RESEARCH PRAXEOLOGIES AND NETWORKING THEORIES

Abstract

In this contribution, we consider the potential offered by the anthropological theory of the didactic (ATD) for addressing the issue of networking between theories through the extension of the notion of praxeology, which is at the core of ATD to research praxeologies. After introducing such an extension, we discuss its implications in terms of networking, giving a crucial role to the notion of didactic phenomena. We then use this language for reflecting on two networking experiences in which we have been involved.

Full text

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RESEARCH PRAXEOLOGIES AND NETWORKING THEORIES
M. Artigue,I M. Bosch,II & J. Gascón III
IUniversité Paris 7 (France), IIUniversitat Ramon Llull (Spain),
IIIUniversitat Autònoma de Barcelona (Spain)
Abstract: In this contribution, we consider the potential offered by the
anthropological theory of the didactic (ATD) for addressing the issue of networking
between theories through the extension of the notion of praxeology, which is at the
core of ATD to research praxeologies. After introducing such an extension, we
discuss its implications in terms of networking, giving a crucial role to the notion of
didactic phenomena. We then use this language for reflecting on two networking
experiences in which we have been involved.
INTRODUCTION
In accordance with the work done by the „theories group‟ in the last two CERMEs,
this contribution joins the efforts made to support a productive cooperation in
European mathematics education research, in order to overcome the framework
compartmentalization that could hinder the capitalisation of knowledge and its
practical exploitation. These efforts have shown that the interaction between
researchers working with different approaches has to go further than the
„communication paradigm‟ that dominates exchange activities in most international
conferences. It needs real „teaching and learning‟ activities to explain what one does
and to understand what the others do. Experiences in „networking theories‟, carried
out in this sense1, have shown that their productive development also requires the
consideration of a shared epistemological model, that is, a common way of thinking
and talking about what scientific work is and how it evolves. In fact any research
activity supposes a particular implicit way of interpreting the nature of problems that
are approached, the empirical field to consider, the kind of methodologies that can be
used and, more generally, what research is and what it is for. When the exchange
between researchers attains the level of the theoretical bases – as is the case in the
„networking theories‟ activities –, then it becomes necessary to question the implicit
epistemological model of each approach, looking for a common language to express
and discuss the respective epistemologies. In this sense, the Anthropological Theory
of the Didactic (ATD) that we use in our research and, more concretely, its central
notion of „praxeology‟ has progressively appeared to us as a useful tool to develop
such a common „language‟ or epistemological model. Here we present the recent
work we have undertaken in this direction.
1 We refer more especially to the so-called „Networking group‟ led by Angelika Bikner-Ahsbahs that emerged from
CERME4 and to the work carried out by the first author in European projects like TELMA and ReMath.

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‘THEORIES’ OR RESEARCH PRAXEOLOGIES?
If, according to ATD, we assume the general anthropological postulate that all
human activities can be described in terms of „praxeologies‟ (Chevallard, 1999,
2006), this must be also the case for research activities. Any kind of research,
including „networking‟ projects, should thus be subsumed under the notion of
research praxeologies. In this perspective, talking about „theories‟ (as in the
expression of „networking theories‟) is the result of a metonymy used to point to the
whole – research praxeologies – by only indicating one part, the theoretical block of
praxeologies. As any other praxeology, research praxeologies are indeed composed
of an amalgam of pieces that can be described by a set of four elements [T///].
The pair [T/] corresponds to the „practice‟ (or know-how) of research, with the types
of problems T that are approached and the techniques  used to approach the
problems. The block [/] forms the technological-theoretical discourse used to
describe, justify and interpret both the research practice and the results obtained.
This theoretical block corresponds to research „knowledge‟ and is often considered
as the representative of the whole praxeology, with the limitations and biases that
this reduction can generate in the approach to and treatment of „networking‟ issues.
We postulate that the notion of praxeology can help overcome these limitations and
that it can also be useful to retrospectively reflect on networking efforts. We also
find it important to stress that research praxeologies, as any other praxeological
form, are „alive‟ entities that evolve and change, which affects at the same time their
four components [T///] and the interaction of these. The evolution of the
practical block [T/] produces new theoretical needs that make the theoretical block
[/] progress and, reciprocally, the evolution of concepts, interpretations or ways of
thinking and the emergence of new results lead to the construction of new techniques
and the formulation of new problems. Research praxeologies can appear as different
kinds of amalgams, more or less organised depending on the maturity of the field. It
is the historical development of the field that helps structure these praxeological
amalgams, making them more coherent and easier to disseminate according to
different didactic and institutional transposition processes. Beyond the static
description of research praxeologies in terms of their practical and theoretical blocks,
processes piloting their dynamics are still to be analysed in depth. Our contribution
consists in considering the notion of „phenomenon‟ and relating it with the
„technological‟ element of praxeologies, which will highlight its crucial role in the
dynamics of praxeologies.
‘PHENOMENA’ AND THE DYNAMICS OF RESEARCH PRAXEOLOGIES
The notion of ‘didactic phenomenon’. Today the notion of „phenomenon‟ does not
happen to have a central function in many didactic approaches. It did however play a
crucial role in the emergence of the theory of didactic situations (TDS) and its vision

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of didactics as a scientific discipline. In the first developments of TDS in the 1980s,
and through several different formulations, Guy Brousseau (1997) defined didactics
of mathematics as the science the essential aim of which is the knowledge of didactic
phenomena, that is, phenomena appearing in the teaching, learning or, more
generally, diffusing of mathematical knowledge in social institutions (including
school ones). Hence, didactic phenomena have to be considered both as a
construction and as an object of study of didactics, in the same way that physics
studies the specific construction „physical phenomena‟, or sociology studies and also
defines social phenomena, etc. –including all the historical controversies about
phenomena delimitation in nature and social sciences.
What is the role played by phenomena in relation to research praxeologies and their
evolution? In a first approach, we can characterise didactic phenomena as empirical
facts, regularities that arise through the study of research problems. Some of these
phenomena enrich the initial theoretical framework to produce new interpretations
and techniques or research methodologies, while others remain at the level of „results
obtained‟ and are reinvested to formulate new problems or to propose new diagnostic
and practice-development tools. In order to clarify the relation between the notion of
phenomenon and the four components of a praxeology, let us start from a very
simple example of mathematical praxeology. Let us consider Pythagoras‟ theorem or,
to be more precise, the phenomenon underlying this theorem, that is, a certain
regularity between the measures of the sides of right triangles. At the beginning we
can consider a type of mathematical problem that could be formulated as the problem
of the characterisation of a right triangle or the graphical representation of a right
angle. The answer to this problem appears as a technological element (the
description of a property of a given set of figures) within the mathematical
praxeology that emerges around this type of problems. This technological element is
not only the description of a regular fact: it also produces new mathematical
techniques, helps formulate new problems and discover new regularities, thus
producing more technological results. In the long run, if the initial regularity appears
to be strong enough, then it comes to integrate the theory of the praxeology as a basic
principle of certain kind of geometries (those with a Euclidian metric).
It is thus an entire mathematical praxeology, with its types of problems, its
techniques and its technological-theoretical discourses that the expression
„Pythagoras theorem‟ refers metonymically to. Behind a technological ingredient
such as a theorem – or any other description of a regular fact or phenomenon – we
can find a whole set of praxeological ingredients (problems, techniques, etc.), which
this technological ingredient contributes to structure. Taking all necessary
precautions, we will briefly establish a parallelism between this example and
research praxeologies in didactics. We will use a concrete example, the phenomenon

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of didactic transposition, to illustrate our proposal2 which is to see the role played by
phenomena in the „structuring‟ of praxeologies, that is to say in their dynamics.
Phenomena and type of problems. As any scientific discipline, didactics of
mathematics aims at identifying and studying a specific kind of phenomena (didactic
ones) in order to attain a greater capacity of action and comprehension. Any research
question or problem thus has to be related – even if this relation is mostly done a
posteriori – to the highlighting of a phenomenon, its delimitation, the conditions
needed for its existence and evolution, etc. Even if research takes as a starting point a
problem emerging from a very concrete teaching or learning practice (as it often
does), an effort is always made to formulate the problem in a more general way,
implicitly considering it as a specimen of a given „type of problem‟. This is a first
step in the work that follows: looking for regularities related to the practical issue
approached, trying to characterise them and, to some extent, „understand‟ or „act
upon‟ them. Let us consider, for instance, the phenomenon of didactic transposition
as it was characterised by Yves Chevallard (1985). Several new problems have been
raised and studied that could not even have been formulated before the identification
of the phenomenon (see Bosch & Gascon, 2006 for a recent review).
Phenomena and technological components In research processes, the results
obtained as an answer to the raised problems generally contribute to enrich the initial
research technology by integrating new characteristics of the studied phenomena or
even new derived phenomena. There always exists a double-direction effect between
the results obtained and the evolution of the technology of research praxeologies,
which can be considered at the core of progress of scientific research.3 For instance,
the study of transpositive processes in different mathematical domains has
highlighted various phenomena that, in turn, have been used as a starting point to
formulate new problems and draw attention to new regularities. A good example is
the phenomenon of the „algebraisation‟ of Calculus at upper secondary school level
(Artigue, 1995), a result that has then been used to analyse the teaching of limits of
functions (Barbé et al., 2005). Other examples coming from the didactic
transposition processes are the derived phenomenon of the „stoppage‟ of didactic
transposition (Assude, 1993) or of „detransposition‟ (Antibi & Brousseau, 2000).
2 We are perfectly conscious of the distance between mathematics and didactics as fields of research. However,
commonalities can be established and can be productive in both senses: sometimes the maturity of mathematics hides
some evolution phenomena that are more visible in the recent and less developed dynamics of didactics.
3 This is less true when the theoretical block of the research praxeology comes from a different discipline. We then
obtain a single-direction effect which „breaks down‟ the dynamics of scientific research: for instance, when a given
notion of cognitive psychology is used to analyse some facts related to the learning of mathematics, because the
„external‟ character of the results obtained, they will have no effect on the development of the initial psychology
notional frame.

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Phenomena and technical components The study of phenomena not only generates
the description of regularities, restrictions or „paradoxes‟ (like those of the „didactic
contract‟, for example). It also leads to new ways of doing research, that is, new
techniques and new methodologies. In the case of transpositive phenomena, the
highlighting of relations and differences, both chronological and diachronic, between
the „scholarly knowledge‟, the „knowledge to be taught‟ and the „(actually) taught
knowledge‟ has now become a technique of didactic analysis by itself. Almost any
problem studied within the ATD or the TDS includes, to some extent, a questioning
about which is the knowledge at stake, where does it come from, what „scholarly
knowledge‟ legitimises its teaching, what changes have been operated on it, what
„noospherian‟ discourses support or hinder its teaching, etc. The notion of didactic
transposition has represented an important enlargement of the field of study of
didactics because it has pointed out the need to also consider the mathematical
activities that exists outside the school (Bosch and Gascón, 2006).
Phenomena and theoretical components. In a praxeology, the „theory‟ component
includes the set of notions and relations that are used to apprehend phenomena
(describe them, formulate questions about them, etc.), to develop them and to
identify new regularities. The „theory‟ appears as the second level of validation of
the activity, as an explanation and justification of the „technology‟. It contains the
assumptions taken, that is, the technological elements that come up being taken for
granted because of their solidity and persistence. At this level we find questions such
as: What phenomena are studied? What is a problem in didactics? Why can this or
that result be assumed as such? The empirical enlargements mentioned before are
also integrated at this level as far as they become basic and implicit assumptions. At
the same time, the unit of analysis that is assumed determines the kind of phenomena
that can be considered and the kind of data that are being collected to bring evidence
to the study. For instance, the existence of transpositive processes between
institutions is a theoretical assumption that is not questioned, nor questionable, in
ATD. The „kind‟ of transpositive processes that are taking place, their main
characteristics and the conditions and restrictions they create on teaching institutions
are, on the contrary, some of the main problems considered by this approach.
The praxeological dynamics we just described may help understand the processes
through which the studied phenomena produce new technological results that
partially become new theoretical tools and produce in turn new research techniques
allowing the identification or construction of new phenomena. It is this praxeological
dynamics that we propose to consider here in order to analyse – and guide –two
networking experiences between European research teams.

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THE EXPERIENCE OF THE GROUP « NETWORKING THEORIES IN
MATHEMATICS EDUCATION »
The working group on „networking theories‟ was created in 2005. It includes 12
researchers from six different countries and its work aims at the exchange,
comparison, and connection between theoretical frameworks. Results obtained have
been presented in previous CERMEs and more recently in a research forum at the
last PME conference (Bikner-Ahsbahs et al., 2010). In this section, we analyse an
episode of its work, already evoked at CERME6 (Artigue, Bosch, Gascon & Lenfant,
2010), with the aim of showing how research questions and theoretical components
of praxeologies influence research methodologies (technical components), the units
of analysis considered pertinent, and the didactic phenomena identified.
The initial work of the group was based on a video realized in a grade 10 Italian
classroom and additional material considered necessary for its analysis by our Italian
colleagues. Each team was asked to analyze the video from its own theoretical
perspective but the data provided was judged insufficient by each except the Italian
team. The video showed two students working in a pair, with little intervention of the
teacher. Additional information provided on the session itself and its context was
quite limited, making an analysis supported by TDS very hypothetical and an
analysis supported by ATD nearly impossible. A questionnaire was then addressed to
the teacher in charge of the classroom, asking for additional information to allow the
different teams to complement the partial analyses already carried out. In the
teacher‟s answers, the attention of several members of the group was especially
attracted by the following excerpt:
I try to work in a zone of proximal development. The analysis of video and the attention we
paid to gestures bring me to become aware of the so called ‘semiotic game’ that consists in
using the same gestures of students but accompanying them with a more specific and
precise language in a relation to the language used by students. Semiotic game, if it is used
with awareness, may be a very good tool to introduce students to institutional knowledge.
This convergence of interests led the group to develop a new strategy for progressing
in the collaborative work undertaken: The TDS team should associate a question
articulated in the TDS framework to this excerpt, and then each of the other teams
should rephrase this question according to its own perspective. We reproduce below
the text introducing the TDS question, which in fact also uses some ATD constructs.
The connection between the mathematics produced by students in what we would label,
using the TDS frame, an adidactic situation through interaction with the adidactic milieu
of this situation on the one hand, and the institutional knowledge aimed at on the other
hand, generally requires at least changes in the ways the mathematics at stake is expressed
in order to progressively tune these it conventional forms of expression. The teacher
considers that he has a specific mediating role to play for making this connection possible
and uses semiotic games as a tool for that. In other terms, semiotic games can be

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considered as components of the praxeology (or more certainly one of the praxeologies)
that he has developed in order to solve this didactic task.
The expression “semiotic game” thus denotes what can be seen as a technique, a
component of a teaching praxeology, resulting from the identification of some
particular phenomena of semiotic mediation. Interpreted that way, it shows how a
theoretical focus (in this case a semiotic focus) can lead to the identification of
specific phenomena, and from that to theoretical constructs or to didactic techniques,
considered as tools for improving the efficiency of learning and teaching processes.
A TDS perspective leads to question the efficiency of such a didactic praxeology for
two reasons at least. The first one is that very often interactions with the adidactic
« milieu » do not guarantee the possibility of establishing a direct connection with
the institutional knowledge aimed at. These limitations are the source of different
didactic phenomena identified as paradoxes of the didactic contract: Topaze effect,
Jourdain effect, meta-cognitive slide. The second reason is that the adidactic
situations most often observed in classrooms are situations of action, not situations
of formulation. In such situations, some linguistic activity generally takes place but it
is not taken in charge in the piloting of the situation through didactic variables.
From this perspective, the video analysis leads to the postulate that, in this particular
context, the distance between what the students have autonomously produced and the
forms of knowledge aimed at by the teacher, as expressed in his answers to the
questionnaire, makes problematic the productive character of such a semiotic game.
Thus the question proposed to the group:
Do the episodes at our disposal allow us to identify characteristics of the semiotic game
technique that would help us to understand their potential for compensating the possible
limits of the interaction with the adidactic milieu for achieving the expected mathematical
goals, and linguistic evolution linked to the needs of institutionalization processes?
In the networking group, each team has rephrased this question from its specific
theoretical perspective. This episode shows how the consideration of a new
theoretical framework, here TSD, can lead to question a didactic praxeology,
legitimately considered as a research result in another didactic culture. For
addressing this question, a new research praxeology has been developed, a research
praxeology which had no reason to emerge in either of these didactic cultures and
only exists because a specific networking activity has been undertaken. In the limited
space of this contribution, we cannot present the results produced by this research
praxeology, nor their elaboration into didactic phenomena. We nevertheless hope to
have shown up to what point the relationships existing between the different
components of research praxeologies and the didactic praxeologies emerging from
research results, deserve our attention. We also hope to have shown that an approach
in terms of praxeologies can be helpful for addressing networking issues.

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THE EXPERIENCE OF THE EUROPEAN PROJECT REMATH
The experience of the European project ReMath (http://remath.cti.gr) offers
complementary insights for putting to the test an approach of networking theories in
terms of research praxeologies. An essential goal of this project was to support the
capitalization of research on digital technologies in mathematics education, through
the development of an integrated theoretical framework, with a focus on the
affordances of digital technologies for mathematics learning in terms of
representations, and more globally of semiotic activities. Six European teams worked
on this project during four years, relying on the previous experience of the European
research team TELMA (Artigue, 2009). A sophisticated methodology was developed
for this project. It relied on a meta-language created in TELMA and a system of
cross-experiments using the same digital technology in different didactical cultures,
whose negotiation, implementation and analysis was taken itself as an object of
research. This project also allowed the researchers involved to insert the networking
activity into a permanent dialogue between the design of digital artefacts and the
design of scenarios for their educational use in different educational contexts.
Looking back at this project through the lens of research praxeologies, it looks clear
that the methodology used allowed the ReMath teams to organize their work around
the collective study of their respective research praxeologies. These research
praxeologies were made explicit enough in their different components for ensuring
the productivity of comparative analyses, and particularly that of the cross-case
studies of the different experimentations carried out with the same digital artefact.
The articulation of common research questions to be addressed by the different
cross-experiments and then the addition by each team of questions reflecting its
specific concerns, the strict organization of interactions between teams all along the
process, from the design of artefacts to the a posteriori analysis of cross-
experimentations, the meta-language of concerns, played an essential role. The
design of artefacts and the cross-experimentations contributed ipso facto to two
different types of research praxeologies: on the one hand, praxeologies inserted in
the didactic culture proper to each team and, on the other hand, a “networking
praxeology” still in development. The problem addressed by the first ones was the
identification of the learning affordances of the systems of representations of
mathematical objects implemented in the six digital artefacts, and of the conditions
for a possible ecology of these in realistic contexts. The problem addressed by the
second one was networking between theories. It situated at a meta-level with respect
to the first ones, and the results it produced have a different nature. Some are
methodological and a priori regard more the practical block of this networking
praxeology, for instance those concerning the technique of cross-experimentation, an
essential ingredient of the networking praxis progressively refined. Some are more
likely to contribute to its theoretical block. This is the case for the boundary objects
identified for facilitating the communication between the theoretical frameworks at

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stake and for the STF (Shared Theoretical Frame). Some results show the possibility
of connections and even offer partial integration of theoretical frameworks while
some others identify limits to such ambitions, but it is worth noticing that, at this
stage, none of these results has been given a clear status of phenomenon. Due to
these characteristics, there is no doubt for us that a systematic a posteriori analysis of
the ReMath project using the notion of research praxeology should be helpful
CONCLUDING REMARKS
The retrospective analysis of the two research projects outlined above seems to
confirm our initial postulate that networking between theoretical frameworks must be
situated in a wider perspective than that consisting of the search for connections
between the objects and relationships structuring these. From this point of view, our
approach is fully coherent with that developed by Radford (2008) who, defining a
theory as a triplet made of a system of principles, a methodology and a template of
research questions, insists on the necessity of considering these three components in
networking activity. Space limits do not allow us to enter into a comparison of our
approach with that presented by Radford conceiving a network of theories as a
semiosphere, but we hope that the discussions in the CERME Working Group will
contribute to clarify similarities and differences. In our approach for instance, the
notion of phenomenon appears as a crucial notion for understanding the dynamics of
research praxeologies and the evolutionary links between their different components,
while the word phenomenon is absent from Radford‟s text. This is an intriguing
difference which certainly needs to be collectively analysed and discussed.
Our reflection tends to show that an approach in terms of research praxeologies can
be productive for networking between theories, especially because it helps address
the essential issue of the functionality of theoretical frameworks, by inserting these
in systems of practices. Networking between theoretical frameworks, if considered as
a task to be solved, requires nevertheless the development of specific praxeological
elements that cannot be separated from research praxeologies. The European projects
evoked above attest the existence of such elements, with emerging techniques and
embryonic technologies made of classifications, structured landscapes, meta-
languages. The model of praxeologies could thus help us compare the different
existing efforts of networking and develop more productive ones. This is
nevertheless only a hypothesis which has not yet been seriously worked out. Finally,
we would like to stress that when adopting such a perspective, one must remain
sensitive to the fact that this approach, as any other one, can also introduce some
limitations. Considering them is indeed part of the epistemological vigilance
required in any research process.
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