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