the teaching.
4. The limitation of complexity on the level of the algebraic solution and
the transfer of the algorithmic part of this approach to independent aided
work. This last choice is imposed by obvious institutional constraints: The
time that can be given over legitimately to this part of the curriculum is lim-
ited; new objects cannot be brought in without some losses. Here, the global
status of the algebraic approach has been rethought: The cases studied
(linear equations, those with separable variables, homogeneous equations)
have been conceived as simple, typical examples that will act as a reference
in the future and will be used as instruments for comparison or approxima-
tion in the study of more complex situations.
Local choices are, of course, subordinate to these global choices and must
be compatible with them. It is at their level that the theory of didactical
situations is really applied.
At this point, it would seem necessary to distinguish between the func-
tioning of the two types of didactical engineering I have identified above:
DIDACTICAL ENGINEERING
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didactical engineering of research and didactical engineering of
production.
The first type constitutes a research methodology. It must therefore allow
for validation following explicit rules. Here, the validation is an internal val-
idation based on the confrontation between the a priori analysis of the sit-
uations constructed and the a posteriori analysis of the same situations.
Keeping in mind that the theory of didactical situations is based on the prin-
ciple that the meaning, in terms of knowledge, of a student's behavior can
only be understood if this behavior is closely related to the situation in
which it is observed, this situation and its cognitive potential have to be
characterized before comparing this a priori analysis with observed reality.
It is clear that such a position on validation is only tenable if the situations
involved in the engineering are strictly controlled regarding the contents
treated, their staging, the role of the teacher, the management of time, and
so forth.
The second type of engineering is more concerned with satisfying the
classical conditions imposed on engineering work: effectiveness, power,
adaptability to different contexts, and so forth.
Obviously, these demands are not equal. Hence, even if it remains
marked by the characteristics of research engineering, production engineer-
ing will, in this phase, take on a certain independence.
In both cases, one starts by searching for a reduced set of classes of situa-
tions that bring into play, in a way that is both suited to the epistemology of
the project and operational, the essential characteristics of the knowledge
targeted in the learning. Even if the concept is still under debate, one cannot
fail to mention the concept of fundamental situation introduced by G.
Brousseau (1986).
These classes of situations make up the structure of the engineering by
defining its key stages. In effect, the criteria that characterize each class al-
low an infinite number of situations to be produced. The researcher will
therefore choose from each class, concentrating on the variables that have
been left free, the specific situation(s) that he or she will integrate into the
engineering, and he or she will have to justify the choices made very pre-
cisely by linking them to the hypotheses underlying the engineering. The
time sequence planned for the situations must also be stated.
Didactical engineers are not expected to provide the same type of
construction. They are expected to highlight the core of the engineering and
to encourage the construction of products that respect this core in a
relatively concise presentation.
This is the type of presentation I attempted, no doubt imperfectly, in
Artigue (1989b). After specifying the global choices made and the reasons
for them, the engineering is presented in a seven-step structure, each step
organized around a few key situations. The seven steps are as follows:
1. What needs does the differential equations tool respond to?
MICHELE ARTIGUE
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lems.
Moreover, each key situation is not described as an isolated object but as
one possible representative of a class of situations specified by certain char-
acteristics. In particular, within each class, one can, depending on the popu-
lation and the time available, adjust the number of situations proposed and
their relative complexity.
As an example, I present the text introducing the key situation of Step 4
(translated):
The key situation retained as a basis for this step is that of forecasting the phase
portrait of an equation that can be integrated explicitly and that presents a certain
number of characteristics chosen in order to avoid putting one setting at a disad-
vantage in relation to another and to allow the dialectic between settings to be
undertaken at the desired level. In particular: (a) Starting a qualitative study must
be easy, as what is at stake in the situation is not located in difficulties at this
level. For example, one could arrange things so that the horizontal isoclinal line is
made up of straight lines, and so that certain particular solutions, which are rela-
tively easy (e.g., isoclinal lines), allow the research to be organized by providing
a regioning of the plane for the solution curves, (b) The algebraic solving, while
it does not give rise to any particular difficulties, must not be too easy; in