Book · January 994 citations 110 reads 2,264 authors

particular, the expressions obtained for the solutions should not be self-evident,
(c) The qualitative solving, although easy at the start, allows broad categories of
solutions to be determined, to foresee in what way they will vary, but must not
allow all the problems set to be solved: for example, the existence of such and
such a type of solution, or the nature of such and such an infinite branch, (d) At
least some of these properties should, however, be accessible to algebraic solving.
This is followed by the presentation of the example used in the research,
showing that it respects the conditions required.
The above description concerns only the mathematical basis of the situa-
tion. It is indispensable but notoriously inadequate if, as in the systemic per-
spective adopted here, one takes up one's position not only on the level of
the contents but also on the level of the didactical situations through which
they are staged.
As far as the researcher is concerned, he or she now has to clarify how the
interaction between the students and this forecasting problem will be orga-
nized in the didactical situation, the consequences that can be inferred from
the anticipated behavior, and how this can be interpreted. In particular, the
researcher must show that the behavior anticipated has a high probability of
appearing and prove that it cannot be induced by interference, for example,
as phenomena related to the didactical contract.
DIDACTICAL ENGINEERING
36
2.
3.
4.
5.
6.
7.
Introduction to the qualitative approach.
Algebraic solving.
The complementarity of the algebraic and qualitative approaches.
Introduction to numerical solving.
The basic tools of qualitative solving.
Integration of the different tools in the solving of more complex prob-

---

This analysis, with the hypotheses on which it is based, is tested through
experimentation. This is organized around a questioning of the a priori
analysis of didactical situations. I cannot describe it in detail here. I shall
simply point out that it brings into play an interrogation on the knowledge at
stake in the situation; on the student and his or her possible relationship to
the problem set; on the role of the teacher; how he or she will intervene; and
on the possible implications of these interventions.
In production engineering, these demands weaken. The questioning re-
mains present in order to guide the conception, to ensure the necessary
didactical vigilance, but it is not directly involved in a process of internal
validation. Moreover, once more, a product that is too rigid is not desirable,
and, while attempting to avoid changing the nature of the situation, one
must take care to leave enough liberty in the management of the situation to
allow for necessary adaptability.
3.4 The Regulation of Didactical Engineering
At this point in the process, a teaching project is proposed. Its viability is
supposed but not guaranteed. In fact, experience has shown that an engi-
neering product is too complex an object to be able to be perfected at the
first attempt. Adjustments will therefore be made during successive experi-
mentations until, in the good cases, one reaches a product that is sufficiently
stable and satisfying to be distributed more widely. My work on differential
equations did not escape this rule. Three years were necessary to develop
the product that is now distributed by the University of Lille 1. In Artigue
(1992), I have analyzed the difficulties encountered and emphasized the in-
terweaving between cognitive difficulties and didactical difficulties. These
difficulties were finally solved, in particular, through the evolution of the
actual teaching contents. In order to face up to the cognitive difficulties en-
countered in the qualitative justification, it was necessary to develop a set of
justifications that operated directly in the graphic setting through relay theo-
rems formulated entirely within this setting. This elaboration allowed
wholly satisfactory results to be obtained, but, nevertheless, posed some se-
rious didactical problems due to the institutional status of the graphic
setting, highlighting the fact that the distribution of such a product, having
nevertheless proved its effectiveness, can only succeed if it takes into
account explicitly the in-depth renegotiation of this status, both with the
teachers and the students.
4. BEYOND THIS EXAMPLE: SOME PROBLEMS TO CONSIDER
After having used an example to try to illustrate how teaching contents are
prepared from a systemic perspective, I would like to return to more general
questions in the last part of this text.
The approach developed aims to take into account the reality of the sys-
tem in which the teaching contents have to exist, and subsequently presents
MICHELE ARTIGUE
37

---

38
the need for an elaboration that is not reduced to the text of the knowledge.
This expresses the wholly reasonable desire to avoid denying the complex-
ity of the didactical aspect. However, it must also be recognized that, at pre-
sent, the application of this approach at the level of production engineering
is not easy, and, moreover, stimulates, through the questions it raises, the
theoretical development of research. Artigue and Perrin (1991) have at-
tempted to analyze these difficulties in the construction of engineerings for
classes mainly containing learning-disabled students. Working with such
classes functioned like a magnifying glass through which the drastical
changes of nature accompanying the transmission become particularly
visible.
Many of these changes are the result of the gaps between the teachers'
beliefs about learning and their role as teacher and the representations un-
derlying the engineering: the teacher's desire to construct a smooth pro-
gression without any breaks, made up of little steps, in which nothing is
proposed to the student that has not already been prepared, to anticipate any
possible errors, which is opposed to the theoretical approaches in terms of
obstacles and cognitive conflicts but allows a comfortable management of
the didactical contract – everything is done so that the student who
cooperates can show the exterior signs of success; if the student fails, the
teacher is not in question. In all good faith, the teachers will therefore twist
the proposed engineering in order to adapt it to their representations and,
while believing that they have altered only a few details, will in fact have
changed its nature.
In fact, these difficulties are indirectly related to failings in the theoretical
framework on which the engineering is based. For too long, the theoretical
framework has not considered the teacher wholly as an actor in the situation
in the same way as the student, and modeling has remained centered on the
relations of the student to the knowledge. This level of modeling is inade-
quate to take into account the problems of engineering outside the strictly
experimental framework, and it is not by chance that, at present, research
concerning the teacher is expanding at a rapid rate.
Finally, besides these questions, designers of an engineering are faced
with delicate problems in writing up their work: What level of description
should they use? How can the underlying epistemology be maintained?
How can conciseness and accuracy be reconciled? How can conciseness and
the presentation of the product be reconciled? These problems, which can
already be seen appearing in any manual that attempts to stray from the
beaten track, are multiplied here, and it must be recognized that, for the
moment, we do not have the means to provide satisfactory answers.
The work accomplished up to now is certainly helpful for a better under-
standing of the problems linked to the preparation of teaching contents, for
the identification of the points on which efforts should be concentrated, and
it has also allowed the creation of a set of functional products that are com-
DIDACTICAL ENGINEERING

---

39
patible with the theoretical frameworks. However, no more than any other
approach, it does not provide a miraculous solution to these highly complex
problems.
REFERENCES
Alibert A., Artigue M., Hallez M., Legrand M., Menigaux J., & Viennot L., (1989).