Three Perspectives on the Issue of Theoretical Diversity

Abstract

Since the birth of the didactics of mathematics in the 1970s, the research community has aimed to build theories that may be used as models for studying phenomena in the teaching and learning of mathematics, within a milieu designed for their teaching. The creativity of researchers giving birth to many theories has created certain problems in the community. Over the last decade, researchers have shared thoughts and views on the relationships between these theories. This chapter proposes three different perspectives on this topic. The first examines the richness of a multidimensional approach based on the mobilisation and networking of various well-identified theories, enabling a segmentation of reality that is well suited to the study of didactic phenomena. The second perspective defends a possible methodology for reducing theoretical diversity. The third perspective examines a social viewpoint of the multiplicity of theories in the didactics of mathematics and the search for connections.

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4.1 Introduction

Since the birth of the didactics of mathematics in the 1970s, the research community has aimed to build theories that may be used as models for studying phenomena in the teaching and learning of mathematics, within a milieu designed for their teaching. A survey of the literature reveals the development and multiplication, both in France and abroad, of a great number of theories for appreciating complex and multifarious phenomena in many different cultures, examined according to a variety of inputs and levels of analysis. These theories involve an interplay of different didactic concepts and tools. The creativity of researchers has created certain problems in the community, such as ‘internal difficulties relating to the communication and capitalisation of knowledge, and external difficulties when holding discussions with other communities, explaining the state of the art on a specific topic to non-specialists, or guiding didactic efforts in a well-argued manner’ (Artigue 2009, p. 307). Over the last decade, researchers have shared thoughts and views on the relationships between these theories at such events as the CERME congress¹ and in European projects such as Technology Enhanced Learning in Mathematics (TELMA) and ReMath. This is clearly reflected in the publication of special journal issues (Morgan and Kanes 2014; Prediger et al. 2008a; Skott et al. 2013) and books such as the Networking of Theories in Mathematics Education (Bikner-Ahsbahs and Prediger 2014). Michèle Artigue’s active participation in all these endeavours warrants the devotion of an entire chapter to this theme.

This chapter proposes three different perspectives on this topic, expressed in the first person. The first of these examines the richness of a multidimensional approach based on the mobilisation and networking of various well-identified theories, enabling a segmentation of reality that is suited to the study of didactic phenomena. To do so, Grugeon-Allys highlights the identity and limitations of didactic theories developed in France during the early days of research in the didactics in mathematics, along with their functionality and complementarity. The second perspective defends a possible methodology for reducing theoretical diversity. Godino does this by drawing on the theory known as EOS (Entidades primarias de la ontología y epistemología). The third perspective is a contribution by Castela, who examines a social viewpoint of the multiplicity of theories in the didactics of mathematics and the search for connections. After considering networking based on the notion of praxeology, she proposes some new perspectives by borrowing from the anthropological theory of didactics and the field theory applied to science respectively.

4.2 The Richness of Didactic Theories and Their Networking for a Multidimensional Approach of Didactic Phenomena

I will begin with a point of view developed by Artigue (2009), whereby ‘a given theory cannot claim to encompass everything and explain everything’ and ‘the coherence and strength of a theory lies primarily in what it relinquishes’ (Ibidem, p. 309). Artigue singles out two guiding principles in her reflections:

• ‘the consideration of the relations between theoretical frameworks cannot be achieved without identifying and respecting their respective coherencies and limitations’,

• ‘[one should] examine theories and their development in terms of functionalities, tracing theoretical objects to the needs that these objects fulfill or at least attempt to fulfill’ (Artigue 2009, pp. 309–310).

In order to determine the identity of theories and their boundaries, Radford (2008) compares the systems of principles, the methodologies and the types of research questions on which they are based.² This constitutes a means of deepening the understanding of theories in relation to their research paradigms, and investigating their limitations and potential connections. It defines the boundary of a theory as ‘the “edge” that a theory cannot cross without a substantial loss of its own identity. (…) behind such an edge, the theory conflicts with its own principles’ (Radford 2008, p. 323). To determine the types of connections between theories, Radford studies the structure and aim of the connection. He considers the theory-networking scale (Prediger et al. 2008b) that distinguishes between several connection types (Ibidem, p. 318). ‘Comparing’ consists of searching for similarities and differences, whereas ‘contrasting’ has to do with emphasising differences. ‘Coordinating theories’ amounts to selecting coherent elements from different theories in order to investigate certain research problems. ‘Combining theories’ tends to involve juxtaposition.

To take this point of view further, I shall hypothesise that a multidimensional approach based on the mobilisation and networking of well-identified theories—each of which uses a particular conceptual filter to segment reality and mobilise its objects of study—can be conducive to the study of complex didactic phenomena. The connection between theories can take several forms and involve varying degrees of integration, culminating with the local level.

To argue in favour of the contributions of a multidimensional approach, I will examine the salient features of the identity and limitations of the founding theories that emerged in the field of research in the didactics of mathematics in France from the 1970s onwards.

4.2.1 Main Theories: The Theory of Conceptual Fields, the Theory of Didactic Situations and the Anthropological Theory of Didactics

These theories aim to study, describe and explain the processes of the teaching and learning of mathematics by assuming that the very nature of mathematical savoir and practices influences these processes. They share the following principles: the fundamental role of mathematical savoir and its epistemology, and the role of problem solving for the learning and teaching of mathematics. These theories developed according to the historical, scientific and cultural conditions of the 1970s as well as the individual background of the researchers, in response to research questions and methodological choices that led the researchers to distinguish their theory from other theories. I will not attempt to present in a few words the fundamental ideas or the concepts developed by each theory, as this would be impossible. I only wish to shed light on certain elements that lie at the core of each of the theories, which constitute their strength and delineate the boundaries between them, at a given stage in the development of research in the didactics of mathematics.

4.2.1.1 A Theory Centred on the Modeling of Knowledge Acquisition

The Theory of Conceptual Fields (TCF) developed by Vergnaud is centred on the modeling of knowledge acquisition. Vergnaud endeavours to bring research in developmental cognitive psychology closer to issues pertaining to teaching and academic learning. Two major concerns form the backbone of TCF.

First, Vergnaud makes a strong hypothesis: that of a link between the creation of connaissance and the structure of mathematical savoir. He defends the idea that it is essential to study mathematical concepts in relation to the situations that will enable their conceptualisation. To him, one of the major challenges in research in didactics in the 1980s was to characterise and classify problems, in the psychological sense of the term,³ that give a concept its meaning and function. Thus, Vergnaud (1990) models a ‘concept’ by means of a ‘triplet of three sets (S, I, L): the set S of situations that give meaning to the concept, the set I of invariants that form the basis of the operationality of schemes (the signified, or signifié), and the set L of linguistic and non-linguistic forms that enable the symbolic representation of the concept, its properties, the situations and the handling procedures (the signifier, or signifiant) (Ibidem, p. 61, personal translation). The conceptualisation of a given concept, built by a given student, at a given moment, corresponds to a set of situations in which the concept is applied, the representations that enable it to be represented, the invariants, rules of action and acting attributes that appear in the course of the associated activity through procedures, as well as the perceived scope of validity; hence the disparities between students’ conceptions and the targeted concepts that are taught.

This point of view leads Vergnaud to consider that one cannot comprehend the development of a concept without setting it in the context of a long time period in a much broader system that is directly linked to the mathematical content of the problems⁴—a conceptual field. A conceptual field encompasses a set of situations that is progressively mastered by reference to several concepts and the set of concepts that contribute to the mastery of the situations. To understand how a student develops and adapts a concept requires a segmentation of mathematical savoir into relatively large domains in order to study the long-term evolution of such processes through a set of relatively diverse situations.

In TCF, students are psychological subjects.⁵ The questions that are studied are connected to the object of study: students’ conceptualisation processes over the long term. The methodology is anchored in cognitive psychology approaches, centred on the study of schemas developed by students during problem solving, which calls for a microscopic scale of analysis. Vergnaud focuses—from the student’s and teacher’s perspective—on the role of the structure of problems and carries out classifications. But at the time of writing, he was not interested in the conditions for putting situations into practice in the classroom, the role of the teacher in the management of situations and interactions, or even the constraints to be taken into account in the educational system.

4.2.1.2 A Theory Centred on the Conditions of the Operation of Situations in an Educational System

In the theory of didactic situations (TDS), Brousseau (1986, 1998) develops conceptual tools for understanding what is at play in the classroom from a mathematical standpoint by focusing the study around the didactic situation, which is initially defined as a system of relationships between some students, a teacher and a mathematical savoir. The object of study in TDS corresponds to the conditions in which a teaching system may bring about optimal development of students’ connaissances in relation to an existing savoir within an educational system. TDS studies didactic phenomena that involve interactions between the savoir, the students and the teacher as a whole. The didactic modelling of situations consists first and foremost in matching, to a given savoir, a minimal class of adidactic situations that make this connaissance appear to be the optimal and independent means of solution in these situations with regard to the milieu. This class of situations can be created by a game involving the cognitive variables⁶ and the didactic variables of a fundamental situation. The ‘Race to 20’ is a prototypical example of this for Euclidian division. Brousseau (1986) defines conceptual tools for understanding the adidactic situations: situations of action, formulation and validation, as well as devolution and institutionalisation. Of utmost importance to him is the belief that meaningful learning of mathematics cannot be achieved if there is too much reliance on the teacher for problem solving. This explains the importance placed on the concepts of devolution and milieu, and on the duality between didactic and adidactic situations.

4.2.1.3 A Theory Focused on Institutions

The Anthropological Theory of Didactics (ATD) shifts the focus on research of didactic transposition, the savoir savant or scholarly knowledge questions and situations towards the institutions in which the situations ‘live’. It studies new and broader questions. Indeed, Chevallard adopts a perspective of epistemological and institutional emancipation in relation to the institutions where the objects of savoir studied the didactics of mathematics ‘live’. A savoir has not always existed: it is the result of human activities and depends on their position and function, according to the place, society and time period. In the educational system, the savoir to be taught should not be considered to be transparent as it is related to the institution in which it is taught. Chevallard (1985) distinguishes, through the process produced by mathematicians from the savoir à enseigner or knowledge to be taught in an institution, the savoir enseigné or knowledge taught by the teacher, and the savoir appris or knowledge learnt by the student. A student’s learning is also determined by the institution where he or she is learning. ATD is a tool for modelling and analysing students’ and teachers’ activities in teaching institutions that enables an appreciation of the implicit constraints (assujettissements) at work. Chevallard (1992, 1999) develops the notion of praxeology to describe the creation and evolution of objects of savoir within an institution, as well as the institutional and personal relationships to these objects. The notion of praxeology encompasses on the one hand the types of tasks and the techniques for accomplishing them, the praxis, and on the other hand the discourse known as technology that justifies a technique and renders it intelligible along with the theory that, in turn, justifies this technology and renders it comprehensible, the logos. This notion provides a tool for analysing the structure of teaching in different institutions, and is particularly useful for understanding the transitions between two institutions in the different stages of didactic transposition. One of the most crucial contributions of ATD is the definition, prior to any study, of the reference epistemological model relating to a given savoir that forms the basis for the analysis of transposition phenomena. As Brousseau points out, ‘this “anthropological” approach is perfectly in line with the theory of situations and completes it. It allows more direct access to a certain number of problems, especially those pertaining to macrodidactics and the relation to savoir’ (Brousseau 2003).

Having reached the end of this first section, I would like to emphasise, apart from the usual issues, the following points based on the fields of action prioritised in these theories: the modelling of students’ acquisition of connaissances in TCF, the modelling of adidactic situations as an optimal means of access to savoir in TDS, the creation and evolution of a savoir in different institutions, and the relationships to this savoir in ATD. These choices give rise to distinct methodologies at different levels of analysis, each with its functionalities—but also its limitations—for addressing new research questions.

4.2.2 Functionalities of Theories

I will now examine the importance and richness of these theories for studying new research questions and take into account different aspects of the didactic phenomena involved by distinguishing between dimensions at different levels of analysis. For any research and segmentation, the consideration of different dimensions can lead to subsegments, each of which is associated with a theoretical framework and an appropriate methodology. I will base my investigation on research pertaining to different ways of connecting elements (Fig. 3.1) derived from TCF, TDS and ATD.

4.2.2.1 Dynamics of a Multidimensional Approach: Complementarity Between Theories and Evolution of Research Questions

Here, I will illustrate the dynamics of a multidimensional approach by contrasting the differences between two theories, then coordinating them, according to two main inputs: the student and the institution.

The investigation focuses on the transition problems and notably the institutional discontinuities that are often involved in these transitions. We consider a study that examines the difficulties faced by 16- and 17-year-old students from ‘lycées professionnels (LP)’, who are among the best academically and yet fail in specially tailored adaptation classes aimed at preparing them for further education in ‘lycées technologiques (LT)’ (Grugeon 1997). One savoir lies at the core of this failure: elementary algebra. The frequent explanations for this failure are of a cognitive kind: the students’ difficulties stem from their standard in knowledge of mathematics. A cognitive approach serves this type of reasoning and its associated conclusions. Adopting an anthropological approach enables us to overcome such negativity, view the cognitive through the institutional filter and put the problem being studied in a wider perspective encompassing all the transition problems. Thus, the contrasting of these two approaches encourages us to explore a new hypothesis: as the two institutions developed different institutional relationships to algebraic objects, the difficulties observed could be explained by the inadequacy of the personal relationships developed under the influence of the first institution, with respect to the expectations of the second institution. In order to carry out the investigation, I considered it necessary to define a ‘sort of reference, independent of the institutions involved, yet positioned within their field of action (…), then build a (multidimensional) analysis framework based on this definition for analysing the institutional and personal relations’ (Grugeon 1997, p. 170). This reference forms the basis of an epistemological analysis of the various relationships to algebra, through the types of problems in the algebraic domain, the algebraic objects and their properties, as well as the modes of representation used to solve them: it has been developed from a summary of research work carried out in the didactics of algebra (Chevallard 1985; Kieran 2007).

This study identified discontinuities in the institutions’ programmes, which a more superficial examination would not only have failed to find, but which could also have led to misapprehensions. The analysis reveals that the dominant institutional relationship to algebra in LP is mainly structured around the use of formulae (for calculating rates or loan repayments), for example for writing equations, whereas in LT, it is focused on equation writing, equations and functions.

The results of this research clearly demonstrate that ‘the relevance and importance of a theoretical construction are closely tied to the manner in which it forces us to change our filters, rendering visible what was previously invisible, forcing us to question our spontaneous interpretations, making apparently erratic, incoherent or counterproductive behaviour rational and understandable, in a word, changing our vision of the world and pointing out where we should direct our energy’ (Artigue 2009, p. 320).

4.2.2.2 Scope and Adaptability of Theoretical Frameworks

I will now deal with the scope of theories and their adaptability, especially with regard to the conditions in which TDS may be used in the study of regular sessions. This involves a ‘coordinating’-type connection between two theories that will be illustrated based on research into the issues surrounding ‘regular’ classes (Perrin-Glorian 1999; Perrin-Glorian and Hersant 2003). I will refer to certain points that were featured in Perrin-Glorian’s contribution at the Artigue Colloquium.⁷ Perrin-Glorian formulated the general problem in these terms: ‘how to define situation when observing a sequence of sessions on proportionality in 6e [i.e. the first year of French secondary school] that one has not prepared?’, unlike what occurs in didactic engineering. Perrin-Glorian and Hersant (2003) propose reconstructing such situations after observing the sessions and analysing them a priori. In order to access the dynamics of the teaching occurring in regular classes, the implemented methodology segments reality by conducting successive zoom-ins that allow an appreciation of different scales of analysis, and aims to link the various levels involved: the microdidactic level of the situation and interactions, the local level of the teacher’s long-term plan, and the macro level of the savoir and the institutional constraints based on programmes and manuals. This segmentation has led Perrin-Glorian and Hersant to call upon TDS concepts and tools at the local or microdidactic level, and ATD concepts and tools at the macroscopic level. ATD is used to analyse savoir and institutional constraints, by making a distinction between the various institutions, based on an epistemological construction of the savoir in question. At the local level of the session, TDS enables us to study the situations that come into play in the sessions considered to be of significance with regard to the didactic aim of the series. The definition of a situation as a game is primarily dependent on three components and their function in the situation: the didactic aim (targeted new connaissances), the material milieu that is or is not set up, the problem and the rules of the game (especially on how to win). The didactic contract is what enables us to interpret the game, whether from the perspective of the teacher or the students, by distinguishing between the milieu-related conditions that can be changed and those that cannot be changed by virtue of being connected to institutional constraints. The teacher’s game consists of organising and regulating the student’s game, and in leading the student to identify and formulate the connaissances that are necessary to win. At the microdidactic level, the analysis of significant episodes relies on the examination of feedback from the milieu and the evolution of the didactic contract. Here, Perrin-Glorian points out the distinction between a psychological subject and a social subject. Indeed, the study of the adidactic situation and the anticipation of procedures for solving it take into account the cognitive dimension of the epistemic subject. The social dimension is brought in at the microscopic level to study the interactions within the classroom, and at the macroscopic level to place didactic issues within social issues. Perrin-Glorian and Hersant (2003) were led to define new dimensions for analysing didactic contract, notably the area of mathematics concerned, the status of the savoir (degree of institutionalisation and familiarity) to the students with regard to tool–object dialectic (Douady 1986), and the properties of the milieu (possibilities of feedback that are open to interpretation by the students).

From this we may observe yet another illustration of the importance, richness and scope of the theories that highlights, at different levels, the dynamics of the coordination between TDS and ATD for dealing with a new research question: ‘TDS manages the local level, especially matters pertaining to didactic engineering, whereas ATD manages the global level and the study of institutional relations by studying official documents, manuals and various teaching resources’ (Artigue 2009, p. 314). Researchers use the complementarities between these theories as levers for conducting research on appropriate segmentations.

4.2.2.3 Different Cultures and Common Sensitivities: Contributions of a Comparison Between Theories

In this section, I would like to demonstrate how the contrasting of theories can enrich the study of a given theme, by bringing in different approaches and varied perspectives. I will refer to the Theory of Semiotic Mediation (TSM) (Bartolini Bussi and Mariotti 2008), which was developed in Italy based on Activity theory. I will examine the respective sensitivities of TDS and TSM to the sociocultural dimension of learning and to its semiotic dimension. Are they different? If the answer is yes, how do the didactic analyses and choices differ and do they enhance the relationship with these dimensions?

To do so, I will draw on certain points discussed in Mariotti’s talk at the Artigue Colloquium. Mariotti refers to crossover experiments conducted between French researchers and Italian researchers in the ReMath project. Didactic analyses and choices differ significantly depending on whether the sociocultural sensitivity leans towards TDS or TSM. TDS is based on the a priori analysis of situations, students’ actions and feedback from the milieu, with students being considered as epistemic subjects (cf. Sect. 4.2.1.2). A posteriori analysis examines the realisation of situations. The focus is, on the one hand, on the relationships between the connaissances that are called upon in the situation as well as the interaction between the actions and feedback of the milieu, and on the other hand, on the institutional techniques and technologies brought into play. By adopting the TSM perspective (Bartolini Bussi and Mariotti 2008), the priority is not to analyse either the productivity of the scenarios built around tools perceived to be sources of feedback, or the institutional constraints. Instead, the focus is on the semiotic mediations that take place in the classroom and the teacher’s role in making them effective; the analysis concerns the indicators given by students that point to the accomplishment of the task, and the way in which these indicators develop.

There is another difference, which has to do with the mediating role of the teacher. TDS mainly focuses on the teacher’s role in engaging students in mathematical activity (devolution) and in formulating and decontextualising, in terms of savoir, the connaissances developed by students and targeted in the situation during institutionalisation. From a TSM perspective, the critical issue is to show how personal significations are connected to the mathematical significations of the targeted and culturally established mathematical savoir. The teacher’s role appears to be essential for enabling students to link the connaissances called upon in the situations to the savoir; this manifests itself in complex interactions occurring over time spans that far exceed those of the institutionalisation phases. This analysis points out the difference in the importance that the TDS and TSM approaches bring to social activities.

In summary, the research work presented in this first perspective illustrates the functionality of theories in their networking for studying new research questions, according to various objectives and levels of integration (Fig. 4.1): a contrasting connection for developing a problem and the associated methodology (Sect. 4.2.2.1), and for revealing the different aspects of an object of study based on the different underlying principles of theories (Sect. 4.2.2.3); a coordination of theories at different levels of analysis (Sect. 4.2.2.2); and a locally integrated construction in the last example born of the necessity to define new concepts for studying new research questions. The next perspective deals with the topic of unification, which will be developed and argued by Juan Godino in contrast to the option of connecting theories.

Fig. 4.1

Landscape of strategies for connecting theoretical approaches (Prediger et al. 2008b, p. 170)

4.3 Hybridisation of Theories: The Case of the onto-Semiotic Approach

As we indicated above, the articulation of theoretical frameworks (networking theories) is receiving special attention. Several authors (Prediger et al. 2008b; Radford 2008; Artigue et al. 2009) consider that the coexistence of the various theories explaining the phenomena of a discipline, such as mathematics education, is to some extent inevitable and enriching, but it can also be a hindrance to its consolidation as a scientific field. As already analysed in the previous section (see Fig. 4.1), Prediger et al. (2008b) describe different strategies and methods for articulating theories, which range from ignoring each other, to their global unification.

Personally I believe that progress in any discipline, particularly in mathematics education, requires that we consider the “Occam’s razor” or parsimony, economy or succinctness principle, used in logic and problem solving. This principle states that among competing hypotheses, those with fewer assumptions are preferable; in other words, the simplest explanation is usually the best. Applying Occam’s razor to mathematics education research justifies the efforts made in the field to compare, articulate and unify theories.

But it is also necessary to consider the phrase attributed to Einstein: “Everything should be kept as simple as possible, but no more,” which can be regarded as a formulation of the “Chatton’s anti-razor” principle: “If an explanation does not satisfactorily determine the truth of a proposition, and is sure that it is true, you should find another explanation”. The multiplicity of theories in mathematics education is a consequence of the implicit application of Chatton’s anti-razor, while efforts of comparison, coordination and unification of theories results from the also implicit application of Occam’s razor. It is important to acknowledge that both principles do not conflict and that a rational position on the multiplicity of theories should be to explore the synergy that exists between them.

4.3.1 Issues in Theories Unification

In this section I argue the necessity and usefulness of articulating (internal and local) mathematics education theories, using the example of four well known theories in “French didactic”: TCF (Theory of Conceptual Fields), Theory of Semiotic Representation Registers (TSRR, Duval 1995, 1996), TDS (Theory of Didactical Situation), and ATD (Anthropological Theory of Didactic). The first two theories focus their attention on the cognitive dimension (individual or subjective knowledge) while the last two basically study the epistemic dimension (institutional or objective knowledge). I, however, believe that the consolidation of mathematics education as a techno-scientific discipline should tackle issues such as:

• What are the problems, principles and methodologies addressed and used in each framework?

• What redundancies exist in the tools used by these frameworks? Are they incompatible?

• Can the cognitive and epistemic tools of different frameworks synergistically coexist?

• Is it useful to construct a theoretical system that takes into account the various dimensions involved (epistemic, cognitive, instructional and ecological), and to avoid redundancies? What would the primitive notions and basic postulates of this new system be?

It is clear that we cannot address these issues here, but only show the relevance and potential utility of moving towards a theoretical system that coherently articulates the epistemic and cognitive approaches, in order to achieve effective instructional designs. To meet this goal I briefly describe some basic notions from these theoretical models whose clarification, confrontation and articulation could be productive. I will briefly mention how these theories conceive knowledge, from the epistemic point of view in TDS and ATD, and from the cognitive point of view in the other theories. This is not the place to make a comparison and possible articulation of these theories and their various components; instead I try to exemplify a networking strategy based on the rational analysis and possible hybridisation of conceptual tools used in each case. The system of results developed by each theoretical framework is not discussed or articulated.

This strategy has given rise to the Onto-semiotic Approach (OSA) in Mathematics Education, which has been developed by Godino et al. (Godino and Batanero 1994; Godino et al. 2007) in an attempt to articulate these and other related theories from an approach they describe as onto-semiotic. These authors conceive the theories under two perspectives:

• 1. In a narrow sense, as “system of tools” (concepts, principles and methodologies) used to answer a set of characteristic questions of an inquiry field; this interpretation may be similar to the triplet given in Radford (2008)—Principles, Methods and Questions.

• 2. In an expanded sense, and in addition to the above components, the “system of results” (knowledge) obtained as result of applying the tools to the questions.

In principle, any theory can produce valuable knowledge for understanding the field and rationally acting upon it. However, various theories may be redundant, inconsistent, insufficient, or more or less effective for the intended work. The clarification, comparison and possible articulation of theories intend to develop a system of optimal conceptual and methodological tools which enhance research in the field. Such an articulation can be performed by rational analysis of the constituent elements of these theories and by developing new conceptual tools when mere amalgam of existing ones is not possible or appropriate. As I aim to demonstrate, this strategy has led to a new theoretical notion, onto-semiotic configuration (Fig. 4.2), which incorporates, in a hybrid or blending way, constituent elements of concept, conception, scheme, mathematical praxeology, and semiotic register of representation.

4.3.2 The Notion of Knowledge in the Theories Analysed

The theoretical contribution of Duval (1995) falls within the line of inquiry, which posits a mental (internal representations) nature for mathematical knowledge, and attributes an essential role in the processes of formation and apprehension of mental representations (noesis) to language and its various manifestations. The availability and use of various semiotic representation systems and their transformations are considered essential in the generation and development of mathematical objects. However, semiosis (production and apprehension of material representations) is not spontaneous and its mastery should be a goal of teaching. Particular attention should be given to the conversion between non-congruent semiotic representation registers. Duval’s cognitive semiotics provides other useful notions for studying mathematical learning, such as types of discourse, and meta-discursive functions of language, functional differentiation and coordination of registers (Duval 1996).

The theory of conceptual fields (Vergnaud 1990, 1994) has introduced a set of theoretical concepts for analysing the construction of knowledge by learners (see Sect. 4.2.2.1). This is why we consider this theoretical model in the cognitive programme, recognising, however, that some theoretical notions (conceptual field) have an epistemic nature. Vergnaud’s basic cognitive notion is that of scheme. The scheme is described as “the invariant organization of behavior for a class of given situations” (Vergnaud 1990, p. 136). The author states that “it is in the schemes where the subject’s knowledge acts; they are the cognitive elements that allow the subject’s action to be operative, and should be investigated.” Each scheme is relative to a class of situations whose characteristics are well defined.

Vergnaud also proposes a notion of concept (Sect. 4.2.1.1) to which he attributes a cognitive nature by incorporating the operative invariants “on which rests the operationality of the schemes.” This notion is different from the concepts and theorems that are found in science; he does not propose an explicit conceptualisation for them. Vergnaud first describes the notion of conceptual field as “a set of situations”, and then clarifies that we should also consider the concepts and theorems involved in solving such situations.

In the TSD the savoir (knowledge to teach) has a separate, preexisting cultural existence and, in a way, is independent of the individuals and institutions interested in its construction and communication. The main objective of the didactic of mathematics is the analysis of communication and reconstruction processes of such cultural knowledge by the subject, in the form of knowing within the didactical systems. The didactic transposition, developed in the ATD framework, recognises the adaptations of this knowledge for its study in the school context, giving rise to different epistemic varieties of the same knowledge.⁸

As for the notions used in the TDS to refer to “subject’s knowledge”, we find ‘representation’ in the sense of internal representation; at other times Brousseau uses the expression “implicit models” for such knowledge and representations. He interprets implicit models as “ways of knowing”, which do not operate in a way completely independent nor totally integrated in controlling the subject’s interactions.

The Anthropological Theory of Didactic has so far focused almost exclusively on the institutional dimension of mathematical knowledge. The notions of mathematical organisation and institutional relationship to the object are proposed to describe mathematical activity and the emerging institutional objects from such activity. The cognitive dimension is described in terms of “personal relationship to the object”, which is proposed as a substitute for the related psychological concepts (such as conception, intuition, scheme, and internal representation).

4.3.3 Towards an Integrative Theoretical System

The brief summary of the concepts used by the four theories to describe mathematical knowledge from the institutional (epistemic) and personal (cognitive) points of view suggests that the simple superposition or the indiscriminate use of them to describe the phenomena of didactic transposition and mathematical learning can only create confusion.

This is one reason why Godino and Batanero (1994) began to lay the foundations of an ontological, epistemological and cognitive model of mathematical knowledge based on anthropological and semiotic bases. With a style reminiscent of axiomatic works in mathematics, these authors began by defining the primitive notions of mathematical practice, institution, institutional and personal practices, institutional and personal object, meaning of an institutional and personal object, and knowledge and understanding of the object. These notions were supplemented in later works with a typology of primary mathematical objects and processes as well as an interpretation of the notion of semiotic function. This notion is conceived as a triadic relationship between two objects, antecedent and consequent, according to a criterion or rule of correspondence, allowing the development of an operational notion of knowledge (meaning, understanding and competence) (Fig. 4.2). These notions may include those related to the epistemological and cognitive approaches used in mathematics education, as described in Godino et al. (2006).

In Fig. 4.2 the notions of practice, object, process (sequence of practices from which the object emerges) and semiotic function (tool which relates the various entities and that takes into account the object referential and operational meaning) are the key elements of the epistemological and cognitive modeling of mathematical knowledge proposed by the OSA. We might think that the onto-semiotic configuration is equivalent to the TCF conceptual triplet or the TAD praxeological quartet, however, the OSA has developed an explicit typology of objects (and processes) that enables more analytical and explanatory descriptions of mathematical activity than the other theoretical notions.

Fig. 4.2

Primary entities of the OSA ontology and epistemology

Specifically, the OSA proposes that in mathematical practices, the following six types of objects intervene: situations–problems, languages, concepts (in the sense of entities which are defined), procedures, propositions and arguments. These primary entities can also be seen from five dual points of view: personal–institutional; ostensive–not ostensive; extensive–intensive; unitary–systemic; and expression–content (Godino et al. 2007).

4.3.4 Concordances and Complementarities

The theories mentioned (TSRR, TCF, TDS, ATD) put different weight on the personal and institutional dimension of mathematical knowledge and its contextual dependence. The OSA postulates that the systems of practices and the emerging objects are relative to the contexts and institutions in which the practices are carried out and the subjects involved in them (i.e., they depend on language games and forms of life, Wittgenstein 1973).

The description of an individual subject’s knowledge about an object O can be undertaken in a comprehensive way with the notion of “systems of personal practices.” Knowledge is also interpreted as the set of semiotic functions that the subject can establish where O is brought into play as an expression or content (signifier, signified). Within this system of practices, when asked to solve a type of problem–situation, we distinguish between those with operative or procedural character and those with discursive nature, and we obtain a construct closely related to the notion of praxeology (Chevallard, 1999), but only if we consider both a personal and an institutional dimension in the notion of praxeology.

We propose the different ways of “solving and communicating the solution” of certain types of problems related to a given object, for example, “function” as the answer to the question “what does the function object mean” for a person (or an institution)? This semiotic modeling of knowledge allows us to interpret the notion of schema as the cognitive configuration associated with a subsystem of practices relative to a class of situations or contexts of use, and the notions of concept-in-act, theorem-in-act and conception, as partial constituent components of such cognitive configurations.

The notion of conception (in its cognitive version) is interpreted in the OSA framework in terms of personal onto-semiotic configuration (systems of personal practices, objects, processes and relationships). In semiotic terms, when we ask for the meaning of a subject’s “conception” about an object O the answer is “the system of operative and discursive practices that the subject is able to express and where the object is involved”. This system is relative to some local and temporal circumstances and is described by the network of objects and processes involved.

Likewise, understanding and knowledge are conceived in their personal–institutional dual nature, involving therefore the system of operative, discursive and normative practices carried out to solve certain types of problem—situations. Subject’s learning of an object O is interpreted as the subject’s appropriation of the institutional meanings of O; it occurs through negotiation, dialogue and progressive linkage of meanings.

The notion of meaning is specified in the OSA framework. Meaning of a mathematical object is the content of any semiotic function and, therefore, according to the corresponding communicative act can be an ostensive or non-ostensive, extensive or intensive, personal or institutional object; it could refer to a system of practices, or a component (e.g., problem situation, notation, or concept). The notion of sense is interpreted as a partial meaning, that is, it refers to the subsystems of practice corresponding to certain frames or contexts of use.

In the OSA framework, Duval’s notions of representation and semiotic register allude to a particular type of referential semiotic function between ostensive objects and (not ostensive) mental objects. The semiotic function generalises this correspondence to any type of objects and also includes other types of dependences between objects. For example, the ostensive expression y = 2x refers to a particular mathematical function (conceptual entity, not ostensive). Between the two entities a representational semiotic function is established. In other situations the function y = 2x can be on behalf of (represent) the class of first-degree polynomial functions, or the general function concept. Now the antecedent and the consequent of the semiotic function are conceptual entities. The mathematical function y = 2x can be used to model certain practical situations, for example, to determine the cost of x kilograms of apples with a unit cost of 2 €. Here prevails the use or pragmatic meaning of the function concept: y = 2x is defined by the system of practices the object participates.

The notion of sense in the TDS is restricted to the correspondence between a mathematical object and the various fundamental situations from which the object emerges and “gives its senses” (it can be described as “situational meaning”). This correspondence is undoubtedly crucial to provide the raison d’être for that object, its justification or phenomenological origin. But it is also necessary to take into account the semiotic functions or correspondences between the object and the remaining operative and discursive components of the system of practices from which the object comes from, understood either in cognitive or epistemic terms.

The TCF extends the notion of meaning as a “response to a given situation” introduced in TDS. This extension is meant to include, in addition to the situational component, the procedural (schemas) and discursive/normative (concepts and theorems) elements. The content considered as “meaning of a mathematical object for a subject” in TCF is nearly the wholeness described in the OSA as the “system of personal practices”. However, the notion of semiotic function and the associated mathematical ontology provides a more general and flexible tool for didactic—mathematical analysis.

An essential aspect that allows us to distinguish between the theoretical models considered in this section is the dialectic between the institutional and personal duality, between epistemological and cognitive approaches, which often appear disjointed, resulting in extreme positions. In some cases the emphasis is on the personal dimension (TCF and TSRR), in others the institutional dimension (ATD and TDS), while in OSA a dialectical relationship between the two dimensions is postulated, so that it can help to coordinate the remaining theoretical models.

4.3.5 Hybridisation and Competition of Theoretical Frameworks

As we have explained, the OSA does not intend to build a “holistic theory that explains everything”, but to advance the construction of a system of conceptual and methodological tools that allows researchers to conduct the macro and micro analysis of the epistemic, cognitive, instructional and ecological dimensions involved in mathematics teaching and learning processes. For the epistemic and cognitive notions analysed, the mere overlap or amalgamation of theoretical tools is not possible, given their heterogeneity and partiality, and the OSA has tried to develop a new framework with a clear hybrid character. The onto-semiotic configuration construct (Fig. 4.2) keeps a “family resemblance” with the notions of concept, conception, semiotic representation register, knowledge, and mathematical praxeology, but is not reducible to any of them, so it requires a specific designation. This notion can be more effective than the original notions, allowing researchers to analyse the micro and macro level of institutional and personal mathematical activity, and to better understand the relationships between both dimensions of mathematical knowledge. To prove this statement, however, would require a deeper analytical and experimental work than that produced in this brief presentation and that provided in Godino et al. (2006).

It is clear that this new entity competes with those already existing, and has to prove its relative effectiveness to solve the paradigmatic issues in the field. Progress is needed in comparing the results obtained from the emerging construct with other theoretical frameworks to test its possible survival.

The ecological analysis outlined of the emerging ideas should be complemented with the corresponding sociological analysis; it is not enough having generated a new potentially strong hybrid entity, it is necessary that social and material circumstances for its development are given. It is needed to attract new researchers involved in the study, to develop understanding and application of the new instruments, and to be able to achieve the necessary resources to conduct research, communicate, discuss and publish the results.

4.4 Social Perspective of the Multiplicity of Theories and the Search for Connections in the Didactics of Mathematics

The final section of this chapter is relatively adventurous in the sense that it offers—if not provides novel answers—to at least introduce new ways of examining the multiplicity of theoretical approaches by proposing to add a sociopolitical dimension to what has hitherto been an essentially epistemological approach. There are three parts to this text. The first part picks up on the key points of Bosch’s contribution at the Artigue Colloquium and is centred on the notion of research praxeology, which is a significant contribution by Bosch, Artigue and Gascón to the reflections on Networking. The next two parts will, as mentioned earlier, attempt to open new doors by borrowing from the anthropological theory of didactics and the field theory applied to science respectively.

4.4.1 Theories as Components of the Praxeological Modelling of Scientific Research

4.4.1.1 Some Reference Points Relating to the European Dynamics of the Networking of Theories

As mentioned in the introduction to this chapter, schemes such as the Working Groups of the CERME congress and the two research projects, Telma and ReMath, have enabled the development of a community of European researchers who are able to understand each other and begin to build a common capital focused on interaction activities between theories.

Most notably, the ReMath project aims to develop certain interactive environments that are tested in conditions as close as possible to real teaching situations. The original version of this project also aimed to reduce the diversity of theoretical frameworks, as such diversity is viewed as one of the reasons why research in digital environments for human learning has a weak influence on practices. However, the initial work carried out quickly led to modifications in the formulation of this aspect of the project. The idea of creating a grand unifying theory that could incorporate contributions from the various views that exist in the field of research in the didactics of mathematics is considered to be a dead end.

ReMath has thus been directed towards the exploration of other ways of connecting theories. In their assessment of the work accomplished, Artigue and Mariotti (2014, p. 350) remark that:

we have obtained practical results in terms of networking situated at different levels of the hierarchic scale presented in section I,⁹ from comparison up to local integration in a few cases; we have also identified limitations to such networking especially when a design perspective is adopted, and coherent design choices must be made.

In several cases, research led to local integration between theory couples, to which Mariotti chose to devote her entire contribution at the ‘Évolution des cadres théoriques’ roundtable (see also Sect. 4.2.2.3). On the whole, however, the networking of theories was of limited scope. On the other hand, in the aforementioned text, Artigue and Mariotti emphasise the productivity of the work accomplished at a meta-level of reflection, a shared effort that could not have been achieved without the development of a specific language and set of tools. At the Artigue Colloquium, Bosch centred her presentation on the contribution of research praxeology to the modelling of activities in the networking of theories and to the building of the necessary metalanguage. I will revisit this contribution in the following section.

4.4.1.2 The Notion of Research Paradigm, a Critical Tool for Thinking About the Complexity of the Networking of Theories

In the standard ATD model, research activities are described in terms of pointwise praxeologies [T/τ/θ/Θ] where the notion of theory appears as the symbol Θ. This very model considers praxeological organisations that contain praxeologies with the same technology, followed by those of with the same theory; one refers to local and regional praxeological organisations. Artigue et al. (2011, 2012) only introduce the expression ‘research praxeology’, but it is clear that this refers to an amalgam of pointwise praxeologies located at the regional level. A praxeological organisation of research may thus be defined by:

• the types of problems that may or may not be addressed, which determines the acceptable research issues and the transformations that they will undergo to define objects of study;

• the corresponding methodologies that are deemed legitimate;

• the technologies of these research techniques, especially the rational arguments that produce them, and that justify and explain methodological choices; and

• the theory—I refer to the definition used in the ReMath group, which is not exactly that in ATD (see below)—of a structured system of knowledge that emerges from research practices and that, in return, conditions the three previous praxeological levels.

In a dynamic vision, such an amalgam can exist prior to the development of a theory. Artigue et al. (2011, 2012) stress the importance of the technological level of praxeologies: it is particularly pertinent when it comes to appreciating the nature and role of knowledge in the emergence phase of an initial amalgam, which will only become a praxeological organisation that is identifiable by a relatively substantial theory as research is carried out. But this lability at the theoretical level does not prevent the emerging organisation from attaining a certain social existence, which materialises at the very least through the development of a community of researchers who refer to it. At this stage of an iterative developmental process, some of the results thus obtained, especially didactic phenomena,¹⁰ create new types of problems and new methodologies. The knowledge produced is integrated into the technology of new praxeologies that come to be included within the growing amalgam. Artigue et al. (2011, 2012) provide the example of didactic transposition: this is a phenomenon that, well before its inclusion in ATD, gave rise to the development of henceforth indispensable praxeologies in research work based on ATD and TDS.

In the above, we may discern the existence of an object that does not appear in the praxeological model: a social organisation, without which praxeologies would neither develop nor organise themselves into an amalgam that could produce a theory. Here, the expression ‘research paradigm’ refers to this epistemic and social pair. A research paradigm is also a social construction, a product of a certain history, the fruit of a search for consensus within a community of researchers specialising in amalgamated praxeologies, which may be conceptualised in ATD by considering a paradigm to be an institution.

The notion of the praxeological organisation of research clearly brings to mind Radford’s proposal (Radford 2008, see 4.2 note 2), which analyses the concept of theory by means of the (P, M, Q) triplet. To these three components the authors of the book Networking of Theories in Mathematics Education have added a fourth, Key constructs.¹¹ They use this new model to present five theoretical approaches that played a major role in the studies reported in their book, the expression ‘theoretical approach’ being preferred to ‘theory’. The two resulting quadruplets are thus brought closer together. If one were to consider that ‘savoirs’, understood as explicit and socially legitimised knowledge, is but little taken into account in Radford’s proposal, then the ‘Key constructs’ component changes everything. Explicit principles and key constructs are elements that the praxeological model considers at the level of the ‘Theory’ component. Is there, then, such a thing as identity? If one considers the theories of mathematical praxeologies, the answer is no, unless the idea of key constructs is very broad. Besides, other types of knowledge are revealed by the ‘Technology’ component of praxeologies. Indeed, this model makes a distinction between two components of methodology: research techniques and their technology. This enables a convenient description of the process that accompanies what is referred to in Radford (2008, p. 322), under the heading ‘Connecting the principles P1 of a theory τ1 and the methodology M2 of a theory τ2’: when one paradigm borrows a research technique from another, the technology is reworked in such a way as to establish technique compatibility with the principles and theory of the borrowing paradigm. If one considers a research paradigm to be an institution, this is a form of the transpositive phenomenon that comes with the interinstitutional circulation of praxeologies (Chevallard 1999, p. 231; Castela and Elguero 2013, p. 132). In summary, the praxeological model still offers greater possibilities for incorporating and analysing the knowledge produced and used by a research paradigm.

Both models take into account—in what appears to me to be the same way—the existence of a field of questions associated with a theory (Q) or a paradigm (set of type T problems). However, praxeological modelling does not allow one to incorporate all the basic principles considered by Radford, since technologies and theories are composed of explicit and legitimised knowledge. Yet a research paradigm is not wholly subsumed by the theoretical component that symbolises it: however developed the latter may be, it cannot reveal everything about the reasons for the choices that constitute the identity of a paradigm—its point of view, to borrow the expression used by Bosch in her talk, or its boundary in the words of Radford (2008, p. 323, see 4.2). Some principles remain implicit. One of the effects of networking activities is to force researchers who call upon the paradigms involved to update the founding principles.

As mentioned above, a research paradigm, and especially the social organisation that it encompasses and that produces it, can begin to exist prior to the existence of a theory in the sense that has been used thus far, i.e., a structured system of knowledge. But in ATD, theory is the technology of technology in the eyes of the community that acknowledges praxeology, and can be very embryonic. In this way, it is possible for the phenomenon of praxeological amalgamation in a scientific community to rapidly manifest itself as the development of a theoretical instance, without being identifiable as a theory in the scientific sense. Initially, the theoretical instance of a paradigm may be strongly characterised by the personality of the researcher or small group of researchers who introduced it: certain philosophical or ideological options serve as a prelude to the priorities given to certain topics, and therefore to the problems studied and the results obtained, which by default leaves its mark on the technologies and subsequently the theory subjected to the scientific process of the assessment. Let us conjecture that as a scientifically recognised theory is being built, there is a tendency to forget the primary reasons that constitute the very foundations of the paradigm and whose signature is borne by all the components of the latter.

Thus, based on this first part, I draw attention to the fact that the work required to appraise the commensurability of two paradigms is not limited to an examination of the conceptual apparatus of each theory. This echoes one of the points highlighted by Artigue and Mariotti in the conclusion of their assessment of the ReMath project:

Despite our maturity as researchers, we all discovered up to what point our knowledge of many of the theoretical frameworks involved in the project was superficial. It had been gained through reading articles, listening to presentations, and discussing with colleagues. It lacked the first hand experience provided by the actual use in a research project. In such conditions, misunderstanding and distortions are frequent. (Artigue and Mariotti 2014, p. 350)

What constitutes the essence, pertinence and efficiency of a paradigm are collectively the objects of research that it favours for reasons that are sometimes implicit; its research practices and its theory. How then, in these conditions, do we connect two paradigms and above all attain a true understanding of a ‘foreign’ paradigm? This is precisely the question to which the ReMath project has brought preliminary answers, a summary of which has been provided by Artigue and Bosch (2014) in terms of networking praxeologies. One might say that the analysis of the amalgamation of research praxeologies developed here belongs to the technology of several of the networking methodologies tested.

At this point, let us turn our attention to one particular outcome of the assessment conducted by the researchers involved in the ReMath project, that is, the fact that they were led to deconstruct their original injunction of reducing theoretical diversity in the didactics of mathematics in order to reconstruct a version that would most certainly be very different. This was accomplished from within the field of research in the didactics of mathematics, alongside a group of didactical studies on the design and testing of sequences by means of software, and is a key component of the networking praxeologies that were developed. I propose to carry on this deconstruction of the postulate on the harmful nature of the multiplicity of research paradigms and the necessity for integrative development in the didactics of mathematics by adopting a complementary approach; indeed, one must, conversely, search for tools beyond didactics in order to consider how it works.

4.4.2 Considering Research in Didactics as Being Externally Determined: A Second Contribution of ATD

The Anthropological Theory of Didactics is not just a theory in didactics: many of its key concepts also apply to other realms of reality. This stems from a certain consistency in the approach by the person to whom its development may be chiefly credited, Yves Chevallard: deconstruct the self-evidences (‘allant de soi’) and to do so, put things into perspective by immersing the realm being studied into another broader realm. Wishing to justify the use of the adjective ‘anthropological’ by clarifying its meaning, Chevallard (1999, p. 223) writes:

Le point crucial […] est que la TAD situe l’activité mathématique, et donc l’activité d’étude en mathématiques, dans l’ensemble des activités humaines et des institutions sociales. Or ce parti pris épistémologique conduit quiconque s’y assujettit à traverser en tout sens –ou même à ignorer- nombre de frontières institutionnelles à l’intérieur desquelles il est pourtant d’usage de se tenir, parce que, ordinairement, on respecte le découpage du monde social que les institutions établies, et la culture courante qui en diffuse les messages à satiété, nous présentent comme allant de soi, quasi naturel, et en fin de compte obligé.

In the rest of the cited paper, the author introduces the postulate that the same praxeological model may be used for all human activities, including mathematics. This stance has enabled didactics developed according to the ATD paradigm to refrain from having viewpoints from the world of research in mathematics imposed on it without question, and to distance itself from the presentations of mathematical knowledge proposed in academic texts.

Thus, when Artigue, Bosch and Gascón use the notion of praxeology to consider research in didactics, they already do so from outside this realm of human activities, which is precisely what I propose to pursue in the following section.

4.4.2.1 Research in Didactics as Being Externally Determined in Its Workings

I will once again adopt the ATD point of view and consider research in didactics as an institution among other elements, determined by other institutions that encompass it or are juxtaposed with it and that it also, in return, determines. Thus, civilisation, society, school and pedagogy appear in the range of institutions that are considered in ATD to influence the discipline of mathematics and its constitutive praxeologies (Chevallard 2007, p. 737). This provides an idea of the diversity of institutional levels involved. We are led to consider the influence exerted by the various societies and geopolitical organisations that are home to research in didactics on the latter, its structure, actors and praxeologies. To appreciate this complex reality requires the consideration of a network of institutions, each possessing a unique identity, at the local level of states, the regional level of geographical or linguistic groups, and finally the global level. Signs of such a structure are easily identified in the organisation of the ICMI as well as in the range of journals devoted to the didactics of mathematics.

On the epistemological front, the theory we have put forward has important consequences, as it amounts to considering that the paradigms of research in didactics, and therefore the results produced, bear the mark of the local sociocultural and institutional contexts in which they appear. Returning to the paradigms involved in the ReMath project, the influence of the French and Italian sociocultural contexts on TDS and TSM is, for example, likely to have affected the role given to the teacher under either theory (see Sect. 4.2.2.3).

4.4.2.2 Research in Didactics as Being Externally Determined by Its Objects

In the above section, we touched on the determinations that affect research institutions, especially the social components of paradigms and via these, therefore, praxeologies. However, we must also consider that these very institutions determine the realm of reality that constitutes the object of study in the didactics of mathematics, that is, all the phenomena of passing down and learning associated with mathematical praxeologies. Noone can dispute the vast distance that separates the following two objects of study: on the one hand, the passing down of arithmetic techniques in the Aymara villages of northern Chile, whose culture developed specific calculation praxeologies, and on the other hand, the use of software in the French (or Italian) education system to promote the learning of algebra. Is the epistemic aim of research in didactics to bring to light universal regularities when the reality that it aims to study is, unlike that of physics for example, so diversified? Assuming that such shared phenomena do exist (didactic contract is often cited as an example), how can they explain the complexity of the two local realities described above? More importantly still, given that research in didactics is as much engineering as it is science, and has both technical and epistemic objectives, what results can they bring about? Can we postulate that the same tools enable us to meet the requirements of the various didactical institutions in the world, to understand any dysfunctions identified, and to develop solutions deemed acceptable by these institutions and their subjects? I propose to consider it more epistemologically justified to adopt the reverse postulate until there is evidence of its erroneous nature. It seems to me that the multiplicity of paradigms is a consequence of the epistemology of a science that aims to take action in the reality being studied in order to improve it, which highlights the local dimension.

One example is the ethnomathematic perspective developed in both South America and Africa in response to what emerged in the period between 1985 and 1990 as a need to ‘“multiculturaliser” les curricula de mathématiques pour pouvoir améliorer la qualité de l’enseignement et augmenter la confiance en soi sociale et culturelle de tous les élèves.’¹² (Gerdes 2009, p. 21). This consisted in coming up with solutions to the widespread failure in the learning of mathematics within an educational system that had not truly done away with the colonial vision of teaching and ‘[presenting] la mathématique comme quelque chose d’“occidental” ou d’“européen”, comme une création exclusive de la race blanche’ ¹³(ibidem, p. 31).

4.4.2.3 How to Meet the Epistemological Need for Mutual Understanding?

Research in the didactics of mathematics is therefore regarded as being determined by both the social and cultural environment in which it is carried out, in terms of its questions and answers as well as its agents and organisations. Such determinations arising from the same source are reflections of each other, which may be considered to be a factor of coherence and efficiency. But this vision contradicts the conception of science as an approach to building objective facts possessing a universal value of truth and, according to Bourdieu, resulting as much from a confrontation between scientists as from a confrontation with reality:

The fact is won, constructed, observed, in and through the dialectical communication among subjects, that is to say through the process of verification, collective production of truth, in and through negotiation, transaction, and also homologation, ratification by the explicit expressed consensus - homologein- (and not only in the dialectic between hypothesis and experiment). (Bourdieu 2001, p. 73)

The extension to higher institutional levels of work leading to homologation, or etymologically speaking to a rational agreement on the same discourse, appears to be an intrinsic component of the scientific approach: ‘The process of depersonalization, universalization, departicularization of which the scientific fact is the product is all the more likely actually to take place the more autonomous and international the field is.’ (ibidem, pp. 75–76)

It is clear that the multiplicity of theories and paradigms of didactical research does not facilitate this change in level as it stands in the way of mutual understanding. This advocates, at the very least, for a verification of the emergence of new theories. Reducing theoretical diversity has been the solution adopted in a certain number of sciences, especially the exact sciences, which are also the oldest. To consider that didactics should follow the same path by replacing all existing theories with integrative wholes is to postulate that the workings of the sciences in question are the only ones possible, to accept the idea that given its youth, didactics should have no other way forward than to align itself with others. This view is illustrated by J. Godino in Sect. 4.3 of this chapter. Conversely, I consider that such a strategy is not relevant and would otherwise cause us to lose much of what has been produced by the various paradigms. The work carried out in the ReMath project tested the fact that the mutual criticism required to build scientific objectivity could be achieved by means that do not presuppose a common theoretical framework. The praxeologies of homologation in didactics could well be different from those of the exact sciences.

In summary, it is my opinion that social and epistemological reasons justify why research in the didactics of mathematics should not be subjected to, as a self-evidence (‘allant de soi’), a principle of reducing theoretical diversity derived from other sciences: the multiplicity of theories and paradigms in this scientific field is seen as being inherent to the eminently social nature of its object and to the intended intervention in the society in which it is found. But it must come up with its own means of attaining scientific homologation.

4.4.3 Research in Didactics as a Power Game

To conclude the work of deconstruction undertaken thus far, I will refer to the field theory by Bourdieu, and more specifically to how he uses it to analyse the workings of science in the aforementioned book, Science de la science et réflexivité.

4.4.3.1 Elements of the Field Theory Applied to Science

The scope of the present text makes it impossible to truly appreciate the force of the work described above. I will, notwithstanding, present certain elements of the field theory, which may not be familiar to all readers.

A field is characterised by a game that is played only by its agents, according to specific rules. The agents are individuals and structured groups; in science they are isolated scientists, teams or laboratories. The conformity of agents’ actions to the rules of the game is partly controlled by objective visible means, but the key point of the theory, through the concept of habitus, is the inculcation of social field rules into the agents’ subjectivity. This individual system of dispositions, partly embodied as unconscious schemes, constitutes an individual’s right of entry into the field.

The field game is twofold. Firstly, it is productive of something that is the field-legitimised goal in the social space. The rules, and therefore the individual dispositions, are established to achieve this goal that every agent considers desirable. In the case of science, the goal is epistemic: tacitly accepting the existence of an objective reality endowed with some meaning and logic, scientists have the common aim of understanding the world and producing true statements about it. As seen above, Bourdieu adds a social dimension to the Bachelardian conception of the construction of scientific fact. Despite this social nature, scientific homologation produces objective statements about the world thanks to specific rules of scientific critical scrutiny, ‘the reference to the real, [being] constituted as the arbiter of research’ (ibidem, p. 69).

Secondly, the game is a competition between agents, which results in an unequal distribution of some specific form of capital—a source of advantage in the game itself and a source of power over the other agents. Thus, a field, including the scientific one, appears to be

a structured field of forces, and also a field of struggles to conserve or to transform this field of forces. […] It is the agents, […] defined by the volume and structure of the specific capital they possess, that determine the structure of the field […This one] defined by the unequal distribution of capital, bears on all the agents within it, restricting more or less the space of possible that is open to them, depending on how well placed they are within the field… (ibidem, pp. 33–34)

Capital includes several species, for instance, in science: laboratory equipment, funding and journal edition. Here, we focus on symbolic capital, especially its scientific modality.

Scientific capital is a particular kind of symbolic capital, a capital based on knowledge and recognition. (ibidem, p. 34)

A scientist’s symbolic weight tends to vary with the distinctive value of his contributions and the originality that the competitor-peers recognize in his distinctive contribution. The notion of visibility, used in the American universitary tradition, accurately evokes the differential value of this capital which, concentrated in a known and recognized name, distinguishes its bearer from the undifferentiated background into which the mass of anonymous researchers merges and blurs. (ibidem, pp. 55–56)

This theory of science as a field challenges an idyllic vision of the scientific community, disinterested and consensual; however through the hypothesis of embodied dispositions, it avoids considering the scientists’ participation in the capital conquest in terms of personal ambition or cynicism.

In summary, we bear in mind that within the field theory, scientific strategies are twofold.

They have a pure – purely scientific- function and a social function within the field, that is to say, in relation to other agents engaged in the field. (ibidem, p. 54) Every scientific choice […] is also a strategic strategy of investment oriented towards maximization of the specific, inseparably social and scientific profit offered by the field. (ibidem, p. 59)

4.4.3.2 The Injunction to Unify Theories Seen as Being Related to a Power Game

Let us return to the main topic. Bourdieu’s work leads us to consider that the production of independent theories, just like the call for their incorporation into wholes where they are engulfed, relates to the game of conquest and contestation of positions of power in the field. To be recognised as the producer of a theory gives both material and symbolic credit to a researcher. This is true of the major didactic theories introduced by persons whose names are ever present. This phenomenon fosters multiplication: there is greater potential, from an individual point of view, in creating one’s own theory than in seeking approval for one’s contributions via an existing theory. Since it is difficult to blend two well-developed paradigms, regulating individual productions appears to be an epistemological necessity.

The same does not necessarily apply to the intermediate structures of research: above all, I believe that the development of a specific paradigm is an asset to an emerging research community, a means of avoiding the domination of older communities whose general tendency is to impose their own paradigms as the only pertinent ones. I have already postulated that the need to unify paradigms could be epistemologically challenged by virtue of the diversity of the didactic reality depending on the societies and countries involved. Now, I question it as an obstacle to an autonomous organisation of didactical research in countries where the latter is just emerging. We have already discussed ethnomathematics, and now I mention socioepistomology, which has been deliberately developed by a group of Mexican researchers with the twofold aim of developing tools adapted to the educational reality of Latin America, and breaking away from what could be felt as a prolongation of colonisation through the dominance of North American and European theoretical frameworks in the didactics of mathematics.

To end this paper, I propose to invert the problem: if developing a paradigm is an empowering factor, then the question lies in the significance—from the point of view of the positions of the various subinstitutions in the field of didactical research, notably regional institutions (characterised, for example, by their geographical location or by a common language)—of the necessity of reducing theoretical diversity, especially when this involves not the integration of several theories into a whole that would preserve the contributions of each but rather a process of selection for the purposes of simplification. Let us recognise that the significance afforded to the issue of the multiplicity of paradigms in didactics is not only related to epistemological reasons, but is also a facet of the social game of the field.