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didactical situations, initiated by G. Brousseau (1986) at the beginning of
the 1970s and developed by several different researchers since that time.
My text is located within this perspective. The first part attempts to clar-
ify how the theoretical frameworks mentioned above shape the approach to
the preparation of mathematics for students by leading it, in particular, to be
placed in a more global systemic perspective than that frequently associated
with approaches in terms of the elementarization of knowledge. Then I shall
use an example to show how these theoretical frameworks become opera-
tional in the development of teaching products through the concept of
didactical engineering. In the conclusion, I shall return to more general
questions that are still largely unanswered.
2. A SYSTEMIC APPROACH TO THE DEVELOPMENT AND
ANALYSIS OF THE CONTENTS OF TEACHING
As pointed out above, this text uses the methodological concepts and tools
provided by two distinct theoretical frameworks, the theory of didactical
transposition and the theory of didactical situations, to study the issue of the
preparation of mathematics for students. Of course, such a short text is un-
able to launch into an explanation of these theories (the reader is referred to
the texts cited in the references); nevertheless, it is clear that these theoreti-
cal frameworks shape and determine, to a certain extent, the current ap-
proach to this issue. This is precisely the point I shall try to clarify first.
The two theoretical approaches mentioned above concern fundamental
but different levels of didactical analysis:
1. The theory of didactical transposition concentrates on the analysis of
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 27-39.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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those processes that are based on reference knowledge, particularly on the
reference knowledge produced by the legitimizing mathematical institution
(scholarly knowledge), that lead to objects of teaching (knowledge to be
taught) that are found in the daily life of the class (taught knowledge). It
naturally tries to go beyond particular studies and highlight certain laws and
regularities in these complex transposition processes.
2. To a certain extent, the theory of didactical situations is situated at a
more local level. It aims to model teaching situations so that they can be de-
veloped and managed in a controlled way.
However, despite their different focuses of interest, these two theories
link up on one essential point related to our topic: They emphasize the need
to envisage the study of didactical phenomena within a systemic approach.
Therefore, in both cases, the preparation of mathematics for students cannot
be perceived as a simple process of the elementarization of knowledge es-
tablished elsewhere, as the simple search for a presentation of some mathe-
matical content adapted to the previous knowledge and cognitive abilities of
students. It is perceived as a didactical task requiring a more global systemic
analysis.
2.1 The Systemic Approach Via the Theory of Didactical
Transposition
If one adopts a "didactical transposition" approach, one introduces an open
system to the analysis that includes, in particular, the institutions at the
source of the knowledge one aims to teach and the institutions targeted by
this teaching. This is done by questioning the constitution and life of this
knowledge, while remaining particularly attentive to the economy and ecol-
ogy of the knowledge to be taught. One questions the possible viability of
the content one wishes to promote while considering the laws that govern
the functioning of the teaching system. One tries to foresee the deformations
it is likely to undergo; one tries to ensure that the object can live and there-
fore develop within the teaching system without too drastically changing its
nature or becoming corrupted.
The reform of modern mathematics has provided excellent ground.for the
study of these phenomena of didactical transposition, and it is, mainly, the
ground chosen by Y. Chevallard in the first reference cited above. The
reader is also referred to Arsac's (1992) review analyzing the evolution of
the theory through studies undertaken both within and beyond the field of
the didactics of mathematics, as well as the following recent doctoral theses:
1. M. Artaud (1993), who studied the progressive mathematization of the
economic sphere, the obstacles encountered, the debates and negotiations
that arose around this mathematization, and their implications for the con-
tents of teaching itself.
2. P. Tavignot (1991), who used a study of the implementation of a new
way of teaching orthogonal symmetry to 11- to 12-year-old students within
DIDACTICAL ENGINEERING
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the French junior secondary school reforms (commenced in 1986) to de-
velop a schema for the investigation of this type of process of didactical
transposition.
I have also used this theoretical framework to study the evolution of the
teaching of analysis in "lycées" (senior secondary school) over the last 15
years, through the evolution of a didactical object, "reference functions,"
which acted as a sort of emblem for the rupture caused by the rejection of
the formalized teaching of modern mathematics (Artigue, 1993).
However, it must also be recognized that, up to the present, the theory of
didactical transposition has mainly been used to analyze transposition mech-
anisms a posteriori. It has hardly ever been involved in an explicit way in
the design of teaching contents or products. For this reason, the rest of this
text will concentrate to a greater extent on the more local approach linked to
the theory of didactical situations and the operationalization of the latter
through didactical engineering.
2.2 The Systemic Approach Via the Theory of Didactical Situations
The present approach will be just as systemic but will concentrate on nar-
rower systems: didactical systems, built up around a teacher and his or her
students, systems with a limited life span, plunged in the global teaching
system, and open, via the latter, to the "noosphere" of the teaching system
and, beyond that, to the society in which the teaching system is located.
The theory of didactical situations, which is based on a constructivist ap-
proach, operates on the principle that knowledge is constructed through
adaptation to an environment that, at least in part, appears problematic to the
subject. It aims to become a theory for the control of teaching situations in
their relationship with the production of mathematical knowledge. The
didactical systems considered are therefore made up of three mutually
interacting components, namely, the teacher, the student, and the
knowledge. The aim is to develop the conceptual and methodological means
to control the interacting phenomena and their relation to the construction
and functioning of mathematical knowledge in the student.
The work involved in the preparation of teaching contents labeled by the
expression didactical engineering, which is the focus of this text, will be
placed in this perspective. Alongside the elaboration of the text of the
knowledge under consideration, this needs to encompass the setting of this
knowledge in situations that allow their learning to be managed in a con-
trolled manner.
2.3 The Concept of Didactical Engineering
The expression "didactical engineering," as explained in Artigue (1991),
actually emerged within the didactics of mathematics in France in the early
1980s in order to label a form of didactical work that is comparable to the
work of an engineer. While engineers base their work on the scientific
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knowledge of their field and accept the control of theory, they are obliged to
work with more complex objects than the refined objects of science and
therefore to manage problems that science is unwilling or not yet able to
tackle.
This labeling was viewed as a means to approach two questions that were
crucial at the time:
1. the question of the relationship between research and action on the
teaching system,
2. the question of the place assigned within research methodologies to
"didactical performances" in class.
This twin function will determine the route that didactical engineering
will take through the didactical establishment. In fact, the expression has
become polysemous, designating both productions for teaching derived
from or based on research and a specific research methodology based on
classroom experimentations.
This text focuses particularly on the first aspect. The reader who is inter-
ested in the second is directed to Artigue (1989a). Nonetheless, it should be
emphasized that didactical engineering for research and didactical
engineering for production are closely interrelated for a variety of reasons.
In particular, there unfortunately does not exist what, at present and at least
in France, could be considered as a body of didactical engineers, and
didactical engineering for production is still essentially carried out by
researchers. It has developed without becoming independent from research:
In production, one simply weakens the methodological constraints of
research by integrating them in the form of questioning that guides the
conception, but the handling of those problems that are not dealt with by the
theory is not mentioned explicitly.
The following section presents an example of how the preparation of
teaching contents can be organized from the perspective of didactical engi-
neering. The example is a reform of the teaching of differential equations
for first-year university students (in mathematics and physics) undertaken in
1986 (Artigue, 1989b; Artigue & Rogalski, 1990). This presentation will try
to bring out the conception of transposition work inferred from the approach
chosen and the role played by its theoretical foundations.
3. PRESENTATION AND ANALYSIS OF A PIECE
OF DIDACTICAL ENGINEERING
The question to be dealt with here concerns the reform of an element of
teaching. The didactician, either a researcher or an engineer, is therefore
faced with a teaching object that has already been implemented. Why
should it be changed? What aims should be included in this reform? What
difficulties can be expected, and how can they be overcome? How can the
field of validity for the solutions proposed be determined? This set of ques-
DIDACTICAL ENGINEERING
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tions must be answered. The work will be made up of various phases. These
phases will be described briefly.
The first, unavoidable phase consists in analyzing the teaching object as it
already exists, in determining its inadequacy, and in outlining the episte-
mology of the reform project.
3.1 The Characteristics of Traditional Teaching: The Epistemological
Ambitions of the Reform Project
In the present case, it had to be noted that, when the study began, the teach-
ing of differential equations for beginners had remained unchanged since at
least the beginning of the century, but that it was also at risk of becoming
obsolete. In order to describe it, I shall refer to the notion of setting intro-
duced in Douady (1984) to diferentiate three essential frameworks for solv-
ing differential equations:
1. the algebraic setting in which the solving targets the exact expression
of the solutions through implicit or explicit algebraic formulae, develop-
ments in series, and integral expressions;
2. the numerical setting in which the solving targets the controlled numer-
ical approximation of the solutions;
3. the geometrical setting in which the solving targets the topological
characterization of the set of solution curves, that is to say, the phase
portrait of the equation, a solving that is often qualified as being qualitative.
French undergraduate teaching was (and still mainly is) centered on alge-
braic solving, with an empirical approach that is characteristic of the initial
development of the theory. This is a stable object that is alive and well in
the teaching system, but it leads students toward a narrow and sometimes
erroneous view of this field. For example, most students are convinced that
there must be a recipe that permits the exact algebraic integration of any
type of differential equation (as they never encounter any others), and that
the only aim of research is to complete the existing recipe book.
If one considers the current evolution of the field, of the growing impor-
tance of numerical and qualitative aspects, such teaching is, despite its long
stability, inevitably threatened with becoming obsolete.
The aim of the work undertaken was to construct a teaching object that
was epistemologically more satisfying, mainly by:
1. opening up the teaching to geometrical and numerical solving and by
managing the connections between the different solution settings in an ex-
plicit way;
2. reintroducing a functionality to this teaching by modeling problems
(internal or external to mathematics) and by tackling explicitly the rupture
necessitated by the transition from functional algebraic models to differen-
tial models (Alibert et al., 1989; Artigue, Ménigaux, & Viennot, 1989).
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Beyond a simple elaboration, the conditions for the viability of such an
object were studied with an experiment carried out in a reformed DEUG
(first two years of university) at the university of Lille I.
3.2 Phase 2 of Engineering: An Analysis of Constraints
In order to better understand and manage the available possibilities, the di-
dactician uses the systemic perspective to view the teaching to be updated as
the equilibrium point of a dynamic system. It is this equilibrium that has to
be studied in order to obtain an idea of its stability and to analyze the rea-
sons for such stability in terms of constraints. By modifying at least some of
these constraints, one may hope to see the system stabilize at another point
of equilibrium that is judged to be more satisfying. An inadequate analysis
of constraints may lead to failure or more certainly (as experiments have a
strong tendency to succeed!) to a more satisfying point of functioning, but
one that only appears viable because it corresponds to a maintained equilib-
rium.
Such an analysis must distinguish between different types of constraint.
Classically speaking, three types of constraint can be distinguished:
1. constraints of an epistemological nature linked to the mathematical
knowledge at stake, to the characteristics of its development, and its current
way of functioning;
2. constraints of a cognitive nature linked to the population targeted by
teaching;
3. constraints of a didactical nature linked to the institutional functioning
of the teaching, especially in the field concerned and in connected fields.
The identification and analysis of constraints gives rise to the further dis-
tinction of constraints that can be qualified as external, which are to a great
extent unavoidable except in the case of exceptional actions, and of con-
straints that appear to be constraints because they have been internalized by
the actors in the didactical relationship, but are no longer such for the
current system. These may be qualified as internal.
If one considers the constraints in the present example that are opposed to
the extension of the teaching contents to a qualitative approach to the solv-
ing of differential equations, the following main constraints can be identi-
fied:
1. On the epistemological level: (a) the long domination of the algebraic
setting in the historical development of the theory; (b) the late emergence at
the end of the 19th century of geometrical theory with the work of H.
Poincaré; (c) the relative independence of the different approaches, which
permits, even nowadays at university level, a certain ignorance regarding
the qualitative approach; and, finally, (d) the difficulty of the problems that
motivated the birth and subsequently the development of the geometrical
theory (the three-body problem, the problems of the stability of dynamic
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systems, etc.) and the resulting difficulty on the level of elementary
transposition processes.
2. On the cognitive level: (a) the permanent existence of mobility between
registers of symbolic expression required by the qualitative approach:
mobility between the algebraic register of the equations, of the formal
expression of the solutions, and the graphic register of curves linked to the
solution (isoclinal lines, curves of points of inflexion, solution curves) –
increased cognitive difficulty being due to having to work on at least two
levels simultaneously: that of functions and that of derivatives; (b) the fact
that teaching is aimed at students for whom the concept of function, the
links between registers of symbolic expression, are, in fact, in the construc-
tion stage; and, finally, (c) the mastering of the elementary tools of analysis
required by qualitative proofs.
3. On the didactical level: (a) the impossibility of creating algorithms in
the qualitative approach, which presents a serious obstacle if one considers
the extent of the recourse to algorithms in teaching; (b) the relative ease of
traditional algebraic teaching, which can give rise to algorithms, and the
status this ease gives it in the DEUG curricula (a time when the pressure
caused by new formal and theoretical demands is relaxed, and when even
momentary success allows didactical negotiation to be taken up again); (c)
the inframathematical status in the teaching of the graphic setting, a
framework that is, however, essential here; (c) the need for the teacher to
manage situations in which, as is generally the case in qualitative solving,
he or she cannot answer all the questions that arise naturally; and (d) the
marginal nature of elementary courses that develop a truly qualitative
approach and the difficulty, consequently, in finding texts that can be used
for reference (currently a text such as Hubbard & West, 1992, could fulfill
this role).
The first two phases constitute an essential component of any serious en-
gineering work, even if this component does not often appear in the finished
products. In fact, this work, which is fundamental for engineering, is only at
its initial stage. It remains constantly present in the background of the con-
ceptual work and will generally be revised after the first experimentation
with the engineering, when the hypotheses and choices that guided the con-
ception have been confronted with "reality." As a counterbalance to the
analysis of constraints, it allows didacticians to define how much freedom
they have, to estimate how much room they have to maneuver: It guides,
therefore, in an essential manner, the subsequent choices that can be made.
3.3 The Actual Conception of the Engineering
In line with the preceding section, the conception of the piece of engineer-
ing is subject to a certain number of choices. In particular, the constraints,
either internal or external, that seem to oppose the viability of the project
have to be displaced, at a reasonable cost.
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These choices can be distinguished as: