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Chapter 9
Instrumental Genesis in the Theory of MWS:
Insight from Didactic Research on Digital artefacts
DOI: 10.1007/978-3-030-90850-8_9
In book: Mathematical Work in Educational Context, Springer
Jean-Baptiste Lagrange* and Philippe R. Richard**
*LDAR, University of Paris, France
** University of Montréal, Canada
Abstract. The central focus of this chapter is the relationship between the no-
tions introduced by Vérillon and Rabardel (1995), ‘artefact’ and ‘instrumental
genesis’, and the homonym notions in the theory of MWS. Following a brief
overview of the Piagetian and Activity Theory perspectives which inspired
Vérillon and Rabardel, we explore how didactic research on digital artefacts
developed two viewpoints on the instrumental genesis: psychological and insti-
tutional. The idea of instrumental genesis in the theory of MWS is consistent
with the concept of instrumented mediation by Vérillon and Rabardel, but is a
more focused notion, since MWS include two additional geneses, semiotic and
discursive. The theory of MWS moreover does not theorize about the media-
tion and as such, in a particular situation where a digital artefact is used, both
viewpoints will be helpful. Regarding the poles of the instrumental genesis, we
propose characterizing a digital artefact by the underlying data representations
and algorithmic treatments, and constructions as data-result models of mathe-
matical configurations, taking advantage of the representations and algorithms
in the artefact.
9.1 Motivation
The authors of this chapter are two researchers in mathematics education with a
long experience in investigating the classroom use of digital artefacts and de-
signing mathematical software environments. Like many other researchers in
this field, we are inspired by Vérillon and Rabardel’s (1995) approach of in-
strumental activity, using the notions of artefact, instrumental genesis and in-
strumented techniques. In parallel, we have drawn on the theory of MWS for
specific research studies (Minh & Lagrange 2016, Richard & al., 2016). This
theory also puts forward the notions of artefact and instrumental genesis, fur-
thermore including two other geneses: semiotic and discursive.
2
The central focus of this chapter is the relationship between notions intro-
duced by Vérillon and Rabardel and the homonym notions in the theory of
MWS. Previous presentations of the instrumental genesis in the theory of MWS
(Kuzniak, Tanguay & Elia 2016) have briefly drawn on the work of Vérillon &
Rabardel, but without taking into account how the two approaches derive from
different research perspectives. The theory of MWS comes from didactic re-
search in geometry, and the work of Vérillon and Rabardel from cognitive er-
gonomics, or in other words, psychology in the workplace. Arguably then,
there is no reason why homonym notions should carry the same meaning. We
consider that in mathematics education research, as in other scientific domains,
it is important to work continuously on basic notions in order to specify the
meanings we seek to convey in a specific theory, and to be cautious with regard
to possible semantic bias when separate theories use a similar vocabulary.
This book is therefore an opportunity to accurately discuss instrumental gen-
esis in the two theoretical perspectives and we expect this discussion to provide
the firm foundations for further research. In order to retain specific focus, a first
choice in this chapter is to consider digital artefacts rather than a broader cate-
gory of artefacts.
The notion of digital artefact applies to a great diversity of devices ubiqui-
tous in human activity and especially in cognition, some of which have been in-
vestigated in Chapter 8. ‘Digital’ fundamentally denotes a choice of data repre-
sentation. For instance, to represent a color, a digital code enables the represen-
tation of a potentially great but finite number of colors, in contrast with an ana-
log representation, which allows any combination. This choice of data repre-
sentation has been adopted because it enables algorithmic treatment, i.e., treat-
ment by an automatic device following given rules: in the example of colors, an
algorithm triggers a display device to activate suitable light emitting diodes;
other algorithms can be used for various processes on a colored picture… In
this way, digital artefacts are characterized by data representation and algorith-
mic treatment. On the one hand, this characterization seems very simple with
regard to the multiple roles digital technologies play in all domains and particu-
larly in mathematics education, especially mediation and communication
(Clark-Wilson, Robutti & Thomas, 2020). On the other hand, a simple charac-
terization by data representation and algorithms proved powerful enough to en-
able Turing (1950) to demonstrate the possibility for digital machines to per-
form processes which seemed restricted to human cognition and which became
a reality more than a half century later. More modestly, we can assume that this
simplicity will be helpful for an in-depth discussion about the notions of arte-
fact and instrumental genesis.
Another choice made for this chapter is to return to founding texts: first the
above-mentioned paper by Vérillon and Rabardel (1995) and also Kuzniak’s
plenary address given at CERME 8 (Kuzniak, 2013) on Geometrical Working
Spaces. The aim of this is to understand how the constructs of artefact and in-
strumental genesis were introduced respectively in cognitive ergonomics and in
3
didactic research about geometry, along with their purpose. We then contrast
the two frameworks, furthermore considering how they evolved after their in-
troduction into the two perspectives. Of course, other works, such as those by
Rabardel and colleagues in cognitive ergonomics, and by Kuzniak and col-
leagues in didactics, could be considered, but we prefer to concentrate on fun-
damental constructs rather than attempt to cover the whole range of aspects in
the two frameworks.
9.2 Genesis and Schemes in Instrumented Activity
First of all, we consider the term "genesis" and its use in "mathematics educa-
tion". Piaget and Szeminska (1941) use "genesis of the number" in their theo-
retical framework to denote the acquisition or construction of numerical knowl-
edge in a developmental perspective (stages). At the same time, the notion of
scheme, is one of the most important in the Piagetian conception of psychologi-
cal activities. For Piaget, a scheme is the structure or organization of actions
such that they transfer or generalize upon repetition of that action in similar or
analogous circumstances.
The genesis is thus that of schemes. Vérillon and Rabardel’s approach is situ-
ated in a Piagetian perspective, drawing on the notions of schemes and genesis.
But Vérillon and Rabardel also stress that classical approaches in the psychol-
ogy of learning overlook situations of knowledge construction where the sub-
ject's action is not directed by an epistemic intention but is rather shaped by a
practical and functional project. For Vérillon and Rabardel, this pragmatic di-
mension of learning situations is important, particularly in science teaching and
in professional fields. Taking some distance from a pure Piagetian approach of
cognition, they refer to an activity theory perspective (Leontiev 1976) consider-
ing that "development results from the progressive appropriation of (...) so-
cially formed attainments”. For them artefacts must be conceived as a compo-
nent of such socially formed attainments: artefacts can be "all the objects of
material culture to which an infant has access during his development". After
the notion of artefact, the main construct of the instrumental approach for
Vérillon and Rabardel is that of instrumented activity. This is illustrated in the
SAI diagram (Fig. 9.1 below) where the "instrument" mediates the subject's re-
lationship to the object of their activity. The authors stress that the "instrument"
exists as an artefact, in the sense mentioned above, but it also has a psychologi-
cal dimension due to its appropriation by the subject in the context of their in-
strumented activity.
4
Fig. 9.1. The SAI diagram, taken from Vérillon and Rabardel (1995).
Vérillon and Rabardel describe processes of genesis in the context of the use
of machines by students, a robot and a lathe. They observe that the geneses are
not reduced to an adaptation (assimilation-accommodation in Piagetian terms)
of the subject to the artefact. The student rather simultaneously constructs both
an understanding of the functioning of the machine and a conception of the
space. They conclude by stressing the consistency of the observations with the
SAI diagram.
“The pupils' data acquisition strategies concerning the artefact, and their representative
and operational activity during its operation (subject/instrument interaction) always
turned out to be interdependent with their conceptions relative to the nature of the
transforming process (instrument/object interaction), with the artefact mediating their
action on the environment but also, in return, mediating their conceptualization of that
environment (subject/object interaction mediated by the instrument).”
In short, Vérillon and Rabardel drew the concepts of scheme and genesis from
Piaget and developed their notion of artefact from activity theory.
9.3 Genesis and Digital artefacts
In this subsection, we build upon Vérillon and Rabardel’s observations in order
to identify characteristics of geneses associated with digital artefacts. We con-
trast geneses of the lathe and of the robot. The lathe is used to cut a material
into a shape of revolution. To create a desired shape, the material is put into ro-
tation and the operator moves a cutting tool along two axes (respectively paral-
lel and perpendicular to the axis of rotation) by way of visible screws. The ro-
bot is used to move objects in a 3D workspace. Externally, the robot is made of
a mechanical device and a control box. The mechanical device is similar to a
human arm: a succession of three mobile segments connect a fixed base to a
5
clamp through four motorized articulations (joints). The control box is made of
three cursors arranged like the three axes of an XYZ orthogonal system. Be-
tween the control box and the device there is a digital processor programmed in
such a way that each position of the three cursors determines a position of the
clamp in a Cartesian representation of the workspace. The consequence of this
is that, in contrast with the lathe, there is no direct visible relationship between
action on a cursor and movement of a part in the device. Above, we character-
ized digital artefacts by data representation and algorithmic treatment. Accord-
ing to this characterization, the robot is, for us, a digital artefact: the control
box is represented as a triplet of coordinates and the robot is represented by the
position of the clamp in the space. Internally, the algorithm implemented on the
digital processor uses another representation of the robot made of an angular
position for each of the four joints, consistent with the mechanical functioning
of the robot: for a given triplet of coordinates, the algorithm computes values of
the four angular positions in order to ensure adequate positioning of the clamp.
Investigating this through experiments with students, Vérillon and Rabardel
observed, in the case of the lathe, a genesis of a geometrical relationship be-
tween the 2D profile created by activating the screws, and the resulting 3D
shape of the rotating material. With the robot, the genesis is marked by a pro-
gressive awareness by the student of different representations (internal and ex-
ternal) of the 3D space. Vérillon and Rabardel distinguish five microgeneses:
(1) The cursors are considered to be associated with movements of the dif-
ferent parts of the arm (joints or segments).
(2) The relationships between the cursors and the parts of the machine vary
depending on the zone where the arm is located.
(3) In some zones, the cursors are also considered to be interacting with
each other.
(4) Pupils become aware of a homomorphism between the geometry of their
actions and the geometry of movements of the clamp.
(5) The relationship between the control space and the working space is no
longer conceived in terms of movement but of positions. The action
schemes, which were previously movement schemes, become schemes
for defining positions inside these two spaces.
For us, the main purpose of the genesis is a better awareness of the algorith-
mic treatment between the control box and the mechanical part, as programmed
by the designer. Paradoxically, because the mechanical part is open to observa-
tion, the internal representation (angular position of the four joints) is more vis-
ible that the external representation required by the user-interface, especially
for students whose understanding of the 3D space has yet to be constructed.
This is why students initially associate actions on cursors with changes in the
angular positions of the joints. In this representation, action on cursors only
works locally or when acting on three cursors together (stages 1 to 3). Students
only gradually identify the fact that the position of the clamp is the object of
6
control, implicitly becoming aware of the algorithm at work (microgenesis 4)
before then understanding the representation of the robot in the 3D space cho-
sen by the designer (microgenesis 5). In keeping with Vérillon and Rabardel’s
analysis, we see the robot as an instrument mediating the students’ relationship
with the 3D space, but we also stress the fact that the artefact becomes an in-
strument for a subject through their developing awareness of both the different
representations and of the algorithm. Awareness is not in the details, but rather
in a global understanding of the input and output, and of the nature of the tran-
sition between these: in the case of the robot, one must understand that some
calculation on positions is made so as to pass from three coordinates to four an-
gles1.
9.4. Psychological and institutional viewpoints on the instrumental
genesis
At the end of the 1990s, researchers in mathematics didactics (Lagrange et al.,
2003) stressed the need for a "multidimensional" approach to digital technolo-
gies in mathematics teaching. Conducting a review of a significant sample of
research papers in this field Lagrange et al. highlighted the fact that a large ma-
jority of researchers adopted either an "epistemological and semiotic" perspec-
tive, centered on the influence of digital technologies on mathematical knowl-
edge and practices and on the representations used, or a "cognitive" perspec-
tive, seeing technologies as opportunities to implement (socio)constructivist
approaches. They proposed to break with unidimensional approaches and to
consider additional dimensions more directly focused on an analysis of the spe-
cific characteristics of the artefacts used. They identified potential new view-
points2 characterized by the indicators of analyses they found in the papers
they reviewed. The indicators of the psychological viewpoint were the analysis
of the possibilities and constraints of digital artefacts for a given subject, and
the evolution of schemes in instrumental geneses. The indicators of the institu-
tional viewpoint were the analysis of the interaction of ICT with tasks and tech-
niques in the culture of a school institution, and the consideration of the role of
instrumented techniques in conceptualization of mathematics.
Compared to the majority of research studies on mathematics education
about digital technologies currently published, this multidimensional approach
has constituted a more nuanced and reflexive approach to the role digital arte-
facts can play, by showing their potentialities, but also the conditions necessary
1 It is tempting to refer here to the idea of a “black box” understanding of the algorithm,
but it would be misleading since in the mediation, the box is never entirely black, but rather pre -
sented in varied “shades of gray”.
2 Lagrange et al refer to instrumental and institutional dimensions rather than psychological
and institutional viewpoints. We adopt the latter formulation to avoid confusions with the use of the
words instrumental and dimension in the theory of MWS.
7
for them to be implemented. This has made it possible to account for the role of
"mathematical" digital artefacts (numerical and formal calculation, dynamic ge-
ometry, spreadsheets, ad hoc programming, etc.) for students’ as well as teach-
ers’ conceptualizations, to show the complexity of the phenomena at hand and
to develop didactic engineering which takes this complexity into account. The
psychological viewpoint derives directly from Vérillon and Rabardel’s pioneer-
ing work reported above. The link between this work and the institutional
viewpoint is less obvious, since it uses the notion of praxeology and associated
concepts (tasks, techniques, theorizations) from the Anthropological Theory of
Didactics (ATD). Nevertheless, Vérillon and Rabardel stress that
“(the) instrumental relation is the subject of real occultation, producing what Y.
Chevallard (1991), in mathematics, called a phenomenon of hypostasiation, through which
the artefact is seen as existing in a sort of ideality independently from the concrete
practices from which it emerges or which constitute it.”
Indeed, according to Chevallard, knowledge itself is the object of hypostasia-
tion when teaching/learning is reduced to visiting a scientific content as a
“monument” overlooking its “raisons d’être”, that is to say the questions and
cultural environments (institutions in ATD) which motivated people to develop
this content. The idea of praxeology is the theoretical tool helping to position a
knowledge content in the development of human culture. Similarly, artefacts,
especially digital artefacts, need not be seen as the result of some serendipitous
invention, but rather the product of culturally (i.e. institutionally) situated activ-
ity. Thus, instruments exist through the praxeologies in which they are set into
operation for a given type of task, and the instrumented techniques are not
given with the artefact, but rather develop in a culture and in institutional prac-
tices. This is the case of the robot: the design with four joints results from me-
chanical constraints, but for tasks like moving objects under the control of a hu-
man operator, moving each joint individually would be a tedious technique. A
technique based on a homomorphism between the control space and the
workspace is more efficient and is therefore adopted in a workplace praxeol-
ogy. The theoretical level of the praxeology consists in the 3D representation of
the space and in the idea of an algorithm able to control the joints from this rep-
resentation. Similarly, mathematical instruments exist through the techniques in
which they are implemented for a given type of task:
“Taking techniques into account is, along with instrumentation, one of the keys to thinking
about the integration of technology in teaching (...) The interest of introducing an
instrument can be seen in the development by the students of new techniques that are
integrated into renewed praxeologies (Lagrange 2000).”
The psychological and institutional viewpoints are arguably complementary
and should not be confused. This contrasts with the argument of Drijvers
(2020), who writes:
"The observable parts of an instrumentation scheme, the concrete interactions between
user and artefact, are called instrumented techniques (...) As such, an instrumentation
scheme consists of one or more observable instrumented techniques...".
8
It is important to carefully distinguish between the subjective dimension of
the scheme (How is it formed by the subject? What mental operations are at
work?) and the institutional and social dimension of the techniques (In which
praxeologies do they fit? In what conditions do these praxeologies develop?)
Schemes and techniques each have their own identity, as do, fortunately, sub-
jects and institutions!
9.4 Work, geneses and artefacts in the theory of MWS
We now examine the basic concepts of the theory of MWS in light of the psy-
chological and institutional viewpoints set out above. The first concept is that
of mathematical work. Vérillon and Rabardel's approach is primarily concerned
with professional and technical domains, as shown by the empirical studies
(lathe, robot) cited in the article, and is based on the idea of activity rather than
work. Nevertheless, turning an object on a lathe, activating a robot for a given
task, are not free activities carried out with the aim of developing abstract geo-
metrical or computational knowledge, but rather activities aiming at the trans-
formation of material entities, and developing in social contexts. In parallel, by
emphasizing work and by considering mathematics both as a science and as a
field of activity, the theory of MWS breaks with the idea of mathematics as an
abstract field disconnected from social practices: like the lathe or robot opera-
tor, the mathematician works with instruments that have a psychological and an
institutional value. Regarding geneses and artefacts, as in the previous sections,
we refer to a text which was a precursor to the theory of MWS. In his plenary
address to CERME 8, Kuzniak (2013) introduces Geometrical Working Spaces
(GWS) and the geneses:
“In our approach, both levels, cognitive and epistemological, need to be articulated in
order to ensure a coherent and complete geometric work. This process supposes some
transformations (...) Here, in order to insist on the developmental process involved in the
constitution of GWS, the notion of genesis is used. For us, a genesis involves not only the
origin but also the development of a process.”
The instrumental genesis, on which we particularly focus here, connects
artefacts, a pole of the epistemological plane, and construction, the correspond-
ing pole in the cognitive plane. Three remarks seem important:
– On the MWS classical diagram and at a first glance, the geneses seem ob-
jective in character, since they connect components of the geometrical activity
in its purely mathematical dimension to observable processes: the instrumental
genesis with a drawing square, for example, would be simply the use of the
square to draw right angles in a given construction. Nevertheless, the words
used - "developmental", "transforms", "provides... status", "gives a meaning" –
all suggest that transformations take place within the subject(s) or institutions
which use the artefacts. Further texts (Chapter 1 of this book) have explained
9
how two views of the instrumental genesis can be considered. The bottom-up
view describes the actions by which the user appropriates the various tech-
niques of use related to the artefact. The top-down view begins from the user's
intended accomplishment, to his or her adequate choice of the artefact with
which to perform the required actions. As such, the instrumental genesis
changes the way the user acts and thinks, and, in return, also transforms the
mathematical knowledge. This is compatible with Vérillon and Rabardel's idea
of instrumental mediation (SAI diagram fig. 9.1), but it is more upstream, be-
cause it does not theorize about dimensions relating to the subject (acting and
thinking) and to institutions (instrumented techniques, impact on praxeologies).
– Vérillon and Rabardel, referring to Vygotsky's psychological instruments,
explain in their conclusion that their remarks
“are not only valid for the activities instrumented by material artefacts (...) the instrumental
dimension also concerns the mastery of language, writing and numbers which may be
considered as technologies intended to communicate, to represent and conserve information and
to calculate."
It is somehow paradoxical to see Vérillon and Rabardel studying psycholog-
ical processes related to technical artefacts which they recognize regretfully to
be overlooked by psychology and then, in their conclusion, to enlarge the scope
to geneses of every kind of artefact. There is arguably therefore a risk that the
idea of instrumental genesis loses some of its potential by being absorbed into
an overly large class of cognitive processes. In contrast, the epistemological
plane of the MWS distinguishes three poles, and each can be considered as
artefacts in the sense of potential psychological instruments. Consequently, the
figurative and semiotic genesis proposed by Kuzniak is an instrumental genesis
in accordance with the work of Vérillon and Rabardel. The same observation
applies to the discursive genesis of proof, definitions and properties of the cor-
pus of reference being artefacts instrumenting reasoning. It seems that the the-
ory of MWS brings clarification, helping us to consider the specificity of instru-
mental genesis, among others. We also note that Vérillon and Rabardel distin-
guish between material and immaterial artefacts, only to then discard this dis-
tinction with regard to instrumentation. We can further observe that in the epis-
temological plane of the MWS, each pole may have a possible material basis,
not only artefacts. This is obvious for signs in semiotic systems. As for the cor-
pus of reference, Richard, Venant and Gagnon (2018) provide evidence that
this is often “materialized” in varied ways for different types of proofs.
– Kuzniak (2013) explains in detail the semiotic genesis and the discursive
genesis. In contrast, instrumental genesis seems to be a blind spot. Consider the
example given at the end of the text. The students (10th grade) are faced with a
task of constructing a triangle with the same angles as a given triangle. The
teacher assumes that the variety of triangles obtained will enable them to infer
the proportionality between the dimensions of the initial triangle and those of
the triangle constructed. He aims to institutionalize proportionality without
demonstration, but he nevertheless wants the students to be aware that it is an
10
unproven conjecture, the imprecision of measurements enabling this awareness.
Kuzniak illustrates these expectations in a first diagram (Fig. 9.2). Although the
students are required to complete a construction task, the diagram mentions no
instrumental genesis. The artefacts and the construction are connected to the
semiotic (figurative) genesis (in blue): they are simply "serving" the visualiza-
tion, and their connection is not discussed. The construction is also supposed to
"trigger" a discursive conjecture-proof genesis (in red).
Fig. 9.2
The second diagram (Fig. 9.3) illustrates what actually happens in the class-
room after the students have not achieved the construction task or have failed
to engage in a process of conjecture: the teacher presents a dynamic geometry
(DG) figure on a computer, makes the software calculate the ratios between the
measurements of the sides for a variety of configurations, and then asks the stu-
dents to observe the invariance of the ratios. The dynamic geometry figure en-
ables the students to pass "directly" from the visualization to an empirical
"proof" based on the observation of the invariance for several configurations.
The precision of the measurements using the software and the plurality of the
configurations make the students think that they have carried out a "true
demonstration", which was not the teacher’s aim. Again, the diagram mentions
no instrumental genesis3.
3The fact that the instrumental genesis is overlooked in this example actually
reflects the teacher’s lack of attention to the role of instruments in designing and con-
ducting the situation. Subsection 9.5.3 herein will come back to this issue and discuss
an alternative design.
11
Fig. 9.3
9.5 Reflecting on key issues about the instrumental genesis in MWS
After situating the instrumental genesis among basic constructs of the theory
of MWS, we look more closely in this section at this notion from both the "psy-
chological" and "institutional" viewpoints. We start by investigating the two
poles of the genesis.
9.5.1 Construction
It is useful to revisit the definition of a construction proposed by Duval (1995):
"(a) construction (...) can work like a model in that the actions on the represen-
tative and the observed results are related to the mathematical objects which are
represented." Construction is thus the elaboration of data-result models of
mathematical configurations. By configuration, we mean that the construction
models a set of interrelated objects, rather than one or more isolated objects.
Models here are not the models of some reality studied in Chapter 11, but they
rather share a common operationality and then proximity with simulation. This
fits well with traditional geometrical figures that model the properties of a set
of geometrical objects. Digital geometry software adds dynamicity, enhancing
the data-result (simulation) character of the model. In algebra, a digital calcula-
tor helps to make a formula work as a model of a function by providing com-
puted values for given inputs. Interpreting Vérillon and Rabardel’s robot study
in the theory of MWS, we must understand that construction is what the stu-
dents do when they operate the robot across the five microgenesis. Like a figure
in geometry is just one of all possible realizations of a configuration, an action
on the cursors is just one of all possible actions. However, it is known that
12
drawing a figure helps to think of all possible realizations, i.e. it is considered a
model of the configuration. Similarly, students’ actions on cursors are a way to
test an implicit model of the robot and to make this model evolve. This sheds
light on one characteristic of the instrumental genesis: it is not a passive appli-
cation of techniques but rather an active process of selecting actions in order to
observe relevant results. Resulting from this active process, the instrumental
genesis can foster cognition and contribute to the “work on techniques”
(Chevallard, 2002).
9.5.2 Digital artefacts, data representation and algorithms
Since Kuzniak’s 2013 plenary, the notion of ‘artefact’ has been vastly investi-
gated.
“In the theory of MWS, mathematical artefacts will generally be associated with material
objects to avoid confusion with other components of the epistemological plane. But these
objects will be intimately linked to rules and techniques of construction or calculation
(Euclidean division, "classical" constructions using a ruler and compass, etc.). These techniques
are based on algorithms, the validation and theoretical status of which are no longer problematic
(This book 1.4.2.1).”
To delve deeper in a reflection on digital artefacts, we return to didactical work
about computer algebra systems (CAS). For the sake of simplicity and consis-
tency with the idea of construction as a data-result model, we restrict the focus
here to symbolic calculators like Maple or Derive conceived as input/output
systems. While these systems presuppose a material basis, this aspect is not de-
cisive. Certainly, the uses of CAS can be different according to whether they
are carried out on a calculator or on a computer, but the stakes for the concep-
tualizations go beyond this material dimension. Computer algebra can be char-
acterized as a set of internal representations of algebraic entities and algo-
rithms, "transparent" to the user, in principle (Elbaz-Vincent 2005). It also of-
fers a user interface with a symbolism which is similar to usual mathematical
symbolism. In mathematical work, the algorithms can be seen as playing an
analogous role to routinized paper-and-pencil techniques mentioned in the cita-
tion above. However, there is a difference concerning one fundamental issue:
controlling (pilotage in French) a mathematical calculation, without which ap-
plying algorithms is useless. In paper-and-pencil techniques, initiation to calcu-
lation algorithms is generally carried out in coordination with the learning of
this subtle control, whereas with formal calculation it is precisely this control
of CAS by the student user that is at stake (Lagrange 2000). How can CAS be
used for a given task? What awareness of the underlying representation and al-
gorithm does this imply? What mastery of the semiotic system specific to the
software is necessary4? How can one control a mathematical calculation as-
4Concerning the necessary awareness of underlying algorithms, consider the example in the
work of Elbaz-Vincent (2005) of the treatment of the expression cos(a)²+sin(a)² by CAS. This expres-
13
sisted by CAS? On what set of reference knowledge can this control be based?
Questions linked to CAS-instrumented techniques have been investigated theo-
retically and empirically (Artigue, 2002; Lagrange, 2005) but this seems to
have little effect, and in countries which have not ruled out the use of CAS for
educational purposes, it is very deceptive (Jankvist, Misfeldt & Aguilar 2019).
The theory of MWS has the potential to make us reconsider and "untangle"
these questions by differentiating between what is properly instrumental -here
the implemented representations and algorithms and the awareness of these by
the user, and what belongs to other geneses, semiotic and discursive. The in-
strumental genesis (in the sense of the MWS) is not isolated, it is rather articu-
lated with other geneses in the working space where the artefact is used (see
Chapters 1 and 2).
We note that the idea of digital artefacts based on data representation and al-
gorithm is not present in the research by Vérillon and Rabardel. In the present
chapter, we see a difference between the lathe and the robot, because action on
the lathe is not mediated via a computerized interface: the genesis is different in
the sense that the link between an action on a screw and a movement of the tool
is not problematic. Indeed, the real problem for a beginner is the effect of a
movement on the shape cut into the material; this effect cannot be thought of as
a process of computation on representations. In mathematics education, there is
a tendency to overlook the representational and algorithmic character of a digi-
tal artefact. Common approaches favor the vision of a digital artefact integrat-
ing the mathematical field for which it was designed without modification. For
instance, publications for teachers present dynamic geometry (DG) as "embed-
ding" elementary Euclidean geometry. They neglect the representations and al-
gorithms underpinning DG and overlook notions which do not exist in classical
geometry, such as the notions of free object and dependent object. These publi-
cations suggest that the practice of dynamic geometry could immediately be in-
tegrated into a usual working space, presenting extended possibilities of action
without transforming its nature. This is a double illusion. On the one hand, the
extended possibilities are in no way spontaneous in the absence of an aware-
ness of the data representation and algorithms at work. On the other hand, by
depriving the students of a working space which enables this awareness and as-
sociation with the semiotic and discursive aspects, teaching also deprives stu-
dents of opportunities to learn. In a suitable working space, attention to the in-
sion it is not evaluated to 1, even if a is declared real. In contrast it is simplified to 1, The reason is that
evaluation and simplification are different algorithms, The first one performs a standard syntactic
treatment in order to get a canonical expression and the second applies heuristically a set of transfor -
mations based on mathematical rules and properties, aiming at decreasing the complexity of the ex -
pression.
As for symbolism, it is very often only partially similar to the usual mathematical symbolism; the
reason for this is that canonical expressions have to be compatible with internal representations and al-
gorithms. For example, summations, limits and integrals must be entered with functional symbolism.
In addition to this, output rarely directly conforms to what a student expects, especially at a secondary
level: an example of this is
x
3
2
when the user expects
x
√
x
.
14
strumental genesis of computer algebra could help students to explicitly learn
how to control a calculation. Taking advantage of the functional aspect specific
to dynamic geometry (identifying dependencies in a dynamic construction)
could support effective approaches to geometry or functions.
Mathematics education is increasingly confronted with diverse and complex
artefacts, among which some based on artificial intelligence algorithms: proof
support software, virtual or augmented reality, data mining, machine learning,
deep learning, large-scale data analysis, etc. (Richard, Vélez Melón & Van
Vaerenbergh, in preparation). Designers are working hard to develop increas-
ingly "accessible" and "transparent" interfaces. Even CAS and DG software
have been augmented with new features and their analysis is not reducible to
that of the preceding paragraph. However, we hypothesize that a similar analy-
sis is possible: the simplicity and efficiency of the theoretical constructs (one
working space, three geneses...) of MWS should make it possible to both guard
against presupposed epistemic effects of the use of a given software and to ask
the right questions: what are the underlying representations and algorithms,
what are the new semiotic systems that the student will be confronted with and
the coordination between objects which they presuppose, what theoretical
frame of reference can be used to guide them, what praxeologies can develop,
how can these aspects and the associated geneses be articulated in a (personal,
suitable, reference) MWS?
9.5.3 Schemes and techniques in the instrumental genesis of MWS
We noted above that the instrumental genesis in MWS is, in a certain way, up-
stream of the approach of Vérillon and Rabardel and then further analyzed
from a psychological viewpoint (development of schemes allowing the use of
an artefact), as well as from an institutional viewpoint (development of instru-
mented techniques and associated praxeologies) in a given MWS. In this sub-
section, we illustrate how the two viewpoints can shed light on the example
(9.4) proposed by Kuzniak (2013).
In this example, students have not integrated the use of an angle transfer
artefact (protractor or tracing paper), or this use does not lead them to the side
measurements on which visualization and reasoning would operate. Instrumen-
tal genesis is nonexistent, no scheme is developed and no technique emerges.
The teacher’s use of DG in the second phase denotes the widespread concep-
tion that we mentioned above, assuming that simply making students look at
DG figures and software calculated ratios will trigger a conceptual reflection.
Focusing on digital artefacts, we propose an alternative situation assuming that
students have the use of DG software (not only the teacher). A strategy often
15
proposed in DG research studies is to let students start by performing a soft
construction5.
Starting from a triangle ABC, a segment DE and a free point F, they have to
guess one or several positions of F, such that DEF has the same shape as ABC
(i.e. corresponding angles have the same value). It can be expected that stu-
dents will mark angles in the two triangles and drag F to a suitable position.
Students could then be advised that their construction is not robust in the sense
that it does not resist dragging of one end of segment DE. The ability of DG to
provide automatic proof (oracle) can also be a reason to reject this construction,
because, in spite of visible angle equalities, the oracle will declare that it is not
proven (Richard, Venant and Gagnon 2018). Students could then be asked to
harden the construction. The assumption is that they will think of transporting
angles, but transporting is incommodious in dynamic geometry. Students will
then have to infer the equality of ratios from various configurations and trans-
pose the procedure for constructing a triangle into DG, knowing its dimensions.
The path that students are expected to follow includes movements along the
instrumental genesis and between the construction pole and the two other cog-
nitive poles (figure).
– A first movement connects the dragging capability in the DG artefact and the
soft construction.
– A second movement involving the construction and the proving poles of the
cognitive plane consists in becoming aware of the limits of this construction.
– In a third movement, the visualizing and construction poles are involved in
order to explore relations between distances.
– The fourth movement connects different capabilities of DG, especially an in-
tersection of circles and a hard construction.
Fig 9.4 The path students are expected to follow in four movements.
5 Leung (2014) presents in more depth how "soft constructions" can be productively im-
plemented in relation to conservation properties.
2
1
3
4
1. Soft construction
using a free point
and dragging.
2. Becoming aware
of limits of this
construction.
3. Visualization of
properties to engage
into hard
construction.
4. Hard construction
using circles and
intersection.
16
From a psychological viewpoint, the instrumental genesis in the first move-
ment consists in appropriating (i.e. build schemes of) the use of a free point F
and the dragging gesture to find an adequate position. In the second movement,
understanding why the construction does not resist and/or why the oracle de-
nies proof implies some awareness of DG (algorithmic) functioning. In the
third movement, students would have to appropriate DG capabilities related to
measures, and in the fourth movement, capabilities related to circles and inter-
section.
From an institutional viewpoint, soft and hard constructions are two instru-
mented techniques for the same task, using different capabilities of DG and be-
longing to different praxeologies: one based on visualization and approxima-
tion, and the second on geometrical properties. It can be expected that students
will not easily abandon a soft construction “that works” to adopt a more intel-
lectually demanding hard construction. It is therefore essential that the teacher
involves students in a discussion on techniques, stressing the interest of adopt-
ing robust procedures consistent with digital software. This discussion is cen-
tral to the second movement6.
9.6 Conclusion
We conclude with a summary of the main points of our contribution. Vérillon
and Rabardel’s instrumental genesis is one of schemes anchored in a Piagetian
perspective. Their analysis implies a fine-grained identification of microgene-
ses specific to the type of artefact used. Focusing on digital artefacts, we ob-
served how schemes develop through the subject’s emerging awareness of the
representations and algorithms which underpin the artefact. Consistent with
Chevallard’s (YEAR) ATD and Leontev’s (YEAR) activity theory, Vérillon &
Rabardel also insist on breaking with an idealistic view of artefacts, consider-
6Lagrange and Erdogan (2009) present a similar case illustrating the impor-
tance of a discussion on techniques. Students in two classes (11th grade, not scientific
majors) have to implement a recurrence relationship on a spreadsheet from a real-life
situation. The teachers in both classes expect students to implement the relationship by
way of a formula copied down along a column. The goal is that they understand the in-
terest of an algebraic formula as a model of the real-life situation. Contrary to this ex -
pectation, students easily compute the requested values by hand, enter these values into
the cells, and decline to enter any formula. Using colors and borders, they are happy to
use the spreadsheet as a tool to display data. Both teachers are disconcerted. One
teacher tries ineffectively to instruct students to make a formula. The other teacher, bet-
ter prepared for situations where technology is used, starts an improvised whole-class
discussion on techniques for calculating values of series, stressing the respective values
of by hand and automatic calculation. Students become aware of the actual potentiali -
ties of the spreadsheet as a calculation tool and of the power of the algebraic symbol-
ism.
17
ing them as emerging from socially situated practices and constitutive of these.
Building on the theoretical elaboration of Vérillon and Rabardel, and of
Chevallard, didactic research introduced two complementary viewpoints or an-
gles from which to consider the instrumental genesis of digital artefacts: the
psychological viewpoint deals with the formation of schemes and associated
mental operations; the institutional viewpoint focuses on instrumented tech-
niques, the praxeologies in which they fit and the social conditions in which
they develop.
As for the instrumental genesis in the theory of MWS, it is first an objective
entity, linking tangible artefacts and observable processes of construction. Be-
cause it also comprises a conceptual dimension transforming both the user and
the mathematical knowledge, it is compatible with Vérillon and Rabardel's idea
of instrumental mediation. However, instrumental genesis in the theory of
MWS is based on different choices compared to Vérillon and Rabardel's no-
tion: on the one hand, it is part of a theorization of mathematical work in edu-
cational settings not limited to the use of instruments, and on the other hand, it
does not theorize about the transformation occurring in the instrumental gene-
sis. The two complementary viewpoints, psychological and institutional (one
providing insight into the cognitive work, the other into how techniques are un-
derstood and implemented), derived from the works of Vérillon and Rabardel,
and of Chevallard, can therefore help to “flesh out” an analysis of the instru-
mental genesis in a particular MWS, on the condition of being cautious not to
merge or confuse ideas drawn from different theoretical perspectives.
Looking at the poles of the instrumental genesis in MWS, we see construc-
tion as data-result models of mathematical configurations. In this modelling
perspective, the instrumental genesis is not a passive application of techniques,
but rather an active process of selecting actions and data in order to observe rel-
evant results. Regarding artefacts, we noted that Vérillon and Rabardel en-
larged the scope, and we stressed the risk that the idea of instrumental genesis
loose some of its potential by being “absorbed” into a too large class of cogni-
tive processes. In contrast, the epistemological plane of MWS distinguishes
three poles, and each of these can be considered as artefacts potentially inte-
grated by the subject through a specific genesis. This clarification is very help-
ful, yet implies that characterization of artefacts in the instrumental genesis of
MWS is not straightforward. Indeed, it requires us to precisely identify capabil-
ities which intervene in the modelling process, beyond their material aspect.
Regarding digital artefacts, these capabilities derive from the representations
and algorithms implemented. We have given an example demonstrating how it
is possible to propose an efficient strategy for using a digital artefact, how an a
priori analysis in the MWS framework gives a central role to the instrumental
genesis and how this analysis can be enriched by considering the genesis from
two complementary viewpoints: psychological and institutional.
Our focus on digital artefacts was productive in the sense that it helped to
question notions in the theory of MWS, and also because it enabled us to better
18
characterize these artefacts: a contribution to didactic research on digital envi-
ronments. As mentioned in this chapter, digital artefacts are ubiquitous, yet
their educational use is not often straightforward in real-life teaching contexts,
meaning that both empirical and theoretical research is more necessary than
ever. We illustrated how a simple characterization of these artefacts, combined
with the potential of the theory of MWS helps us to gain a more precise percep-
tion of situations of use. Our focus is certainly also a limitation with regard to
other artefacts such as the mathematical machines (Bartolini-Bussi & Maschi-
etto, 2006) mentioned in Chapter 8. We nevertheless believe that the ideas pre-
sented in this chapter will be useful to analyze other types of artefacts in a
MWS framework.
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