Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France

Abstract

This chapter analyses specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts and following influences on, and interactions with the international communities of research. This complex genesis is traced in four sections: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education. Beyond historical and cultural specificities, the chapter aims to evidence the potential of interactions between teachers and researchers, as well as interactions between researchers in mathematics and mathematics education for improving our understanding of learning and teaching issues in mathematics.

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1 Introduction

In this chapter, I analyse specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts, and following influences on and interactions with the international communities of research. Questioning the dynamics of the theoretical frameworks , that we bear and that leads us, is complex, as each theory is a result of individual and collective pathways (Trouche, 2009), which meet a set of sometimes critical facts and are subject to multiple influences. I have organised this chapter in four sections, giving voice to some main actors¹ involved in these complex geneses: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education.²

2 The Emergence of Didactics of Mathematics in France as the Result of an Exceptional Conjunction of Phenomena

It is impossible to summarise in a few lines a tumultuous history, made of contrary motions. We will try to underline here some major facts and trends: the position of mathematics teaching in the French educational debate, the role that mathematicians took in this debate, the joint action of psychologists, mathematicians, and teachers themselves in it, and finally, the creation of the IREM as a ‘total social fact’.

2.1 A Strong and Questioned Position of Mathematics as a Subject of Teaching

The position of mathematics in French curricula appears to be getting stronger over time, if we consider, for setting the scene, three key moments: 1802, 1902 and 2002 (change of centuries of course, but also of economic, social and political conditions ). This position appears, however, largely questioned if we consider, beyond these benchmarks, the discontinuity of the curricular development.

1802: Napoleon’s ordinance of 19 frimaire of Year XI (December 10, 1802) stated: ‘The lycées will essentially teach Latin and mathematics’. For Gispert (2014, p. 230), ‘In placing mathematics at the same level as Latin in the male secondary curriculum, [this ordinance] took into account the new situation following the French Revolution, in which mathematics had become a core aspect of an intellectual education combining theory and practice’.

1902: a new reform, following a great survey launched by the French Parliament, reasserted these two structuring aspects of mathematics education (Gispert, 2014, p. 233):

• The educational importance of mathematics and science: ‘It was, for a time, the end of the monopoly on classical humanities by the lycées, through the creation of a modern curriculum that was on par—at least in theory—with the classical curriculum. It also furthered the development of new disciplines such as the living languages, sciences, and mathematics’.

• The importance of the experience for learning mathematics and connecting them to sciences: ‘[In the first cycle], it was recommended to use the concrete experience and induction as the first step necessary before the transition to deductive reasoning. In the second cycle, it was necessary to introduce the new and connected notions of functions and their variations: teaching has to be now linked to physics and its need’.

2002: a report of the CREM³ Commission (appointed in 1999 by the French Ministry of education for rethinking the teaching of mathematics for the new century) stated ‘La mathématique est la plus ancienne des sciences et celle dont les valeurs sont les plus permanentes’ (Kahane, 2002). It situates the mathematics among the other sciences and underlines the necessity of connecting their teaching in combining rigor and imagination.

Beyond this apparent continuity, the situation is more contrasted. First of all, in two centuries, the school system underwent a true metamorphosis, from a school for males and elite to a school for (almost) all, with compulsory education until 16 years of age. Secondly, there is often a large gap between the prescribed and the real curriculum. For example, after the ordinance of 1802, Gispert (2014, p. 230) notices that ‘actually the real teaching, after this ordinance, continues to favour Latin and the classical humanities until the end of the nineteenth century, and to separate theory and practice’: it appears that two kinds of mathematics teaching existed, according to the social class and schooling structure (lycées vs. primary schools): formation of the mind on one side, training for the practice on the other side. Thirdly, questions at the heart of mathematics teaching appeared very sensitive to social and political events (and the twentieth century was fertile in such major events). Gispert (2014, p. 235) indicates, for example, ‘that the reform of the beginning of the twentieth century was accused of being inspired by the German model of the Realschule to the detriment of the specificity of a ‘French spirit’ based on Latin and the classical humanities. In 1923, the Chamber, strongly dominated by conservatives, voted for a new reform that revoked the 1902 programs and principles. Secondary instruction, including mathematics, was again dominated for decades by a theoretical and abstract conception’. Contrary evolutions happened at the end of the 1930s, under the left-wing regime of the Popular Front.

Last but not least, mathematics teaching appears very sensitive to intellectual and scientific pressure. A major event was the constitution, after the Second World War, of the Bourbaki group of mathematicians, who wrote the manifesto The Architecture of Mathematics , characterising mathematics as follows: ‘In the axiomatic conception, mathematics appears all in all as a reservoir of abstract forms—the structures of mathematics; and one finds—without knowing well why—that certain aspects of experimental reality mold themselves in some of the forms, as a sort of pre-adaption’ (Bourbaki, 1962, p. 46, our translation). This theoretical construct resonates with the structuralist movement: for example, Weil, a member of Bourbaki, closely collaborates with the anthropologist Lévy-Strauss on the structure of parenthood (Lévy-Strauss, 1949). As Brousseau (2012, p. 104) states:

Avec les espérances d’une après-guerre et l’aisance financière des trente glorieuses, des propositions d’origines diverses, concernant entre autres, l’éducation (Langevin-Wallon), la psychologie (Piaget), « la » mathématique (Bourbaki), la linguistique (Chomsky), etc. se rassemblent sous une même bannière épistémologique : le structuralisme.

This synergy puts forward the position of mathematics in society ‘The new mathematics and its structures were recognised not only by mathematicians but even by scholars in other fields, in particular in the humanities, as a language and scientific tool that were essential for having access to any knowledge’ (Gispert, 2014, p. 236) and led to a deep reform of mathematics teaching, the so-called ‘modern mathematics’. This reform concentrates on all the objectives of the society: to be modern, to be in line with the development of science and to be democratic:

In December 1966, in this context of profound institutional changes, the National Education Ministry gave in to the demands of mathematics teachers and created a commission for the study of teaching mathematics, led by André Lichnerowicz […] The program of the Commission was clear. It had to first work out new guidelines for teaching mathematics in primary and secondary school and assess their viability by pilot experiments. […] In force as of the 1969 school year, the reform based itself on a critique of traditional mathematics teaching (symbolized by classical geometry), considered too far removed from living mathematics, that is to say mathematics as taught and done in universities since the mid-1950s, with algebra of sets, probability theory, and statistics. Euclidian geometry and calculus were no longer taught as such.

‘Convinced that mathematics has to act as a driving force in the development of hard sciences and of human and social sciences as well, in citizens’ daily lives, and, beyond that, in the modernisation of society, the proponents of the reform saw in mathematics above all a new language that allowed all citizens to understand its functioning. One of the principal challenges of these reformers was to offer to all children, no matter what their academic future, the most modern mathematics’ (Gispert, 2014, p. 238).

This led to a very abstract teaching of mathematics and, in primary schools, to the use of manipulatives (e.g. the famous Dienes blocks, cf. Dienes, 1970), and also put less emphasis on classical mathematical instruments . Ten years later, after considerable discussion, enlisting a large part of the society, this ambitious reform was abandoned but mathematics remains as ‘the decisive discipline discriminating between student academic orientations […], a true subject of selection’ (Gispert, 2014, p. 237).

Despite, or perhaps because of these upheavals, mathematics teaching remains, till this time, at the heart of the educational debate in France. One of the characteristics of this debate was the place that mathematicians took in its animation that is the purpose of the following section.

2.2 An Important Role of the Mathematicians in the Society, and Their Interest for Mathematics Teaching

To take the same span than the previous section, we could underline that from the time of Napoleon to the current period, the interest and influence of French mathematicians for education was important: Monge, in the first case, creating new institutions (as Ecole Normale and Ecole Polytechnique in 1794) and programmes of teaching; Villani, in the second case, Fields medal 2010, frequently advocating a renewing of mathematics teaching in the French media. Between these two examples, we could follow a real continuity of mathematicians’ interventions in France, on three complementary aspects: their international engagement for discussing the issues of mathematics teaching, their engagement in the national educational institutions for designing new curricula and their interventions related to the use of tools.

The birth and the development of ICMI (International Commission on Mathematical Instruction) evidences the engagement of—not only French—mathematicians on teaching questions: ICMI was created in Roma in 1908 by the IV International Congress of Mathematicians. ‘Its first president was Felix Klein, an eminent mathematician and promoter of an important reform for teaching of mathematics in Germany. A substantial role in establishing the commission was played by David Eugene Smith, a professor at Teachers College of New York, who was deeply interested in education and the history of mathematics. Thus the commission was born of the closest collaboration between mathematicians and educators’ (Menghini et al., 2008, p. 1). To be noticed: Henri Fehr (a Swiss mathematician) and Charles Laisant (a French politician and mathematician) had created the international research journal written in French L'Enseignement Mathématique in 1899, and from early on this journal became the official organ of ICMI in 1908. The use of the French language in this journal indicates the importance of this language, at this time, as an international means for scientific communication. The 11th edition of this journal⁴ gave the composition of the commission, including 3 delegates from France among 34 members. Four French mathematicians were elected president during the history of ICMI: Jacques Hadamard (1932–1936), André Lichnerowicz (1963–1966), Jean-Pierre Kahane (1983–1999) and Michèle Artigue (2007–2009), the first woman to occupy such a position. The first three were mathematicians deeply interested in questions of education; the last one, Artigue, is a didactician of mathematics, with a strong mathematics background⁵; we will meet these names again below. We could consider such an evolution as symbolic of the emergence of didactics of mathematics as a new recognised field of research, we will go back to this point further.

We retain from this short evocation of the ICMI birth the interest of mathematicians, not only for discussing general questions of teaching, but also for implementing new curricula in their country. It was the case in Germany with Klein, and the case in France for three essential moments (see Sect. 10.2.1): the reform of 1902, the reform of the ‘Modern Math’ in 1967, and the reflection for a new curriculum in 2002:

• Regarding the reform of 1902, the commission for designing the curricula in sciences was chaired by a mathematician (Gaston Darboux). Poincaré, Borel and Hadamard made lectures in the ‘Musée pédagogique’⁶ for supporting its implementation (Belhoste, 1990)

• The second moment was the reform of ‘Modern Math’ in 1967, led by a commission chaired by André Lichnerowicz. We have seen in the previous section that the implementation of such a reform was a true catastrophe. In 1967, as in 1902, the driven idea was to bring closer real mathematics and mathematics taught,⁷ with the illusion that ‘closer to the real mathematics, closer to the real need of their teaching’

• The third moment was the reflection of the CREM, chaired by Jean-Pierre Kahane (Sect. 10.2.1). This commission gathered mathematicians, teachers, but also didacticians (among them Michèle Artigue). Retaining the lesson of the ‘Modern math’, this commission did not want to reform the whole curricula at once, but offer some perspectives for thinking the teaching of mathematics on the long term, conceiving its work in relation to the experience of the teachers on the field:

  La réflexion sur l'enseignement des mathématiques est donc, par nature, une réflexion à long terme […]. Elle prend point d'appui sur ce que nous savons du mouvement des sciences, et sur une vision implicite de l'avenir à long terme: des possibilités sans nombre, des dangers déjà identifiés, et une multitude de problèmes auxquels l’humanité ne pourra faire face qu'en mobilisant toutes les ressources de l'imagination, de la curiosité, de la créativité, des capacités d’analyse critique et de raisonnement, et des connaissances engrangées par les générations précédentes. La réflexion doit prendre en compte le mouvement actuel de la science comme son histoire et tout ce qui doit être revisité de son passé. Elle doit être ambitieuse, audacieuse, et en même temps tenir compte des contraintes de terrain. Elle doit marier les analyses épistémologiques et didactiques. Au sein de la commission elle a bénéficié d'une grande variété d'expériences et de sensibilités. Elle doit se poursuivre à l'extérieur, et de façon permanente (Kahane, 2002).

This last moment differs from the two previous ones, in the way of thinking the distance between the mathematics currently developed by the mathematicians and the mathematics to be taught. This distance has been conceptualised further as the didactical transposition, a major concept, developed by the didactician Chevallard (see Sect. 10.4). To be noticed: another difference between the reforms of 1902 and 1967 on one side, and the reform of 2002 on the other side is that the propositions of the CREM… had not been really applied, probably due to their financial as well as didactical cost.

A last aspect of the mathematicians’ interventions concerns the use of tools and the context of their use, reflecting a constructive point of view on mathematics and its teaching.⁸ Poincaré (1904, p. 275) insisted on this aspect, supporting the incessant use of mobile instruments in geometry teaching : ‘J’ai dit que la plupart des définitions mathématiques étaient de véritables constructions. Dès lors, ne convient-il pas de faire la construction d’abord, de l’exécuter devant les élèves, ou, mieux, de la leur faire exécuter de façon à préparer la définition ?’. Maschietto and Trouche (2010, p. 34) evidence how this issue of tool use runs through most of the issues of L’enseignement mathématique. They underline also (p. 39) the productive notion of mathematics laboratory, as places to learn mathematics from experiments (see Chap. 3), proposed by French mathematicians at the beginning of the twentieth century, particularly Borel (1904) and rediscovered one century later: for Kahane (2006), ‘The main feature of math laboratories is that they are places for experiments. Experiments in mathematics need time and freedom. The pupils should be provided with subjects to explore, they should not have a task to stick to. They should feel free, not under pressure’. This proposition is clearly in line with the pedagogical and philosophical tradition of active methods for learning (cf. Dewey in USA, Freinet in France, Montessori or Pestalozzi in Italia), but also with the spirit of Klein’s propositions in the first international movement for reforming mathematics teaching, supporting it by the use of geometrical models and artefacts (Schubring, 2010). In doing this, the mathematicians wished to bring closer the mathematicians’ practices and the mathematics learning and teaching practices. We will see further how this notion of mathematics laboratory meets the notion of a-didactical situation of Brousseau (Sect. 10.3).

We have evidenced in this section the engagement of mathematicians to reform mathematics teaching, bringing them closer to the contemporary mathematics and the conditions of their production, with a growing awareness of the necessary distance between ‘mathematics for mathematicians’ and ‘mathematics for teachers and students’. This awareness is probably due to the lessons of history, particularly the lesson of ‘Modern math’ , and also to the interactions of a range of actors in the field of mathematics teaching. This is the purpose of the following subsection.

2.3 A Joint Action of Teachers, Psychologists and Mathematicians for Rethinking Mathematics Education

We will evidence, in this section, the growing and convergent views of teachers, psychologists and pedagogues, particularly in the francophone area, in the debate on mathematics to be taught. Among scholars, a double movement took place in the second half of the twentieth century.

The first movement was a convergence of interest between some mathematicians, psychologists and philosophers, leading in 1950 to creation of a new organisation, CIEAEM (Commission Internationale pour l’Etude et l’Amélioration de l’Enseignement des Mathématiques [International Commission for the Study of and Improvement of Teaching Mathematics]), ‘in which French mathematicians played an important role’ (Gispert, p. 236), due to the Bourbaki influence. The first book coming from this new organisation is in French: L’Enseignement des mathématiques (Piaget et al., 1955). It is a collection of chapters written by a psychologist (Piaget, first author, from Geneva), a logician (Beth, from Amsterdam), three—French-mathematicians (Choquet, Dieudonné et Lichnerowicz) and a pedagogue (Gategno, from London). The introduction makes clear the goal of the book, aiming to enlighten what is possible to teach (the psychologist point of view), what has to be taught (the mathematicians’ point of views) and how to teach it (the pedagogue point of view). It should be pointed out that the book is a succession of chapters, not really articulated; this reflection is driven without the inputs of the teachers, but wishes ‘to attract the interest of high school teachers supposed to be thought-provoking to them in a way that could renovate their teaching’ (Piaget et al., 1955, p. 8). Teachers, however, will soon invade the scene…

The second movement was a specialisation of some mathematics scholars towards the issues of education, taking them as a main interest of research (cf. Sect. 7.2). Hans Freudenthal, chairing ICMI from 1967 to 1970, was the first president considering mathematics education as a field of research in its own right. It corresponded to his view of mathematics, seen ‘not primarily as a body of knowledge, but as a human activity’ (Bass, 2008, p. 12), his view on mathematics education, seen essentially as a development from the concrete to the general, and his view on research on mathematics education to be developed as a new field, not restricted to a statistical or psychological point of view. He drew all the consequences of such a position in creating a journal, a conference, and an institute: he founded in 1968 a journal dedicated to this question (Educational studies in mathematics); he launched the first ICME (International Congress on mathematical education) in 1969 in Lyon, as a manifestation of independence from IMU:

Freudenthal’s bold and adventurous launching of ICME 1 (in Lyon) was essentially, and characteristically, a unilateral action, for which he sought approval and authorization from no one, not the Executive Committee of ICMI, nor that of IMU. And this provoked some anger and concern about ICMI inside ILU. But ICME 1 was a great success… (Bass, 2008, p. 12).

Finally, he founded in 1971 the Institute for the Development of Mathematical Education (IOWO) at Utrecht University, which, after his death, was renamed the Freudenthal Institute. Such creativity cannot be fully understood independently of the social and political context of 1968, as we will see soon.

The intervention of teachers in the debate was certainly decisive. The French APMEP (Association des Professeurs de Mathématiques de l’Enseignement Public) [The Association of Mathematics Public School Teachers], was created in 1910, in a moment of academic, social and political upheaval (marked by the creation of trade unions and associations⁹). A former president of this association, Eric Barbazo, dedicated his Ph.D. (Barbazo, 2010) to its history, and evidenced the engagement of this association, from its beginning, in the public debate, reflecting the position of its members and the organisation of schooling. For example, in 1912 (the teaching of mathematics concerned at this mome nt an elite), in response to a survey initiated in 1912 by the Chamber on the implementation of the reform of 1902 (see Sect. 10.2.1), ‘the Association expressed reservations concerning the method of relying on the concrete in mathematics. It highlighted the potential dangers of such a method and the harm that it could do if it was used to substitute experience for proof more generally. Students should not be deprived of the advantages they could obtain from the study of mathematics, which had always been a school of logic’ (Gispert, 2014, p. 234).

Convinced that the Bourbaki’s ideas were a means for promoting ‘mathematics for all’, The APMEP created a commission named ‘axiomatic and re-discovery’, evidencing the double teaching necessity of ‘learning logic and structures’ and ‘favouring students activity’. Convinced that teacher education were a critical issue, ‘APMEP, together with the Société Mathématique de France, organised between 1955 and 1963, lectures for secondary school teachers on the notions of structures that, for the most part, they had not seen in their studies’ (Gispert, 2014, p. 236). The conjunction of the influence of the group Bourbaki, of the structuralist intellectual spirit and of the pressure of APMEP towards the ministry of education leads en 1967 to the constitution of the official commission Lichnerowicz, giving birth, in 1971, to the Modern Math reform (Sect. 10.2.1) and launching the ideas of constituting new institutes, the IREM (Institute for research on mathematics teaching) for supporting this radical reform. Actually the creation of the IREM is a more complex story, concentrating on all the features of this period, as we will show in the following section.

2.4 The IREM as a Total Social Fact, and the Incubator of the French Didactics of Mathematics

The creation of the IREM can be considered indeed as a total social fact (Mauss, 1966), i.e. a fact that informs and organises seemingly quite distinct practices and institutions.

A first institution (and associated practices) is the APMEP. In 1958, the president of this association, Gilbert Walusinski, already proposed the creation of Institutes for training and pedagogical research, dedicated to the development of theoretical as well as practical pedagogical research, and to the development of interactions between teachers of different levels (from schools to university) for improving teacher training, in the perspective of the ‘Ecole unique’.¹⁰

A second (emerging) institution is the collective (still not a community) of research on mathematics education. In 1964 a young primary teacher of Bordeaux, having a bachelor in mathematics, Guy Brousseau (who will be the ‘hero’ of the following section of this chapter) asks Lichnerowicz for a question to be studied on mathematics teaching (cf. Sect. 10.3). In order to gather the conditions of an answer, he created, with the support of Lichnerowicz, a Center for research on mathematics teaching, becoming later the COREM (Center of Observation and Research on Mathematics Teaching). In the text describing the organisation of this centre, established in Bordeaux, directed by two mathematicians and involving himself, Brousseau writes:

Avant de chercher à influencer l’enseignement, il convient d’abord de l’observer et de le comprendre en n’agissant que de façon limitée, contrôlée a priori par des connaissance scientifiques et a posteriori par des expériences reproductibles. L’important et le difficile était de rendre possible l’établissement de rapports appropriés entre des chercheurs mathématiciens et un système d’enseignement.¹¹

The link between research and teacher training , the link between research and experimentation, as well as the link between teachers of different levels and between researchers from different scientific fields appear as the major features of these centres that the Commission Lichnerowicz will retain later in its recommendations:

It is necessary to progressively create, in each university, Institutes for Research on mathematics teaching with the double objective of performing teacher training at all the levels, to organize necessary experiments, in order to implement their conclusions as facts, in a progressive way. The commission estimates that the IREM have to facilitate or provoke the teamwork and to weave a network of teams in each academic region.¹²

It is well known that there is a long way from an institutional proposition, to its implementation. What makes this implementation possible was certainly the social and political pressure of 1968: the third institution involved in the creation of IREM could be considered as the social movement, including teachers and trade unions. Bass (2008, p. 13) confirms the essential role of the social pressure, of the APMEP and its president at this time, Maurice Glaymann:

In 1968, at the time of the student demonstrations, Glaymann asked, and received an audience with the new Minister of National Education, Edgar Faure. Faure has decided to move things and Glaymann reiterates the APM proposition of creating IREMs. Faure, after one week of reflection, to evaluate the cost of such an operation, proposed to create an IREM in Paris and says that he has evaluated the coast to be 3 million francs. Glaymann answers him that with the same budget, the APM thinks that three IREMs could be created. This came to pass, Glaymann was named first director of the IREM at Lyon, and this, just in time, put him in a position to offer to host the international congress on mathematics education [in 1969] proposed by Frendenthal.

That is a point where a set of actors already evoked met.

Once created, the IREM developed as a network of Institutes in each university in France. The centre of Brousseau, became the Centre of observation and research on mathematics teaching, associated with primary schools, depending on the IREM of Bordeaux. The IREM became then an incubator for a new field of research. The dynamic of the research in IREM is well summarised by Artigue and Douady:

This evolution is due to the running of these institutes. IREM gather indeed teachers of several levels and, thus they forced the research which was born in them to not be isolated in an academic ghetto, but on the contrary to keep in touch with the schooling institution, the classes, the teachers. The difficulties created by the new curricula […] evidenced the inadequacy of the points of view leading to the reform. They evidenced also the limits of the research centred on action and innovation, the necessity for the didactics of mathematics, taking into account the neighbouring fields (psychology, epistemology, sociology, linguistic, sciences of education) to constitute a theoretical field specifically fitted to its problematic and to the methods of research that it developed (Artigue & Douady, 1986, p. 70).

It is thus an exceptional conjunction of phenomenon (mathematical, pedagogical, scientific, intellectual, social and political) that leads, in this country and this period, to the emergence of the French didactics of mathematics, as ‘a fragment of the history of the IREM’ (Rouchier, 1978, p. 153). This emergence of didactics is not a French exception: Biehler et al. (1994) evidence the richness of the interactions between several national communities, facilitated by a network of international scientific conferences and commissions, steering, at an international level, Didactics of Mathematics as a Scientific Discipline.¹³ The interaction with the German community has been particularly intense (Strässer, 1994). But there is no doubt that there is a French specificity: the context of IREM, the number and diversity of persons involved in this network, the particular status of mathematics in France (where such institutes, exist only for mathematics) and the mathematical Bourbaki context where the field was born, leads probably to a more theoretical structured field. I propose, in the following parts of this chapter, a visit of this field (keeping in mind the place of artefacts), aiming to evidence both its diversity and its unity, confirming Kilpatrick’s point of view (1994, p. 90): « Aux yeux des américains, la didactique des mathématiques française possède une remarquable unité ».

3 Two Founding Theoretical Frameworks: Brousseau’s Theory of Didactical Situations; Vergnaud’ Theory of the Conceptual Fields

From the 1970s, the growth of the French community of didactics of mathematics was very rapid (Artigue & Douady, 1986, p. 71): first master teaching in 1975, a national seminar in 1978, a new journal (Recherches en didactique des mathématiques) and the first summer school in 1980, a first group of research recognised by the CNRS¹⁴ in 1981.¹⁵ Referring to Brousseau and Vergnaud as the founders of the field does not come only from a personal choice. The French community of didactics of mathematics, 20 years after its birth, acknowledged this role in a collective book: Vingt ans de didactique des mathématiques en France. Hommage à Guy Brousseau et Gérard Vergnaud (Artigue et al., 1994).¹⁶ It is, indeed, impossible, in the frame of this chapter section, to summarise the scientific contributions of these two preeminent researchers. I will only try to enlighten some major aspects of their works in line with the tool focus of this book.

3.1 Guy Brousseau and the Critical Notions of ‘Situation’ and ‘Milieu’

The question Lichnerowicz asked to Brousseau was: ‘You ought to study the limiting conditions for an experiment in the pedagogy of mathematics’ (Brousseau et al., 2014, p. 172). Brousseau described his reaction: ‘my questions were not of the type of ‘how many experimental and model classes should the administration set up, and what would be the budget for that?’ But rather how to reconcile the flexibility necessary in order to adapt the project to a class with a respect for conventional conditions common to a whole cohort of schools—which notions were indispensible and how to make them accessible’ (Brousseau et al., 2014, p. 173). The decision resulting from this reflection was the creation of the COREM (Sect. 10.2.4), which was linked to a primary school, ‘during 25 years the most advanced laboratory of experimental didactics of mathematics’.¹⁷ Brousseau’s theoretical framework draws its substance from the work in this laboratory.¹⁸ His major work, The theory of didactical situations, has been translated into English in 1997 (Brousseau, 1997), and its fundamental idea, Teaching through situations, regarding the theme of fractions, gave matter to a recent book (Brousseau et al., 2014). The fundamental idea of ‘situations’ is defined by Warfield (2014), in a short book ‘inviting to didactics’: ‘A Situation describes the relevant conditions in which a student uses and learns a piece of mathematical knowledge. At the basic level, these conditions deal with three components: a topic to be taught, a problem in the classical sense and a variety of characteristics of the material and didactical environment of the action’.¹⁹

In this section, I would like to deepen this idea, focusing on three structuring notions of Brousseau’s theory : a-didactical situations, didactical situations, and the milieu, and I will do that through an example, then, a quotation from Brousseau. The example is the emblematic situation of the puzzle, described by Brousseau et al. (2014, p. 51). (Fig. 10.1)

Fig. 10.1

One example of the Brousseau’s puzzles. ‘Instruction: Here are some puzzles. You are going to make some similar ones, larger than the ones I am giving you, according to the following rules: (a) The segment that measures 4 cm on the model must measure 7 cm on your reproduction. (b) When you have finished, you must be able to take any figure made up from pieces from the original puzzle and make the exact same figure with the corresponding pieces of the new puzzle. (c) I will give a puzzle to each group of four or five students, and every student must either do at least one piece or else join up with a partner and do at least two. Development: After a brief planning phase in each group, the students separate to produce their pieces. The teacher puts (or draws) an enlarged representation of the complete puzzle on the chalkboard’

Let me now introduce Brousseau’s main concepts (Brousseau et al. 2014):

A-didactical situations occur in the classroom, and have the goal of reproducing the conditions of a real mathematical activity dealing with a determined concept: i.e. a mathematical situation. In the course of an a-didactical situation, the students are supposed to produce a correct and adequate action or mathematical text without receiving any supplementary information or influence.

With this definition in hand, a didactical situation can be defined as the actions taken by a teacher to set-up and maintain an a-didactical situation designed to allow students to develop some goal concept(s). In particular, the teacher sets up the milieu, which includes the physical surroundings, the instructions, carefully chosen information, etc. The milieu may, or may not include a material element (for example Cabri geometry), and other cooperating or concurrent students, etc. but it does at the least include the savoirs ²⁰ of the subject, and certain of her current connaissances. ²¹ It is essential that the mi : lieu is designed in a way that it only obeys “objective” necessities, and that the student be convinced of that fact. Once that design is in place, the teacher’s mandate is limited to making sure the students focus on the milieu and not on the teacher” (p. 203).

The previous example allows me to illustrate the three fundamental notions proposed here by Brousseau:

• The a-didactical situation is constituted in the classroom by the problem of enlarging a puzzle according to a given constraint (adding 3 cm at a given dimension); the determined concept the students are dealing with is the concept of proportionality

• The didactical situation is constituted by the actions taken by the teacher to set up and maintain this a-didactical situation. This setting up lies on very subtle adjustments (the initial dimensions of the puzzle pieces, the organisation of students’ collective work, the cutting of time in successive phases…) are not randomly chosen, but the result of a very careful analysis of numbers of experiments in the frame of the COREM

• The milieu of the situation is all the ‘things’ the students are acting on, and all the ‘things’ which are providing feedbacks to the students. In the situation at stake, the puzzle, the ruler, the ‘savoir’ of the students on numbers, their ‘connaissance’ of the additive model… are part of the milieu

The feedbacks of the milieu allow the development of the a-didactical situation: a student does not need a validation from the teacher, as the feedback from the milieu (see Fig. 10.2) is enough to convince her that her method is wrong. And finally the targeted knowledge—the proportionality—is the necessary way for solving the problem.

Fig. 10.2

The result of students’ action. One of the strategies and behaviours observed. ‘Almost all the students think that the thing to do is to add 3 cm to every dimension. Even if a few doubt this plan, they rarely succeed in explaining themselves to their partners and never succeed in convincing them at this point. The result, obviously, is that the pieces are not compatible’. [The authors detailed some other students’ strategies: adding 3 cm to each segment on the outside square, leading to obtain a rectangle measuring 17 cm × 20 cm; multiplying each measurement by 2 and subtracting 1, as they observe that 4 × 2 − 1 = 7…]. The results: ‘All the children have tried out at least one strategy, and most have tried two. By the end of the class, they are all convinced that their plan of action was at fault, and they are ready to change it so they can make the puzzle work. But no one group is bored or discouraged. At the end of the session, they all are eager to find the right way’ (Brousseau et al., 2014, p. 53)

It is not possible, in the frame of this section, to further develop the other essential concepts constituting this theory (situation of devolution, of institutionalisation , didactical contract…). But what I have presented here is, to me, the heart of Brousseau theory, modelling the learning of mathematics as a social game, with specific rules, the targeted knowledge constituting the optimal way for winning, individually, and with the other students, the game.

The game develops through the interaction with a milieu. This notion of ‘milieu’, that Brousseau did not translate in English, is very interesting to be analysed: a milieu seems to be always, and it will not be a surprise for the reader of this book, ‘full of artefacts’. But it contains more, and is permanently fed by interactions with other students. For me, a possible English translation of this term, in line with further conceptualisations (Sect. 10.5.1), should be ‘the student’s resources in the situation’.²²

3.2 Gérard Vergnaud, and the Conceptualisation Seen as a Cognitive Mediated Process

Brousseau came from the community of mathematicians, Vergnaud came from the community of developmental psychologists. But there were a number of connections. Brousseau mentions that ‘the experimental designs imagined by Piaget [were] directly inspired by his exchanges with the mathematician Gonseth’ (Brousseau et al., 2014, p. 192), and Piaget, situated at the borders of several fields (epistemology, biology, logic) was, together with prestigious mathematicians, a founding member of the CIEAEM (Sect. 10.2.3). But this original difference between the two researchers could explain some major differences in point of view, Vergnaud saw conceptualisation as a developmental process, and according to a crucial importance to the connections between the operational form of knowledge and the predicative one. I will present his theory focusing on these two aspects.²³

First of all, Vergnaud shared with Piaget the idea that even highly structured concepts develop from the most elementary actions of a subject. These actions apply in situations (opening a door, solving an equation, climbing stairs…), and facing these situations lead the subject to develop schemes (the more or less flexible ways of opening a variety of doors, etc.). Studying the processes of mathematics learning leads Vergnaud to specify this notion of scheme:

The function of schemes, in the present theory, is both to describe ordinary ways of doing, for situations already mastered, and give hints on how to tackle new situations. Schemes are adaptable resources: they assimilate new situations by accommodating to them. Therefore the definition of schemes must contain ready-made rules, tricks and procedures that have been shaped by already mastered situations; but these components should also offer the possibility to adapt to new situations. On the one hand, a scheme is the invariant organisation of activity for a certain class of situations; on the other hand, its analytic definition must contain open concepts and possibilities of inference. From these considerations, it becomes clear that schemes comprise several aspects defined as follows:

• The intentional aspect of schemes involves a goal or several goals that can be developed in sub-goals and anticipations.

• The generative aspect of schemes involves rules to generate activity, namely the sequences of actions, information gathering, and control s.

• The epistemic aspect of schemes involves operational invariants , namely concepts-in-action and theorems-in-action. Their main function is to pick up and select the relevant information and infer from it goals and rules.

• The computational aspect involves possibilities of inference. They are essential to understand that thinking is made up of an intense activity of computation , even in apparently simple situations; even more in new situations. We need to generate goals, sub-goals and rules, also properties and relationships that are not observable.

The main points I needed to stress in this definition are the generative property of schemes, and the fact that they contain conceptual components, without which they would be unable to adapt activity to the variety of cases a subject usually meets. I also feel the need to add several comments in what follows. The dialectical relationship between situations and schemes is so intricate that one sometimes uses an expression concerning situations to refer to a scheme, for in-stance high jumping, or solving equations with two unknowns, as well as an expression concerning schemes to refer to a situation, for instance ‘rule of three’ situations (the rule of three is a scheme, not a situation) (Vergnaud, 2009, p. 88).

What is certainly crucial for mathematical learning, is the conceptual component of schemes, namely the operational invariants : concepts-in-action and theorems-in-action, that is implicit properties, that are not necessarily true, but appear as relevant in a certain domain. For example, when learning to multiply two integers, students used to develop a strong theorem-in-action as ‘the product of two numbers is a number bigger than the two initial numbers’; and a strong concept-in-action as ‘the multiplication is a machine for increasing numbers’. Other examples are given in the case of symmetrical figures (Fig. 10.3), and further for the use of graphing calculators (Sect. 10.5.1). Such operational invariants are relevant in a certain domain (that is a reason for their resistance), and turn into obstacles as soon as the mathematical context exceeds this domain.

Fig. 10.3

Two figures associated to symmetry

A concept is, for Vergnaud (2009), related to a given subject and to a moment of her conceptualisation, and it is defined by a triplet: a set of situations, a set of operational invariants and a set of representatives. For a given student, the concept of symmetry exists as soon as she is able to associate to this word a set of situations (in or out of school), a set of operational invariants (for example, ‘the figure and its image are separated by an axis’) and a set of representatives (figures drawn on paper, objects that can be moved from either side of a rule, sentences for describing such situations…). A concept is, in this frame, always associated to a set of artefacts allowing it to be set in different situations. Noticeably, Vergnaud uses the same word of ‘concept’, for designating something well recognised by a scientific community, and for designating a subject’s temporary construction: Vergnaud justified this ambiguity in arguing that a concept is always a living entity, engaged in a genesis, personal or collective. A concept never lives in isolation, but takes sense in the frame of a conceptual field, that Vergnaud (2009, p. 86), giving the example of the conceptual field of the additive structures, defined as ‘a set of situations and a set of concepts tied together’.

The second main idea of Vergnaud addresses the different forms of knowledge. For him, there is a gulf between the operational form of knowledge, which allows one to do something, and the predicative form of knowledge, which allows one to state/justify what has been done, or what is to be done, as explained in the following extract.

Some researchers even consider that the difficulty of mathematics is mainly a linguistic difficulty. This view is wrong, because mathematics is not a language, but knowledge. Still, understanding and wording mathematical sentences play a significant role in the difficulties students encounter. To illustrate this point, let us take two situations (Fig. 10.3) in which students have to draw the symmetrical shape of a given figure. These situations contrast with each other, both from the point of view of the schemes that are necessary for the construction and from the point of view of the sentences that one may have to understand or produce on these occasions. The first figure corresponds to a situation likely to be presented to 8- or 10-year-old students, in which they have to complete the drawing of the fortress symmetrically from the vertical axis

The second one could be typically given to 12- or 14-year-olds in France: construct a triangle symmetrical to triangle ABC in relation to d (‘d’ here refers to the dotted line).

In the first case, there are some coordination difficulties because the child needs to draw a straight line just above the dotted line, neither too high nor too low, and everybody knows that it is not that easy with a ruler; there is the same kind of awkwardness for the departure point and the arrival point. There are also conditional rules. For example, ‘one square to the left on the part already drawn, one square on the right on the part to be drawn,’ or else ‘two squares down on the figure on the left, two squares down on the right,’ or else ‘one square to the right on the left figure, one square to the left on the one on the right,’ starting from a reference point homologous to the point of departure on the left. These rules are not very complex. Nevertheless they rely upon several concepts-in-action and theorems-in-action concerning symmetry and conservation of lengths and angles. As all angles are right angles and lengths are expressed as discrete units (squares), the difficulty is minimal.

In the second case, drawing the triangle A′B′C′, symmetrical to triangle ABC in relation to line d, is much more complex, with the instruments usual in the classroom (ruler, compass, set square). Even the reduction of the triangle to its vertices as sufficient elements to complete the task is an abstraction that some students do not accept easily because they see the figure as a non-decomposable whole. One step further, using d as the axis of symmetry for segments AA′, BB′, CC′, is far from trivial. Why draw a circle with its centre in A, and why should we be interested in the inter-sections of that circle with line d? One can also use a set square and draw a perpendicular line from A to d, measure the distance from A to d, go across line d to construct A′ at the same distance of A to d. But how can I think of the distance to be the same when there is no line yet?

The epistemological jump from the first to the second situation is obvious. But there are also big jumps between different sentences that are likely to be articulated on these occasions. I will use French rather than English because of the syntax of definite articles in French:

• 1. La forteresse est symétrique (‘The fortress is symmetrical’)

• 2. Le triangle A′B′C′ est symétrique du triangle ABC par rapport à la droite d (‘Triangle A′B′C′ is symmetrical to triangle ABC in relation to line d’).

• 3. La symétrie conserve les longueurs et les angles (‘Symmetry conserves lengths and angles’).

• 4. La symétrie est une isométrie (‘Symmetry is an isometry’).

Between sentence 1 and sentence 2, there is already a qualitative jump: the adjective symétrique moves from the status of a one-element predicate to the status of a three-elements predicate (A is symmetrical to B in relation to C).

Between sentence 2 and sentence 3, the predicate symétrique is transformed into an object of thought, la symétrie, which has its own properties: it conserves lengths and angles. Nominalisation (i.e., to form a noun from another word class or a group of words) is the most common linguistic process used to transform predicates into objects. In sentences 1 and 2, the idea of symmetry is a predicate (propositional function); in sentence 3, it has become an object (argument). Lower-case ‘s’ is the kind of symbol used by logicians for arguments, whereas upper-case ‘S’ is used for predicates. The two new predicates, conserving lengths and conserving angles, are thus properties of this new object ‘s’.

When we move from sentence 3 to sentence 4, a new transformation takes place; the retention of lengths and angles then becomes an object of thought: isometry. This time the predicate is the inclusion relationship between two sets: the set of symmetries S and the set of isometries (Vergnaud, 2009, p. 90).

For Vergnaud, the predicative form of knowledge is a necessary means for building knowledge, but not the main one: the operative form is more subtle, richer than the predicative one. For him, solving a problem is the source and the criterion of knowledge. Schemes appear thus at the centre of the Vergnaud theoretical framework, as an essential link between gestures and thought. This importance given to gestures and artefacts situates the work of Vergnaud at crossroads of influences: Piaget of course, but also Vygotsky for the structuring place of mediations (Vergnaud, 2000) and Bourdieu for the social founding of psychology (Bronckart & Schurmans, 1999). Looking further, I could relate the notion of scheme to Eastern culture, considering the dialectic interaction between hand and mind , as in the following quote, which describes the gradual synthesis of ‘proper gestures’ to a very complex scheme:

Entre force et douceur, la main trouve, l’esprit répond. Par approximations successives, la main trouve le geste juste. L’esprit enregistre les résultats et en tire peu à peu le schème du geste efficace, qui est d’une grande complexité physique et mathématique, mais simple pour celui qui le possède. Le geste est une synthèse (…). L’adulte ne se rend plus compte qu’il lui a fallu accomplir un travail de synthèse pour mettre au point chacun des gestes qui forment le soubassement de son activité consciente, y compris de son activité intellectuelle (Tchouang Tseu, in Billeter, 2002).

Brousseau/Vergnaud: two different views on mathematics teaching, the first one focusing on a micro-level (the interactions student-milieu through very finely tuned didactical situations), the second one on a macro-level (the process of conceptualisation, through the encountering of various situations, most of them at school, drawing attention on intermediate forms of knowledge). Both views share an understanding of learning mathematics through mathematical situations, from interactions with specific resources (milieu vs. mediations).

4 Chevallard and the Anthropological Theory of Didactics

Roughly speaking, as the main source of inspiration of Brousseau (resp. Vergnaud) was mathematical (resp. psychological), it could be said that the main source of inspiration of Chevallard²⁴ was anthropological. I will present in this section his theory, focusing on two essential points: the concept of praxeology, and the importance of tools. Then I will evidence some convergence and tensions, contrasting Chevallard’s approach with Brousseau’s and Vergnaud’s ones.

4.1 A View on Mathematical Activity Through Artificial Praxeologies, Products of Human Cultures

Chevallard, in developing his theoretical framework, the so-called ‘Anthropological approach of didactics’ (ATD in the following), often evokes the work of the French anthropologist Marcel Mauss (1872–1950), who introduces the notion of total social fact for designating essential social phenomena:

These phenomena are at once legal, economic, religious, aesthetic, morphological and so on. They are legal in that they concern individual and collective rights, organized and diffuse morality; they may be entirely obligatory, or subject simply to praise or disapproval. They are at once political and domestic, being of interest both to classes and to clans and families. They are religious; they concern true religion, animism, magic and diffuse religious mentality. They are economic, for the notions of value, utility, interest, luxury, wealth, acquisition, accumulation, consumption and liberal and sumptuous expenditure are all present… (1966, pp. 76–77).

For Chevallard, referring to the English anthropologist Douglas (1986), a given institution constitutes a total social fact. The word institution stands here for each social structure which allows—and impose to—its members, occupying various positions in this structure, different ‘ways of doing’: a classroom, in this sense, constitutes an institution, as well as a school, as well as the schooling system, in a given country and at a given period.

Chevallard defines, in a given institution, a didactic fact:

What I shall henceforth call a didactic fact is any fact that can in some way be regarded as the effect of a socially situated wish to cause someone to learn something. Let me add—this is a more difficult point, on which I shall not dwell any longer—that a didactic fact is , considered to be so only to the extent that it is effective in influencing the learning process (2005, p. 22).

This definition of a-didactical fact is very powerful, and leads to a very general definition of didactics: ‘Didactics should, in my view, be defined as the science of the diffusion of knowledge in any social group, such as a class of pupils, society at large, etc.’ (Chevallard, 2005, p. 22).

Didactics, as a science, analyses didactical facts in a structured way, as elements of local or global praxeologies. Chevallard defines a praxeology in the following:

What exactly is a praxeology? We can rely on etymology to guide us here—one can analyse any human doing into two main, interrelated components: praxis, i.e. the practical part, on the one hand, and logos, on the other hand. “Logos” is a Greek word which, from pre-Socratic times, has been used steadily to refer to human thinking and reasoning—particularly about the cosmos. Let me represent the “praxis” or practical part by P, and the “logos” or noetic or intellectual part by L, so that a praxeology can be represented by [P/L]. How are P and L interrelated within the praxeology [P/L], and how do they affect one another? The answer draws on one fundamental principle of ATD—the anthropological theory of the didactic—, according to which no human action can exist without being, at least partially, “explained”, made “intelligible”, “justified”, “accounted for”, in whatever style of “reasoning” such an explanation or justification may be cast. Praxis thus entails logos which in turn backs up praxis. For praxis needs support—just because, in the long run, no human doing goes unquestioned. Of course, a praxeology may be a bad one, with its “praxis” part being made of an inefficient technique—“technique” is here the official word for a ‘way of doing’—, and its “logos” component consisting almost entirely of sheer nonsense—at least from the praxeologist’s point of view! (2005, p. 23).

A praxeology is made of four comp onents: a set of tasks, a set of techniques, as a way of accomplishing these tasks, a set of technologies, as discourses justifying the techniques, and a theory accounting for these technologies. Let me illustrate this with an example drawn from Chevallard (2005) (Fig. 10.4).

Fig. 10.4

A task carried out in the frame of a given praxeology (Chevallard, 2005, p. 24). The task consists in writing a given number under the form $a+b\sqrt{3}$. The technique consists in seeing each number $x=a+b\sqrt{3}$ as a root of the equation ${\left(x-a\right)}^2=3{b}^2$. The technology consists in knowing that $\mathrm{Q}+\mathrm{Q}\sqrt{3}$ is a field. The theory is that of algebraic structures

This deep idea of a socially and culturally built mathematics world is essential to understand Chevallard’s frame, as evidenced in the following quotation:

Why do mathematicians seem so attracted to triangles for example? Why does geometry tell us about angles, lines and rays, or about crossing lines and parallel lines? Why does geometry make room for the notions of acute angle, obtuse angle, and reflex angle? If you are tempted to answer: “Mathematicians are interested in all these entities simply because there do exist crossing lines, rays, acute angles, reflex angles, etc., that is, just because these ‘things’ are out there, in the natural world, waiting for us to study them”, then you have been infected with the evil “monumentalistic” doctrine that pervades contemporary school epistemology. If indeed you accept such a poor, unspecific reply, it is more than likely that you have secretly espoused a naturalistic view of the human world—including the mathematical world—, forgetting that almost everything out there, as well as everything in our minds, is socially contrived. A straight line is a concept, not a reality outside us. It is something created in order to make sense of the outside world and to allow us to think and act more in tune with that reality (Chevallard, 2005, p. 26).

This anthropological point of view enlightens the seminal Chevallard’s work on didactical transposition as a social construct:

The transition from , knowledge regarded as a tool to be put to use, to knowledge as something to be taught and learnt, is precisely what I have termed the didactic transposition of knowledge […] Although long-established, teaching, or the project to have someone learn some knowledge and know it, is therefore a peculiar undertaking. The very first predicament that faces this undertaking is related to its definition as a social reality. In defining itself, teaching must draw on culturally accepted concepts. Essentially it defines itself as a process by which people who do not know some knowledge will be made to learn it, and thereby come to know it. Such is the social contract by which the teaching institution, whatever its concrete institutional forms, binds itself to society (Chevallard, 1988, p. 6).

This point of view leads also Chevallard, with Marianna Bosch, to pay attention to tools in/for mathematics doing, learning as well as teaching activity.

4.2 The Tools at the Heart of Mathematical Activity

In the following quotation, Bosch and Chevallard evidence, in contrast to the dominant ‘western cultural axiology’, the importance of tools, materia , ls, visual, audible or tactile, conditioning mathematics activity. Far from being isolated, they constitute a complex of working tools, at the heart of the mathematicians as well as of the students’ activity:

Writing, symbols, words, speech, gestures and graphic objects used in mathematical activity—or what we call, due to their material and perceptible characteristics, ostensive objects—are reflected in very different ways in mathematics education research work, according to the concept of mathematical activity that is implicitly or explicitly assumed by researchers. In the framework of the anthropological approach, ostensive objects, in dialectical interaction with non-ostensive objects (notions, concepts, ideas, etc.), appear as the raw material of tasks, techniques, technologies and theories of the different praxeological organisations (praxeologies) mathematical knowledge is made of. This conceptualisation, which highlights the instrumental value of ostensive objects side by side with their semiotic value, allows us to evidence how specific praxeologies may be affected by generic constraints concerning the ostensive dimension, for example the difficulties of writing or the supposed transparency attributed to verbal discourse. Similarly, the problem of ‘loss of meaning’ which affects certain types of ostensive manipulations is easier to approach in this framework when it is considered with regard to the technological and theoretical needs of the corresponding praxeological organisations.

Our inquiry allows a presentation in more simple terms. It starts from the premise that Western culture establishes, in the range of human practices, a structural opposition between activities considered to be ‘manual’ and activities considered to be ‘intellectual’. This opposition is not neutral. Western cultural axiology prioritises activities of ‘the spirit’ (in English ‘mind’, in Spanish ‘mente’) over the work ‘of the hand’, that is to say, the work that involves the body—with the exception of those ‘body parts’ that are located ‘in the head’ …

It goes without saying that what is regarded as ‘mathematics’ is considered to be of the first type of activities, that is, working ‘with the head’ with notional tools, reasoning, ideas, insights and very little material elements. In fact, the few material instruments used in school mathematics (paper and pencil, blackboard and chalk, ruler and compass, calculator, computer) are generally regarded as simple ‘aids’, sometimes as an indispensable aid but not actually a part of the activity itself. Other objects, if not material at less sensitive (writings, formalisms, graphics, words, speeches, etc.), activated by mathematicians can sometimes play a specific role in the activity, but they are assumed to play the role of ‘signs’, replacing other objects they are supposed to represent.

We now understand that mathematics does not spontaneously appear as an activity in the true (economic) sense of the word; an act or intervention which involves actors and material objects, as instruments which extend the human body to increase its capacity (in strength, accuracy, etc.), or as external objects against which the action is realised. The current conceptualisation of mathematical activity tends to repress the place of material tools which are part of the activity and, if it takes into account particular objects such as discourse, writing and graphs, the focus is not on these objects themselves (and how to handle them), but on what they are supposed to refer to, what they ‘represent’ or ‘signify’, in short, their meaning To do mathematics it is necessary to have words, writings, figures and symbols, but what is important would be beyond words and writing. The dominant point of view in this respect can be considered as idealistic in the sense that it seems to only take into account one aspect of the concrete observable mathematical activity: its signifying function, the production of concepts.

Detaching ourselves from this common vision we suggest considering how mathematical activity is conditioned by the material, visual, audible and tactile instruments it puts into play. It is known that the absence of a concept can block the development of mathematical ‘thinking’, at both the historical level of a community and at the individual levels of a researcher or a student. One may ask to what extent is this absence merely an absence of an idea, a way of ‘thinking’ or ‘conceiving’ the world, or the absence of a complex of work tools (which are, for the most part, material in nature), the availability or absence of which could change in a ‘catastrophic’ manner the performance of the activity. We believe that the didactic analysis of the development of mathematical knowledge—in history as well as in the life of a person or a class—cannot consider this dimension as secondary, assigning it to a purely instrumental function in the construction of concepts”²⁵ (Bosch & Chevallard, 1999, pp. 89–90).

What is a ‘complex of working tools’ is to be explained in more depth. Chevallard distinguishes two kinds of objects: ostensive objects (which can be concretely manipulated), and non-ostensive objects. For example the notation ‘log’, the word ‘logarithm’, as well as the graphical representation of the function logarithm are ostensive objects; the notion of logarithm is a non-ostensive object.

Looking at an example (Fig. 10.5) to illustrate the difference of these objects and their joint mobilisation in the mathematical activity:

Fig. 10.5

A co-activation of ostensive and non-objective objects for transforming an algebraic expression. The technique used for transforming the first expression into the second one needs to mobilise: (a) Several ostensive objects, belonging to different registers: written ones (parentheses, figures, letters…), oral ones (small discourses like ‘x plus 4x equals 5x’), sign languages (for gathering the terms of the same degree (The notion of ostensive objects is close to the notions of representatives introduced by Vergnaud (Sect. 10.3.1). For a discussion on this point, see Chevallard (2005, p. 31))…. (b) Non-ostensive objects guiding the usage of the ostensive ones: the notions of: ‘terms having the same degree’, ‘factorisation’, ‘reminders of order 2’… Chevallard (1995)

For Chevallard, ostensive and non-ostensive objects are intrinsically articulated, at each level of the mathematical activity (expressing a task, using a technique, explaining a technique, integrating a technology in the frame of a theory). In this approach, tools are not relegated to a lower level of mathematical activity: manipulative, concepts and theorems are permanently and jointly involved in this activity.

4.3 Some Convergence and Tensions, and a Productive Atmosphere

These three theoretical frameworks share some fundamental ideas, grounding the French community of mathematics didactics: each teaching and learning analysis starts from the mathematical content of what is to be l earnt; student learning is viewed as an individual and a social activity; and interactions with objects (milieu, tools, instruments…) are viewed as crucial for developing this activity.

However some essential tensions can be distinguished:

• While Brousseau analyses what should be done for teaching mathematics, Chevallard analyses what can be done, or cannot be done, according to institutional constraints; I could illustrate this tension through the pair didactical contract (Brousseau)/social contract (Chevallard); a-didactical situation (Brousseau)/tasks and techniques (Chevallard).

• While Vergnaud situates the knowledge as an individual and social cognitive construct in progress, Chevallard relates to knowledge as a social and historical construct; I could illustrate this tension through the pair conceptual fields (Vergnaud)/praxeologies (Chevallard); schemes (Vergnaud)/techniques (Chevallard).

For the three approaches, addressing the interrelations between tools and knowledge appears as a node of complexity: they a re caught by Brousseau with the notion of milieu, by Vergnaud with the notion of schemes (and the dialectic relationship between gestures and operational invariants), by Chevallard with the notion of ostensive objects (and the dialectic relationship between ostensive and non-ostensive objects).

Noticeably, these interrelations between tools and knowledge motivate several conceptualisations in this productive atmosphere of an emerging scientific emerging field:

• Douady (1987) defined the tool-object dialectic as ‘a cyclic process organising the role of the teacher and the pupils, in which mathematical concepts appear successively as tools for the solution of a problem and as objects with a place in the construction of an organised knowledge’.

• Duval (2006) defines a semiotic register as a system of signs allowing a transformation of representations (Fig. 10.6).

  Fig. 10.6

  

  Classification of four types of registers that can be mobilised in mathematical processes (Duval, 2006, p. 110). He distinguishes four types of semiotic registers (see Fig. 10.6) and analyses the problems of understanding in mathematics learning, from a cognitive point of view, through the difficulties of using these registers. Using semiotic registers means either doing a treatment (i.e. moving from one representation in a given register to another one in the same register, or a conversion (i.e. moving from one register to another one). Doing conversion is often a source of difficulties, because no two distinct semiotic registers will have the same structure

Working with registers is especially important in mathematics as they are the only way for accessing mathematics objects. For Duval, internal representations come from a process of internalisation of external representations. The notions of semiotic registers seem to be close to the Chevallard’s notion of ostensive objects (see Sect. 10.4.2), but where Duval analyses difficulties in terms of structure of semiotic registers, Chevallard relates to the structure of mathematics, i.e. praxeologies²⁶;

• Balacheff (1996), studying the new problems arising from the computational tools, evidences the necessity to take into account a new transposition (after the Chevallard’s didactical transp osition, see Sect. 10.4.1) the computational transposition: ‘Let us call computational transposition the process which leads to the specification and then the implementation of a knowledge model. Computational transposition refers to the work needed to fit the requirements of symbolic representation and computation’.

• Finally, I would like to mention the work Robert and Rogalski (2002), the so-called double approach (ergonomics and didactics), taking into account both the teacher’s didactical goals (aiming to organise students’ activity towards the learning of mathematics) and professional constraints (linked to the Activity theory, see Sect. 9.4). This approach leads to determine five components of teachers’ activity: cognitive, mediative, institutional, social and personal, and to analyse teachers’ activity both in class and out of class (preparing lessons, discussing with colleagues).

These interconnected conceptualisations opened the way for further theoretical developments, both for these theories themselves, and for new frames to emerge:

• Brousseau’s description of a didactical situation as ‘the actions taken by the teacher to set up and maintain an a-didactical situation’ (Sect. 10.3.1) inspired me for introducing the notion of instrumental orchestration (Sect. 15.4).

• The variety o f concepts introduced for taking into account teachers’ resources suggest the need for developing a unified framework; this is discussed in the next chapter.

Intellectual geneses are also individual geneses: Vergnaud was the supervisor of the Ph.D. of Rabardel, who developed the notion of scheme in the context of instrumented activity, and proposed a new approach with regard to artefacts, as we will see in the following section.

5 The Instrumental Approach as a Search of New Theoretical Tools for Analysing Tools in Mathematics Education

In this section, I would like to evidence how the proliferation of very new tools in mathematics education motivated the emergence of a new theoretical framework in France, nourished by the soil of the existing interrelated frameworks we have just described. Then, zooming out, I will relate these ideas to an international survey to show how the French situation resonates with wider international trends.

5.1 The Emergence of the Instrumental Approach

The proliferation of graphic calculators imported in classrooms by students themselves aroused a lot of debates in society, and analyses in the community of mathematics education (cf. Sect. 13.2). The study of students’ mathematical activity, using such material, brings up (what appears as) new phenomena, evidencing the influence of tools on conceptualisation. In my Ph.D. dedicated to this question (Trouche, 1997), I pick up several such phenomena, among these the influence of images on conceptualisation (see Fig. 10.7).

Fig. 10.7

A standard view of a function on a graphic calculator screen (Guin & Trouche, 1999, p. 198)

This phenomenon and other similar examples (Guin & Trouche, 1999) led me to look for new concepts, taking into account the potential of tools for mathematics education. In a living scientific community, a new approach rarely emerges from the initiative of a single researcher: it emerges for answering to practical needs. Noticeably, at a French summer school on the integration of complex calculators in 1997, Michèle Artigue and myself (Artigue, 1997; Trouche, 1997), independently, borrowed the same essential concepts of artefacts and instruments, instrumentation and instrumentalisation (Vérillon & Rabardel, 1995), for our respective communications, installing the first milestones of what will become ‘the instrumental approach of mathematics didactics’. Guin and Trouche (1999) describe this first attempt for developing a new frame:

Verillon and Rabardel’s studies focusing on learning processes involving instruments in the area of cognitive ergonomy are based on this idea. If cognition evolves through interaction with the environment, accommodating to artefacts may have an effect on cognitive development, knowledge construction and processing, and the nature itself of the knowledge generated (Vérillon & Rabardel, 1995, p. 77). They suggest models and concepts to analyse the instrumented activity of children confronted with tasks involving artefacts.

Verillon and Rabardel stress the difference between the artefact (a material object) and the instrument as a psychological construct : “The instrument does not exist in itself, it becomes an instrument when the subject has been able to appropriate it for himself and has integrated it with his activity” (Vérillon & Rabardel, 1995, p. 84). The subject has to develop the instrumental genesis and efficient procedures in order to manipulate the artefact. During this interaction process, he or she acquires knowledge, which may lead to a different use of it. Similarly, the specific features of instrumented activity are specified: firstly, the constraints inherent to artefacts; secondly, the resources artefacts afford for action; and finally, the procedures linked to the use of artefacts. The subject is faced with constraints imposed by the artefact to identify, understand and manage in the course of this action: some constraints are relative to the transformations this action allows and to the way they are produced. Others imply, more or less explicitly, a prestructuration of the user’s action.

The reorganisation of the activity resulting from the introduction of instruments also affords new possibilities of action which are offered to the user; they may provide new conditions and new means for organising action. Thus, it can be argued that, because the instrument is not given but must be worked out by the subject, the educational objectives stated above require the analysis of the instrumented activity of artefacts involved in the learning processes.

It seems quite natural that mathematics education has borrowed from cognitive ergonomics ways of thinking appropriation processes of artefacts. These concepts basically distinguish on one hand what was given to the subject (artefacts, historically and culturally situated) and on the other hand, what was built by the subject (the instruments) during its finalized activity finalized. To be taken into account the long and complex process (the genesis) supporting the construction, combination of two developments, instrumentation and instrumentalisation .

What appears as essential, in this preliminary construction:

• The distinction between an artefact (a product of human activity,²⁷ that a subject can appropriate for performing a given task) and an instrument (resulting from this appropriation process).

• The distinction of two processes, structuring the instrumental genesis, from an artefact to an instrument: a process of instrumentation, directed from the artefact towards the subject, and a process of instrumentalisation, directed from the subject towards the artefact.

This distinction leads to a careful analysis of artefacts, their constraints and potentialities (to be related to affordances, see Chap. 7): in which way does the computational transposition (Sect. 10.4.3) transform the knowledge involved? What are the gestures favoured by the artefacts and in which way do they influence the student’s knowledge in progress? The discussion was deepened through discussions in the French community of mathematics education, particularly during the summer school of this community in 1999, through a lecture by Rabardel (2000). This allowed scholars to establish links between this approach and other existing approaches (Fig. 10.8).

Fig. 10.8

A representation of an instrumental genesis (Trouche, 2004, p. 289)

This link was made easier, as Rabardel himself situates his work in the thread of Vergnaud (Sect. 10.3.2), framed by the couple scheme/situation. Defining an instrument as a mixed entity with two components (artefact and schemes) leads to describe the instrumented action schemes, and the operational invariants involved in them, and to think the dialectics between technical and conceptual work (Artigue, 2002). We had also to re-think the design of situations, particularly a-didactical ones (Sect. 10.3.1) according to the constraints of the artefact and the targeted knowledge. At least, we had to take into account the institutions in which the artefacts were integrated, and, beyond experimental classes, re-think the conditions for an integration of artefacts in ordinary settings. In ordinary settings, artefacts rarely live alone: several artefacts are engaged in students’ activity, making necessary to study the system of instruments they develop.

Finally, the instrumental approach appears as a new frame fully integrated in a network of theoretical approaches, marked by the theory of didactical situations, the anthropological approach of didactics and the theory of the conceptual fields. These interrelations appeared also in the theoretical developments arising in the French community at the end of the last century, giving more importance to the teacher’s role, even in the a-didactical situation: Chevallard (1997) elaborates on the ‘familière et problématique’ figure of the teacher, Margolinas (2002), working in the frame of Brousseau’s theory, analyses the different situations and milieus of a teacher, and Trouche proposed the concept of orchestration (Sect. 15.3) for modelling the teacher’s role in rich technological environments.

But a theoretical approach never develops in an isolated national frame. The instrumental approach was discussed in the Third Computer Algebra in Mathematics Education Symposium held in Reims in 2003, and this discussion gave birth to two papers (Hoyles et al., 2004; Trouche, 2004). This led a number of scholars to give more importance to the instrumentalisation process, i.e. to the creative power of students developing their own instruments from the available artefacts. After focusing on the importance of artefacts as supports of activity and mediators of knowledge (i.e. the instrumentation processes), it implies to rebalance the relationships constituting the instrumental geneses. It resonates with an Engeström’s remark, revisiting the work of Vygotsky and his colleagues: ‘it seems to be all but forgotten that the early studies led by Vygotsky, Leont’ev, and Luria not only examined the role of artifacts as mediators of cognition, but was also interested in how children created artifacts of their own to facilitate their performance’ (Engeström et al., 1999, p. 26). This discussion was further developed in a journal issue dedicated to the work of Celia Hoyles (Trouche & Drijvers, 2014).

After having grounded the instrumental approach in the ‘French field’, I would like to link, beyond the local interaction during the CAME symposium, French and international trends in the field of mathematics education, and that is the purpose of the following section.

5.2 Zoom Out, Where the National Characteristics Join International Trends (and Vice Versa)

In this section I draw on a meta-study of a comprehensive corpus of publications, driven by a French team (Lagrange et al., 2003), supported by the French Ministry of research. This study addresses the field of research and innovation in the world-wide field of the integration of ICT in mathematics education from 1994 to 1998. In contrast with classical meta-studies,²⁸ this study did not focus only on the findings of publications, but considered also characteristics like the questions addressed, approaches, cognitive theoretical background, etc. The authors expected that ‘analysing this material would help to identify as many as possible aspects of the complexity of the integration, some of them widely addressed and others less considered by the literature’ (p. 241). I draw on this study in order to have a means for comparing the evolutions in the French and the international community of mathematics education in the field of information and communication technology (ICT).

The study proceeded in four stages. The first stage consisted in gathering all the references related to ICT and mathematics education in a variety of international sources (the Zentralblatt für Didaktik der Mathematik database, four international journals on mathematical education, seven international journals on computers for mathematics learning, books with chapters on technology and mathematics education, etc.) as well as French works (professional and research journals, dissertations, research and official reports, etc.); this resulted in a corpus of 662 published works. The second stage consisted in a first sorting of this large corpus, distinguishing the type of analysis (presentation of a technological product, experimentation-innovation, research report or general reflection), the mathematical field, the type of technology, and the country of the first author. Figure 10.9 shows that: research publications were not in a majority. The literature about ICT includes classroom innovation and pure speculation as well as research studies; a number of papers did not specify a mathematical field, focusing on the support of technology in ‘general’ mathematical learning.

Fig. 10.9

Elements of description of the first corpus (Lagrange et al., 2003, p. 242)

The third stage consists in a reduction of the corpus, removing papers which had insufficient substance (for example: technical descriptions , or simple description of an innovative classroom activity), whilst keeping a large enough selection to respect the diversity of approaches and to avoid biases, resulting in a corpus of 79 papers. The fourth stage was dedicated to an in-depth analysis of this corpus. For each of these papers, one participant of the project established a detailed review showing the following characteristics: problematic,²⁹ theoretical background, details of the questions addressed, methodology used, specific findings and an appreciation by the reviewer. This analysis led the authors to distinguish seven different orientations for characterising these papers, named ‘dimensions’. For each of these dimensions, a set of indicators was designed, resulting into a grid (Fig. 10.10).

Fig. 10.10

The dimensions of analysis and their indicators (Lagrange et al., 2003, p. 247)

Using a statistical procedure based on a cluster analysis, we obtained clusters of papers sharing specific indicators. The procedure also selected one or two papers at the centre of each cluster, which were the publications statistically best represented. In each cluster, French and international papers were represented.

Globally, the 1994–1998 literature appeared to restrict its analysis to potentialities of ICT itself (easier and more varied representations, new aspects of mathematical knowledge, etc.) rather than questions raised by its insertion into the ‘ordinary’ mathematics teaching. The general picture of ICT in the teaching and learning of mathematics emerging from this analysis is that of a field where publications about innovative use or new tools and applications dominated. Research studies differed with regard to the way they considered the potentialities of ICT, but they converged in a focus on the student, the cognitive role of ICT (third dimension) and in an emphasis on epistemological and semiotic aspects (first and second dimensions). The other dimensions appeared as emergent:

• The institutional dimension gathers papers focusing on the difficult viability of technology in schools. Papers with a pioneer spirit started from today’s difficulties to motivate the use of tomorrow’s technology, while others looked for reasons in more permanent characteristics of technology and of the educational institutions

• The instrumental dimension gathers papers analysing constraints, evidencing complexity of appropriation processes, conjecturing relationships between gestures and conceptualisation

• The situational dimension gathers a few papers taking into account new economy of problem solving, and new situations to be though for integrating technologies; changes to be made in the curriculum, and the need for thinking about the use of technology together with other learning situations

• Very few papers considered the teacher dimension

It is amazing to realise that the ‘dimensions’ distinguished by the French team were close to the theoretical frameworks they were close to: the cognitive dimension with Vergnaud, the institutional dimension with Chevallard, the instrumental dimension (the ‘French instrumental approach’) and the situational dimension with Brousseau. But this shows that the French papers were not isolated: they share in each of these dimensions elements of analyses, and through the cross references, they feed other theoretical frames as well as being fed by them.

It is also interesting to analyse evolutions along the period covered by the survey. The authors ‘discerned a long-term motion towards awareness of a more complex integration and the subsequent necessity of new dimensions of analysis. It is confirmed by what we know of the institutional and instrumental dimensions in today’s research studies and of the emerging reflections on the teacher’ (p. 260). They noticed a growing interest for the Brousseau theory of situations (Sect. 10.3.1) which could ‘help when looking in depth into changes in the learning situations and when showing precisely what is at stake in these new situations’ (Sutherland & Balacheff, 1999, p. 259). They note that ‘elements of evolution appear in the convergence towards dialectical approaches to issues like visualisation and contextualisation. These approaches contribute to the development of new dimensions by helping to better consider the institutional contextualisation of knowledge as well as the schemes attached to the use of a technological tool in their instrumental dimension’ (p. 259). Evoking Mariotti (2002), the authors provide evidence that, just after this period, the instrumental dimension experienced a strong development. Regarding the complexity of teaching and learning situations with ICT, researchers became more cautious. The authors note that ‘Interesting research studies start from the observation of teachers struggling to integrate ICT into the real teaching’ (Monaghan, 2001, p. 259).

Thus the period 1992–1998 appeared as a period of transition, from a naive point of view on integration to a more balanced point of view, to which the instrumental approach gives means of expression: for preparing the ICMI study on mathematics education and technology, the introductory document (Hoyles & Lagrange, 2006) proposed as one of the key publications, the book by (Guin et al., 2005) which constitutes precisely both a presentation and a discussion of this approach at an international level.

6 Elements for Discussion

In this chapter, I have tried to explain the exceptional conjunction of phenomena leading, from 1970, to a strong development of theoretical frameworks in mathematics didactics in France. This productivity had been acknowledged by international distinctions (the Felix Klein medal to Guy Brousseau and Michèle Artigue, and the Hans Freudenthal medal to Yves Chevallard). Beyond this national context, we have shown that these constructs draw their sap from various theoretical traditions, and resonate with international trends.

The instrumental approach appears as a theoretical construction, starting from taking into account of ‘artefacts for doing mathematics’, and developing thanks to the theoretical ground provided by the other frames. In the field of ICT in mathematics education, interaction between the French community and the international one have strongly developed, due probably to the novelty of phenomena arising with the rapid evolution of technologies.

The instrumental approach, in the thread of these interactions, has deeply evolved:

• The evolution can be related to a concept, for example the concept of orchestration , enriched by Drijvers et al. (2010), see also Chap. 15.

• The evolution can also touch the relationships between concepts: the upheavals of digital resources and of Internet led in 2007 to expand the vision beyond technology, to the set of resources that sustain the activities of teachers, including textbooks (Sträßer, 2009). It leads Gueudet and Trouche (2009, see also chap. 15) to substitute the dialectic resources-documents to the dialectic artefact-instrument. This new model has resulted in the documentational approach that also induces other openings: considering the documentary work of the teacher in a variety of places and long time involving new methodological developments; new concepts are emerging, for example the notion of the teacher resource system , paving the way for new theoretical fruitful interactions (for example, for analysing the structure of the resource system, a successful track seems to be the study in terms of praxeologies, Sect. 10.4.1).

The interactions between frameworks allow more generally to deepen the concepts at stake. It was the case for the interactions between: the instrumental approach and the semiotic approach (Maschietto & Trouche, 2010); the instrumental approach and the ontological semiotics approach (Drijvers, Godino, Font, & Trouche, 2012); the documentational approach and the double approach (Sect. 10.4.3, Gueudet & Vandebrouck, 2011). This work was, at an international level, theorised as a set of possible strategies to grow the theories themselves (Prediger, Arzarello, Bosch, & Lenfant, 2008), see Chap. 11.

The documentational approach has developed a methodology of reflective investigation (Chap. 15), involving the gaze of the teacher on its own resources, and providing new tools to the researcher for analysing teachers’ work. We could all struggle through this self-confrontation with our own resources in research. The representation that I have, at this point, of my own resource system is pretty close to that of the worker Demarcy, looking at his working place at the factory, as ‘heterogeneous supports, improvised vices for stalling pieces…’ (Linhart, 1978). What is true for a researcher is also probably for a community. Research communities in mathematics education, especially the part thereof that look to technology have long been committed to this cross questioning of theoretical tools, far beyond the French research communities (Trouche & Drijvers, 2014). A history far from being finished…