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Chapter 2
The French Didactic Tradition
in Mathematics
Michèle Artigue, Marianna Bosch, Hamid Chaachoua, Faïza Chellougui,
Aurélie Chesnais, Viviane Durand-Guerrier, Christine Knipping,
Michela Maschietto, Avenilde Romo-Vázquez and Luc Trouche
Abstract This chapter presents the French didactic tradition. It first describes the
emergence and development of this tradition according to four key features (role of
mathematics and mathematicians, role of theories, role of design of teaching and
learning environments, and role of empirical research), and illustrates it through
M. Artigue (B)
University Paris-Diderot, Paris, France
e-mail: michele.artigue@univ-paris-diderot.fr
M. Bosch
University Ramon Llull, Barcelona, Spain
e-mail: mariannabosch@iqs.edu
H. Chaachoua
University of Grenoble, Grenoble, France
e-mail: hamid.chaachoua@imag.fr
F. Chellougui
University of Carthage, Tunis, Tunisia
e-mail: chellouguifaiza@yahoo.fr
A. Chesnais ·V. Durand-Guerrier
University of Montpellier, Montpellier, France
e-mail: aurelie.chesnais@umontpellier.fr
V. Durand-Guerrier
e-mail: viviane.durand-guerrier@umontpellier.fr
C. Knipping
University of Bremen, Bremen, Germany
e-mail: knipping@math.uni-bremen.de
M. Maschietto
University of Modena e Reggio Emilia, Modena, Italy
e-mail: michela.maschietto@unimore.it
A. Romo-Vázquez
Instituto Politécnico Nacional, Mexico City, Mexico
e-mail: avenilderv@yahoo.com.mx
L. Trouche
IFé-ENS of Lyon, Lyon, France
© The Author(s) 2019
W. Blum et al. (eds.), European Traditions
in Didactics of Mathematics, ICME-13 Monographs,
https://doi.org/10.1007/978-3-030-05514-1_2
11
12 M. Artigue et al.
two case studies respectively devoted to research carried out within this tradition
on algebra and on line symmetry-reflection. It then questions the influence of this
tradition through the contributions of four researchers from Germany, Italy, Mexico
and Tunisia, before ending with a short epilogue.
Keywords French didactics ·Mathematics education ·Anthropological theory of
the didactic ·Theory of didactical situations ·Theory of conceptual fields ·
Didactic research on line symmetry ·Didactic research on algebra ·Didactic
research on proof ·Didactic interactions ·Influence of French didactics
This chapter is devoted to the French didactic tradition. Reflecting the structure
and content of the presentation of this tradition during the Thematic Afternoon at
ICME-13, it is structured into two main parts. The first part, the three first sections,
describes the emergence and evolution of this tradition, according to the four key fea-
tures selected to structure the presentation and comparison of the didactic traditions
in France, Germany, Italy and The Netherlands at the congress, and illustrates these
through two case studies. These focus on two mathematical themes continuously
addressed by French researchers from the early eighties, geometrical transforma-
tions, more precisely line symmetry and reflection, and algebra. The second part
is devoted to the influence of this tradition on other educational cultures, and the
connections established, in Europe and beyond. It includes four sections authored
by researchers from Germany, Italy, Mexico and Tunisia with first-hand experience
of these interactions. Finally, the chapter ends with a short epilogue. Sections 2.1
and 2.8 are co-authored by Artigue and Trouche, Sect. 2.2 by Chesnais and Durand-
Guerrier, Sect. 2.3 by Bosch and Chaachoua, Sect. 2.4 by Knipping, Sect. 2.5 by
Maschietto, Sect. 2.6 by Romo-Vázquez and Sect. 2.7 by Chellougui.
2.1 The Emergence and Development of the French
Didactic Tradition
As announced, we pay specific attention to the four key features that structured
the presentation of the different traditions at ICME-13: role of mathematics and
mathematicians, of theories, of design of teaching and learning environments, and
of empirical research.
e-mail: luc.trouche@ens-lyon.fr
2 The French Didactic Tradition in Mathematics 13
2.1.1 A Tradition with Close Relationship to Mathematics
Mathematics is at the core of the French didactic tradition, and many factors con-
tribute to this situation. One of these is the tradition of engagement of French mathe-
maticians in educational issues. As explained in Gispert (2014), this engagement was
visible already at the time of the French revolution. The mathematician Condorcet
presided over the Committee of Public Instruction, and well-known mathematicians,
such as Condorcet, Lagrange, Laplace, Monge, and Legendre, tried to respond to the
demand made to mathematicians to become interested in the mathematics education
of young people. The role of mathematicians was also prominent at the turn of the
twentieth century, in the 1902 reform and the emergence of the idea of scientific
humanities. Darboux chaired the commission for syllabus revision and mathemati-
cians strongly supported the reform movement, producing books, piloting textbook
collections, and giving lectures such as the famous lectures by Borel and Poincaré.
Mathematicians were also engaged in the next big curricular reform, that of the
New Math period. Lichnerowicz led the commission in charge of the reform. Math-
ematicians also contributed through the writing of books (see the famous books by
Choquet (1964) and Dieudonné (1964) offering contrasting visions on the teaching
of geometry), or the organization of courses for teachers, as the APMEP1courses by
Revuz. Today mathematicians are still active in educational issues, individually with
an increasing participation in popularization mathematics activities (see the activi-
ties of the association Animath2or the website Images des Mathématiques3from the
National Centre for Scientific Research), and also through their academic societies as
evidenced by the role played nationally by the CFEM,4the French sub-commission
of ICMI, of which these societies are active members.
The Institutes of Research on Mathematics Teaching (IREMs5) constitute another
influential factor (Trouche, 2016). The creation of the IREMs was a recurrent demand
from the APMEP that succeeded finally thanks to the events of May 1968. Indepen-
dent from, but close to mathematics departments, these university structures welcome
university mathematicians, teachers, teacher educators, didacticians and historians
of mathematics who collaboratively work part-time in thematic groups, developing
action-research, teacher training sessions based on their activities and producing
material for teaching and teacher education. This structure has strongly influenced
the development of French didactic research, nurtured institutional and scientific rela-
tionships between didacticians and mathematicians (most IREM directors were and
1APMEP: Association des professeurs de mathématiques de l’enseignement public. For an APMEP
history, see (Barbazo, 2010).
2http://www.animath.fr (accessed 2018/01/08).
3http://images.math.cnrs.fr (accessed 2018/01/08).
4CFEM (Commission française pour l’enseignement des mathématiques) http://www.cfem.asso.fr
(accessed 2018/01/08).
5IREM (Institut de recherche sur l’enseignement des mathématiques) http://www.univ-irem.fr
(accessed 2018/01/08).
14 M. Artigue et al.
are still today university mathematicians). It has also supported the strong sensitivity
of the French didactic community to epistemological and historical issues.
With some notable exceptions such as Vergnaud, the first generation of French
didacticians was made of academics recruited as mathematicians by mathematics
departments and working part time in an IREM. Didactic research found a natu-
ral habitat there, close to the terrain of primary and secondary education, and to
mathematics departments. Within less than one decade, it built solid institutional
foundations. The first doctorate programs were created in 1975 in Bordeaux, Paris
and Strasbourg. A few years later, the National seminar of didactics of mathematics
was set up with three sessions per year. In 1980, the journal Recherches en Didactique
des Mathématiques and the biennial Summer school of didactics of mathematics were
simultaneously created. Later on, in 1992, the creation of the ARDM6complemented
this institutionalization process.
2.1.2 A Tradition Based on Three Main Theoretical Pillars
The didactics of mathematics emerged in France with the aim of building a genuine
field of scientific research and not just a field of application for other scientific fields
such as mathematics or psychology. Thus it required both fundamental and applied
dimensions, and needed specific theories and methodologies. Drawing lessons from
the innovative activism of the New Math period with the disillusions it had generated,
French didacticians gave priority to understanding the complex interaction between
mathematics learning and teaching in didactic systems. Building solid theoretical
foundations for this new field in tight interaction with empirical research was an
essential step. Theories were thus and are still conceived of first as tools for the
understanding of mathematics teaching and learning practices and processes, and
for the identification of didactic phenomena. It is usual to say that French didactics
has three main theoretical pillars: the theory of didactical situations due to Brousseau,
the theory of conceptual fields due to Vergnaud, and the anthropological theory of the
didactic that emerged from the theory of didactic transposition, due to Chevallard.
These theories are complex objects that have been developed and consolidated over
decades. In what follows, we focus on a few main characteristics of each of them.
More information is accessible on the ARDM and CFEM websites, particularly video
recorded interviews with these three researchers.7
6ARDM (Association pour la recherche en didactique des mathématiques) http://ardm.eu/ (accessed
2018/01/08).
7The ARDM website proposed three notes dedicated to these three pioneers: Guy Brousseau, Gérard
Vergnaud and Yves Chevallard. On the CFEM website, the reader can access long video recorded
interviews with Brousseau, Vergnaud and Chevallard http://www.cfem.asso.fr/cfem/ICME-13-
didactique-francaise (accessed 2018/01/08).
2 The French Didactic Tradition in Mathematics 15
2.1.2.1 The Theory of Didactical Situations (TDS)
As explained by Brousseau in the long interview prepared for ICME-13 (see also
Brousseau, Brousseau, & Warfield, 2014, Chap. 4), in the sixties, he was an ele-
mentary teacher interested in the New Math ideas and having himself developed
innovative practices. However, he feared the deviations that the implementation of
these ideas by elementary teachers without adequate preparation might generate.
Brousseau discussed this point with Lichnerowicz (see Sect. 2.1.1) who proposed
that he investigate “the limiting conditions for an experiment in the pedagogy of
mathematics”. This was the beginning of the story. Brousseau conceived this inves-
tigation as the development of what he called an experimental epistemology to make
clear the difference with Piagetian cognitive epistemology. According to him, this
required “to make experiments in classrooms, understand what happens…the condi-
tions of realizations, the effect of decisions”. From that emerged the ‘revolutionary
idea’ at the time that the central object of didactic research should be the situation
and not the learner, situations being conceived as a system of interactions between
three poles: students, teacher and mathematical knowledge.
The theory was thus developed with the conviction that the new didactic field
should be supported by methodologies giving an essential role to the design of sit-
uations able to make mathematical knowledge emerge from students’ interactions
with an appropriate milieu in the social context of classrooms, and to the observa-
tion and analysis of classroom implementations. This vision found its expression in
the COREM8associated with the elementary school Michelet, which was created in
1972 and would accompany the development of TDS during 25 years, and also in
the development of an original design-based methodology named didactical engi-
neering (see Sect. 2.1.4) that would rapidly become the privileged methodology in
TDS empirical research.
As explained by Brousseau in the same interview, the development of the theory
was also fostered by the creation of the doctorate program in Bordeaux in 1975 and the
resulting necessity to build a specific didactic discourse. The core concepts of the the-
ory,9those of adidactical and didactical situations, of milieu and didactic contract,
of devolution and institutionalisation, the three dialectics of action, formulation and
validation modelling the different functionalities of mathematics knowledge, and the
fundamental distinction made in the theory between “connaissance” and “savoir”
with no equivalent in English,10 were thus firmly established already in the eighties.
From that time, the theory has been evolving for instance with the introduction of the
hierarchy of milieus or the refinement of the concept of didactic contract, thanks to
the contribution of many researchers. Retrospectively, there is no doubt that the use
8COREM: Centre pour l’observation et la recherche sur l’enseignement des mathématiques.
9In her text Invitation to Didactique, Warfield provides an accessible introduction to these concepts
complemented by a glossary: https://sites.math.washington.edu/~warfield/Inv%20to%20Did66%
207-22-06.pdf (accessed 2018/01/08).
10“Connaissance” labels knowledge engaged by students in a situation while “savoir” labels knowl-
edge as an institutional object.
16 M. Artigue et al.
of TDS in the analysis of the functioning of ordinary classrooms from the nineties
has played an important role in this evolution.
2.1.2.2 The Theory of Conceptual Fields (TCF)
Vergnaud’s trajectory, as explained in the interview prepared for ICME-13, was quite
atypical: beginning as a student in a school of economics, he developed an interest in
theatre, and more particularly for mime. His interest for understanding the gestures
supporting/expressing human activity led him to study psychology at Paris Sorbonne
University, where the first course he attended was given by Piaget! This meeting was
the source of his interest for analyzing the competencies of a subject performing a
given task. Unlike Piaget however, he gave a greater importance to the content to
be learnt or taught than to the logic of the learning. Due to the knowledge he had
acquired during his initial studies, Vergnaud chose mathematical learning as his field
of research; then he met Brousseau, and engaged in the emerging French community
of didactics of mathematics.
In the same interview prepared for ICME-13, he emphasizes a divergence with
Brousseau’s theoretical approach regarding the concept of situation: while Brousseau
is interested in identifying one fundamental situation capturing the epistemological
essence of a given concept to organize its learning, Vergnaud conceives the pro-
cess of learning throughout “the confrontation of a subject with a larger and more
differentiated set of situations”. This point of view led to the development of the the-
ory of conceptual fields, a conceptual field being “a space of problems or situations
whose processing involves concepts and procedures of several types in close con-
nection” (Vergnaud, 1981, p. 217, translated by the authors). The conceptual fields
of additive and multiplicative structures he has especially investigated have become
paradigmatic examples. In this perspective, the difference between “connaissance”
and “savoir” fades in favour of the notion of conceptualisation. Conceptualisation
grows up through the development of schemes; these are invariant organisations of
activity for a class of situations. The operational invariants, concepts-in-action,or
theorems-in-action, are the epistemic components of schemes; they support the sub-
jects’ activity on the way to conceptualizing mathematical objects and procedures. In
fact, the theory of Vergnaud has extended its influence beyond the field of didactics of
mathematics to feed other scientific fields, such as professional didactics, and more
generally cognitive psychology.
2.1.2.3 The Anthropological Theory of the Didactic (ATD)
As explained by Chevallard in the interview prepared for ICME-13, the theory of
didactic transposition emerged first in a presentation he gave at the Summer school
of didactics of mathematics in 1980, followed by the publication of a book (Cheval-
lard, 1985a). Questioning the common vision of taught knowledge as a simple ele-
mentarization of scholarly knowledge, this theory made researchers aware of the
2 The French Didactic Tradition in Mathematics 17
complexity of the processes and transformations that take place from the moment it
is decided that some piece of knowledge should be taught, to the moment this piece
of knowledge is actually taught in classrooms; it helped researchers to make sense
of the specific conditions and results of these processes (Chevallard & Bosch, 2014).
The anthropological theory of the didactic (ATD) is an extension of this theory. In
the interview, Chevallard makes this clear:
The didactic transposition contained the germs of everything that followed it […] It showed
that the shaping of objects for teaching them at a given grade could not be explained only
by mathematical reasons. There were other constraints, not of a mathematical nature […] In
fact, the activity of a classroom, of a student, of a scholar is embedded in the anthropological
reality. (translated by the authors)
With ATD the perspective became wider (Bosch & Gascón, 2006; Chevallard & Sen-
sevy, 2014). Institutions and institutional relationships to knowledge became basic
constructs, while emphasizing their relativity. A general model for human activi-
ties (encompassing thus mathematics and didactic practices) was developed in terms
of praxeologies.11 ATD research was oriented towards the identification of praxe-
ologies, the understanding of their dynamics and their conditions of existence and
evolution—their ecology. In order to analyse how praxeologies ‘live’ and ‘die’, and
also how they are shaped, modified, disseminated, introduced, transposed and elim-
inated, different levels of institutional conditions and constraints are considered,
from the level of a particular mathematical topic up to the level of a given culture,
a civilization or the whole humanity (see the concept of hierarchy of didactic code-
termination12). Such studies are crucial to analyse what kind of praxeologies are
selected to be taught and which ones are actually taught; to access the possibilities
teachers and students are given to teach and learn in different institutional settings, to
understand their limitations and resources, and to envisage alternatives. During the
last decade, ATD has been complemented by an original form of didactic engineer-
ingintermsofstudy and research paths (see Sect. 2.1.4). It supports Chevallard’s
ambition of moving mathematics education from what he calls the paradigm of visit-
ing works by analogy with visiting monuments, here mathematics monuments such
as the Pythagorean theorem, to the paradigm of questioning the world (Chevallard,
2015).
11A praxeology is a quadruplet made of types of tasks, some techniques used to solve these types
of tasks, a discourse (technology) to describe, explain and justify the types of tasks and techniques,
and a theory justifying the technology. Types of tasks and techniques are the practical block of the
praxeology while technology and theory are its theoretical block.
12The hierarchy of didactic codetermination categorizes these conditions and constraints according
to ten different levels: topic, theme, sector, domain, discipline, pedagogy, school, society, civiliza-
tion, humanity.
18 M. Artigue et al.
2.1.3 Theoretical Evolutions
The three theories just presented are the main pillars of the French didactic tradition,13
and they are still in a state of flux. Most French researchers are used to combining
them in their theoretical research frameworks depending on their research problé-
matiques. However, today the theoretical landscape in the French didactic commu-
nity is not reduced to these pillars and their possible combinations. New theoretical
constructions have emerged, reflecting the global evolution of the field. Increasing
interest paid to teacher practices and professional development has led to the dou-
ble approach of teacher practices (Robert & Rogalski, 2002,2005; Vandebrouck,
2013), combining the affordances of cognitive ergonomics and didactics to approach
the complexity of teacher practices. Increasing interest paid to semiotic issues is
addressed by the Duval’s semiotic theory in terms of semiotic registers of represen-
tations (Duval, 1995) highlighting the decisive role played by conversions between
registers in conceptualization. Major societal changes induced by the entrance into
the digital era are met by the instrumental approach to didactics, highlighting the
importance of instrumental geneses and of their management in classrooms (Artigue,
2002), and by the extension of this approach in terms of a documentational approach
to didactics (Gueudet & Trouche, 2009).
A common characteristic of these constructions however is that they all incorpo-
rate elements of the ‘three pillars’ heritage. The theory of joint action in didactics
developed by Sensevy is strongly connected with TDS and ATD (Chevallard & Sen-
sevy, 2014); the instrumental and documentational approaches combine ATD and
TCF with cognitive ergonomics. The double approach—ergonomics and didactic of
teacher practices—reorganizes this heritage within a global activity theory perspec-
tive; this heritage plays also a central role in the model of mathematical working
spaces14 that emerged more recently (Kuzniak, Tanguay, & Elia, 2016).
These new constructions benefit also from increasing communication between the
French didactic community and other research communities in mathematics educa-
tion and beyond. Communication with the field of cognitive ergonomics, facilitated
by Vergnaud, is a good example. Rabardel, Vergnaud’s former student, developed
a theory of instrumented activity (Rabardel, 1995) just when the didactic commu-
nity was looking for concepts and models to analyse new phenomena arising in
the thread of digitalization, more exactly when symbolic calculators where intro-
duced in classrooms (see Guin & Trouche, 1999). Thus, the notion of scheme of
instrumented action developed in a space of conceptual permeability giving birth
to the instrumental approach. As this theoretical construction responded to major
13Other constructions, for instance the tool-object dialectics by Douady (1986), have also been
influential.
14The purpose of the MWS theory is to provide a tool for the specific study of mathematical
work engaged during mathematics sessions. Mathematical work is progressively constructed, as a
process of bridging the epistemological and the cognitive aspects in accordance with three different
yet intertwined genetic developments, identified in the model as the semiotic, instrumental and
discursive geneses.
2 The French Didactic Tradition in Mathematics 19
concerns in mathematics education (Lagrange, Artigue, Laborde, & Trouche, 2003),
soon its development became an international affair, leading to the development of
new concepts, such as instrumental orchestration (Drijvers & Trouche, 2008), which
allowed rethinking of the teachers’ role in digital environments. The maturation of
the instrumental approach, the digitalization of the information and communication
supports, the development of the Internet, led to consider, beyond specific artefacts,
the wide set of resources that a teacher deals with when preparing a lesson, motivat-
ing the development of the documentational approach to didactics (Gueudet, Pepin,
& Trouche, 2012).
2.1.4 Relationship to Design
Due to the context in which the French didactics emerged, its epistemological foun-
dations and an easy access to classrooms provided by the IREM network, classroom
design has always been conceived of as an essential component of the research
work. This situation is reflected by the early emergence of the concept of didac-
tical engineering and its predominant methodological role in research for several
decades (Artigue, 2014). Didactical engineering is structured into four main phases:
preliminary analyses; design and a priori analysis; realization, observation and data
collection; a posteriori analysis and validation. Validation is internal, based on the
contrast between a priori and a posteriori analyses, not on the comparison between
experimental and control groups.
As a research methodology, didactical engineering has been strongly influenced
by TDS, the dominant theory when it emerged. This influence is especially visible in
the preliminary analyses and the design phases. Preliminary analyses systematically
include an epistemological component. In the design of tasks and situations, particu-
lar importance is attached to the search for situations which capture the epistemolog-
ical essence of the mathematics to be learnt (in line with the concept of fundamental
situation, Sect. 2.1.2.1); to the optimization of the potential of the milieu for stu-
dents’ autonomous learning (adidactic potential); to the management of devolution
and institutionalization processes. Didactical engineering as a research methodology,
however, has continuously developed since the early eighties. It has been used with
the necessary adaptations in research at all educational levels from kindergarten to
university, and also in research on teacher education. Allowing researchers to explore
the potential of educational designs that cannot be observed in ordinary classrooms,
it has played a particular role in the identification and study of the learning and teach-
ing potential of digital environments. The development of Cabri-Géomètre (Laborde,
1995) has had, from this point of view, an emblematic role in the French commu-
nity, and at an international level, articulating development of digital environment,
development of mathematical tasks and didactical research.
In the last decade, ATD has developed its own design perspective based on what
Chevallard calls the Herbartian model, in terms of study and research paths (SRP).
In it, particular importance is given to the identification of generating questions with
20 M. Artigue et al.
strong mathematics potential, and also to the dialectics at stake between inquiry and
the study and criticism of existing cultural answers. This is expressed through the
idea of a dialectic between medias and milieus. A distinction is also made between
finalized SRP whose praxeological aim is clear, and non-finalized SRP corresponding
to more open forms of work such as interdisciplinary projects. More globally, the
evolution of knowledge and perspectives in the field has promoted more flexible and
collaborative forms of didactical engineering. A deep reflection on more than thirty
years of didactical engineering took place at the 2009 Summer school of didactics of
mathematics (Margolinas et al., 2011).
Design as a development activity has naturally taken place, especially in the action
research activities developed in the IREMs and at the INRP (National Institute for
Pedagogical Research, now IFé15 French Institute of Education). These have been
more or less influenced by research products, which have also disseminated through
textbooks and curricular documents, but with a great deal of reduction and distor-
tion. However, up to recently, issues of development, dissemination or up-scaling
have been the focus of only a few research projects. The project ACE (Arithmetic
and understanding at primary school)16 supported by IFé, and based on the idea of
cooperative didactical engineering (Joffredo-Le Brun, Morelatto, Sensevy, & Quilio,
2017) is a notable exception. Another interesting evolution in this respect is the idea
of second-generation didactical engineering developed by Perrin-Glorian (2011).
2.1.5 The Role of Empirical Research
Empirical research has ever played a major role in the French didactics. It takes
a variety of forms. However, empirical research is mainly qualitative; large-scale
studies are not so frequent, and the use of randomized samples even less. Statistical
tools are used, but more for data analysis than statistical inferences. Implicative
analysis (Gras, 1996) is an example of statistical method for data analysis which
has been initiated by Gras, a French didactician, and whose users, in and beyond
mathematics education, organize regular conferences.
Realizations in classrooms have always been given a strong importance in empir-
ical research. This took place within the framework of didactical engineering during
the first decades of research development as explained above. However, since the
early nineties the distortion frequently observed in the dissemination of didactical
engineering products led to increasing attention being paid to teachers’ representa-
tions and practices. As a consequence, more importance was given to naturalistic
observations in empirical methodologies. The enrolment of many didacticians in the
IUFM (University institutes for teacher education) from their creation in 1991,17 and
15http://ife.ens-lyon.fr/ife (accessed 2018/01/08).
16http://python.espe-bretagne.fr/ace/ (accessed 2018/01/08).
17IUFM became in 2013 ESPÉ (Higher schools for teacher professional development and for
education, http://www.reseau-espe.fr/).
2 The French Didactic Tradition in Mathematics 21
the subsequent move from the work with expert teachers in the IREMs to the work
with pre-service and novice teachers, also contributed to this move.
The influence of the theory of didactic transposition and subsequently of ATD
also expressed in empirical research. It led to the development of techniques for the
analysis of transpositive processes, the identification of mathematics and didactic
praxeologies through a diversity of sources (textbooks, curricular documents, educa-
tional material…). Empirical research is also influenced by technological evolution
with the increasing use of videos, for instance in studies of teacher practices, and by
increasing attention being paid to semiotic and linguistic processes requiring very
detailed micro-analyses and appropriate tools (Forest, 2012). The involvement of
teachers working with researchers in empirical research is one of the new issues that
have been developed in French didactic research. It is central in the research stud-
ies considering collective work of groups of teachers and researchers, and working
on the evolution of teacher practices; it is also central for the exploration of issues
such as the documentational work of teachers mentioned above (Gueudet, Pepin, &
Trouche, 2013).
The next two sections illustrate this general description through two case studies,
regarding the research carried out on line symmetry-reflection and on algebra over
several decades. These themes have been selected to show the importance of math-
ematics in the French didactic tradition, and also because they offer complementary
perspectives on this tradition.
2.2 Research on Line Symmetry and Reflection
in the French Didactic Tradition
Line symmetry and reflection18 appear as a fundamental subject, both in mathematics
as a science and in mathematics education. In particular, as Longo (2012) points out,
We need to ground mathematical proofs also on geometric judgments which are no less solid
than logical ones: “symmetry”, for example, is at least as fundamental as the logical “modus
ponens”; it features heavily in mathematical constructions and proofs. (p. 53)
Moreover, several characteristics of this subject make it an important and interesting
research topic for the didactics of mathematics: line symmetry is not only a mathe-
matical object but also an everyday notion, familiar to students, and itis also involved
in many professional activities; it is taught from primary school to university; among
geometric transformations, it is a core object as a generator of the group of isome-
tries of the Euclidean plane; it has played a central role in the (French) geometry
curriculum since the New Math reform. This curriculum and the subsequent ones,
18In French, the expression «symétrie axiale» refers both to the property of a figure (line/reflection
symmetry) and the geometric transformation (reflection). Therefore, these two aspects of the concept
are probably more intertwined in French school and research than in English speaking countries.
The word “réflexion” might be used in French but only when considering negative isometries in
spaces of dimension greater than 2.
22 M. Artigue et al.
indeed, obeyed the logic of a progressive introduction of geometrical transforma-
tions, the other subjects being taught with reference to transformations (Tavignot,
1993)—even if this logic faded progressively since the eighties (Chesnais, 2012).
For all these reasons, the learning and teaching of reflection and line symmetry has
been the subject of a great deal of didactic research in France from the eighties up
to the present.
From the first studies about line symmetry and reflection in the eighties,
researchers have explored and modelled students’ conceptions and teachers’ deci-
sions using the theory of conceptual fields (Vergnaud, 1991,2009) and TDS
(Brousseau, 1986,1997), and later the cK¢ model of knowledge19 (Balacheff & Mar-
golinas, 2005; Balacheff, 2013), in relation to questions about classroom design, in
particular through the methodology of didactical engineering (see Sect. 2.1). Since the
nineties and the increasing interest in understanding how ordinary classrooms work,
however, new questions have arisen about teachers’ practices. They were tackled
in particular using the double didactic-ergonomic approach mentioned in Sect. 2.1.
In current research, as the role of language in teaching and learning processes has
become a crucial subject for part of the French research community, line symmetry
and reflection once again appear as a particularly interesting subject to be inves-
tigated, in particular when using a logical analysis of language (Durand-Guerrier,
2013) or studying the relations between physical actions and verbal productions in
mathematical activity.
This example is particularly relevant for a case study in the French didactic tradi-
tion, showing the crucial role of concepts and of curriculum developments in research
on the one hand, and the progressive capitalization of research results, the evolution
of research questions and the links between theoretical and empirical aspects on the
other hand.
2.2.1 Students’ Conceptions, Including Proof and Proving,
and Classroom Design
The first Ph.D. thesis on the teaching and learning of line symmetry and reflection
was defended by Grenier (1988). Her study connected TCF and the methodology of
didactical engineering relying on TDS. She studied very precisely students’ concep-
tions of line symmetry, using activities where students were asked to answer open
questions such as (Fig. 2.1).
In addition, various tasks were proposed to students of constructing symmetri-
cal drawings and figures that were crucial in identifying erroneous conceptions and
19cK¢ (conception, knowing, concept) was developed by Balacheff to build a bridge between
mathematics education and research in educational technology. It proposes a model of learners’
conceptions inspired by TDS and TCF. In it conceptions are defined as quadruplets (P, R, L, )
in which P is a set of problems, R a set of operators, L a representation system, and a control
structure. As pointed out in (Balacheff, 2013) the first three elements are almost directly borrowed
from Vergnaud’s model of conception as a triplet.
2 The French Didactic Tradition in Mathematics 23
What do these figures have in common?
Fig. 2.1 Example of question to study students’ conceptions (Grenier 1988, p. 105)
key didactical variables of tasks. This constituted essential results on which further
research was based (see below). In her thesis, Grenier also designed and trialled
a didactical engineering for the teaching of line symmetry and reflection in mid-
dle school (Grade 6). Beyond the fine grained analysis of students’ conceptions
and difficulties presented in the study, her research produced the important result
that to communicate a teaching process to teachers, it is necessary to consider not
only students’ conceptions, but also teachers’ representations about the mathematic
knowledge at stake, the previous knowledge of the students, and the way students
develop their knowledge.
Relying on the results of Grenier, Tahri (1993) developed a theoretical model
of students’ conceptions of line symmetry. She proposed a modelling of didactical
interaction in a hybrid tutorial based on the micro-world Cabri-Géomètre in order
to analyse teachers’ decisions. Tahri’s results served then as a starting point for
the research of Lima (2006). Lima (2006) referred to the cK¢ (conception, knowing,
concept) model of knowledge in order to identify a priori the controls that the students
can mobilize when solving problems related to the construction and the recognition
of symmetrical figures. In the cK¢ model, a conception is characterized by a set of
problems (P), a set of operators (R), a representation system (L), and also a control
structure (S), as explained above. Lima’s study showed the relevance of the modelling
of the structure of control in the case of line symmetry; in particular, it allowed the
author to reconstruct some coherent reasoning in cases where students’ answers
seemed confused. This refinement of the identification of students’ conceptions of
line symmetry was then used to study teachers’ decisions when designing tasks
aimed at allowing students to reach adequate conceptions of line symmetry. Lima
concluded that a next step was to identify problems favouring the transition from a
given student’s conception to a target conception, both being known.
CK¢ was also used in the Ph.D. of Miyakawa (2005) with a focus on validation.
Considering with Balacheff that «learning proof is learning mathematics», Miyakawa
studied the relationships between mathematical knowledge and proof in the case of
line symmetry. His main research question concerned the gap between pragmatic
validation, using pragmatic rules that cannot be proved in the theory (e.g. relying
on perception, drawings or mental experience) and theoretical validation relying on
rules that can be proved in the theory (Euclidian geometry). Miyakawa especially
focused on rules that grade 9 students are supposed to mobilize when asked to solve
either construction problems or proving problems. He showed that, while the rules
at stake were apparently the same from a theoretical point of view, students able
24 M. Artigue et al.
to solve construction problems using explicitly the appropriate rules might not be
able to use the corresponding rules in proving problems. The author concluded that
although construction problems are playing an important role to overcome the gap
between pragmatic and theoretical validation, they are not sufficient for this purpose.
A new thesis, articulating the TCF and the geometrical working spaces approach
mentioned in Sect. 2.1 was then defended by Bulf (2008). She studied the effects
of reflection on the conceptualization of other isometries and on the nature of geo-
metrical work at secondary school. As part of her research, she was interested in
the role of symmetry in the work of stone carvers and carpenters. Considering with
Vergnaud that action plays a crucial role in conceptualization in mathematics, she
tried to identify invariants through observations and interviews. The results of her
study support the claim that the concept of symmetry organizes the action of these
stone carvers and carpenters.
2.2.2 The Study of Teachers’ Practices and Their Effects
on Students’ Learning
As Chevallard (1997) points out, it took a long time to problematize teaching prac-
tices:
Considered only in terms of his weaknesses, […] or, on the other hand, as the didactician’s
double […], the teacher largely remained a virtual object in our research field. As a result,
didactical modelling of the teacher’s position […] is still in its infancy. (p. 24, translated by
the authors)
Results of research in didactics in France—and, seemingly, in the rest of the
world—progressively led researchers to problematize the question of the role of
the teacher in the teaching and learning process. Questions arose in particular from
the “transmission” of didactical engineering, as illustrated by Grenier’s Ph.D. thesis.
For example, teachers would not offer sufficient time for the students to explore prob-
lems. Instead they would give hints or answers quickly. It also appeared that when
implementing a situation, teachers would not necessarily institutionalize with stu-
dents what the researcher had planned. Along the same lines, Perrin-Glorian, when
working with students in schools from disadvantaged areas (Perrin-Glorian, 1992),
identified that the teachers’ decisions are not necessarily coherent with the logic of
the situation (and knowledge at stake). Some hypotheses emerged about the factors
that might cause these “distortions”. Mainly, researchers explained them by the need
for teachers to adjust to the reality of classrooms and students:
But it also appears that control of problem-situations cannot guarantee the reproducibility
of the process, because some of these discrepancies result from decisions that the teacher
takes to respond to the reality of the class. (Grenier, 1988, p. 7, translated by the authors)
A second important hypothesis was that the researchers and teachers had different
conceptions of what learning and teaching mathematics is. Hence, in the eighties,
2 The French Didactic Tradition in Mathematics 25
many researchers decided to investigate the role of the teacher in the teaching process
and to study ordinary teachers’ practices. This led to new developments within exist-
ing theories and to the emergence of a new approach. A structured model of milieu was
developed in TDS (Brousseau, 1986; Margolinas, 2004); modelling of the teacher’s
position was initiated in ATD by Chevallard (1999). Robert and Rogalski developed
the double approach of teachers’ practices mentioned in 1.3 based on Activity Theory
and the socio-historical theory of Vygostky, and connecting didactic and ergonomic
points of view on teachers’ practices. Research in this approach is driven by the inves-
tigation of regularities and variability of ordinary teaching practices depending on
contexts, mathematical subjects, grades, teachers, etc. This investigation then allows
the identification of causes and rationales underlying teachers’ practices. Another
focus is to investigate the effects of teaching practices on students’ learning. Ches-
nais’ Ph.D. thesis about the teaching and learning of reflection and line symmetry
(Chesnais, 2009) is a good example of this evolution. Relying on previous research
and using the didactic and ergonomic double approach, she compared the teaching
and learning of these subjects in the 6th grade between a school situated in a socially
disadvantaged area and an ordinary one. The research was based on both ‘natural-
istic observation’ (during the first year) and an experiment (during the second year)
which consisted of the transmission of a teaching scenario about reflection and line
symmetry elaborated by one teacher from the ordinary school to another one from
the socially disadvantaged school. The results showed that socially disadvantaged
students could perform “as well as” ordinary ones provided that certain conditions
are fulfilled. In particular, the following crucial conditions were identified: an impor-
tant conceptual ambition of the teaching scenario, its coherence and “robustness”
and the fact that the teacher receiving it is sufficiently aware of some specificities
of the content and students’ learning difficulties. Moreover, the research showed
that multiple reasons drove the teachers’ choices, and explained some differences
identified between them: the fact that they taught to different audiences had proba-
bly a great influence but also their experience in teaching, determining their ability
to identify the important issues about the teaching of line symmetry and reflection
(in this case, it appeared that the experienced teacher of the ordinary school had a
more coherent idea about the teaching of line symmetry and reflection because she
had experienced teaching with older and more detailed teaching instructions). The
research also suggested that collaborative work between teachers might be a good
lever for professional development under certain conditions (in this work, the role
of the researcher as an intermediary was crucial because the first teacher was not
clearly conscious of what made her scenario efficient).
2.2.3 Current Research
A consistent part of recent French research heads toward a thorough investigation of
the role of language in the teaching and learning of mathematics. Research globally
considers language either as an object of learning (as part of concepts), as a medium
26 M. Artigue et al.
for learning (its role in the conceptualisation process) and for teaching, and/or as a
methodological means for researchers to get access to students’ and teachers’ activity.
Line symmetry and reflection represent once again an interesting subject with regard
to the role of language. Indeed, a logical analysis (Vergnaud, 2009; Durand-Guerrier,
2013) shows that symmetry can be considered as a property of a given figure but also,
via reflection, as a ternary relation involving two figures and an axis, or as a geometric
transformation involving points, or even as a binary relation—when considering
two figures and questioning the existence of a reflection transforming one into the
other. Studying how these “variations of the meanings of words” may be expressed
in the French language shows an incredible complexity, in particular because of
the polysemy of the words “symétrie” and “symétrique”. This makes symmetry a
good subject to study the relationships between action and language in mathematical
activities, and how teachers and students deal with this complexity. For example
Chesnais and Mathé (2015) showed that 5th grade20 pupils’ conceptualisation of the
“flipping over property”21 of reflection results from the articulation of several types
of interactions between students and milieu mediated by language, instruments like
tracing paper, and by the teacher: for example, they showed that the manipulation of
the tracing paper (flipping it to check if the initial figure and its image match) needs
to be explicitly identified and that it is complementary to the use of an adequate
vocabulary. It also appeared that teachers identified these issues differently.
Questions about teacher education and the development of teaching resources also
play an important role in recent developments of research in the field. Here again,
line symmetry and reflection were chosen as subjects for research. For instance,
Perrin-Glorian elaborated the concept of second generation didactical engineering
mentioned in Sect. 2.1 in the context of a long term and original research project
regarding the teaching and learning of geometry, in which reflection and line sym-
metry play a crucial role (Perrin-Glorian, Mathé, & Leclercq, 2013). Searching for
the construction of coherence from primary to secondary school, this project also
led to a deepening of reflection on the role that the use of instruments can play in a
progressive conceptualisation of geometrical objects.
We cannot enter into more details about this important set of research, but through
this case study, we hope to have made clear an important feature of the development
of didactic research in the French tradition: the intertwined progression of research
questions, theoretical elaborations and empirical studies, coherently over long peri-
ods of time.
20The 5th grade corresponds to the last grade of primary school in France.
21We refer here to the fact that reflection is a negative isometry.
2 The French Didactic Tradition in Mathematics 27
2.3 Research on School Algebra in the French Didactic
Tradition. From Didactic Transposition
to Instrumental Genesis
Research on school algebra started at the very beginning of the development of the
French didactic tradition in the 1980s with the first studies of the didactic transposition
process (Chevallard, 1985a,b,1994). Since then, and for more than 30 years, school
algebra has been the touchstone of various approaches and research methodologies,
which spread in the French, Spanish and Italian speaking communities, thanks to
numerous collaborations and common research seminars like SFIDA (see Sect. 2.5).
Fortunately, all this work has been synthesized in two very good resources. The
first one is a special issue of the journal Recherches en Didactique des Mathématiques
(Coulange, Drouhard, Dorier, & Robert, 2012), which presents a summary of recent
works in the subject, with studies covering a wide range of school mathematics,
from the last years of primary school to the university. It includes sixteen papers
grouped in two sections: a first one on teaching algebra practices and a second one
presenting cross-perspectives with researchers from other traditions. The second
resource is the survey presented by Chaachoua (2015), and by Coppé and Grugeon
(2015) in two lectures given at the 17th Summer school of didactics of mathematics
in 2013 (see Butlen et al., 2015). They focus on the effective and potential impacts
between didactic research, the teaching profession and other instances intervening in
instructional processes (curriculum developers, textbooks authors, teacher educators,
policy makers, etc.). The discussion also deals with transfers that have not taken
place and highlights various difficulties that seem to remain embedded in the school
institution.
We cannot present this amount of work in a few pages. Instead of focusing on
the results and the various contributions of each team, we have selected three core
research questions that have guided these investigations in the field of secondary
school mathematics. First, the analysis of the didactic transposition process and the
associated questions about what algebra is and what kind of algebra is taught at
school. Second, research based on didactical engineering proposals which address
the question of what algebra could be taught and under what kind of conditions.
Finally, both issues are approached focusing on ICT to show how computer-assisted
tools can modify not only the way to teach algebra but also its own nature as a
mathematical activity.
2.3.1 What Algebra Is to Be Taught: Didactic Transposition
Constraints
To understand the role played by research on school algebra in the development of
the field of didactics of mathematics, we should look back to the 1980s and Cheval-
lard’s attempt to approach secondary school mathematics from the new perspective
28 M. Artigue et al.
opened by the theory of didactic situations (TDS). At this period, TDS was mainly
focused on pre-school and primary school mathematics. Its first enlargements to sec-
ondary school mathematics gave rise to the analyses in terms of didactic transposition
(Chevallard, 1985a). Algebra was one of the case studies that received more atten-
tion. Contrary to the majority of investigations of this period, studies on the didactic
transposition processes directly adopted an institutional perspective anchored in deep
epistemological and historical analyses.
The first studies (Chevallard, 1985b) pointed out the nature and function of alge-
bra in scholarly mathematics since the work of Vieta and Descartes, and its fading
importance in secondary school curricula after the New Maths reform. What appears
is a certain lack of definition of algebra as a school mathematical domain (in France as
well as in many other countries), centred on the resolution of first and second degree
equations. Many elements of what constitutes the driving force of the algebraic work
(use of parameters and inter-play between parameters and unknowns, global mod-
elling of arithmetic and geometrical systems, etc.) have disappeared from school
curricula or only play a formal role. Part of this evolution can be explained by a cul-
tural difficulty in accepting the primarily written nature of algebra, which strongly
contrasts with the orality of the arithmetical world (Chevallard, 1989,1990; see also
Bosch, 2015).
A first attempt to describe the specificities of algebraic work was proposed by con-
sidering a broad notion of modelling that covers the modelling of extra-mathematical
as well as mathematical systems (Chevallard, 1989). Analyses of algebra as a mod-
elling process were later developed in terms of the new epistemological elements
proposed by the Anthropological Theory of the Didactic based on the notion of
praxeology. This extension brought about new research questions, such as the role
of algebra and the students’ difficulties in school institutional transitions (Grugeon,
1995) or the constraints appearing in the teaching of algebra when it is conceived as
a process of algebraisation of mathematical praxeologies (Bolea, Bosch, & Gascón,
1999). In this context, algebra appears linked to the process of modelling and enlarg-
ing previously established praxeologies.
2.3.2 Teaching Algebra at Secondary School Level
While the aforementioned studies are more focused on what is considered as the exter-
nal didactic transposition—the passage from scholarly knowledge to the knowledge
to be taught—the question of “what algebra could be taught” is addressed by investi-
gations approaching the second step of the didactic transposition process, the internal
transposition, which transforms the knowledge to be taught into knowledge actually
taught. This research addresses teaching and learning practices, either from a ‘nat-
uralistic’ perspective considering what is effectively taught and learnt as algebra at
secondary level—as well as what is not taught anymore—or in the design and imple-
mentation of new proposals following the methodology of didactical engineering.
Research questions change from “What is (and what is not) algebra as knowledge
2 The French Didactic Tradition in Mathematics 29
to be taught?” to “What can be taught as algebra in teaching institutions today?”.
However, the answers elaborated to the first question remain crucial as methodolog-
ical hypotheses. Experimental studies proposing new conditions to teach new kinds
of algebraic activities rely on the previous elaboration of a priori epistemological
models (what is considered as algebra) and didactic models (how is algebra taught
and learnt).
The notion of calculation program (programme de calcul), used to rebuild the
relationships between algebra and arithmetic in a modelling perspective, has been at
the core of some of these instructional proposals, especially in the work carried out
in Spain by the team led by Gascón and Bosch. Following Chevallard’s proposal to
consider algebra as the “science of calculation programs”, a reference epistemologi-
cal model is defined in terms of stages of the process of algebraisation (Bosch, 2015;
Ruiz-Munzón, Bosch, & Gascón, 2007,2013). This redefinition of school algebra
appears to be an effective tool for the analysis of curricula and traditional teach-
ing proposals. It shows that the algebraisation process in lower secondary school
mathematical activities is very limited and contrasts with the “fully-algebraised”
mathematics in higher secondary school or first year of university. It also provides
grounds for new innovative instructional processes like those based on the notions
of study and research activities and study and research paths (see Sect. 2.1.4) that
cover the introduction of negative numbers in an algebraic context and the link of
elementary algebra with functional modelling (Ruiz-Munzón, Matheron, Bosch, &
Gascón, 2012).
2.3.3 Algebra and ICT
Finally, important investigations of school algebra address questions related to the
integration of ICT in school mathematics. They look at the way this integration might
influence not only students’ and teachers’ activities, but also the nature of algebraic
work when paper and pencil work is enriched with new tools such as a spreadsheet,
a CAS or a specific software specially designed to introduce and make sense of
elementary algebraic manipulations. This research line starts from the hypothesis
that ICT tools turn out to be operational when they become part of the students’
adidactic milieu, thus focusing on the importance and nature of ICT tools feedback
on the resolution of tasks. However, the integration of ICT tools in the adidactic milieu
cannot be taken for granted as research on instrumental genesis has shown. Many
investigations have produced evidence on new teachers’ difficulties and problems
of legitimacy to carry out such integration, pointing at didactic phenomena like the
double reference (paper and pencil versus ICT tools) (Artigue, Assude, Grugeon, &
Lenfant, 2001) or the emergence of new types of tasks raised by the new semiotic
representations produced by ICT tools. In fact, the epistemological and didactic
dimensions (what is algebra and how to teach it) appear so closely interrelated that
a broad perspective is necessary to analyse the didactic transposition processes.
The instrumental approach of technological integration emerged, especially in the
30 M. Artigue et al.
case of the use of CAS and technologies not initially thought for teaching. This
approach was then extended to other technologies and has experienced significant
internationalization (Drijvers, 2013).
In respect to technologies specifically designed for teaching, particular software
devices have been designed on the basis of didactic investigations. They thus appear
as a paradigm of connection between fundamental research, teaching development
and empirical validation of ICT didactic tools (Chaachoua, 2015). The first one is
Aplusix,22 a micro world especially designed for the practice of elementary alge-
bra that remains very close to the students’ paper and pencil manipulations, while
providing feed-back based on a detailed epistemological and didactic analysis of
potential learning processes (Trgalová & Chaachoua, 2009). Aplusix is based on
a model of reasoning by equivalence between two algebraic expressions which is
defined by the fact that they have the same denotation (Arzarello, Bazzini, & Chiap-
pini, 2001). Therefore, Aplusix enables students to work on the relationship between
sense and denotation, which is essential to effectuate and understand the transfor-
mation of algebraic expressions. The second device is Pépite, a computer-based
environment providing diagnostics of students’ competences in elementary algebra,
as well as tools to manage the learning heterogeneity with the proposal of differen-
tiating teaching paths (Pilet, Chenevotot, Grugeon, El Kechaï, & Delozanne, 2013).
Both Aplusix and Pépite are the fruit of long term and continuous research work in
close collaboration with computer scientists.
On the whole, all these investigations have contributed to a fundamental epistemo-
logical questioning on the nature of algebraic work and its components, sometimes
enriched by linguistic or semiotic contributions. They all share the research aim of
better identifying the universe of didactic possibilities offered by today’s school sys-
tems and of determining the local conditions that would allow a renewed teaching of
the field, directly aligned with mathematical work in primary school and connected
with the world of functions and calculus for which it is preparatory
The longitudinal dimension of algebra and its predominance in higher secondary
education and beyond show that one cannot approach school algebra without ques-
tioning the rationale of compulsory education as a whole and how it can supply the
mathematical needs of citizens. We are thus led to the initial project of Brousseau and
the fundamental problems that motivated the development of TDS: to reconstruct the
compulsory mathematical curriculum based on democratic and effective principles
that can be discussed in an objective and non-authoritarian way.
Let us finish this section with a tribute to Jean-Philippe Drouhard, one of the
founders of the Franco-Italian Seminar in Didactics of Algebra (SFIDA, 1992–2012,
see Sect. 2.5), who took special leadership in the development of the research on
school algebra and devoted his research life in didactics to the study of the semio-
linguistic complexity of elementary algebra (Drouhard, 2010).
After these two cases studies, we enter the second part of this chapter which is
devoted to the influence of the French didactic tradition on other educational cultures,
and the connections established, in Europe and beyond. These connections have a
22http://www.aplusix.com.
2 The French Didactic Tradition in Mathematics 31
long history as evidenced in Artigue (2016), and the IREMs have played an important
role in their development, together with the many foreign students who have prepared
their doctorate in France or been involved in co-supervision programs since the early
eighties. The authors of the four following sections, from Germany, Italy, Mexico
and Tunisia were such doctorate students who have now pursued their careers in
their own country. These four case studies illustrate different ways through which
a research tradition may diffuse, influencing individual researchers or communities,
and the source of enrichment that these interactions are for the tradition itself.
2.4 View of the French Tradition Through the Lens
of Validation and Proof
In this section I (Christine Knipping) present my view of the French didactic tradition.
I take the position of a critical friend (the role given to me at ICME-13), and I
consider this tradition through the lens of didactic research on validation and proof.
This topic is at the core of my own research since my doctoral work, which I pursued
in France and Germany. I will structure the section around three main strengths that
I see in the French tradition: Cohesion, Interchange, and Dissemination. The first
strength is Cohesion as the French community has a shared knowledge experience
and theoretical frameworks that make it possible to speak of a French Didactique. The
second strength is an open Interchange within the community and with others. The
third strength allows the dissemination of ideas in the wider world of mathematics
education. These three strengths, which have also influenced my own research, will
be illustrated by examples from a personal perspective.
2.4.1 Cohesion
As a student, having just finished my Masters Degree in Mathematics, Philosophy and
Education in the 1990s in Germany, I went to Paris, and enrolled as a student in the
DEA-Programme in Didactics of Mathematics at University Paris 7 (see Sect. 2.1).
Courses were well structured and introduced students to key ideas in the French
didactic tradition: the theory of didactical situations (TDS) developed by Brousseau,
the theory of conceptual fields due to Vergnaud, and the anthropological theory of the
didactic that emerged from the theory of didactic transposition, conceptualised by
Chevallard. The courses were taught by a wide range of colleagues from the research
group DIDIREM of Paris 7, which is now the LDAR (Laboratoire de Didactique
André Revuz). Among others Artigue, Douady, Perrin-Glorian, as well as colleagues
from several teacher training institutes (IUFM) were involved. These colleagues not
only introduced us to the theoretical pillars of the French tradition in mathematics
education, but they also showed us how their own research and recent doctoral work
32 M. Artigue et al.
was based on these traditions. This made us aware of the power of French conceptual
frameworks and demonstrated vividly how they could be applied. It also showed us
some empirical results these frameworks had led to and how French research in
mathematics education was expanding quickly at that time. The specific foci of
research and research questions in the French community were an obvious strength,
but phenomena that were not in these foci were not captured. A few of our professors
reflected on this and made us aware that classroom interactions, issues of social
justice and cultural contexts were more difficult to capture with the given theoretical
frameworks. Also validation and proof, a topic I was very interested in, was hardly
covered by our coursework in Paris, while in Grenoble there was a clear research
focus on validation and proof since the 1980s (Balacheff, 1988), that probably was
reflected in their DEA programme at the time.
The French National Seminar in Didactics of Mathematics, which we were invited
to attend, was another experience of this phenomenon of cohesion. Many of the pre-
sentations at the National Seminar were based on the three pillars of French didactics
and also French Ph.D.’s followed these lines. But there were topics and approaches
beyond these frameworks that seemed to be important for the French community;
validation and proof was apparently one of them. Presentations in this direction were
regularly and vividly discussed at the National Seminar and quickly became public
knowledge within the community. For example, Imre Lakatos’ striking work Proofs
and Refutations was not only translated by Balacheff and Laborde into French in
1984, but also presented to the French community at one of the first National Semi-
nars. Looking through the Actes du Séminaire National de Didactique des Mathéma-
tiques shows that validation and proof is a consistent theme over decades. In this area
of research French colleagues recognise and reference each other’s work, but cohe-
sion seems less strong in this context. Besides Balacheff’s first school experiments
with Lakatos’ quasi-empirical approach (Balacheff, 1987), Legrand introduced the
scientific debate as another way to establish processes of validation and proof in the
mathematics class (Legrand, 2001). Coming from logic, Arsac and Durand-Guerrier
approached the topic from a more traditional way, attempting to make proof acces-
sible to students from this side (Arsac & Durand-Guerrier, 2000). Publications not
only in the proceedings of the National Seminar, but also in the journal Recherches
en Didactique des Mathématiques (RDM), made the diverse approaches in the field
of validation and proof accessible. So in this context Interchange seemed the strength
of the French community.
2.4.2 Interchange
Such interchange, within the community as described above but also with researchers
from other countries, is in my view another striking strength. The Colloque Franco-
Allemand de Didactique des Mathématiques et de l’Informatique is one example of
this and valued by its publication in the book series associated with Recherches en
Didactique des Mathématiques (Laborde, 1988). Many other on-going international
2 The French Didactic Tradition in Mathematics 33
exchanges, as described for example in the following sections of this chapter, also
illustrate this strength of the French community. I received a vivid impression of
the passion for interchange at the tenth Summer school of didactics of mathematics
held in Houlgate in 1999. The primary goal of such summer schools is to serve as a
working site for researchers to study the work of their colleagues. Young researchers
are welcome, but not the focus. Consistently colleagues from other countries are
invited. Here Boero (Italy), Duval (France) and Herbst (USA) contributed to the
topic “Validation, proof and formulation”, which was one of the four themes in
1999. Over the course of the summer school their different point of views—cognitive
versus socio-cultural—became not only obvious, but were defended and challenged
in many ways. These were inspiring debates. This interchange was also reflected in
the International Newsletter on the Teaching and Learning of Proof ,23 established
in 1997 whose first editor was Balacheff. Innovative articles and also the careful
listing of recent publications made this Newsletter a rich resource for researchers in
the area at the time. Colleagues from diverse countries published and also edited this
online journal, which continues with Mariotti and Pedemonte as editors. Debates on
argumentation and mathematical proof, their meaning and contextualization, as well
as cultural differences are of interest for the Newsletter. Sekiguchi’s and Miyazaki’s
(2000) publication on ‘Argumentation and Mathematical Proof in Japan’ for example
already explicitly addressed this issue. One section in the Newsletter also explicitly
highlights working groups and sections of international conferences that deal with the
topic of validation and proof. References to authors and papers are listed and made
accessible in this way. Interchange is therefore highly valued and also expressed
by the fact that some articles are published in multiple languages (English, French,
Spanish).
2.4.3 Dissemination
Interchange is in multiple ways related to dissemination, another strength of the
French didactic tradition. Looking for vivid exchange also implies that perspectives
and approaches from the community become more widely known and disseminated.
The research work of Herbst is prominent in this way; French tradition had a visible
impact on his own unique work, which then influenced not only his doctoral students
but also other colleagues in the US. This is evident as well in the many bi-national
theses (French-German; French-Italian) in mathematics education, including quite a
few on the topic of validation and proof, my own Ph.D. thesis among them (Knipping,
2003). French universities were in general highly committed to this kind of double
degree and had international graduate programs and inter-university coalitions with
many countries. As Ph.D. students we were highly influenced by French research
traditions and incorporated ideas from the French mathematics education community
into our work. Pedemonte’s Ph.D. thesis (2002) entitled Etude didactique et cogni-
23http://www.lettredelapreuve.org (accessed 2018/01/08).
34 M. Artigue et al.
tive des rapports de l’argumentation et de la démonstration dans l’apprentissage
des mathématiques (Didactic and cognitive analyses of the relationship between
argumentation and proof in mathematics learning) is an example of this. Her work
and ideas were then further disseminated into the international validation and proof
research community.
The working group on argumentation and proof, which has been meeting since
the Third Conference of the European Society for Research in Mathematics Educa-
tion (CERME 3, 2003) in Bellaria has been a vivid place not only for discussion and
exchange, but also a site where ideas from the French community have continuously
been prominent. French colleagues are always present, serving in guiding functions,
and scholars like me who are familiar with ideas and approaches of the French tra-
dition and have actively used them in our own work spread these ideas further into
the international community. Interesting crossover work has also emerged through-
out the years between different disciplines. Miyakawa, who also did his Ph.D. work
in Grenoble with Balacheff (Miyakawa, 2005), is another interesting example of a
colleague who stands for dissemination of French ideas and is interested in the kind
of interdisciplinary work that I see as characteristic for French research in the field
of validation and proof. He is now well established in Japan but he continues to
collaborate with French colleagues and reaches out to other disciplines. Recently he
presented at CERME 10 a paper with the title Evolution of proof form in Japanese
geometry textbooks together with Cousin from the Lyon Institute of East Asian Stud-
ies (Cousin and Miyakawa, 2017). Collaboratively they use the anthropological the-
ory of the didactic (ATD) to study the didactic transposition of proofs in the Japanese
educational system and culture and to better understand proof taught/learnt in this
institutional context. Reflecting on the conditions and constraints specific to this
institution Miyakawa and Cousin help us also to see in general the nature of difficul-
ties for students in the context of proof-and-proving from a new perspective. French
theoretical frameworks are again fruitful for this kind of inter-cultural-comparative
work.
In closing, from my perspective as a critical friend working in the field of validation
and proof, these three strengths, Cohesion, Interchange, and Dissemination, have
contributed to the success of the French Didactique. As a critical friend, I should
also observe that each of these strengths comes with costs. Theoretical cohesion
can limit the research questions that can be addressed, and research groups strongly
focused on one area inevitably neglect others. This is reflected in some limitations in
interchange and dissemination. French voices are clearly heard in some contexts, but
hardly at all in others, and interchanges are sometimes unbalanced. Overall, however,
it is clear that France has been fortunate to have a strong community in didactics,
supported by a range of institutions that foster interchange and dissemination, of
which this set of presentations at ICME is yet another example.
2 The French Didactic Tradition in Mathematics 35
2.5 Didactic Interactions Between France and Italy.
A Personal Journey
Didactic interactions between Italy and France have a long history. For instance,
Italian researchers participated in the French Summer schools of didactics of math-
ematics from the first. In this section, after pointing out some structures that have
nurtured these interactions, I (Michela Maschietto) present and discuss them through
the lenses of my personal experience, first as a doctoral student having both French
and Italian supervisors, then as an Italian researcher regularly involved in collabora-
tive projects with French researchers. I also approach them in a more general way,
considering both the cultural and institutional conditions in which the research has
developed in the two countries.
2.5.1 Opportunities for Collaboration: SFIDA, Summer
Schools and European Projects
Among the different institutional structures that provided opportunities for collabo-
ration between French and Italian researchers, SFIDA24 (Séminaire Franco-Italien de
Didactique de l’Algèbre) certainly played a crucial role. The idea of this seminar arose
from the interest in teaching and learning algebra shared by the researchers of the
Italian teams at the University of Genova and Turin (respectively, directed by Boero
and Arzarello) and the French team at the University of Nice (directed by Drouhard).
SFIDA sessions were organized twice per year from 1993 to 2012 (SFIDA-38 was
the last edition), alternatively by the three research teams, and held in their respective
universities. A unique feature of this seminar was that everyone spoke his/her own
language, as the programs of each session show. This attitude to overcome language
constraints fostered the participation of researchers from other universities, and also
students, both Italian and French. SFIDA was not only a place for sharing projects
or work in progress, but the seminar functioned also as a working group that allowed
the emerging of new ideas in this field of research, as attested by the articles devoted
to this seminar in the second part of the special issue of Recherches en Didactique
des Mathématiques on didactic research in algebra (Coulange, Drouhard, Dorier, &
Robert, 2012) already mentioned in Sect. 2.3.
Other scientific events allowed the two communities to meet each other and col-
laborate, like the French Summer school of didactics already mentioned and the
conferences of the Espace Mathématique Francophone. The participation and con-
tribution of Italian researchers and teacher-researchers to those events, the involve-
ment of Italian researchers in their scientific and organizing committees strengthened
the relationships between the communities. The team of the University of Palermo
(directed by Spagnolo) was for itself especially involved in the regular scientific
24https://sites.google.com/site/seminairesfida/Home/ (accessed 2018/02/10).
36 M. Artigue et al.
meetings of the group on Implicative Analysis (Gras, 1996), and even organized two
of them (in 2005 and 2010). Spagnolo, supervised by Brousseau was also one of
the first Italian students to get his Ph.D. in a French university (Spagnolo, 1995),
together with Polo supervised by Gras (Polo Capra, 1996). Furthermore, around
the years 2000, several Italian students carried out their doctoral thesis in different
French universities.
French and Italian research teams were also involved in several European projects
on the use of technologies, on teacher training and on theoretical perspectives. For
instance, the ReMATH project (Representing Mathematics with Digital Media) has
focused on the analysis of the potentiality of semiotic representations offered by
dynamic digital artefacts (Kynigos & Lagrange, 2014). Adopting a perspective of
networking among theoretical frameworks, this project has fostered the development
of specific methodologies for such networking like the idea of cross-experiment
(Artigue & Mariotti, 2014). In recent times, other research teams collaborated within
the FASMED project on formative assessment.25
Despite those collaborations, relevant differences exist between the Italian and
French traditions in mathematics education: they do not only have to deal with
differences in theoretical frameworks, but also with cultural differences of the two
communities in which research is carried out, as explained in Chap. 4. A critical
perspective on them has been proposed by Boero, one of the promoters of SFIDA,
who highlighted some difficulties to establish collaborations between French and
Italian researchers since the beginning of SFIDA. For Boero (1994), they were due
to:
•Italian researchers had been more interested in studying the relationships between
innovative didactical proposals and their development in classes than modelling
didactical phenomena, as in the French tradition;
•The experimental parts of the Italian research involving classes were not situations
the researcher studied as an external observer, but they were an opportunity to make
more precise and test the hypotheses about the Italian paradigm of “research for
innovation”.
•The presence of several teacher-researchers in Italian research teams.
By a cultural analysis of the context in which researchers work Boero deepens
his reflection in a more recent contribution: he claims that he is “now convinced that
these difficulties do not derive only from researchers’ characteristics and personal
positions, but also (and perhaps mainly) from ecological conditions under which
research in mathematics education develops” (Arcavi et al., 2016, p. 26). Among
these conditions, he especially points out: the features of the school systems (i.e., the
Italian national guidelines for curricula are less prescriptive than French syllabuses
and primary school teachers in Italy usually teach and follow the same students
for five years); the economic constraints of research and the weight of the cultural
environment (i.e., the cultural and social vision of mathematics, the spread of the
25Improving Progress for Lower Achievers through Formative Assessment in Science and Mathe-
matics Education, https://research.ncl.ac.uk/fasmed/ (accessed 2018/02/10).
2 The French Didactic Tradition in Mathematics 37
idea of mathematics laboratory, the development of mathematics research, and the
influence of sociological studies that Boero considers weaker in Italy than in France).
I would like to give now another vision of the didactic relationships between Italy
and France by reflecting on my personal scientific journey, from the perspective of
boundary crossing (Akkerman & Bakker, 2011), as my transitions and interactions
across different sites, and boundary objects, as artefacts, have had a bridging function
for me.
2.5.2 A Personal Scientific Journey
This journey started at the University of Turin, where I obtained a one-year fellowship
to study at a foreign university. The University of Bordeaux I, in particular the
LADIST (Laboratoire Aquitain de Didactique des Sciences et Techniques) directed
by Brousseau was my first destination, my first boundary crossing. At the LADIST
I became more deeply involved in the Theory of Didactical Situations (Brousseau,
1997) that I had previously studied. A fundamental experience for me as a student
in mathematics education was the observation of classes at the École Michelet,the
primary school attached to the COREM (see Sect. 2.1). The activities of the COREM
allowed me to compare the experimental reality with the factual components that
Brousseau highlighted in his lessons to doctoral students at the LADIST. My first
personal contact with French research was thus characterized by discussions, passion
and enthusiasm for research and, of course, by the people I met. At the end of my
fellowship, I moved to Paris to prepare a DEA.26 There I met Artigue and other French
colleagues, and I read Boero’s paper (Boero, 1994) quoted above that encouraged
me to become aware of the potential of my boundary crossing.
After the DEA, I continued my doctoral studies within an institutional agreement
between the University of Paris 7 and the University of Turin. I had two supervisors
(Artigue and Arzarello) from two didactic cultures who had not yet collaborated.
Retrospectively, I can claim that the dialogue between these cultures was under
my responsibility. Ante litteram, I looked for a kind of networking strategy (Predi-
ger, Bikner-Ahsbahs, & Arzarello, 2008) appropriate to the topic of my research:
the introduction of Calculus in high school with graphic and symbolic calculators
(Maschietto, 2008). From the French culture I took the methodology of didactical
engineering (Artigue, 2014) with its powerful a priori analysis, the idea of situation
and a-didacticity, and the instrumental approach (see Sect. 2.1). From the Italian cul-
ture, I took the strong cognitive component following embodied cognition, a semiotic
focus with the attention to gestures and metaphors, and the cognitive roots of con-
cepts. The experimental part of my research was a didactical engineering carried out
in some Italian classes. I had to negotiate the planned situations with the teachers of
these classes who were members of the research team of the University of Turin. In
that process, a relevant element was the a priori analysis of the planned situations.
26Diplôme d’Etudes Approfondies.
38 M. Artigue et al.
It was a powerful tool for sharing the grounded idea of the didactical engineering
with those teachers who did not belong to the French scientific culture, and became
a boundary object.
At the end of my doctoral period, I moved to the University of Modena e Reggio
Emilia with a research fellowship within the European project on mathematics exhi-
bitions Maths Alive. Finally, I got a research position in that university some years
later, and I currently work there. In Modena, I met the framework of the Theory of
Semiotic Mediation (Bartolini Bussi & Mariotti, 2008) (see Chap. 4), and the math-
ematical machines. These are tools mainly concerning geometry, which have been
constructed by teacher-researchers (members of the team of that University directed
by Bartolini Bussi, now Laboratorio delle Macchine Matematiche27) following the
Italian tradition of using material tools inspired by the work of Castelnuovo in mathe-
matics laboratory. In my first designs of laboratory sessions, I tried to locally integrate
the TDS and the Theory of Semiotic Mediation by alternating a moment of group
work with a-didactical features and a moment of collective discussion with an insti-
tutional component, with the aim of mediating mathematical meanings. Another
boundary crossing for me.
The collaboration with French colleagues was renewed when I moved to the
INRP28 in Lyon for two months as visiting researcher in the EducTice team. My new
position as a researcher and not as a student, changed the conditions, making more
evident the potential of boundary crossing. The work with the EducTice team was the
source of new insights and productive exchanges about: the notion of mathematics
laboratory and the use of tools in mathematics education (Maschietto & Trouche,
2010); the notion of resources and collaborative work among teachers from the per-
spective of the Documentational Approach (see Sect. 2.1); and the role of digital
technologies in learning, teaching and teacher education. The second and third ele-
ments contributed to the MMLAB-ER project I was involved in Italy in terms of
planning and analysis of the teachers’ education program (Maschietto, 2015). The
first and third elements have been essential in the development of the original idea of
duo of artefacts based on using a material artefact and a digital one in an intertwined
way in the same learning project (Maschietto & Soury-Lavergne, 2013). We jointly
planned the design of the digital counterpart of a material artefact,29 the tasks with the
duo of artefacts, resources and training for primary school teachers, and the idea of
duo of artefacts developed as a boundary object for all of us, bridging our respective
practices.
The challenge of boundary crossing is to establish a continuity across the dif-
ferent sites and negotiated practices with other researchers and teachers. This story
tells my personal experience of boundary crossing between the French and Italian
didactic cultures. This is a particular story, but not an isolated case. It illustrates how
27MMLab, www.mmlab.unimore.it (accessed 2018/02/10).
28Institut National de Recherche Pédagogique, now Institut Français de l’Éducation (http://ife.ens-
lyon.fr/ife) (accessed 2018/02/10).
29In this case, the material artefact is the “pascaline Zero + 1”, while the digital counterpart is the
“e-pascaline” constructed in the new Cabri authoring environment.
2 The French Didactic Tradition in Mathematics 39
the repeated boundary crossing of individual researchers has contributed and still
contributes to the dissemination of the French didactic culture and, in return, to its
enrichment by other didactic cultures.
2.6 Didactic Interactions Between France
and Latin-America: The Case of Mexico
In Latin America, as in other regions of the world, Mathematics Education emerged
in the 1970s, and its development was marked by a strong relationship with the
French community of educational mathematicians. Initially, the Department of Math-
ematics Education at the Centro de Investigación y de Estudios Avanzados (DME-
CINVESTAV, created in 1975), and the network of French IREMs have made excep-
tional contributions to this process by establishing relations between specialists in
Mexico and France. Over time, other institutions emerged and the French theoretical
approaches were adapted to respond to different educational needs of the continent,
which resulted in a greater theoretical development: production of new tools, broad-
ening of notions, questioning of scope, study of new issues, etc. Analyzing the case of
Mexico, the focus of this section, we show other research communities how the rela-
tionships between institutions and research groups can generate scientific advances
with social impact. Also, it leads to important reflections on current challenges of
Latin America reality, especially that of achieving quality education for all, a key
element in securing peace and social development.
2.6.1 The DME at CINVESTAV and Its Relation to “the”
French School
CINVESTAV’s Department of Mathematics Education (DME) was created by Filloy
and Imaz. Its founding reflected the urgent need to modernize study plans and pro-
grams for mathematics in harmony with an international movement which demanded
that the mathematics being taught be brought more in line with current theories in the
discipline. In that context, two French researchers, Brousseau and Pluvinage, were
invited to Mexico City in 1979, thus initiating relations between Mexico and “the”
French Didactic School.
Since its creation, the DME has maintained a well-defined scientific vocation.
Because it is part of a centre devoted to research, its institutional conditions propitiate
developing research projects, organizing and participating in congresses, colloquia
and spaces for disseminating science, teacher-training courses, and Master’sand Doc-
toral programs, among other scientific and academic activities. Also, it has a copious
production of materials, both research-based and didactic (designed for teachers and
40 M. Artigue et al.
students). Thus, it is an ideal space for establishing relations with research groups in
other latitudes.
The first research projects developed at the DME centred on rational num-
bers, algebra, probability and functions, within theoretical frameworks that included
epistemology, the cognitive sciences, and computation. Today, however, the fields
explored have diversified broadly. Its methods for developing these activities and
disseminating their results have been presented at international meetings (ICME,
CIEAEM, PME) that led to forging contacts with research groups in France. The
fact that two members of the DME—Hitt and Alarcón—did their doctoral studies
in France, followed a few years later by the post-doctoral study periods of Cantoral
and Farfán, have clearly influenced the development of Mathematics Education in
Mexico through, for example, the creation of the Escuela de Invierno en Matemática
Educativa (Winter School for Mathematics Education, EIME), reflecting the expe-
rience of French Summer School. A second key creation was the Reunión Lati-
noamericana en Matemática Educativa (Latin American Forum for Mathematics
Education, RELME), which is held annually. The year 2017 has witnessed the 31st
edition of this congress, which is opening up new channels for relations between
France and Latin America, including a collection of studies framed in the Math-
ematical Working Space approach (see Sects. 2.1 and 2.2). Another event of this
nature—propelled mainly by Brazilian researchers—is the Simposio Latinoameri-
cano de Didáctica de la Matemática (Latin American Symposium on Mathematical
Didactics, LADIMA30), which was held in November, 2016, with a strong pres-
ence of researchers from France and Latin America. Also, researchers at the DME
were founding members of the Comité Latinoamericano de Matemática Educativa
(Latin American Committee for Mathematics Education, CLAME), whose achieve-
ments include founding the Revista Latinoamericana de Matemática Educativa
(Latin American Journal of Mathematics Education, RELIME), which appears in
such indexes as ISI Thomson Reuters. Indeed, the journal’s Editorial Board includes
French researchers and its list of authors reflects joint studies conducted by French
and Latin American scholars.
Finally, academic life at the DME is characterized by the enrolment of students
from Mexico and several other Latin American countries in its Master’s and Doctoral
programs. These graduate students enjoy the opportunity to contact “the” French
School through specialized literature, seminars, congresses and seminars with French
researchers.
2.6.2 The DIE-CINVESTAV: A Strong Influence on Basic
Education Supported by TDS
Since its founding, the Department of Educational Research (DIE for its initials
in Spanish) has maintained a close relationship with Mexico’s Department of Pub-
30http://www.boineventos.com.br/ladima, accessed January 2018.
2 The French Didactic Tradition in Mathematics 41
lic Education that has allowed it to participate in producing textbooks, developing
teaching manuals, and formulating study plans for math courses and processes of
curriculum reform for basic education (students aged 6–15). The TDS has been the
principal theoretical reference guiding these activities, as we show in the following
section.
The 1980s brought the development of a project called “from six years” (de los seis
años), one of the first TDS approaches, and one that would influence other programs,
mainly in the curriculum reform for the area of basic education driven by the Depart-
ment of Public Education in 1993. That reform introduced new textbooks, manuals
and didactic activities strongly influenced by the French School, and especially TDS.
Block, who would later complete his Doctorate (co-directed by Brousseau) partici-
pated actively in elaborating material for primary school, while Alarcón did the same
for secondary school. That reform program was in place for 18 years (1993–2011),
during a period that also saw the implementation of the Program for Actualizing
Mathematics Professors (1995) and the introduction of several additional reforms
that reflected the impact of TDS in Mexico’s Teachers Colleges (Escuelas Normales),
the institutions entrusted with training elementary school teachers. In Block’s words,
“after over 20 years of using those materials, it is likely that there are still traces of
the contributions of TDS in the teaching culture”.
2.6.3 The PROME at CICATA-IPN: A Professionalization
Program for Teachers that Generates Relations
Between France and Latin America
In the year 2000, the CICATA at Mexico’s Instituto Politécnico Nacional created
remote, online Master’s and Doctorate Programs in Mathematics Education (known
as PROME) designed for mathematics teachers. This program has generated multiple
academic interactions between France and Latin America because, while students
come mostly from Latin American nations (Mexico, Argentina, Chile, Colombia
and Uruguay) some instructors are French. Also, their teacher training programs
include elements of TDS and ATD, exemplified by the Study and Research Periods
for Teacher Training (REI-FP).
The Learning Units (LU) on which these programs are based have been designed
with the increasingly clear objective of functioning as a bridge between research in
Mathematics Education and teaching practice. To this end, the organizers encour-
age a broad perspective on research and its results by including LUs designed by
instructors at PROME and from other areas of the world. French researchers such
as Athanaze (National Institute of Applied Sciences of Lyon, INSA-Lyon), Geor-
get (Caen University), and Hache (LDAR and IREM-Paris), have participated in
designing and implementing LUs, while Kuzniak (LDAR and IREM-Paris) was
active in developing a Doctoral-level seminar, and Castela (LDAR and Rouen Uni-
versity) has participated in online seminars, workshops, the Doctorate’s Colloquium,
42 M. Artigue et al.
and PROME’s inaugural Online Congress. These interactions allow the diffusion of
research while recognizing teachers’ professional knowledge and opening forums
to answer urgent questions and identify demands that often develop into valuable
research topics.
2.6.4 Theoretical Currents, Methodologies and Tools
Two theoretical currents that arose in Latin America—ethnomathematics and
socioepistemology—have generated new ways of doing research and problematizing
teaching and learning in the field of mathematics. The term ethnomathematics was
coined by the Brazilian, D’Ambrosio, to label a perspective heavily influenced by
Bishop’s cultural perspective on mathematics education (Bishop, 1988). Socioepiste-
mology, meanwhile, was first proposed by the Mexican scholar, Cantoral, and is now
being developed by several Latin American researchers (see Cantoral, 2013). This
approach that shares the importance attached to theoretical foundations in French
research and the need for emancipation from the dominant traditions in this disci-
pline, considers the epistemological role of social practices in the construction of
mathematical knowledge; the mathematical object changes from being the focus of
the didactical explanation to consider how it is used while certain normative practices
take place. Since all kinds of knowledge matter—everyday and technical knowledge
for example—Socioepistemology explains the permanent development of mathemat-
ical thinking considering not only the final mathematical production, but all those
social circumstances—such as practices and uses—surrounding mathematical tasks.
Another theoretical development, though of narrower dimensions, is the extended
praxeological model that emerged from Romo’s doctoral dissertation (co-directed
by Artigue and Castela), which has been widely disseminated in two publications
(Romo-Vázquez, 2009; Castela & Romo, 2011). This approach makes it possible
to analyse mathematical models in non-school contexts in order to transpose them
to mathematics teaching by designing didactic activities shaped primarily for engi-
neers and technicians; for example, the cases of electrical circuits and laminated
materials (Siero & Romo-Vázquez, 2017), and blind source separation in engineer-
ing (Vázquez, Romo, Romo, & Trigueros, 2016). This model, accompanied by TDS
tools, has also been applied in analyses of the practices of migrant child labourers in
northern Mexico (Solares, 2012).
In terms of methodologies, it could be said that, as in the French School, research
conducted in Mexico is largely qualitative in nature, though data-gathering and
implicative analyses using the CHIC program developed by Gras are now being
utilized. The methodology of didactical engineering (see Sect. 2.1) is still one of the
most often employed, though rarely in its pure form. Socioepistemology is recog-
nized by its use of Artigue (1990) and Farfán (1997).
2 The French Didactic Tradition in Mathematics 43
2.6.5 Areas of Opportunity and Perspectives
The institutions introduced herein, and the brief historical profiles presented, reveal
how interactions between France and Mexico have strengthened the development
of Mathematics Education in Latin America. However, as Ávila (2016) points out,
many challenges still need to be addressed, especially in terms of ensuring that all
students have access to high-quality instruction in mathematics that will allow them
to better understand and improve the world. This entails participating in specific types
of education, including the following: in indigenous communities, with children of
migrant workers, child labourers in cities, and children who lack access to technology,
and in multi-grade schools, to mention only a few. In this regard, the research by
Solares (2012) and Solares, Solares, and Padilla (2016) has shown how elements of
TDS and ATD facilitate analysing and “valuing” the mathematical activity of child
labourers in work contexts in Sonora and metropolitan Mexico City, and the design of
didactic material that takes into account the knowledge and needs of these population
sectors.
Finally, we consider that another important area of opportunity is generating
didactic proposals for the Telesecundaria system (which serves mainly rural areas
in Mexico) that, in the 2014 educational census escolar, represented 45.3% of all
schools involved in teaching adolescents aged 12–15. This system requires a “tele-
teacher” (classes are transmitted by television) and a teacher-monitor whose role
is mainly to answer students’ questions and resolve their doubts; though they were
recently granted more autonomy and are now expected to develop didactic material
on “transversal” topics for various subjects (e.g. physics, history, mathematics and
Spanish). In this educational setting, many scholars feel that designing SRPs (see
Sect. 2.1 on ATD) may be an optimal approach, since SRPs are characterized by their
“co-disciplinary” nature. Designing and implementing such materials will make it
possible to better regulate teaching practice and the formation of citizens who are
capable of questioning the world. It is further argued that there are more general
areas, such as the role of multimedia tools in students’ autonomous work, large-
scale evaluations, online math education, the study of cerebral activity associated
with mathematical activities, and the nature of mathematical activities in technical
and professional contexts, among other fields, that could lead to establishing new
relations between France and Mexico to conduct research in these, so far, little-
explored areas.
There is no doubt that both French and Mexican research have benefitted from this
long term collaboration. Mexican research has enriched the perspectives of French
didactics, especially with the development of socioepistemology and ethnomathe-
matics, which have shown—beyond European logic—how autochthonous and native
American communities produce mathematical knowledge, while highlighting the
role of social practices in the social construction of knowledge. These approaches
have produced studies in the ATD framework (Castela, 2009; Castela & Elguero,
2013). Likewise, studies of equality, and of education-at-a-distance, primarily for
the professionalization of teachers, have developed very significantly in Latin Amer-
44 M. Artigue et al.
ica, and may bring about new relations that seek a greater impact on innovation in
the teaching of mathematics in the classroom.
2.7 Didactic Interactions Between France and African
Countries. The Case of Tunisia
The past decade has seen an important development of research in mathematics edu-
cation across Francophone Africa, and the collaboration with French didacticians
has played a decisive role in this development. Collaborations in mathematics edu-
cation between Francophone African countries and France started in fact early after
these countries got their independence, in the New Math period, with the support of
the recently created IREM network and the INRP. Some IREMs were even created
in African countries, for instance in Senegal as early as 1975 (Sokhna & Trouche,
2016). When doctorate programs in the didactics of mathematics opened in French
universities in 1975 (see Sect. 2.1), African students entered these programs and pre-
pared doctoral theses under the supervision of French researchers. The data collected
for the preparation of the thematic afternoon at ICME-13 show that by 1985, eleven
such theses had been already defended by students from four countries, and twenty
more at the end of the 20th century, with nine countries represented. One can also
observe the progressive development of co-supervision with African researchers,
especially within the institutional system of co-tutoring doctorate. This is the case
for sixteen of the twenty-five theses defended since 2000.
Research collaboration was also nurtured by the regular participation of African
didacticians in the biannual Summer schools of didactics of mathematics (see
Sect. 2.1) and, since the year 2000, by the tri-annual conferences of the Francophone
Mathematical Space (EMF) created on the initiative of the French sub-commission
of ICMI, the CFEM. One important aim of the creation of the EMF structure was
indeed to favour the inclusion of Francophone researchers from non-affluent coun-
tries into the international community of mathematics education. EMF conferences
alternate South and North locations, and three have already been held in Africa, in
Tozeur (Tunisia) in 2003, Dakar (Senegal) in 2009, and Algier (Algeria) in 2015.
As evidenced by the four case studies regarding Benin, Mali, Senegal and Tunisia
included in Artigue (2016), these collaborations enabled the creation of several mas-
ter and doctorate programs in didactics of mathematics in Francophone African
countries, and supported the emergence and progressive maturation of a community
of didacticians of mathematics in the region. A clear sign of this maturation is the
recent creation of ADiMA (Association of African didacticians of mathematics),
the first conference of which was held at the ENS (Ecole Normale Supérieure) of
Yaounde in Cameroon in August 2016; the second one is planned at the Institute of
Mathematics and Physical Sciences of Dangbo in Benin in August 2018.
Tunisia is a perfect example of such fruitful interactions. In the next paragraphs,
I (Faïza Chellougui) review them, from the first collaborations, in the seventies,
2 The French Didactic Tradition in Mathematics 45
involving the Tunisian association of mathematical sciences (ATSM), the French
association of mathematics teachers (APMEP) and the IREM network until today. I
show how these collaborations have nurtured the progressive maturation of a Tunisian
community of didacticians of mathematics, today structured in the Tunisian associa-
tion of didactics of mathematics (ATDM), and enriched the research perspectives in
both countries. More details can be found in (Chellougui & Durand-Guerrier, 2016).
2.7.1 The Emergence of Didactic Interactions Between
France and Tunisia
As just mentioned above, didactic interactions between Tunisia and France in math-
ematics stem from the relationships between the APMEP and the ATSM created
in 1968, the oldest association of mathematics teachers in the Arab world and in
Africa. The two associations indeed have a long term tradition of cross-invitation
and participation in their respective “National days” meetings and seminars, of reg-
ular exchange of publications, etc. As early as 1977, Brousseau was invited to the
national days of the ATSM. He presented the didactics of mathematics, its questions,
concepts and research methods to a large audience of teachers, illustrating these
with the research work he was developing on the teaching of rational and decimal
numbers.
The IREM network then played an important role, especially the IREM of Lyon,
thanks to its director, Tisseron. Tisseron had taught mathematics to pre-service math-
ematics teachers at the ENS in Tunis in the seventies, and also didactics at the ENS
in Bizerte in the eighties when a didactic course was introduced in the preparation of
secondary mathematics teachers. The decisive step for the development of didactics
of mathematics as a research field in Tunisia occurred in fact in 1998, with the accred-
itation of a graduate program in the didactics of mathematics (DEA) at the Institute
for higher education and continuous training in Tunis (ISEFC).31 This accreditation
had been prepared by Abdeljaouad, mathematician and historian of mathematics at
the University of Tunis, who has played a crucial role in the emergence and develop-
ment of didactic research in Tunisia, and Tisseron (see Abdeljaouad, 2009 for more
details).
2.7.2 Development and Institutionalization
In order to set-up this program, a fruitful collaboration developed between the ISEFC
and four research teams in French universities having doctorate programs in the
didactics of mathematics: the LIRDHIST team (now S2HEP32) in the University
31http://www.isefc.rnu.tn/home.htm (accessed 2018/01/08).
32https://s2hep.univ-lyon1.fr (accessed 2018/01/08).
46 M. Artigue et al.
of Lyon 1 of which Tisseron was a member, the DIDIREM team (now LDAR33)
in the University Paris 7, the Leibniz team in the University Grenoble 1, and the
LACES34 at the University of Bordeaux 2. Initially the DEA courses were taught
by researchers from these teams, but gradually Tunisian scholars took them partially
in charge. In 2006, the DEA turned into a Master program in the framework of the
LMD (License-Master-Doctorate) reform of university. This institutional change led
to modification of the organization of the program, and from the fall of 2010, ISEFC
was empowered to offer a Master of research in didactics of science and pedagogy.
The overall objective is ensuring a high level of training taking into account the
multiple components of careers in science education. Eventually, a new Master in
didactics of mathematics was set-up in October 2015, aiming to be innovative and
open to the international community.
Another important step for the institutionalization of the field was the creation of
the Tunisian association of didactics of mathematics (ATDM) in 2007. This asso-
ciation provides an institutional status to the young community of didacticians of
mathematics and supports the dissemination of its research activities and results. For
instance, it organizes an annual seminar to which contribute both well-known inter-
national and Tunisian researchers, to allow the regular diffusion of new or on-going
research and to promote exchanges and debates within the didactic community.
Since the establishment of the DEA, fifteen doctoral theses and more than forty
DEA or Master dissertations have been defended, most of these under the co-
supervision of French and Tunisian researchers. Research reported in these theses
and dissertations mainly addresses the teaching and learning of specific mathematics
concepts, from elementary grades up to university. Important attention is paid to
the epistemology and history of the concepts and domains at stake. The existence
of Tunisian researchers specialized in the history of mathematics especially Arabic
mathematics, such as Abdeljaouad, certainly contributes to nurture and instrument
this attention. The main theoretical frameworks used are those of the French didac-
tics, and especially ATD, TDS and TCF (see Sect. 2.1). One specificity however,
is the number of theses and research projects that concern higher education and
the transition from secondary to tertiary mathematics education (see for instance
the theses by Chellougui on the use of quantifiers in university teaching (Chel-
lougui, 2004), by Ghedamsi on the teaching of Analysis in the first university year
(Ghedamsi, 2008), and by Najar on the secondary-tertiary transition in the area of
functions (Najar, 2010). In fact, these theses and more global collaboration between
French and Tunisian researchers have substantially contributed to the development
of research in the area of higher education in France and Francophone countries in
the last decade. This is also the case for logical perspectives in mathematics edu-
cation, as shown by the thesis of Ben Kilani, Chellougui and Kouki (Chellougui,
2004; Ben Kilani, 2005; Kouki, 2008). Tunisian researchers have also pushed new
lines of research, such as those related to the teaching and learning of mathematics
in multilinguistic contexts, poorly addressed by French didacticians. In that area, a
33https://www.ldar.website (accessed 2018/01/08).
34http://www.laces.univ-bordeauxsegalen.fr (accessed 2018/01/08).
2 The French Didactic Tradition in Mathematics 47
pioneering work was the thesis of Ben Kilani who used logic to show the differences
of functioning of the negation in the Arabic and French language, and to understand
the difficulties induced by these differences in the transition between Arabic and
French as language of instruction in grade 9.
2.7.3 Some Outcomes of the French-Tunisian Didactic
Collaboration
This long term collaboration has enabled the emergence, progressive development
and institutionalization of didactics of mathematics as a field of research and practice
in Tunisia. The majority of Ph.D. graduates have found a position in Tunisian higher
education; they constitute today a community with the capacity of taking in charge
the didactic preparation of primary and secondary mathematics teachers, and most
Master courses. In 2017 moreover, the two first habilitations for research supervi-
sion35 have been delivered to Tunisian didacticians (Ghedamsi, 2017; Kouki, 2017),
which is a promising step for this community.
The French-Tunisian collaboration has certainly played a role in the increasing
regional and international visibility and recognition of Tunisian researchers in the
didactics of mathematics observed in the last decade. International visibility and
recognition expresses through contributions to Francophone international events in
the field such as the Summer school of didactics of mathematics to which the Tunisian
delegation is regularly the largest foreign delegation, the EMF and ADiMa confer-
ences. Recently it has also expressed through contributions to CERME conferences
organized by the European Society for Research in Mathematics Education or the
recently created INDRUM network, federating mathematics education research at
university level. International recognition also expresses through invited lectures at
seminars and congresses, and diverse scientific responsibilities. My personal case is a
good illustration. I had an invited lecture at ICME-13 in 2016, and am a member of the
International program committee of ICME-14. I have been a member of the scientific
committee of two Summer schools, in 2007 and 2017, of EMF 2015 and INDRUM
2016. I was co-chair of the Topic Study Group entitled “Pluralités culturelles et
universalité des mathématiques: enjeux et perspectives pour leur enseignement et
apprentissage” (Cultural diversity and universality of mathematics: stakes and per-
spectives) at EMF2015, and of the group “Logic, numbers and algebra” at INDRUM
2016.
There is no doubt that there exists today a Tunisian community of didacticians
of mathematics, dynamic and mature, open to the world beyond the sole frontiers of
the Francophone world. While maintaining privileged links with the French didactic
community, which has supported its emergence and development, it is creating its
35Habilitation for research supervision is a diploma compulsory to compete for full professor
position at university in Tunisia as is the case in France.
48 M. Artigue et al.
own identity, and more and more offers challenging perspectives and contributions
to this French community.
2.8 Epilogue
In this chapter reflecting the contributions at the ICME-13 thematic afternoon, we
have tried to introduce the reader to the French didactic tradition, describing its emer-
gence and historical development, highlighting some of its important characteristics,
providing some examples of its achievements, and also paying particular attention
to the ways this tradition has migrated outside the frontiers of the French hexagon
and nurtured productive relationships with researchers in a diversity of countries,
worldwide. This tradition has a long history, shaped as all histories by the conditions
and constraints of the context where it has grown and matured. Seen from the outside,
it may look especially homogeneous, leading to the term of French school of didac-
tics often used to label it. The three theoretical pillars that have structured it from its
origin and progressively developed with it, with their strong epistemological founda-
tions, the permanent efforts of the community for maintaining unity and coherence,
for capitalizing research knowledge, despite the divergent trends normally resulting
from the development of the field, certainly contribute to this perception. We hope
that this chapter shows that cultivating coherence and identity can go along with
a vivid dynamics. The sources of this dynamics are to be found both in questions
internal and external to the field itself and also, as we have tried to show, in the
sources of inspiration and questions that French didacticians find in the rich connec-
tions and collaborations they have established and increasingly continue to establish
both with close fields of research and with researchers living in other traditions and
cultures. The first sections, for instance, have made clear the important role played
by connections with psychology, cognitive ergonomics and computer sciences. The
last four sections have illustrated the particular role played by foreign doctorate stu-
dents, by the support offered to the creation of master and doctorate programs, and
also by institutional structures such as the IREM network or the INRP, now IFé,
since the early seventies; this is confirmed by the eight case studies presented in
(Artigue, 2016). Looking more precisely at these international connections, there
is no doubt that they are not equally distributed over the world. Beyond Europe,
Francophone African countries, Vietnam, and Latin America are especially repre-
sented. For instance, among the 181 doctoral theses of foreign students supervised
or co-supervised by French didacticians between 1979 and 2015, the distribution is
the following: 56 from Francophone Africa, 54 from America, all but one from Latin
America among with 28 for Brazil, 15 from East Asia among with 12 from Vietnam,
15 from Middle East among with 15 from Lebanon, and 36 from Europe.36 For Fran-
cophone African countries, Vietnam and Lebanon, the educational links established
36Data collected by Patrick Gibel (ESPE d’Aquitaine) with the support of Jerôme Barberon (IREM
de Paris).
2 The French Didactic Tradition in Mathematics 49
in the colonial era are an evident reason. For Latin America, the long term cultural
connections with France and the place given to the French language in secondary
education until recently, and also the long term collaboration between French and
Latin American mathematics communities, have certainly played a major role. These
connections are dynamic ones, and regularly new ones emerge. For instance, due to
the will of the French Ecoles normales supérieures (ENS) and to the East China Nor-
mal University (ECNU, Shanghai) to develop their collaboration in various domains
of research (philosophy, biology, history, … including education), and to the pres-
ence on each side of researchers in mathematics education, new links have emerged
since 2015 and, currently, 4 Ph.D. are in preparation, co-supervised by researchers
from the two institutions.
All these connections and collaborations allow us to see our tradition from the
outside, to better identify its strengths and weaknesses, as pointed out by Knipping
in Sect. 2.4, and to envisage ways to jointly progress, at a time when the need of
research in mathematics education is more important than ever.
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