HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 27-43, 2023

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FACING THE CHALLENGE OF THEORETICAL DIVERSITY:
THE DIGITAL CASE
Michèle Artigue
Université Paris Cité, France

Abstract
The presentation of TSG57 at ICME-14 outlined that mathematics education is a scientificfield in which many theoretical cultures coexist, and that “this diversity can be regarded as richness
but it also challenges research as well as communication and cooperation in the field”. It wasadded that “how the scientific community can cope with this diversity with scientific integrity
remains an open question”. In this paper, we propose to contribute to the reflection on thischallenging issue by considering research on technology-based teaching and learning. First, we
present a brief overview of the theoretical landscape in this area of research, highlighting its
diversity. We then introduce two conceptual tools that have proven their effectiveness in addressing
issues of theoretical diversity: the scale of networking strategies and the concept of researchpraxeology, before focusing on two case studies. These regard the instrumental approach and the
documentational approach to didactics, the emergence and development of which illustrate well
the global dynamic of the field towards increasing theoretical diversity, the questions raised bythis dynamic and the insightful efforts made to deal with it.
Keywords: mathematics education, digital technology, research praxeology, scale of networking
strategies, instrumental approach, documentational approach to didactics

1. INTRODUCTION
Increasing theoretical diversity is a general phenomenon in mathematics education, and research
focused on technological issues is no exception, as we already showed it in our plenary lecture at CERME5
fifteen years ago (Artigue, 2007). In fact, in that lecture we pointed out that, for decades, this area of research
has reflected the general trends and major developments in the field but has also been a source of inspiration
for it, as shown for instance, by the contribution of research in this area to constructivist and situatedperspectives as well as to semiotic and embodied approaches. We also argued that such a situation made thisarea of research an interesting window through which to look at theoretical diversity and at its practical
implications. This is all the more so as the development and the use of digital technologies, since theiremergence, have been given a transformative aim in mathematics education.
A similar vision is proposed in the volume resulting from ICMI Study 17, the second ICMI Study
dedicated to the teaching and learning of mathematics with digital technologies (Hoyles & Lagrange, 2010).
In fact, in this volume, Chapter 7 (Drijvers et al., 2010) is especially devoted to theoretical perspectives. The
Corresponding author: Michèle Artigue

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Michèle Artigue

authors point to the influence on this area of research of the global evolution of the field from constructivist
to socio-cultural perspectives, and they illustrate this with three examples: the webbing and situatedabstraction frame in reference to (Noss & Hoyles, 1996), the theory of didactical situations and the related
notion of “antagonist milieu” (Brousseau, 1998) and the perceptuo-motor activity frame developed by
Nemirovski for use in research on modelling environments involving physical apparatus, such as the wellknown water wheel (Rasmussen & Nemirovski, 2003). They also highlight the theoretical creativity of thisarea of research. Two “current developments” are then presented with more details: the instrumental approachon the one hand, and mediation and semiotic mediation on the other hand. In the first case, the authors rightly
point out the fact that the theoretical combination at stake between cognitive ergonomics and theanthropological theory of the didactic has led to different variations reflecting the particular weight given tothese two perspectives, the first one cognitive, the second one institutional (see Section 3 for more details),
although all variations share fundamental distinctions and concepts, such as the distinction between artefact
and instrument, and the concept of instrumental genesis. For the second, the researchers involved share theclaim that the epistemological nature of mathematical objects makes them accessible only through themediation of representations, and that technological devices substantially alter these mediation possibilities.
However, once again, as the authors point out, these premises are dealt with through different theoreticalconstructions, for instance the concept of human-with-media introduced by Borba and Villareal (2005),
which emphasizes the unity between humans and tools, or the theory of semiotic mediation (Bartolini Bussi& Mariotti, 2008), which emphasizes the essential role of the teacher in the transition from personal meanings
rooted in the context of the artefact to mathematical meanings. Despite the fact that the chapter begins withthe assertion that, due to technological evolution, “communication has become a more integrated part oftechnology use”(p. 98), the theoretical influence of this evolution does not seem very strong at the time,rather a line for future research. Looking to the future, the authors write that the key word should be“connectivity”, because connectivity is a key word for approaching a technological evolution that isfundamentally changing our modes of communication and learning opportunities in and out-of-school. In
fact, this issue is mainly addressed in Chapter 11 (Beatty & Geiger, 2010), where it is emphasized thattechnological evolution calls for theoretical frameworks that allow researchers to approach mathematicallearning as a collaborative endeavor in technologically enhanced communities of practice. Various examples
are presented where the influence of theoretical constructs associated with communities of practice (Wenger,
1998) is clear. However, there is no doubt that at the time of the ICMI Study 17 research on these newer
technological affordances was not so much developed.
The chapter on technology in the book published for the 20th anniversary of ERME (Dreyfus et al.,2018) confirms the theoretical diversity observed in the ICMI Study volume (Hoyles & Lagrange, 2010), andorganizes it in relation to a didactic tetrahedron with knowledge, technology, students and teacher as edges(Trgalová et al., 2018). However, it does not show an obvious movement of research towards the latest
technologies. In fact, the evolution observed in the last decade, at least until recently, seems to be mainly due
to the increase of research on teachers, their knowledge and their practices in digital environments, their
preparation and professional development, their documentational work. For example, the documentational
approach to didactics began to develop about fifteen years ago, motivated both by this attention to teachers
and by the evolution of their documentational work induced by technological advances, as well explained in

Theoretical diversity: the digital case29

(Gueudet & Trouche, 2010). This increased attention is especially addressed in the book The Mathematics
Teacher in the Digital Era (Clark-Wilson et al., 2014) that makes clear the variety of theoretical constructs
developed to address these issues. These include the adaptation of Ball’s mathematical knowledge forteaching model ((MKT) to the technological pedagogical and content knowledge model (TPACK) by Mishra
and Koehler (2006), Ruthven’s structures features of classroom practice framework (Ruthven, 2007), and theextension of the instrumental approach to the teacher, leading to specific constructs such as instrumental
orchestration (Trouche, 2005), double instrumental genesis (Haspekian, 2011) and genesis of use (Abboud& Vandebrouck, 2013). They also include, of course, the specific constructs associated with the
documentational approach, such as the concepts of documentational genesis, resource system anddocumentational trajectory (for an updated vision, see Trouche, Gueudet & Pepin, 2019). Of particularinterest for a reflection about networking is the chapter written by Ruthven, which explores the similarities,
complementarities and contrasts between instrumental orchestration, TPACK and his own framework(Ruthven, 2014). Another point worth highlighting is the place given to the instrumental approach and its
extensions in these various syntheses. For instance, in (Clark-Wilson et al., 2014) an entire section with six
chapters is dedicated to related research.
In recent years, however, the context has been transformed by major technological changes: themassification of mobile multi-touch technologies, the development of virtual or augmented reality devices,and the growing influence of social networks on practices. The context has also been transformed by the
disruptions caused by the pandemic situation, with the abrupt shift to online or hybrid forms of teaching at
all levels of education, and the efforts made to limit the growth of educational inequalities by investing
massively in the most widely accessible technologies, such as mobile phones. Actions and research are
multiplying in response to the new challenges encountered, which will undoubtedly add to the existing
theoretical diversity.
In the aforementioned CERME lecture (Artigue, 2007), we argued for the collective development of
networking activities in order to limit the growing risk of fragmentation of the field. This need is also stressedin the ICMI Study 17 volume where it is written that the key word for the future should be “connectivity”,
not only for reasons of technological evolution, but also because efforts should be made to better connect theexisting diversity of theoretical perspectives. This is even more necessary today. Since then, however, thesituation has changed with the undeniable advances of the networking enterprise (Bikner-Ahsbahs &Prediger, 2014; Kidron et al., 2018). These are both conceptual, methodological and practical. To what
extent, then, are we now better equipped to meet the challenge of theoretical diversity in relation totechnology-based mathematics teaching and learning? Our aim in this paper is to contribute to the reflection
on this question. To this end, we introduce two conceptual tools in the next section before exploiting them totwo case studies. Over the last decade, we have indeed experienced the strong potential offered by the
combined use of these two conceptual tools to address the challenges posed by theoretical and linguistic
diversity (Artigue, 2019, 2021; Artigue & Bosch, 2014; Mesiti et al., 2021).

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2. INTRODUCING TWO CONCEPTUAL TOOLS
2.1. The scale of networking strategies
As explained in (Bikner-Ahsbahs & Prediger, 2010), this construct aims to show the variety of strategies
that can be used to create connections between theories. The scale distinguishes eight strategies, paired andordered between two extreme positions: “ignoring other theories” and “unifying globally”, both of which
considered not desirable. However, as the authors themselves point out, this idea of a linear order must be
taken flexibly, as the degree of integration depends on the precise work done, not just on the strategy.

Figure 1: Scale of networking strategies (Bikner-Ahsbahs, 2016, p. 34, adapted from (Bikner-Ahsbahs &Prediger, 2010, p. 492) (CC BY 4.0))

The authors provide precise definitions for these different strategies. For example, on page 495, they
explain that coordinating means that “a conceptual framework is built by well-fitting elements from different
theories” and that this strategy therefore presupposes the complementarity of the theoretical approaches
involved, whereas combining means that “the theoretical approaches are only juxtaposed according to a
specific aspect”. The combining strategy can thus involve theories with some conflicting basic assumptions.
On page 496, we are told that integrating locally and synthesizing label a pair of strategies that focus “on thedevelopment of theories by putting together a small number of theories or theoretical approaches into a newframework”, and the authors differentiate between the two strategies by considering the dissymmetry/symmetry of the theories involved in terms of scope and degree of development. As explained in (Artigue2019), we find it more appropriate to differentiate between these two strategies according to the dissymmetrical/
symmetrical contribution of the theories involved in the resulting construction, and to consider that any time
there is a significant dissymmetry in the connection, even for theories having a similar state of development,
there is local integration. This is the case, for example, when a broad theory is enriched by theoreticalconstructs from another well-established theory in order to build an integrated theoretical framework thattakes into account research findings in a particular area produced in a different theoretical culture.
2.2. The concept of research praxeology
The notion of research praxeology introduced in (Artigue et al., 2011) extends to research practices theconcept of praxeology at the heart of the anthropological theory of the didactic (hereafter ATD) (Chevallard,

Theoretical diversity: the digital case312019). Indeed, a basic principle of ATD is that all human practices can be modelled in terms of praxeologies.
These consist of a praxis block and a theoretical block in dialectical interaction. A priori, thus, this is also thecase for research practices. By definition, the praxis block of research praxeologies includes the different
types of tasks that the research activity requires, and the associated techniques of study. Their theoretical
block consists of the discourse used to describe, justify and interpret research techniques (technologicaldiscourse in ATD language), and a theory consisting of “statements of a more general and abstract character,with a generally strong justifying and generating power.” (Bosch & Chevallard, 2020). At the simplest levelof point praxeologies, dealing with a single type of task, a typical research praxeology in mathematics
education is associated with a research question and a technique of study for this question. These form its
praxis block. This block is dialectically linked to a theoretical block consisting of technological andtheoretical discourses. The theoretical block includes at least some methodological discourse that explains
and justifies the technical choices; some research background, namely existing knowledge regarding thequestion at hand or related questions, and associated constructs that are considered of interest, plus at least
one theory that underlies the whole construction.
Of course, this is a very simple case. Most research questions cannot be solved by using a single
technique, and the theoretical block of research praxeologies, even point praxeologies, often, does notmobilize a single theory but some theoretical combination. Most often, research praxeologies emerge fromquestions about teaching and learning processes, or more globally about the functioning of didactic systems
or institutions, that is, through their praxis block. But the questions and their formulation are influenced by
the “theoretical” already present in the researcher’s or the research team’s environment. The study techniques
are guided by those that are implemented in close research praxeologies, or that are familiar to the researcher.
In this way, the praxis and the theoretical blocks interact from the outset. Moreover, the study of a researchquestion generally requires more than a single point praxeology. It is necessary to develop a coherent set ofresearch praxeologies that share, at least in part, the same theoretical block. Local and regional praxeologies
thus emerge. New questions also arise from research results; new elements enter the technological discoursein terms of distinctions, categorizations, didactic phenomena. If they become reasonably shared, they will
enrich the theory itself, thus contributing to the dynamic of research praxeologies.
We claim that approaching the relationships between theories through this praxeological lens can help
researchers to take better account of the functional role of theories in the research activity. The interest in
adopting such a functional stance towards theories became clear to us while working in the European project
TELMA whose aim was to capitalize European knowledge about technology enhanced learning inmathematics (Artigue, 2009). At that time, however, the concept of research praxeology had not yet been
introduced. In the context of research on the educational use of digital technologies, its first use took place
in a retrospective analysis of the theoretical activities carried out in the ReMath European project that had
emerged from TELMA (Artigue & Mariotti, 2014).
In the following two sections, we use these tools to present and discuss two case studies. These regard
the instrumental approach and the documentational approach to didactics, respectively. Both approaches
have obvious connections and both have generated a multiplicity of theoretical interactions, given thediversity of researchers who have contributed to their development or simply used them. They provide alimited, but sufficiently rich, ground to support the reflection aimed at in this article.

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3. A FIRST CASE STUDY: THE INSTRUMENTAL APPROACH
3.1. Overview of the instrumental approach
The instrumental approach (IA in the following) emerged in France in the mid-1990s in a specifictechnological context, that of CAS technology (Artigue, 2002), and with two theoretical pillars: the ergonomic
perspective on contemporary tools developed by Rabardel and Vérillon (Rabardel, 1995; Vérillon &Rabardel, 1995) and ATD1. In fact, this approach quickly spread beyond the small community in which it wasborn. The rapid publication of books, both in French and in English, and of many articles certainly favoredthis dissemination. All over the world, researchers began to incorporate IA constructs into their theoreticalframeworks. This phenomenon was already evident at the time of the ICMI Study 17 mentioned above, andin the last decade the theoretical diversity has even increased. All constructions share the reference to the
Rabardel’s and Vérillon’s ergonomic perspective, in particular:●

the distinction between artefact and instrument;●

the concept of instrumental genesis with its two movements dialectically connected: instrumentalizationfrom the user to the artefact and instrumentation from the artefact to the user;●

and the conceptualization of instrumental genesis in terms of the elaboration or appropriation ofschemes.
However, these constructs are combined with or integrated into different theories, which leads tosignificantly different research praxeologies. This phenomenon was already visible in the first two doctoral
theses based on IA, those of Defouad (2000) and Trouche (1997), the latter having the theory of conceptual
fields (Vergnaud, 1991, 2009) as its main theoretical component. The impact on the different components of
their respective research praxeologies is clear, with obvious consequences on the results obtained. In
Defouad’s thesis, in line with the ATD conceptualization of human activities in terms of praxeologies, moreemphasis is placed on the instrumented techniques developed, the material signs (ostensives in ATD
language) used in them, and the discourse explaining and justifying the techniques (the technologicaldiscourse in the praxeological model), rather than on the schemes underlying them. Furthermore, special
attention is paid to the relationships between the students’ instrumental geneses and the classroom instrumental
genesis, and their management by the teacher. The results are mainly expressed in terms of regularities
identified in the students’ instrumental geneses and of didactic phenomena that show links between thecharacteristics of the students’ instrumental geneses and the institutional conditions and constraints thatshape the ecology of instrumented techniques in the classroom. In Trouche’s thesis, the results are mainly
expressed in terms of schemes of instrumented action and of their evolution in the transition from graphic tosymbolic calculators.
A few years later, in the ReMath project (Kynigos & Lagrange, 2014), thanks to the development ofspecific methodological tools, we were able to analyze the effect on research praxeologies of the integrationof instrumental perspectives in theories born outside the French didactic tradition such as the theory ofsemiotic mediation (Bartolini Bussi & Mariotti, 2008) or constructionism (Papert, 1980). New local
integrations have also resulted from the extension of IA to the teacher, such as its integration in the dual
1 More detailed information about the instrumental approach, its emergence and development, can be found in the Michèle
Artigue’s Unit of the ICMI AMOR project: https://www.mathunion.org/icmi/awards/amor/michele-artigue-unit

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ergonomic and didactic approach of teachers’ practices (Robert and Rogalski, 2002), well illustrated by the
GUPTEN project in France (Lagrange, 2013).
3.2. Networking challenges: the scheme/technique case
There is no doubt that all these connections have contributed to the wealth of knowledge and results
produced under the IA umbrella, but they have also raised serious challenges. We will illustrate thesechallenges with an example, the scheme/technique case, which has been the source of intense debate foralmost a decade (see the synthesis offered in (Monaghan, 2007)). IA emerged as a synthesis between
Rabardel’s and Vérillon’s instrumental perspective and ATD. In such a process, ATD categories anddiscourses were intertwined with those proposed by cognitive ergonomics. Instrumental geneses were
thought in terms of the development of praxeologies, and thus approached in terms of instrumented techniques
and associated technological discourses, rather than only in terms of schemes, as was the case in ergonomic
publications. This duality was discussed extensively in the first decade of IA. The collective work on thisissue helped to clarify the different points of view and, in particular, to reject simplistic assimilations such as
those that reduced techniques to gestures or to the observable part of schemes. In fact, thinking in terms ofpraxeologies means that techniques cannot be isolated from the technological discourse that describes,
explains and justifies them. In a sense, reducing techniques to gestures is akin to reducing schemes to theirobservable characteristics without considering the essential component of schemes that the operational
invariants underlying the observed regularities are. Indeed, the many contributions to the scheme/techniquedebate have made it clear that schemes and techniques correspond to two different and complementary waysof approaching instrumental issues, both insightful but irreducible to each other.
However, a new element deserves to be taken into account in today‘s reflection on this topic. In fact,
while maintaining its institutional anchorage, ATD has progressively incorporated the concept of personal(mathematics) praxeology, following research by Crozet and Chaachoua (2016)2. Like institutional
praxeologies, personal praxeologies are modelled in terms of quadruplets that dialectically articulate a praxis
block consisting of types of tasks and techniques and a theoretical discursive block. However, this personal
dimension, the very nature of the students‘ verbalizations used to identify the theoretical block of personal
praxeologies, should allow for more productive connections between the language of schemes and thepraxeological language.
3.3.The ReMath contribution
The European ReMath project also had important spin-offs in terms of networking involving IA, as itsoon became clear that, despite the diversity of their theoretical cultures, five of the six teams involved weresensitive to Rabardel’s and Vérillon’s conceptualizations and had incorporated them in some way into their
theoretical background. Thanks to a carefully designed methodology, we were able to document and analyze
the effects of these incorporations both in the design and development of digital dynamic artefacts(DDAs) – six DDAs were actually created or improved as part of this project – and in the use of these DDAs
by two different teams, the team in charge of the DDA design and another team from another country with a2 As explained in Module 9 of the Artigue’s unit of the AMOR project, the idea of personal praxeology had been alreadyintroduced in Defouad’s thesis, but it did not disseminate at the time and was soon forgotten.

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different theoretical background. In this paper, we focus on the case of constructionism, and a comparison
involving the Greek team in the Educational Technology Lab (ETL), for which constructionism was the main
theoretical support, and the French team in DIDIREM, now the Laboratory of Didactics André Revuz(LDAR), whose theoretical background combined IA with the theory of didactic situations (Brousseau,
1998) and Duval’s semiotic theory (Duval, 1995), both very influential in French didactic research on digitaltechnologies (Artigue & Trouche, 2021). As explained in the special issue (Kynigos & Lagrange, 2014), in
ReMath ETL was in charge of two DDAs, Cruislet and MaLT, and DIDIREM was in charge of the DDA
Casyopée. Their constructionist culture made the Greek researchers particularly sensitive to the expressive
power of digital artefacts, which they had theorized in terms of “half-baked microworld”, that is to say“pieces of software explicitly designed so that their users would want to build on them, change them or decompose parts of them in order to construct an artifact for themselves or one designed for instrumentation byothers” (Kynigos, 2007, p. 336). This obviously influenced their (local) integration of Rabardel’s andVérillon’s ergonomic perspective into constructionism, leading them to emphasize the instrumentalization
dimension of the process of instrumental genesis. Indeed, instrumentalization captures the vision that
artefacts can be transformed by their users in ways anticipated by the designers, but also in ways that are notanticipated (catachresis phenomena). Instrumental genesis results from the dialectic interaction between
instrumentalization and instrumentation processes, and a constructionist perspective leads to emphasize therole of instrumentalization in this dialectic interaction. In the design of Cruislet and MALT, and in the tasks
they proposed to students, the ETL researchers were particularly sensitive to the need to support this
instrumentalization dimension of instrumental geneses and the students’ creativity associated with it. In linewith the constructionist culture, programmable simulations using the Logo language, open and challenging
tasks with strong potential for the students’ expressiveness were essential tools for this. The contrast with the
DDA Casyopée was clear. This DDA was very innovative in terms of dynamic connections between
representations – for example, it offered advanced tools for the functional modelling of geometric situations
involving covariation of lengths and areas –, but the focus was on institutional graphic and symbolic
representations, and the support offered by the DDA for their treatment within a given semiotic register orconversion from one register to another one, in line with Duval’s semiotics. In terms of design priorities,
therefore, more emphasis was placed on supporting instrumentation processes than on supporting
instrumentalization processes, and the learning potential offered by the emergence and progressive
transformation of alternative representations to the institutional ones was not considered. Such comparisons
sharpened our vision of instrumentalization, of the dialectical games between instrumentalization andinstrumentation, and made clear the influence on our vision of the theories with which we had networked
ergonomic perspectives.
The cross-case study regarding the DDA Cruislet, involving ETL and DIDIREM, was also particularly
insightful. It highlighted the profound differences between scenarios based on the theory of didactic situations
and constructionism, although both theories share socio-constructivist principles. The concept of fundamental
situation, the aim of optimizing the adidactic potential of situations through the selection and managementof didactic variables, and the attention paid to the organization of the dual processes of devolution andinstitutionalization, are at the very heart of the theory of didactic situations and of didactic engineering
designs based on this theory. Such constructs are alien to constructionist scenarios. In these scenarios, theTheoretical diversity: the digital case35

aim of ensuring that the tasks designed and their technological and social environment offer strong potential
for supporting the students’ expressiveness is more important than the aim of achieving a precise learning
goal and optimizing the possible trajectories. The Cruislet software, which offers various possibilities for
simulating airplane flights over a map of Greece, was perfectly adapted to a constructionist approach tolearning. For the French teachers and researchers, experimenting with it was a real challenge, and ultimately
only possible within a specific project device, less constrained than the usual lessons. However, thisexperience was also a source of questioning and enrichment for us, as shown in (Le Feuvre, Meyrier &Lagrange 2010). Today, the evolution of the concept of didactic engineering carried out by ATD through theconcept of study and research path and the related methodological tools such as question-answer graphs(Barquero & Bosch, 2015), undoubtedly provides a more appropriate framework for exploiting the potentialof such constructionist DDAs in the French didactic culture.
Beyond this particular case, the ReMath cross-experiments showed that the productive networking oftheories does not obey the same constraints when theories are networked, a posteriori, to analyze and interpretcollected data, and when they are networked to build a conceptual framework to support didactic design, be
it the design of digital artefacts or the design of tasks and learning situations involving them. The
experimentation of Cruislet mentioned above showed that the logics underlying constructionist design and
TDS design were so far apart that it was very difficult, if not impossible, to coordinate them in a conceptual
framework. Significant distance was also observed in the conceptual frameworks supporting the design ofsituations involving the DDA Casyopée by the French DIDIREM team and the Italian UNISI team from the
University of Siena. The UNISI design was guided by the theory of semiotic mediation (Bartolini Bussi &Mariotti, 2008), while the DIDIREM design, once again, was guided by TDS and Duval’s semiotics. In linewith their theoretical background, the Italian researchers conceived Casyopée as an instrument of semiotic
mediation, and, in their design, they carefully organized the transition from semiotic signs attached to theartefact to mathematical signs detached from the artefact. Much time was devoted to this transition, incollective discussions led by the teacher without access to Casyopée. For the French researchers, in line with
the TDS, Casyopée was a central element of the a-didactic milieu. Their design, while paying attention to theconnection to be made between paper-pencil and instrumented techniques, did not conceive this connectionin terms of transition. This was clear in the institutionalization phase where both instrumented and noninstrumented forms of knowledge were combined. Conversely, the various cross-case studies carried out in
ReMath have shown the potential offered by the combination of constructionism, TDS and the theory ofsemiotic mediation in the a posteriori analysis of the experiments (see, for example, Maracci et al., 2013). In
summary, the a posteriori combination of complementary and even conflicting theoretical perspectives in theanalysis and interpretation of data can be very productive, while the combination of theoretical perspectives
that are too far apart can make a coherent design impossible. More compatibility is needed. This is consistent
with the distinction made between the combining and coordinating strategies in the scale of networking
strategies (see Section 2) and confirms the pertinence of this distinction. Design based on theoretical
networking requires coordination, and not just combination, of theories or theoretical constructs.

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4. A SECOND CASE STUDY: THE DOCUMENTATIONAL APPROACH TO DIDACTICS
The documentational approach to didactics (hereafter DAD) was born in a course given at the French
Summer school of didactics of mathematics in 2007 (Gueudet & Trouche, 2009). It was motivated by thesignificant changes in the documentational work of teachers brought about by technological evolution, theincreasing number of digital and online resources available to and used by teachers, and the growing role ofprofessional and social networks in their professional activity. Digital technologies are therefore at the heartof DAD. As Gueudet (2009) explains in her analysis of the development of DAD, its source was her andTrouche’s earlier work, which was mainly concerned with digital technologies. For Trouche, it was thecontribution to the development of IA and to the SFODEM project, “a project of in-service teacher training,
mainly at distance, aiming to support the integration of ICT in the teachers’ practices by a rich offer ofresources on a platform” (Gueudet, 2019, p. 19). For Gueudet, it was her research on learning processes with
online exercises, first at university level and then extended at all levels of schooling, in particular inconnection with the association of teachers Sésamath, an association that has produced very much usedonline resources, since its creation more than 20 years ago (https://www.sesamath.net).
The development of DAD in just fifteen years is impressive, as is impressive the number of researchers
from different cultures and backgrounds who have contributed to it, as shown in (Trouche, Gueudet & Pepin,2019). Another interesting element for our reflection is that, from the beginning, DAD has combined different
sources of inspiration: IA and it can be seen as an extension of IA to the documentational work of teachers,
but also many others. For instance, Adler’s conception that “resources for school mathematics extend beyond
basic material and human resources to include a range of other human and material resources, as well as
mathematical, cultural and social-resources” (Adler, 2000, p. 210) is explicitly used in the extended definitionof resources proposed in the DAD. Remillard’s participative approach to the use of curriculum resources(Remillard, 2005), or Wenger’s community of practice approach (Wenger, 1998), to mention just but a few,
have also been very influential.
Over the last decade, we can observe a progressive evolution of the connections, both on the side of thefounders of DAD and on the side of its contributors and users. Analyzing the first book published on thisapproach (Gueudet & Trouche, 2010), we have shown in (Artigue, 2019) that Gueudet and Trouche mention
many authors in the two chapters they co-author, but that few theoretical connections are really elaborated
beyond the foundational one with IA. For example, we observe only one example of local integration beyond
the already mentioned one with Adler’s conception of resources. It occurs when the authors link teachers’
systems of documents and systems of activities by introducing a categorization of teachers’ activities
explicitly inspired by the study moments of ATD (Chevallard 2002). The foundational connection with IA
leads to the distinction between resource and document, analogous to the distinction between artefact andinstrument. Indeed, a document is defined as a hybrid entity consisting of updated/recombined resources andof a scheme of utilization of these. This connection also leads to the concept of documentational genesis withthe dual processes of instrumentation and instrumentalization, and the development of schemes.
An interesting point is the awareness of researchers, since this emerging state of DAD, of the need forspecific methodologies that allow them to approach the teachers’ documentational activity in-class and outof-class, with a long-term follow-up, and a close involvement of the teacher in the collection of data leading

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to a reflective stance. Very soon, this methodology was given a name: “reflective investigation” and a specific
tool was associated with it: the SRRS (schematic representation of resource system), and both became
emblematic of the research praxeologies based on DAD. In the chapters of the same book written by otherresearchers, one can observe many cross-references but, once again, rather few advanced forms of networking,which is not surprising in this nascent state of DAD. In most chapters, the main purpose seems to be to makeanother approach understandable, or to contrast it with DAD. Only one chapter really goes further and opens
the way to local integration (Trgalová, 2010). As explained in (Artigue, 2019, p. 100), “the author connects
DAD with the model of teacher activity developed by Margolinas (2002) and Balacheff’s cK¢ model ofconceptions (Balacheff 1995), two constructions already connected by these two authors to analyze teachers’
didactic decisions in (Balacheff and Margolinas, 2005)”. It is interesting to note that, in this particular case,the data used comes from a doctoral thesis whose methodology has stimulated some form of documentational
work, which certainly makes the networking more practicable.
As shown in (Artigue, 2019), the theoretical core of DAD research praxeologies stabilized quickly,
which contributed to the coherent structuring of these praxeologies, together with the privileged role given
to the methodology of reflective investigation and the SRRS tool. This did not prevent the diversification ofresearch problématiques, to which the increasing cultural diversity of the researchers contributing to DAD
certainly contributed. When preparing our plenary lecture at the Re(s)sources 2018 conference, we asked
Luc Trouche for a selection of references to get an updated vision of the theoretical connections in DAD, andwe received a list of theoretical crossings with one or two references for each. The list mentioned thirteentheories and theoretical approaches. Some of them were already present in the first book such as informationand communication sciences, Remillard’s approach or communities of practice, but also were new such as
cultural historical activity theory, the theory of social creativity, constructionism or the meta-didacticaltransposition approach. Even if it remains partial, the analysis of the connections carried out in (Artigue,2019) shows a clear progression in their quality. The collaborative work that has been developed, evidenced
by the fact that most publications are co-authored, has obviously made this progress possible. There is no
doubt that DAD researchers have been able to create communication devices and forms of collaborative
work that help to meet the challenges raised by the increasing diversity of theoretical connections involvedin the development of the theory, and also by the linguistic diversity of their community as shown by the
DAD-MULTILINGUAL project (https://hal.science/DAD-MULTILINGUAL/).
In terms of research praxeologies, DAD today unifies, at a regional level, a diversity of researchpraxeologies aimed at better understanding of the transformations of teachers’ activity induced by the use ofdigital technologies, and at developing innovative uses of the increasing diversity of these technologies. As
already mentioned, DAD research has created a methodological tool, the methodology of reflective
investigation. This methodology, which provides a study technique well adapted to the research questions
addressed by DAD, has become emblematic of its research praxeologies, and has been progressively enrichedas shown in (Trouche, Gueudet & Pepin, 2019). The theoretical block of DAD research praxeologies has also
been progressively enriched in dialectic interaction with the enrichment of their praxis block. For example,
the study of documentational genesis over longer periods of time than was initially the case has led to theconcept of documentational trajectory, which in turn has led to new methodological and representational
tools (see, for example, Rocha, 2018).

38

Michèle Artigue

In this second case study, the combined use of the networking and praxeological lenses helps us tounderstand a dynamic particularly rich in theoretical connections that has developed over the last fifteen
years, to understand why we observe such a theoretical diversity in it, and how this diversity has gradually
been better managed through more advanced forms of networking, and also thanks to the unifying
characteristics of the DAD research praxeologies.

5. CONCLUSION
In this paper, we have shared some elements of the knowledge we have gained through our experienceof networking between theories, focusing on research on teaching and learning in digital environments andthe ways in which these influence teaching and learning processes in mathematics. If we return to the questionthat has guided the reflection developed in this paper: “To what extent are we now better equipped to face thechallenge of theoretical diversity in what regards technology-based mathematics teaching and learning?”,
our reflection makes clear that conceptual and methodological tools have come a long way since the issues
raised by theoretical diversity were seriously put on the agenda of the international mathematics education
community at the beginning of this century. As a result, as we have tried to show, knowledge has advanced
both in terms of understanding the diversity challenge and in terms of developing strategies to address it. All
the forms of networking identified in the scale of networking strategies have been used productively,
involving a number of theories with different state of development, aims and scope. The connections
established and the knowledge built now form a solid background for supporting research in this area.
Networking research has also shown the interest of developing specific research praxeologies. These allow
us to examine our respective research practices, their theoretical frameworks and their results without
denaturing the theories and theoretical constructs involved, in order to build significant connections, andmake clear their potential and limitations, as has been done, for example, in ReMath and in (Bikner-Ahsbahs& Prediger, 2014). More globally, our reflection confirms the interest of adopting a praxeological lens toapproach the diversity challenge.
The theoretical landscape of research on technology-based teaching and learning is a dynamic one. On
the one hand, the development of multi-touch mobile technologies, of virtual and augmented reality devices,
is the source of new interesting theoretical connections, such as the connection between IA and radical
embodied cognitive science approaches presented in (Shvarts et al., 2021). On the other hand, the massive
move towards online and hybrid mathematics teaching provoked by the pandemic situation has abruptly
placed technology-based teaching and learning at the center of the educational agenda. This situation, andthe resulting increase in educational inequalities, is also a source of new theoretical needs and connections,
as illustrated by Borba (2021), for example. In this contribution to the special call on mathematics education
research in pandemic times launched by the journal Educational Studies in Mathematics, Borba indeed
reworks the theoretical construct of humans-with-media that he introduced years ago to emphasize that the
production of knowledge results from the collective agency of humans and media (Borba & Villareal, 2005)
to take into account the impact and agency of non-living things such as COVID-19. He also suggests
combining this theoretical perspective with that offered by critical mathematics education (Skovsmose,

Theoretical diversity: the digital case391994) to address the crucial issue of educational inequalities exacerbated by the pandemic. These are only
two examples but there is no doubt that networking needs will continue to grow in this area of research aswell as in mathematics education more generally.

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Theoretical diversity: the digital casedoi.org/10.1017/CBO9780511803932

Michèle Artigue
Université Paris Cité, Paris, France
Email: michele.artigue@univ-paris-diderot.fr
https://orcid.org/0000-0002-8176-8243

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