Towards a theory of instrumental
2.3. Towards a theory of instrumental
conflict The didactical and pedagogical
traditions which have arisen owing to the laws and regulations of teaching and consequently since the industrialisation of teaching
determined how the majority of content should be presented in order for it to be assimilated by the greatest number of learners. It is a fact that these traditions are not always a great help when one wishes to introduce ICT into a learning and teaching situation.
In fact, things get even more complicated
pedagogical artefacts are associated with technical artefacts. For example, a software which teaches multiplication in elementary school is a technical artefact, which, to become a technical instrument has to become a learning object which, in turn, depends upon the instrumentalisation and instrumentation of the user. But as much as it may be a technical artefact, this VLE also brings into play the aforementioned didactical and pedagogical artefacts, which, in their turn, have to be suitably instrumentalised and instrumented in order to become real instruments. That which Peraya [13] terms techno‐semio‐pragmatic appears similar to what is referred to here as an overlay of three artefactual layers:
didactical, pedagogical and technical. (cf. fig 3).
Thus, the introduction of a technical system may provoke a disturbance of the balance between didactical and pedagogical artefacts, to the extent that the formalisms representation and/or the representation scenarios which were pertinent beforehand are found no longer usable. These disturbances to equilibrium may be termed instrumental conflicts, suggesting
that
the
processes of instrumentalisation and instrumentation of the various artefacts in question can interfere with each other.
In an instrumented teaching and learning situation, the learner‐subject is not only a physical, cognitive or social entity in interaction with a technical system, he is equally a subject who is intentionally engaged in the undertaking of his tasks [16]. In the realisation of these tasks, the learner carries out activities which can be both productive and constructive, to the extent that the subject produces a response to the situation and where the task concerned confers upon him an additional cognitive development. Each time one introduces a technical system, one takes the risk that the different levels of instrumental genesis may interfere with one another and deprive the learner at times of the possibility to respond to the situation and of constructing the didactical instrument as envisaged in the particular situation.
Fig. 3 Relationship of artefacts as sources of instrumental conflict
It is in this combination and usage learning: the environment allows one to see that learner‐users make of didactical,
and therefore to understand. (b) That of a pedagogical and technical artefacts that the
naturally positive contribution to teaching: optimisation and efficiency of learner
this illusion is based upon the principle that activity comes about, be it productive or
the introduction of a VLE results in reducing constructive and the hypothesis is advanced
the cognitive loading upon learners in the here that failure to produce the anticipated resolution of mathematical problems. In responses results from what may be called
instrumental conflict, in other words, the effect, the computerised artefacts would lift unfortunate association between one
the technical operations off the learners’ orseveral artefacts which produces a failure
shoulders and thereby allow them to focus in the instrumental genesis of at least one of
upon the mathematical objects. the three artefacts.
For the IT specialists, putting in place
a training structure can be achieved by the
III. IT‐BASED
DIDACTICAL AND successive addition of ‘digital building
PEDAGOGICAL OBJECTS IN THE
blocks’ of different shapes and sizes, which
TEACHING OF MATHEMATICS
can vary from a simple document right
A significant portion of VLEs have through to an entire training programme. been developed by ICT specialists with a
This vision arises from the engineering of view to autonomous learning with the
pedagogical objects based upon an object‐ benefit of pre‐existing knowledge or
centred approach, also used in the competencies [17]. VLE designers not
development of software. What is important necessarily being teachers or specialist
in this conception is the internal coherence trainers in tutoring learners, the question of
of the technical artefacts operating in the the integration and adaptation of these
learning context. But in the particular technical artefacts to classic teaching and
context here technical artefacts constitute training practices cannot fail to be posed.
the third element, giving rise to the This question is as much, if not more,
emergence of instrumental conflicts. What, important than the conception of learning
then, does the didactical object actually objects, which in the present context of ICT
signify for Mathematics teachers and the development covers matters which run far
pedagogical object for computer specialists? beyond the realm of pedagogical
engineering. In effect, it is all about
3.1 Computerised didactical objects as
normalising teaching content in order to be
viewed by teachers of Mathematics
able to organise it by function relating to the
A study of the usage of Dérive intended learning objectives, given VLE
1 software programme by Lagrange and functionalities, the language in which they
Drouhard [20] has shown that the pupils did are presented, the area of knowledge they
not automatically manage the transition address, etc. This paper holds to the logic of
from the technical to the conceptual and conception and representation of didactical
that they did not directly access the and pedagogical objects.
didactical objects which could be Mathematics is the school discipline
manipulated by the software. In reality, this area where the introduction of VLEs is the
process did not work to solve every most evident, from the very inception of
problem and its operation could only be which one often finds a quite animated
technical because resorting to Dérive did discourse [18]. According to Guin and
Trouche [19], this discourse draws its
1 This software programme is just part of the panoply legitimacy from Piaget’s constructivism and
of digital calculation programmes. The originality of is in fact characterised by a twin illusion: (a)
Dérive lies in the fact that it was conceived as the That of a naturally positive contribution to system covering the broadest possible range of formal calculus. It did not really address a specific teaching need.
not mean by its very use that a better that for the pupils it is a case of bringing explanation of processes would be achieved.
forward ideas of factorisation in the classic From this observation Artigue [18]
paperpencil model with the help of Dérive, deduced the existence of two phenomena,
and the other in which the pupils would that of double‐reference and pseudo‐
produce results of factorisation coming transparency, in order to explain the
from Dérive. In both cases, the pupils integration complexity of a VLE, in this case
encounter difficulties. These difficulties Dérive, in the context of teaching and
result from the computer transposition as learning in Mathematics.
Balacheff [21], expressed it, and from constraints associated with such a transfer.
3.1.1 Double reference This will be addressed further later. The situations observed by
The second observation relates to Lagrange and Drouhard [20] occurred in
trigonometric calculus with the help of two environments: that of the software
Dérive. It resulted in the same conclusions programme and that of paper‐pencil and
being drawn according to which pupils are consisted of factorisations of the polynomial
confronted with simplification difficulties Xn‐1 and of trigonometrical calculation. The
with the software programme. Although the phenomenon of double reference thus
simplifications that Dérive enables are arises from a confrontation of the
based upon the formulae of classical traditional environment, paper‐pencil, and
trigonometry, not least there remains the that of Dérive [19].
problem that Dérive’s simplifications are Artigue [18] thus takes account of the
difficult to put into practice. rational factorisation of the polynomial in
It would appear to be very clear that the penultimate year of highschool: “In the
the integration of a VLE in learning does not paper/pencil environment the factorisation
make any easier or better teaching and of the polynomial is linked, at this
learning situations in mathematics, as the educational level, to research into real roots
proponents of this discourse would […], to techniques of polynomial division
suggest. On the contrary, the computer […]. Dérive’s algorithm in the internal
transposition often comes with constraints workings of the machine worked by
which can constitute real handicaps to intermediary factorisations in Z/pZ.
learning. These constraints thus weaken the Evidently these two levels were not
mediation capacity of the technical artefact. accessible to these 17 year old pupils,
Under these conditions the software because Dérive was to function as a ‘black
programme no longer plays its role in box’ producer of various results which
epistemological mediation such that would be valid a priori” (p. 20). In reality,
achieving the didactical objective (in this the fact that the technical system was
case the cognitive mathematical objective) similar to a ‘black box’ is not unusual. For
is no longer possible.
example, when pupils use a calculator, they In her thesis concerning the do not have access to the way in which the
integration of spreadsheets in algebraic machine does its calculations. The
calculations, Haspékian [22] cites evidence difference between the factorisation of
of the difficulty teachers have in moving polynomials situation and that of doing
traditional paper/pencil simple calculations with a basic calculator
from
the
environment to the electronic spreadsheet. lies essentially in the difference in degree of
The difficulty lay in integrating a tool with complexity, which is determined by the
such variable functions as this. She teacher.
introduced the notion of instrumental The author shows that in this case
distance which she summed up as “the there are two possible interpretations: one
stronger the degree of instrumentation stronger the degree of instrumentation
phenomenon as the gap between what is the greater the distance from ‘habitual
written by the student and that which is scholarly practice’, the more the tool will
shown on the screen: “to enter (a+2)/5, seem difficult to grasp” (p. 296). She
certain pupils, having correctly added the demonstrated that in such a situation, a
pair of brackets around the (a+2), were teacher who is not an expert user of the tool
astonished to find their screen showing the can present an additional complexity to the
data without brackets and asked if what organisation and management of teaching,
they had done was right or not. The because the introduction of the spreadsheet,
appearance and disappearance of brackets as in this case, implies that new teaching
seemed, to some of the students, to be and learning practices be put in place which
playing a rather mysterious game which take full account of the constraints and
they little understood such that they could properties of the spreadsheet.
not work out what brackets were supposed It is crucial to assert that instrumental
to be about”. (p. 64)
distance as measured by the greater or Artigue [18] points out that the Dérive lesser degree of difficulty in integrating the
interface did not at a stroke enable students spreadsheet may be translated as the term
to alter the length of the line between the instrumental conflict used herein, as the
upper and lower elements of a fraction notion of difficulty makes reference to the
which they could do all too easily by hand. problematic combining of didactical,
And yet this information is necessary as it pedagogical and technical instruments. In
allows students to know where the line in a effect the use of spreadsheets implies the
fraction should go. There is in this a introduction within the teaching and
constraint linked to the fact that the learning system of new objects, of a new
keyboard only provides for one keystroke representation, of new functions and
for division. There is unarguably a significance, thus new symbolisms. The
discrepancy produced by the transition to period necessary to master these new
computer between the traditional didactical capacities is inevitably going to be one of
object and the computerised didactical upset and tension: one of disequilibrium in
object: this is the phenomenon of pseudo‐ the teaching and learning process.
transparency.
The double reference appears very This situation represents an obstacle similar to the fact that didactical objects
to the identification of mathematical such as defined by teachers according to the
symbols whose function is precisely to posture of non‐dissociation between noesis
enable pupils to develop their capabilities and semiosis are transposed by the student:
in Mathematics. The lack of the facility to be such as paper‐pencil for the VLE. The fact
able to produce these lines with the Dérive even that the notion of double reference
software programme is an example of a should be necessary to explain usage
situation in which the introduction of a VLE difficulties encountered by students
a demonstrates the consequences of
is
responsible
for introducing
disequilibrium in the learning process. As nondissociation whenever didactical objects
symbolic representations, the lines in a are computerised.
fraction, which here are taken as pedagogical objects, only have one role,
3.1.2 Pseudo‐transparency which is to assist in the resolution of the In order to provide an illustration of
mathematical problem. They are also a the notion of pseudo‐transparency an
means of more clearly identifying a example drawn from the work of Guin and
object critical to Trouche [19] will be adopted, as borrowed
mathematical
conceptualisation [23]. This aspect is very conceptualisation [23]. This aspect is very
may have been.
considered as a rupture of cognitive development. This research, however,
3.2. Computerized pedagogical objects as
considers such a disequilibrium caused by
viewed by computer specialists
the Dérive environment as being It is interesting to note that computer instrumental conflict, as the failure in the
specialists’ thinking regarding pedagogical implementation of the pedagogical artefact
objects emerged at the same time as the advent of VLEs. The term pedagogical
and the line between the two components of object, synonym of learning object, only
a fraction act against the pupil’s way of makes sense in relation to the latter. This working and thus prevent him from coming object‐oriented approach has gone through to an understanding or of appreciating
three successive phases, which were significance, i.e. from the didactical artefact.
crystallised in norms: LOM [24], SCORM This example of pseudo‐transparency
[25] and EML [26]. It should be recalled provides the opportunity to confirm the
before moving on that these three models of existence of a semiotic non‐conformity
pedagogical artefact correspond to three between the traditional and VLE
drivers (respectively economic, technical environment. The fact that showing
and pedagogical) which preside over object brackets was simply not possible on the
Moreover, what Dérive interface or that the keyboard could
conceptualisation.
computer specialists call objects are in not be given specific functions enabling the
reality artefacts in as much as they are not embodied in a VLE and in interaction with a
writing of differentiated signs of lines duly
remain symbolic adapted to a perfect and complete constructions fixed by digital processes. representation of the mathematical contents Pernin [27] highlights the lack of of division served to disturb the majority of
userlearner,
they
consensus as to the definition of a pupils. Such ambiguities could also arise
pedagogical object, and this despite the without the use of a computer, but they are
definition given to it by the work group normally well dealt with by teachers who
IEEE‐Learning Technology Standards can most easily resolve the disequilibrium
Committee. In effect for the IEEE‐LTSC between the formalisms of representation,
a pedagogical/learning object is defined as that is to say between the semiotic registers
“any entity, digital or otherwise, which can and cognitive objects. Dérive here creates a
be used or referenced in training provided disequilibrium
by a means of technological support”. pedagogical artefact (in this case the line in
Looked at more closely, the definition which computer specialists give of the pedagogical
the fraction or division) and thus the object is not too far from this. For David [28] formalism of representation of the
a pedagogical object is a digital document didactical artefact which can also create
allowing the learner to get engaged in an difficulties for the teacher without good
autonomous learning activity regardless of anticipation on his part.
the context of object utilisation. Put another It would seem that, beyond the
way, it has to be reusable in all learning perspective offered by Artigue [18] in
contexts.
proposing this notion of pseudo‐ But in order for a digital object to transparency as a means to study semiotic
stake a claim to being a pedagogical object, non‐conformity between the traditional
its conception has to integrate the paper‐pencil context and that of a VLE in a
recommendations of pedagogical activity. transposition to a computerised situation,
The model object to which he makes the real problem is to take account of
reference is that which complies with the LOM norm specifications, the structure of
the possible deformation of didactical which is based upon four levels comprising
objects as it arises from the use of the course, the lesson, the curriculum and objects as it arises from the use of the course, the lesson, the curriculum and
intermediary. It constitutes of a coherent granular structure in all technological
grouping of basic digital resources capable learning environments. What is central to
of being shared amongst learners on a the conception of this model is its
distance learning platform. At this level the characteristic of reusability. It is very much
system allows control of the carrying out of
a vision which gave rise to the concept of learning activities. It makes possible the the inter‐operability of VLEs, according to
provision of information on resources which digital resources have to be able to be
utilisation and the carrying out of activities compatible with the technical structures
on the platform by the key players. (c)The where they are likely to be used. However,
third level is that of the bringing together of the LOM model has not enabled convenient
the content. This provides a coherent and ‘universal’ inter‐operability to be
structuring of content at the core of an achieved.
entity deemed of higher level, like a course, Another
very
computing‐based
chapter or module.
conception of the concept of the pedagogical The LOM and SCORM models, let it be object is provided by Contamines, George
remembered, serve to facilitate the and Hotte [7]. It must be borne in mind all
orientation and indexation of pedagogical the time that these authors did not use the
objects, and precisely apply this role of the term pedagogical object but that of
pedagogical object to very diverse entities. educational resource, covering a great
The principal consequence of this is that one variety of learning objects. Beyond the
cannot discern between a pedagogical indisputable relevance and interest that can
object and a didactical object, such that this
be accorded to their work, it is no less well research is left to attempt to do it by founded than the meaning ‐ of the rest
separating that which relates to the borrowed from Klassen [29] focusing upon
disciplinary content taught from the four points ‐ which they give to pedagogical
formalism of representation or presentation objects, which serves to increase the
for teaching purposes. This lack of confusion which reigns around the
discernment would appear to reside in the definition of pedagogical objects. For them,
fact that the central aspect of the object‐ an educational resource is an ‘atomic’ entity,
oriented approach relies less upon learning
a video clip or a web page for example. It is activity than upon computing artefacts. In also of a composite nature and refers to a
effect, these models consider elementary non‐dissociable
artefacts to be just as much pedagogical multimedia) or an assembly of learning
whole
(didactical
objects (although they are located at objects (p. 161). It is appropriate to note
different levels), such as images, web pages, that this ambiguity concerning pedagogical
content structures, courses, lessons and objects can on the part of learners
modules. Yet it would seem necessary to themselves lead to a situation in which
make a distinction between pedagogical they have altogether different ideas of
artefacts which can be considered as what constitutes a pedagogical object.
scenarios and formalisms which serve to If the construction of the LOM model
present the didactical artefacts which are has not offered much satisfaction in respect
the contents of learning. of its own expected constituted functions, that is to say the reutilisation of pedagogical
IV. RELEVANCE OF THE CONCEPT OF
objects in all VLEs, one can nevertheless
INSTRUMENTAL CONFLICT
recognise that the SCORM model represents The concept of instrumental conflict progress in the computing conception of
draws its relevance from the generalisation pedagogical objects. It concerns a model by
of the use of ICT in teaching. As it has been Pernin [27] composed of three well‐
noted earlier in this paper, the introduction identified levels: (a) The first is that of the
of a VLE might disturb the very equilibrium basic digital resource, such as an image:
of a classical teaching situation, in which the JPEG or GIF, a WAV or MP3 sound file, a
didactical artefacts can be conveniently didactical artefacts can be conveniently
represent mathematical objects by several instrumented by the learners, and so that
semiotic systems, what is clear from the they thus become socially useful
classical form of teaching can reveal itself to instruments. But the evidence provided
be that much more difficult, even impossible herein would seem to indicate clearly that
when a technical artefact is introduced. the two scientific communities interested in
For computer specialists, the notions ICT in teaching are coming up against
of granularity and inter‐operability enable difficulties in identifying didactical objects
the LOM and SCORM indexation norms to and pedagogic objects when they are in
deal with the variety of pedagogical objects computerised form.
that they would seek to describe, but also to conceptualise the difficulty brought about
4.1. Further notions concerning related
by the absence of a distinction between the objects pedagogical object as such and its
For teachers of mathematics, the integration within a technical system. notions of double‐reference and
Everything happens as if (and this would pseudotransparency take account of the fact
seem both accurate and to be the norm) the that accessible didactical objects in some
mathematics teachers could not easily software programmes do not always work
computerise certain of their didactical for their pupils whether it be relating to
objectives, for lack of ability to their paper‐pencil representation or in
conceptualise the dissociation between the accommodating the constraints imposed by
taught content and its formalism of the user‐interface. From an instrumental
representation or its presentation for perspective, the difficulties encountered by
teaching purposes, and as if the computer pupils are an inadequacy in the combination
specialists could not suitably put of didactical artefacts which are the
pedagogical objects into a teaching mode by mathematical objects and pedagogical
reason of also not being able to make the objects, i.e. their formalisation by
same distinction.
mathematical signs in a computerised
Fig. 4 Distinction between didactical, pedagogical and technical objects according to existing
approaches .
In a way, teachers of mathematics and Modelling Language) developed by Kopper the computer‐specialists are giving two
[26, 30] which is at the origin of the IMSLD 2 different names to the same objects and are
[31] presents a real leap forward in the in need of further objects to account for the
pedagogical realm when compared to the difficulties posed by their respective
LOM and SCORM models already nomenclatures (cf. fig 4). Instrumental
addressed above. This language for theory and the separation that has been
pedagogical modelling identifies several introduced here between didactical objects,
types of activity amongst which are learning pedagogical objects and technical objects
activities, student support activities and provides the opportunity to unify these two
instrumentation activities [27]. conceptions of the integration of ICT in
This refocusing upon the activity is teaching.
becoming common while designing distance The distinction between didactical
learning platforms. The majority of VLE objects, pedagogical objects and technical
platforms draw upon a representation of objects is not just an exercise in rhetoric