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2.1 Introduction

The French tradition of research on the learning and teaching of mathematics, often referred to simply as ‘didactique’, has developed a range of theoretical tools. These tools share a common intellectual and professional hinterland, and although each is honed to analysis of particular types of didactical question, they have increasingly been used by French researchers in coordinated ways. As this movement towards a more systematically articulated didactique has developed within France, ideas from didactique have become sources of inspiration or points of reference for researchers outside France. Fresh questions have naturally arisen about the central concepts of didactique as they encounter new professional cultures and their associated intellectual traditions. At the Artigue colloquium, the last two authors of this chapter convened a round-table discussion to explore this issue. Each of the first four authors contributed by bringing a particular perspective inspired both by their professional contacts with Michèle Artigue and their own interests and experience.

The first two of these contributions looked, in rather different ways, at correspondences and contrasts between French didactique and some of the central theoretical frameworks influential in English-speaking research on the teaching of mathematics. Abraham Arcavi launches our discussion by comparing didactical engineering with design research and the didactical contract with sociomathematical norms.

2.2 Towards a Dialogue Between Traditions and Approaches in Mathematics Education

  • Abraham Arcavi 

I am honored and pleased to be part of this homage to Michèle Artigue. It is a most appropriate opportunity to express my personal gratitude to her. I learned much from Michèle’s writings and much more from the long and lively conversations I was very lucky to have with her on many occasions. The luck refers to the many opportunities we have had to encounter each other (including a mini-course in Sweden that we co-taught). However, the depth and the breadth of our conversations have less to do with luck and more to do with Michèle’s friendly predisposition to engage in dialogue, with her special ways of communicating, and above all with her willingness to address my questions and comments, even if naïve—always with unusual patience, care and deep respect for her interlocutor. I was consistently impressed by her devotion as a teacher, as a “bridge builder” (between the knowledgeable and the less knowledgeable, between the French tradition and other schools of thought, between mathematicians and mathematics educators), and by the scope of her vast knowledge and wisdom. Inspired by Michèle’s work and personality, I would like to argue for a comprehensive effort to enhance and extend the dialogue among traditions and communities in mathematics education, with depth, care, respect and patience. In particular, I think it is of special importance to continuously address, for example, the following question: Do the approaches of the French didactique overlap those of the other traditions, and if so, in what sense? Or maybe the issues are not the same after all?

I will briefly dwell on some aspects of this question by using examples and by reporting on a conversation with a colleague who works in different traditions of research and practice.

My departure point is the very title: “Didactique goes travelling”. The travel metaphor implies moving away from one’s comfortable and well-known territory to other, less known, places, to meet people who speak different languages and/or have different perspectives, and with whom we may talk about our places and ways of doing things. As a traveler for short, intermediate and long periods, including emigration and adaptation to a new place, I know that changing places often compels us not only to meet new perspectives, but also to profoundly revisit and make visible our own ideas and basic assumptions, which we tend to take for granted.

In this “travel”, I will rely on informal conversations with Michèle Artigue and on some of her lectures I attended.

2.2.1 Contrasting Traditions

In the 1980s, both the French didactique and the Anglo-Saxon traditions of mathematics education seemed to be at a similar juncture:

  1. (a)

    Dissatisfaction with the methods of study (external methodologies such as tests and questionnaires, and overemphasis on statistical comparisons between experimental and control groups) as the main avenues to “scientificity”; and

  2. (b)

    A need to move away from cognitive research mostly held in laboratories in order to capture “the complex intimacy of classroom functioning”. The Anglo-Saxon tradition started to focus on the socio-cultural aspects of teaching and learning, and the French didactique launched the Theory of Didactical Situations, the idea of Didactical Engineering and more (perhaps there is some chronological misrepresentation implicit here, since the roots of French didactique can be traced back to the 60s, therefore the ideas were not only a reaction to unsatisfactory extant theories and methods, however, their visibility and influence became apparent in the 80s). In a certain sense, the departure point and the ultimate goal of both traditions were similar: to capture the complexity of learning in classroom settings, and to understand the “ecology” beyond the “individual’s biology”.

In the following, I would like to consider two examples of ideas/constructs from both traditions, to juxtapose them in order to ponder points of tangency, the extent of the overlap and the character of the differences. Such juxtaposition is rather simplistic, and thus it may do some injustice to both the depth and the breadth of the ideas. Nevertheless, it is proposed as a starting point to launch a much needed dialogue, even if it only serves the purpose of a “balloon to shoot at”.

First example: Design research—Theory of Didactical Situations

Design research (Brown 1992)

Theory of didactical situations (Brousseau 1997; Artigue 2000)

• From work sites/assigned tasks towards communities of learners, where students take charge of their own learning

• Engineer educational environments and simultaneously conduct experimental studies of those innovations

• Study simultaneous changes in the system, concerning the role of students and teachers, the type of curriculum, the place of technology, and so forth This is intervention research to inform practice and which enables migration from experimental classrooms to average classrooms operated by and for average students

• Autonomous learners

• Work toward a theoretical model of learning and instruction rooted in a firm empirical base

• From cognising subjects to didactic situations: set of interactions between students, teachers and mathematics at play in classrooms

• The didactic situation shapes and constrains the knowledge constructed

• Without understanding the situation it is not possible to interpret the students’ behaviors in cognitive terms

• To understand teaching and learning processes and the ways they interact

• To develop rational means for controlling and optimising didactical situations

• Didactic, a-didactic situations and devolution processes

• Confrontation between a priori analysis of engineering design and a posteriori analysis of data collected

Second example: Socio-mathematical norms—Didactical contract

Socio-mathematical norms (Yackel and Cobb 1996)

Didactical contract (Brousseau 1986; Artigue 2000; Sierpinska 1999)

• Normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient and mathematically elegant in a classroom

• What counts as an acceptable mathematical explanation and justification

• Influence the learning opportunities for both the students and the teachers

• The students and the teachers may interactively elaborate the norms

• Clarify how students develop beliefs and values

• Concept developed by Brousseau as a possible explanation for ‘elective failure’ in mathematics

• Expectations of the teacher with respect to the students, and conversely, regarding the mathematical content

• “Rules” pertinent to the knowledge taught

• The rules of the didactic contract are implicit: the teacher and the students do not sign a chart of ‘rights and obligations’ but they are there and we know that they are there when they are broken

2.2.2 Reflections of a “Traveler”

In my pursuit of understanding and reflecting upon the differences between traditions, I thought that it may also be appropriate to bring vivid testimonies of a “traveler” among traditions. I was fortunate to meet Takeshi Miyakawa in Tsukuba University in Japan during 2005. Takeshi is currently a faculty member in the Department of Mathematics at the Joetsu University of Education. What is special about Takeshi’s background is that he did his PhD in Grenoble, France, under the guidance of Nicolas Balacheff, and his post-doctoral fellowship with Patricio Herbst at the University of Michigan in the USA. So Takeshi has first-hand experience in three traditions of mathematics education. In preparation for the Artigue colloquium panel, I contacted Takeshi and he kindly agreed to reply to my questions. Here are some of them:

  1. 1.

    As a person who worked in both the French tradition of didactique and the Anglo-Saxon research tradition in mathematics education, what would you say are the main differences between them?

  2. 2.

    Did you find theoretical “points of tangency” between the two traditions?

  3. 3.

    Do you have any explanation as to why the French tradition and the Anglo-Saxon tradition do not engage in a deeper dialogue than seems to be the case?

Takeshi’s answers are extensive and interesting. I include here some of the main points, taking into account what I think would best serve the promotion of the dialogue between traditions.

The French research tradition in mathematics education is relatively homogeneous. This is consonant with Laborde’s (1989) description of research as based on the setting up of an original theoretical framework developing its own concepts and methods and satisfying three criteria: relevance in relation to observable phenomena, exhaustivity in relation to all relevant phenomena, and consistency of the concepts developed within the theoretical framework. It is described as a national program to be carried out neither by single researchers, nor by single teams, but by several institutional sites. The American tradition is much more heterogeneous: “Research in mathematics education is diverse. Very often, the theories from other fields of research, like linguistics, sociology, ethnomethodology, psychology, cognitive science, educational science, philosophy of mathematics, etc. are used. And the research work focuses on different aspects according to the adopted outside theory. It also seems, in my view, due to this diversity, that there is no sharp line between the research in math education and the educational research in general” (Miyakawa 2012).

Another difference in Takeshi’s view is the centrality of the role and nature of mathematical content knowledge in the French tradition, whereas this may not always be the case in the USA. Even when there may be points of tangency regarding the objects of study, it is often quite difficult to “put them together” because the goals, the foci and the approaches to research are so different. Such differences should be pursued, captured and made explicit, as a pre-requisite for a fruitful dialogue.

I would like to conclude on the basis of a reflection by Takeshi, which may also contribute to pinpointing why the rapprochement between traditions is not happening as intensively as one might have expected: it is not easy to understand the French research tradition if you are in the American tradition, and vice versa. Understanding the other’s perspective is not a simple endeavour (Arcavi 2007), and among its difficulties is undertaking a deep revision of one’s own implicit assumptions, beliefs, biases and predispositions in one’s own research tradition. It also implies the explication of the implicit—and this may require more than simply reviewing published research results.

However, says Takeshi, there have been several appeals followed up by actual research which seriously undertakes this rapprochement. The CERME working group on theory may be one of these examples.

I encourage the furthering of these attempts, and I also encourage the research community to be inspired by the intellectually sound and personally empathic approach so nicely pursued by Michèle.

2.3 First Entr’acte

As Miyakawa notes above, French didactique has traditionally been more strongly affiliated with mathematics in its disciplinary culture and social organisation than with mainstream educational studies and core social sciences, in contrast with English-speaking mathematics education. Jeremy Kilpatrick continues this discussion by examining the commensurability of these two traditions of study of the teaching of mathematics, and the conceptual frameworks that they have developed.

2.4 Lost in Translation

  • Jeremy Kilpatrick 

“La syntaxe française est incorruptible. C’est de là que résulte cette admirable clarté, base éternelle de notre langue. Ce qui n’est pas clair n’est pas français; ce qui n’est pas clair est encore anglais, italien, grec ou latin.”Footnote 1

Antoine de Rivarol 1783

English-language speakers struggle to express what is meant by didactique des mathématiques. “Didactic of mathematics” sounds stilted, even when pluralised, and “mathematics education” is simply wrong. A common solution is to use the French words and let the Anglophone reader adopt the meaning he or she chooses. L’ingénierie didactique raises further hurdles. Connecting didactique to Anglophonic research in mathematics education guarantees that something will be lost—but what?

Almost two decades ago, in June 1993, a conference on 20 years of didactique des mathématiques was held in Paris. The report of that conference (Artigue et al. 1994), which focused on the contributions of Guy Brousseau and Gérard Vergnaud, marked a milestone in the development of didactique as a research enterprise. Brousseau’s theory of didactical situations and Vergnaud’s theory of conceptual fields, supplemented by the semiotic approach of Raymond Duval and the anthropological approach of Yves Chevallard, have in the ensuing years provided four important theoretical frameworks on which much of the field of French didactique has been built (Winsløw 2005).

During the past decades, a major role in bringing didactique onto the world stage has been played by Michèle Artigue, who through her many presentations and publications in French, English and Spanish has elucidated for numerous audiences the theory of didactical situations, didactical engineering, and the anthropological theory of didactics. Her interest in the integration of computer technologies into mathematics education, particularly at the university level, has led her to combine and elaborate the anthropological approach in didactics with the theory of instrumentation in cognitive ergonomics (Artigue 2002), a combination that has been especially productive in studying learning in computer algebra system environments. Over the past decade, in particular, Artigue has been a highly visible champion of all things didactique.

2.4.1 Didactique as Traveller

The theme La didactique en voyage (Didactique goes traveling) provides a challenging metaphor: Where does didactique choose to go? What does it do along the way? And what does it take along on its travels? A brief consideration of each of these questions serves to introduce my main concern: What is lost when didactique leaves its Francophonic nest?

2.4.1.1 Destinations: Places Travelled

The report of the conference (Artigue et al. 1994), held when didactique was just reaching adulthood, suggests that whereas during its youth it had reached many places within France, it had not managed to travel very far beyond the country’s borders. Most of the affiliations of the contributors to the volume were with institutions in France, and only a handful were outside—a few European countries, the United States and Canada. Now that didactique is on the verge of middle age, that situation has greatly changed.

Didactique has ventured around the world, aided in large part by translations of important works into other languages. For example, the 1997 Spanish translation of Chevallard’s (1985) book on didactic transposition was enormously influential in bringing his ideas to Latin America. Publications in English (e.g., Artigue and Winsløw 2010; Barbin and Douady 1996; Herbst and Chazan 2009a; Laborde and Perrin-Glorian 2005; Winsløw 2005) brought work on didactique to a much wider audience, with research reports still coming primarily from France and its neighbors Spain and Italy, but increasingly from places such as Argentina, Denmark, Palestine, South Africa and Vietnam. The emphasis on studying teaching situations in mathematics classrooms using approaches anchored in the theoretical frameworks of didactique has given the work broad appeal.

2.4.1.2 Itineraries: Some Accomplishments

During its travels, didactique has expanded its reach (see Caillot 2007, for a brief history of that expansion across disciplines). At first, it seemed a simple traveler: Arising from the theory of didactical situations, it has been defined as “a research field whose central goal is the study of how to induce a student to acquire a piece of mathematical knowledge” (Warfield 2007, p. 86).

As a field of research, didactique seemed only part of the larger field of research, study, theory, and practice that is called, in English, mathematics education, and in fact, seemed only part of the research of other scholarly endeavors in that field. But over time, as didactical engineering and anthropological theory were added to its arsenal, didactique grew beyond the realm of research: “In French, the term ‘didactique’ does not mean the art or science of teaching. Its purpose is far more comprehensive: it includes teaching AND learning AND school as a System, and so on” (Douady and Mercier 1992, p. 5).

Even a modest survey of the accomplishments of didactique over the past four decades would be beyond the scope of this chapter, so I mention only a few examples. Discussing the ways in which theories can provide a language to describe practice, Silver and Herbst (2007) note how Vergnaud’s theory of conceptual fields has been used by US researchers: “Much of the empirical work [on the learning of additive and multiplicative structures] done by scholars like Carpenter, Fuson, Behr, and their associates concurs with Vergnaud’s observations” (p. 52).

Silver and Herbst (2007) also give several examples of how research done in the US “can be seen as illuminating the origins of some [didactical] contractual difficulties and entanglements” (p. 55). They show how the construct of the didactical contract fits with the work of Mary Kay Stein and her colleagues (Stein et al. 2000) on the negotiation of task demands in the mathematics classroom. Silver and Herbst also connect the didactical contract to much other research on the work of the teacher. “The notion of didactical contract has turned out to be a particularly helpful notion to turn a descriptive theory of instruction based on [a] relational conception of teaching into an explanatory theory” (Silver and Herbst 2007, p. 63; see also Herbst 2003, 2006).

Sriraman and English (2010) term didactique “the French tradition” (p. 19), and they outline both the theory of didactical situations (which they term TDS) and the anthropological theory of didactics, noting that both are major theories of mathematics education. Linking TDS to other research on mathematics teaching, they assert:

Taken in its entirety, TDS comprises all the elements of what is today called situated cognition. The only difference is that TDS is particularly aimed at the analysis of teaching and learning occurring within an institutional setting (p. 23).

2.4.1.3 Baggage: The Burden of Terminology

Despite the broad dispersion and wide-ranging accomplishments of didactique over the past decades, it has not had the influence outside the Francophone world that one might have expected, given the field’s shift in focus to classroom situations:

During the past 30 years, researchers in mathematics education from English-, French-, and Castilian-speaking regions have been giving increased attention to classroom instruction…. Communication among researchers across those language differences has however been scarce. Different theoretical traditions as well as cultural differences in how to write and edit scholarship have often contributed to exacerbate the obvious differences in language competence and thus discouraged mutual acknowledgment. (Herbst and Chazan 2009b, p. 13)

In my view, part of the communication problem is that didactique carries some heavy baggage stemming largely from the language it employs and its metaphors in particular. Pimm (1988) has noted what he calls “a fundamental metaphoric structure in English” (p. 31), which I would claim is present in French as well, in which one links together an adjective and a noun, with the adjective pointing “to a new context of application, and sometimes considerable effort is required to create a meaning for the whole expression” (p. 31). He gives some examples from science and mathematics, with the mathematical examples including spherical triangle, complex plane, and differential geometry. In a footnote, he notes two examples from didactique:

Colette Laborde has pointed out to me that two terms central to the discipline of didactique des mathématiques, namely contrat didactique and transposition didactique…have had a difficult reception, in part precisely because of crossed interpretations. At one level, the last thing that a contrat didactique is is a contract (because it is completely tacit and unspoken); a transposition didactique usually involves a far greater alteration of material, structure and form than does a musical transposition. (p. 33)

In such metaphors, as Pimm (2010) later observed, “The semantic pressure is always piled onto the noun—it is never the adjective that has to shift or expand its meaning” (p. 613).

Another example is l’ingénierie didactique, which readily translates as didactical engineering. Although the adjective poses some problems that are discussed below, the more severe problem is posed by the noun. Didactical engineering is not engineering in the same sense as mechanical or electrical engineering. But, as Artigue (1994) observes, it is meant to “label a form of didactical work that is comparable to the work of an engineer” (p. 29). As she correctly notes, engineers “manage problems that science is unwilling or not yet able to tackle” (p. 30), and so didactical engineering is intended to handle such problems in didactics. Didactical engineering “has become polysemous, designating both productions for teaching derived from or based on research and a specific research methodology based on classroom experimentation” (p. 30). Whereas apparently, in French, l’ingénierie didactique has made the journey from metaphor to accepted, well-formulated usage, in English, didactical engineering still sounds awkward. Anglophones rarely use engineering when speaking of activities in education, and when they do, it is often used in a pejorative sense, suggesting the treatment of learners as material to be managed rather than educated.

Didactique, in creating a precise vocabulary for its work, has made extensive use of the fundamental metaphoric structure identified by Pimm, generating terms that need careful exegesis before they are used. Anglophones may find that English versions of those terms come laden with extra baggage that makes them difficult to interpret correctly.

2.4.2 Didactique in Translation

Like any metaphor, la didactique en voyage is limited in applicability. Didactique is not literally a traveler. So let me switch to another formulation and look at didactique as a body of work being translated from French to English. It was heartening to learn recently that, along with its neutral sense of “intended to instruct,” didactic has much the same pejorative sense in French that it does in English and that neither sense was the one originally intended:

About a quarter of a century ago, when some of us decided to use didactique as we do now, we were faced with two main obstacles. First, didactique was used in French essentially as an adjective, not as a noun. Second, the word had only two received meanings in common parlance, neither of which—understandably—tallied with what we had in mind.Footnote 2 (Chevallard 1999, p. 6)

Although the creators of didactique chose to give it a third—scientific—meaning, as discussed above, that did not solve the problem of the reader unfamiliar with that meaning.

For more than a decade, the editors of Recherches en Didactique des Mathématiques (RDM) have given me the opportunity, for many of the manuscripts accepted for publication, to edit the abstract in English that accompanies the resumen in Spanish and résumé in French. Occasionally, I am given the résumé and asked to produce an abstract, but usually my task is to polish the English of an abstract that an author has constructed. The unpolished abstracts run the gamut from elegant English prose to collections of English words that look as though they might have been produced by a computer-assisted translation program. With the latter kind of abstract, I struggle to find a good compromise between my understanding of the résumé and intelligible English. Perhaps if I were more fluent in French, I would be less sensitive to the difficulties that some authors apparently have in finding a reasonable English equivalent for the ideas they have expressed in French.

The challenge of polishing these RDM abstracts has made me acutely aware that translation from French to English, although at times fairly straightforward, is far from being a matter of simple equivalence. There is translation, and then there is interpretation, where it may be necessary to explain the intended meaning using words rather different from what a literal word-for-word translation would produce. French is known for its clarity, precision and rigor; English has other characteristics. A translation from French is bound to lose shades of meaning, particularly when one is dealing with the difficulties and complexities of didactique. Warfield (2007) put it well:

Over the past decade I have made a sequence of translations [from French to English], each time feeling cleverer than the last, and each time discovering afterwards some nuance that has nonetheless disappeared. (p. 92)

Robert Frost once said, “Poetry is what gets lost in translation.” When didactique is translated from the French milieu to that of English, it loses not only poetry but also clarity and nuance.

2.5 Second Entr’acte

One could imagine, then, that the nuance of didactique might be more easily grasped by speakers of other Romance languages. Certainly, didactique seems to have proved particularly influential in Spanish-speaking research on mathematics education. However, from his reflections on difficulties encountered in the dialogue between French didactique and an emerging Italian approach to research on mathematics education, Paolo Boero draws the lesson that research on mathematics education is strongly shaped by many (other) features of the local ecology.

2.6 Some Reflections on Ecology of Didactic Research and Theories: The Case of France and Italy

  • Paolo Boero 

2.6.1 Introduction

Beginning with lively discussions at the French Summer Schools of Mathematics Didactics during the 80s, and her translation into French of my plenary lecture at PME-XIII (1989), Michèle Artigue has played an intensive and, in my opinion, crucial, role of ‘mediator’ in order to further the dialogue between the already well constituted tradition of French research in Didactics of Mathematics and the newly developing Italian research in the field.

While reflecting on the difficulties in establishing collaboration between the communities of French and Italian mathematics educators, I am 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. By comparing the cases of France and Italy, I will attempt to identify and describe some variables that are related to local conditions, and in my opinion, that influence research developments and their outcomes (cf Bartolini Bussi and Martignone 2013): the features of the school system (teacher’s “mission” and degree of freedom); the economic constraints of research (conditions for funding), particularly in the initial stage; and especially the weight of the cultural environment (particularly, but not only, as concerns the field of mathematics). Discussing the influence of these variables may contribute to identifying potential meeting points and possibilities of productive collaboration between mathematics educators from different countries, and also to better understanding the roots of different theoretical elaborations and scientific productions developed in different countries.

2.6.2 Institutional and Economic Constraints

In the Italian reality, many of the studies performed concern educational (frequently, long term) innovation, which is conceived within theoretical frameworks that evolve according to the needs emerging from the analyses of what happens in the experimental classrooms. Needs may concern more advanced framings and/or the integration of new tools to improve teaching sequences and their analyses.

Such an evolutionary process, named ‘research for innovation’ by Arzarello and Bartolini Bussi (1998), also provides the researcher with an environment where basic research may take place. (An example of the relationships between the educational context shaped by research for innovation and didactical specific research is presented in Boero et al. 2009). The differences in the most typical research performed in France compared to Italy arise for various reasons. Later, I will consider some cultural factors; here I would like to deal with some important differences between France and Italy, which concern the institutional features of the school system, the teacher’s role in the classroom, and the economic constraints that have influenced the development of Italian research in mathematics education (in particular, in the beginning, in the late70s and in the early80s).

Firstly, the Italian school system is much less rigidly organised than the French one; national programs and, more recently, the Indicazioni nazionali per il curriculum (National guidelines for curricula) are much less detailed and prescriptive than French programs and guidelines for teaching; and nothing exists in Italy comparable with the guidance and control functions exercised in France by inspectors (cf Bartolini Bussi and Martignone 2013). The Italian situation allows some teachers to violate not only official prescriptions, but also universally accepted didactical and pedagogical principles. However, the situation allows other teachers to engage in large scale innovative projects and to gradually develop their competence as true researcher-teachers who participate in research teams, while maintaining their teaching functions in their classrooms (Malara and Zan 2008).

Another aspect of the Italian situation concerns the role of the teacher: the teacher is conceived in official documents as well as in current mentality, within the school and outside the school, not only as a specialist in the teaching of one (or several) discipline(s), but also as an educator. This conception probably depends on the great interest of Catholic culture in educational issues, and on its strong influence, in the past, on teacher education for kindergarten and primary school. During past debates concerning how many years should be spent by a teacher in the same classroom (with the same students), the great importance of the educational side of the profession always resulted in the choice of long periods. Usually, a primary school teacher teaches the same students for 5 years, while in secondary school, a teacher teaches the same students for 3 years in lower secondary school and for 2–3 years in high school.

In such an institutional and educational context, broad long-term teaching projects may be developed and tested, frequently opened to extra-school reality, frequently extending over more than one year of teaching. These projects provide teachers and researchers with the opportunity to identify and appreciate the long term changes that educational choices produce, in comparison with other more or less traditional choices. As a consequence, when a teacher realises how the innovation resulting from collaborating with the researcher results in a different mastery of concepts and skills, in comparison with previous teaching cycles, then the teacher feels motivated to further engage in research work. Furthermore, both the teacher and the researcher are motivated to attempt to identify and analyse the mechanisms that have improved the teaching results, thus contributing to further improving the classroom work and develop interesting research work.

But such within-the-school reasons (which make the Italian situation different from the French) are not sufficient to explain: the importance of research for innovation in Italy; the strong presence of teachers in the Italian research teams; and the fact that the development of theoretical knowledge and toolkits is strictly connected with a perspective of didactical innovation (cf Bartolini Bussi and Martignone 2013), and not so much aimed at modeling the teaching process and what happens in ordinary classrooms (an aim so relevant in French didactics of mathematics).

Starting from the middle of the sixties, we find several mathematicians who engaged (from different positions—in favor of or against modern mathematics) as promoters of the reform of national programs, and then (since the beginning of the seventies) as organisers of projects for an alternative teaching of mathematics, in collaboration with teachers. Indeed, some Italian mathematics teachers—the best known was Emma Castelnuovo—were already engaged in the movement for the reform of mathematics teaching at the national and international level. Starting from the middle of the seventies, research grants by the National Research Council (including the first research fellowships to prepare researchers in the field of mathematics education) were specifically aimed at elaborating upon and experimenting with innovative educational projects, based on different methodological and epistemological assumptions. When some of the mathematicians and the teachers engaged in research in mathematics education served as members of ministry commissions for the development of new national programs (in the years 1977/79, for grades VI to VIII; and in the years 1983/85, for grades I to V), they were able to offer the principles and the results of their didactical innovations to many classrooms. Also in more recent years (starting from the year 2000) some mathematicians engaged in mathematics education research and some researcher-teachers from our research teams took part in the development of the national guidelines for curricula for all school levels. As a result, present official texts (particularly on the methodological and cultural side) and accompanying materials sponsored by the Italian Mathematical Union (UMI)—the Italian association of mathematicians—particularly the examples of good practice, largely rely upon the research for innovation work performed by the university-school research teams.

Such a history of the development of research for innovation in Italy, its funding (particularly in the first period), and its influence on the evolution of national programs and guidelines for curricula, may make it possible to better understand why Italian researchers in mathematics education prefer to engage in research with immediate or at least eventual innovations in the teaching of mathematics. It may also help to better understand why they prefer to engage in long-term teaching experiments and innovative projects, based on wide-ranging hypotheses—in spite of the dangers of such a choice for the scientific validity of results.

2.6.3 The Influence of the Cultural Context

While comparing research in mathematics education developed in France and in Italy, we may ask ourselves why the problématique of the mathematics-reality relationships and of the relationships between mathematics and other disciplines (like physics) is so relevant in Italian research, and why, on the contrary, research on the sociological and institutional sides is so little developed in Italy in comparison with France. My hypothesis is that those differences largely depend on the different cultural contexts in which research in mathematics education developed, and still develops, in France and in Italy.

As concerns mathematics, the lively debate in Italy among mathematicians about mathematics and its teaching during the sixties and the seventies resulted in a plurality of positions which corresponded rather well to the plurality of positions about Modern Mathematics among teachers engaged in teaching reform. During the same years, when many Modern Mathematics textbooks were distributed in Italian schools under the pressure of the in-service teacher preparation organised by the Italian Ministry of Education, according to OECD orientations, the Italian Mathematical Union promoted the translation into Italian of the School Mathematics Project (SMP). SMP was a project originating in the UK and linked to the tradition of mathematics teaching in that country. In the same period, Emma Castelnuovo’s textbooks and volumes on mathematics teaching were popular among teachers engaged in elaborating new ways of teaching. Those positions, initiatives and materials opposed to Modern Mathematics contributed to developing experiences of lively relationships in the classroom between motivation of students, construction and development of mathematical knowledge, and knowledge of natural and social phenomena. Progressively those relationships were further developed and resulted in different educational orientations: those inspired by Freudenthal, based on the idea of vertical and horizontal mathematizing; more radical positions (mathematics develops both on the conceptual side and on the ways of thinking side according to the development of knowledge in suitable fields of experience) as in the projects of the Genoa research team led by me since the middle of the seventies; or positions resulting in the classroom construction of mathematical knowledge through problem solving activities within mathematics, but constantly open to applications to extra-mathematical problems.

By comparing what happened in France and in Italy we may identify deep differences which may be attributed to the influence of the Bourbaki team in the debate on mathematics and its teaching in France during the fifties and sixties. In fact, those differences originated far back in French history (since Descartes) and are still strong even today. In the French cultural tradition, mathematics is consistent, and its structural and formal features and its rigorous, coherent and unitary organisation are very relevant. Moreover, mathematical modeling (as well as statistics) belonged for long periods of the twentieth century to the field of applied mathematics, outside mathematics. While those positions have been dominant in France for long periods, in Italy some streams of research in geometry developed, based on intuition and spatial organisation, up to the middle of the twentieth century; and applied mathematics always belonged in Italy to the field of mathematics, in parallel with geometry, algebra and mathematical analysis.

Concerning sociological studies and the interest in the institutional aspects of schooling and their influences on teachers’ choices, Italian research in mathematics education (and also in the sciences of education) is rather weak. When we consider phenomena related to sociological or institutional aspects, we usually refer to elaborations and tools taken from abroad (like Brousseau’s construct of Didactical Contract, and Chevallard’s Theory of Didactical Transposition). We may acknowledge the consequence of an important lack in Italian culture: nothing can be compared in Italy, as concerns the importance and the resonance of the cultural environment, with P. Bourdieu’s or M. Foucault’s constructions.

2.6.4 A Healthy (or Unhealthy?) Eclecticism

Should we borrow some tools or constructions from abroad? If we consider Italian scientific production in mathematics education, we may identify a strong tendency to use or adapt theoretical tools of different origins (borrowed from different disciplines, or derived from different theories in the field of mathematics education). The only constraints concern local coherency (when dealing with application to a given research problem). Some original contributions and constructs (like Mathematical Discussion by Bartolini Bussi 1996; or Semiotic Bundle by Arzarello et al. 2009; or Field of Experience Didactics by Boero and Douek 2008; or Semiotic Mediation by Bartolini Bussi and Mariotti 2008; or Cognitive Unity of Theorems by Boero et al. 2007) have a local scope and are usually integrated with other theoretical constructs borrowed from different theories. In the general framework of research for innovation (a kind of meta-theoretical framework), this kind of eclecticism is not only legitimate, but also favored according to the needs of didactic innovation and of an in-depth, comprehensive interpretation of what happens in the experimental classrooms. In Italy, we frequently compare mathematics education to the engineering sciences, and I find that this analogy may be rather well justified, in the Italian case. Here again we may identify the influences of an institutional context and a cultural context. Particularly at the beginning, and during the seventies and the eighties, the policy of funding oriented mathematics educators towards the improvement of mathematics teaching and the analysis of experimental situations. Moreover, in the Italian cultural context, the importance attributed to great coherent and autonomous theoretical constructions (in human sciences as well as in natural sciences) is not so strong as to orient research in that direction.

2.7 Third Entr’acte

Boero identifies an eclecticism in Italian research in mathematics education, in which ideas and tools are borrowed from disparate theories, even disciplines. Perhaps the greater maturity of didactique means that, as systematic relations between its constructs have come to the fore, they have taken on both an independent life and a new—almost objective—character, their roots in other disciplines transcended. In his contribution, Luis Radford reminds us of key epistemological approaches on which the youthful didactique drew to develop its analysis of the genesis of new knowledge. While both are, in some sense, historically informed, Radford draws our attention to two very different strands of historical epistemology, one emphasising the internal logic of a discipline as the motor of its development, another urging attention to the broader socio-cultural framing of the discipline.

2.8 Epistemology as a Research Category in Mathematics Teaching and Learning

  • Luis Radford 

2.8.1 Introduction

In a seminal text, Artigue (1990) discusses the function of epistemological analysis in teaching. In 1995 she returns to this issue in her plenary conference delivered at the annual meeting of the Canadian Mathematics Education Study Group/Groupe canadien d’études en didactique des mathématiques. In my presentation, I draw on Artigue’s ideas and inquire about the role of epistemology in mathematics teaching and learning. In particular, I ask the question about whether epistemology might be an element in understanding differences and similarities between current mathematics education theories.

As we know very well, mathematics came to occupy a predominant place in the new curriculums of the early 20th century in Europe. It is, indeed, at this moment that, in industrialised countries, the scientific training of the new generation became a social need. As Carlo Bourlet—a professor at the Conservatoire National des Arts et Métiers— noted in a conference published in 1910 in the journal L’Enseignement Mathématique:

Notre rôle [celui des enseignants] est terriblement lourd, il est capital, puisqu’il s’agit de rendre possible et d’accélérer le progrès de l’Humanité toute entière. Ainsi conçu, de ce point de vue général, notre devoir nous apparaît sous un nouvel aspect. Il ne s’agit plus de l’individu, mais de la société.Footnote 3 (Bourlet 1910, p. 374)

However, if the general intention was to provide a human infrastructure with the ability to ensure the path towards progress (for it is in technological terms that the 20th century conceived of progress and development), it remains that, in practice, each country had to design and implement its curriculum in accordance with specific circumstances. Curriculum differences and implementation resulted, indeed, from internal tensions over political and economic issues, as well as national intellectual traditions and the way in which the school was gradually subjected to the needs of national capitalist production. These differences resulted also from different concepts of education. To give but one example, in North America, over the 20th century, the curriculum has evolved as it is pulled on one hand by a “progressive” idea of education—i.e., an education centered on the student and the discovery method—and, on the other, by ideas which organise the teaching of mathematics around mathematical content and the knowledge to be learned by the student. While proponents of the second paradigm criticise the first for the insufficiency of their discovery methods used to develop students’ basic skills in arithmetic and algebra, proponents of the first paradigm insist that, to foster real learning, children should be given the opportunity to create their own calculation strategies without instruction (Klein 2003). We see from this short example that the differences that underlie the establishment of a curriculum are far from circumstantial. They are, from the beginning, cultural. Here, they relate to how we understand the subject-object relation (the subject that learns, that is to say the student; and the object to learn, here the mathematical content) as mediated by the political, economical, and educational context. And it is within a “set of differences” in each country that the increasingly systematic reflection on the teaching and learning of mathematics resulted, in the second half of the 20th century, in the establishment of a disciplinary research field now called “mathematics education”, “didactique des mathématiques”, “matemática educativa”, “didattica della matematica”, etc.

As a result of its cultural determinations (which, of course, cannot be seen through deterministic lenses: they are determinations in a more holistic, dialectical, unpredictable sense), this disciplinary field of research cannot present itself as something homogeneous. It would be a mistake to think that the different names through which we call a discipline merely reflect a matter of language, a translation that would move smoothly from one language to another. Behind these names hide important differences, possibly irreducible, in the conception of the discipline, in the way it is practiced, in its principles, in its methods. They are, indeed, as the title of this panel indicates, research traditions.

The work of Michèle Artigue explores several dimensions of the problem posed by the teaching and learning of mathematics. In this context, I explore two of these dimensions.

The first dimension consists in going beyond the simple recognition of differences between the research traditions in mathematics teaching and learning. Artigue has played, and continues to play, a fundamental role in creating bridges between the traditions found in our discipline. She is a pioneer in the field of research that we now call connecting theories in mathematics education (e.g., Prediger et al. 2008). Artigue’s role in this field is so remarkable that there was, at the Artigue conference, a panel devoted to this field.

A second dimension that Artigue explores in her work is that of epistemology in teaching and more generally in education. She has also made a remarkable contribution to the point that there was also a panel on this topic at the conference. In what follows, I would like to briefly focus on the first dimension in light of the second. In other words, I would like to reflect on epistemology as a research category that provides insight in understanding differences and similarities in our research traditions.

2.8.2 Epistemology and Teaching

The recourse to epistemology is a central feature of the main theoretical frameworks of the French school of didactique des mathématiques (e.g., Brousseau 1983; Glaeser 1981). The recourse to epistemology, however, is not specific to mathematics. There is, I would say, in French culture in general, a deep interest in history. An inquiry into knowledge cannot be carried out without also raising questions about its genesis and development. In this context, one could hardly reflect on mathematical knowledge without taking into account its historical dimension. I can say that it is this passion for history that surprised me in the first place when I arrived in France in the early 1980s. In Guatemala, my native country, and perhaps in the other Latin American countries, as a result of the manner in which colonisation was conducted from the 16th century to the 19th century, history has a deeply ambiguous and disrupting meaning: it means a devastating rupture from which we will never recover and that continues to haunt the problem of the constitution of a cultural identity. In France, however, history is precisely that which gives continuity to being and knowledge—a continuity that defines what Castoriadis (1975) calls a collective imaginary. From this collective imaginary emanates, among other things, a sense of cultural belonging that not even the French revolution disrupted in France. Immediately after the French revolution, men and women certainly felt and lived differently from the pre-revolutionary period; however they continued to recognise themselves as French. With the disruption of aboriginal life in the 15th century (15th century as reckoned in accordance with the European chronology, of course, not to the aboriginal one), the aboriginal communities of the “New World” were subjected to new political, economical, and spiritual regimes that changed radically the way people recognised themselves. One may hence understand why the passion for history that I found in France was something new for me, as was also the idea of investigating knowledge through its own historical development.

The function of epistemology, however, is not as transparent and simple as it may first appear. And this function is even less transparent in the context of education. The use of epistemology in the context of education cannot be achieved without a theoretical reflection on the way in which epistemology can help educators in their research. It is precisely this reflection that Michèle Artigue undertakes in her 1990 paper in RDM and to which she returns in her plenary lecture delivered at the annual meeting of the Canadian Mathematics Education Study Group/Groupe canadien d’études en didactique des mathématiques (Artigue 1995). Indeed, in these papers she discusses the function of epistemological analysis in teaching and identifies three aspects.

Firstly, epistemology allows one to reflect on the manner in which objects of knowledge appear in the school practice. Artigue speaks of a form of “vigilance” which means a distancing and a critical attitude towards the temptation to consider objects of knowledge in a naive, a naive non-historical way.

A second function, even more important than the first one, according to Artigue, consists of offering a means through which to understand the formation of knowledge. There is, of course, an important difference when we confront the historical production of knowledge and its social reproduction. In the case of educational institutions (e.g., schools, universities), the reproduction of knowledge is achieved within some constraints that we cannot find in the historical production of knowledge.

Les contraintes qui gouvernent ces genèses [éducatives] ne sont pas identiques de celles qui ont gouverné la genèse historique, mais cette dernière reste néanmoins, pour le didacticien, un point d’ancrage de l’analyse didactique, sorte de promontoire d’observation, quand il s’agit d’analyser un processus d’enseignement donné, ou base de travail, s’il s’agit d’élaborer une telle genèse.Footnote 4 (Artigue 1990, p. 246).

The third function, which is not entirely independent of the first, and which is the one that gives it the most visibility to epistemology in teaching, is the one found under the idea of epistemological obstacle. Artigue wrote in 1990 that it is this notion that would come to an educator’s mind if we unexpectedly asked the question of the relevance of epistemology to teaching.

Finally, the historical-epistemological analysis has undoubtedly refined itself in the last twenty years, both in its methods and in its educational applications (see, for example, Fauvel and van Maanen 2000; Barbin et al. 2008). We understand better the theoretical assumptions behind the notion of epistemological obstacle, its possibilities and its limitations.

My intention is not to enter into a detailed discussion of the notion of epistemological obstacle that educators borrow from Bachelard (1986) and that other traditions of research have integrated or adapted according to their needs (D’Amore 2004). I will limit myself to mentioning that this concept relies on a genetic conception of knowledge, that is to say a conception that explains knowledge as an entity whose nature is subject to change. Now, knowledge does not change randomly. Within the genetic conception that informs the notion of epistemological obstacle, knowledge obeys its own mechanisms. That is why, for Bachelard, the obstacle resides in the very act of knowing, it appears as a sort of “functional necessity”. It is this need that Brousseau (1983, p. 178) puts forward when he says that the epistemological obstacles “sont ceux auxquels on ne peut, ni ne doit échapper, du fait même de leur rôle constitutif dans la connaissance visée”.Footnote 5

This conception of knowledge as a genetic entity delimits the sense it takes in the different conceptual frameworks of the French school of didactique des mathématiques. More or less under the influence of Piaget, knowledge appears as an entity governed by adaptive mechanisms that subjects display in their inquisitive endeavours. These mechanisms are considered to be responsible for the production of operational invariants: this is the case in the theory of conceptual fields (Vergnaud 1990). As a result, this theory looks at these invariants from the learner’s perspective. But the adaptive mechanisms can also be understood differently: they can be considered as forms of action that show “satisfactory” results in front of some classes of problems. “Satisfactory” means here that they correspond to the logic of optimum or best solutions in the mathematician’s sense. This is the case in the theory of situations that looks at these forms of actions under the epistemological perspective. Beyond the boundary that defines the class of problem where knowledge shows itself to be satisfactory, these forms of action generate errors. That is to say, they behave in a way that is no longer suitable in the sense of optimal, mathematical adaptation. Knowledge encounters an obstacle. The crossing or overcoming of the obstacle ineluctably requires the appearance of new knowledge.

How far and to what extent do we find similar conceptions of knowledge in other educational research traditions? I would like to suggest that it is here where we can find a reference point that can allow us to find differences and similarities in our research traditions—sociocultural theories, critical mathematics education, socio-constructivist theories, and so on.

I mentioned above that in the genetic perspective on knowledge, the obstacle appears with a “functional necessity”. However, there are several ways to understand this need. In what follows I give two possible interpretations.

The first interpretation, and perhaps the most common, is to see this need as internal to mathematical knowledge. This would involve conceiving of mathematical knowledge as being provided, in a certain way, with its own “internal logic.” This interpretation justifies how, in the epistemological analysis, the centre of interest revolves around the content itself. Social and cultural dimensions are not excluded, but they are not really organically considered in the analysis (D’Amore et al. 2006). To use an analogy, these dimensions constitute a “peripheral axiom” which we can use or not, or use a bit if we will, without compromising the core theorems (or results) of the theory.

In the second interpretation, the development of knowledge appears intimately connected to its social, cultural and historical contexts. So we cannot conduct an epistemological analysis without attempting to show how knowledge is tied to culture, and without showing the conditions of possibility of knowledge in historical-cultural layers that make this knowledge possible. It is here that we find Michel Foucault’s conception of knowledge, whose influence in the French tradition of mathematics education has remained, surprisingly, relatively marginal.

What is important to note here is that behind these two interpretations of knowledge and its development are two different conceptions of the philosophy of history. In the first interpretation, history is intelligible in itself. In the second interpretation, history is not necessarily intelligible. To be more precise, in the first interpretation, in which the theoretical articulation goes back to Kant (1991), the conception of the history revolves around the idea of a reason that develops by self-regulation. History is reasonable in itself. There are aberrations and ruptures, of course, but if you look more closely, history appears intelligible to reason. Here, “history is a slow and painful process of improvement” (Kelly 1968, p. 362). In the second interpretation, in which theoretical articulation goes back to Marx (1998), history and reason are mutually constitutive. Their relation is dialectical. There is no regulatory, universal reason. The reason is historical and cultural. Their specific forms, what Foucault calls epistemes, are conditioned in a way that is not causal or mechanical, by its nesting in the social and political practices of the individuals. It is precisely the lack of such a nesting in the rationalist philosophies that Marx deplores in The German Ideology: “the real production of life appears as non-historical, while the historical appears as something separated from ordinary life, something extra-superterrestrial” (1998, pp. 62-63). He continues further: those theoreticians of history “merely give a history of ideas, separated from the facts and the practical development underlying them” (1998, pp. 64–65). In the Hegelian perspective (Hegel 2001) of history that Marx prolongs in his philosophical works, it is, indeed, in the socio-cultural practices that we must seek the conditions of possibility of knowledge, its viability and its limits. Reason is unpredictable and history, as such, is not intelligible in itself. It cannot be, because it depends on the reasons (always contextual and often incommensurable between each other) that generate it.

In this philosophical conception of history, what shape and role could the epistemological analysis have? And what could be its interest in different traditions of research on the teaching and learning of mathematics? Concerning the first question, one possibility is the use of a materialist hermeneutic (Bagni 2009; Jahnke 2012) that emphasises the cultural roots of knowledge (Lizcano 2009; Furinghetti and Radford 2008). Concerning the second question, the reasons already given by Artigue in the early 1990s seem to me to remain valid. These reasons can undoubtedly be refined. This refinement could be done through a reconceptualization of knowledge itself, reconceptualization that might consider the political, economical and educational elements that, as suggested previously, come to give their strength and shape to knowledge in general and to academic knowledge in particular. The topicalisation of epistemology in the different theoretical frameworks and the different traditions of research would be an anchor point to better understand their differences and similarities.

2.9 Concluding Comments

Abraham Arcavi, with the support of Takeshi Miyakawa, convincingly makes the point that establishing connections between theoretical frameworks is important for mathematics education as a scientific domain but is also very difficult, especially if these frameworks have arisen in different cultures and responded to different problématiques. A major reason for this difficulty lies in the implicit assumptions underlying the work of researchers and the questions they ask. Hence, extensive and intensive dialogues are needed to make progress. Abraham has shown a direction for such dialogue, and Takeshi has experienced it in the practice of his research in France, in the USA and in Japan. Nevertheless, such communication remains fraught with potential misunderstandings.

Jeremy Kilpatrick highlights these communicative difficulties from the point of view of “translation” in his contribution, but he shows how such translation must reach far deeper than language. A translation between cultures is involved cultures that incorporate different views of schooling and education, as well as different views about the role of theory in mathematics education research, as Paolo Boero has expounded eloquently and exemplified clearly in his contribution.

Radford takes a further step when he encourages us to follow Michèle Artigue’s lead (of 25 years ago) in investigating the role of epistemology in mathematics teaching and learning. He explains how epistemology has the potential to lead beyond the mere recognition of the differences and difficulties of translation: refining the analysis of the epistemological foundations underlying different theories in different cultural contexts can lead to a deeper understanding of the differences and similarities and hence support building bridges.

The four contributors to this chapter point out that one needs a deep understanding of both cultures, the one translated from and the one translated into, in order to be able to build bridges, and they all point to Michèle as having developed such deep understanding in her own and foreign contexts of mathematics education research. In particular, Michèle’s deep epistemological questioning has made an essential contribution to her being exemplary in connecting researchers from different cultures working in different paradigms.

The CERME working group on theory was mentioned repeatedly, and indeed a sustained effort at establishing deep bridges between theories has sprung from that working group and prompted a group of researchers to not only lead dialogues between theories but to look at different aspects of a classroom lesson by means of different theoretical frameworks, and to compare and connect these frameworks while trying to formulate and answer research questions. A comprehensive description of this effort has recently been published in book form (Bikner-Ahsbahs and Prediger 2014). Not surprisingly, one of the leaders in these efforts over the past decade has been Michèle.

All contributors have pointed to the central role Michèle has been playing and continues to play in many facets of mathematics education research (and practice—but that’s for other chapters in this book). We cannot express it better than Abraham Arcavi does in his piece, so we join him and, in the name of all authors of this chapter, repeat how impressed we are by her as a devoted teacher, as a “bridge builder” (between the knowledgeable and the less knowledgeable, between the French tradition and other schools of thought, between mathematicians and mathematics educators); and by the vast scope of her knowledge and wisdom.