The association, which is registered in France, regroups French researchers and numerous foreign ones from all continents interested in the development and the improvement of research in didactics of mathematics. ARDM intends to :
A prominent researcher in a central field for mathematics education André Rouchier ^{[*]} |
Guy Brousseau’s career is inscribed in the history of the past forty years concerning the changes in mathematical education. Il is linked to the creation of the great paradigms that have structured fundamental research in this field. This becomes obvious when retracing the steps of his academic path, his contribution to research in mathematics education, his involvement in collective projects and international exchanges and finally the various dimensions of his influence.
An exceptional career
Guy Brousseau began his career as a student of a teachers training college in order to become a primary teacher. He remained a primary teacher for a couple of years before joining thanks to a secondment, all people that were involved, at the beginning of the sixties, in the launching of the general movement for change in mathematical education. With the support of the governing body, he completed his university career before being employed as an assistant at Bordeaux I University. It was at this same university, at the IREM^{[1]}, and with the continuing support of Professor Jean Colmez, that he carried out most of his research on mathematics education in compulsory education. He submitted his thesis in 1986. With the support of the academic authorities, he set up the COREM^{[2]} that he was in charge from 1973 to 1998, before creating the LADIST^{[3]} a laboratory linked to the COREM. In the meantime the creation of the IUFM enabled him to become a University Professor in 1992, until his retirement in 1998. He then became professor emeritus at the IUFM^{[4]} which permitted him to continue his scientific work (supervising theses) in a new laboratory affiliated with Victor Segalen Bordeaux II University, the DAEST^{[5]}.
Guy Brousseau’s first published work appeared in 1961 at the CIEAEM^{[6]} conference in La Chataigneraie (Switzerland). This was followed by a text book aimed for first year elementary school (Grade1) (1965) and soon after came a succession of regular publications in the field of didactics from 1968 to the present day. The great intricacy of his personal work in teacher training at the IREM, also the specificity and originality of his research lead to the publication in university produced literature (Journal of the IREM from 1969 to 1978) of some essential articles to understand the fundamental theoretical instrument that is the theory of didactical situations. One can find these papers as well as others that were previously published in various journals such as RDM^{[7]} in a volume published in English by Kluwer and entitled “Theory of Didactical Situations in Mathematics”.
Profound and original scientific choices
Guy Brousseau’s passion for mathematics education came from a double fascination: fascination for mathematics on the one hand, its explanatory ability and capacity to train the process of thought, and, on the other, fascination for the transfer and spread of knowledge, as well as the study of conditions that make it possible. Throughout his scientific career, in serving this double passion, he knew how to summon up constant and inexhaustible energy, faultless determination, limitless curiosity and extreme precision that led him to develop and propose the most thorough and coherent theory of the past thirsty years.
This approach and the way of thinking emerge, in their force and particularity, in the second half of the sixties. Brousseau at that time took an original and decisive theoretical decision that is presented in a fundamental paper: “The process of mathematization” given at a talk during the annual conference of the APMEP^{[8]} in 1970. This paper was an important and major contribution. Its typicality and relevance cannot be denied.
Considering the student and the teacher as essential actors of teaching and learning, it would also be necessary to first of all focus one’s attention to a third instance, “the silent actor”: the situation in which they evolve and in which the student’s and teacher’s activity unfurls according to their respective goals: to learn and to teach. The situation is organized by one and experienced by the other, it evolves through the interplay of their interaction according to the rules, most often tacit, activated at the very heart of the didactical contract. The situation is designed as a model of the knowledge to be taught. It is at the same time, the condition for establishing a didactical relationship specific to the knowledge in question and the preferred instrument in the teaching-learning process. If one wants the situation to allow the learning of mathematics, it must not be arbitrary in the modalities of action it offers to the student.
One might characterize the irruption of the notion of situation as a central topic of research from two points of view:
- The first being that it consists in adopting, in a certain way, a dual position, compared to the experimenter who approaches and questions the students, with the help of specially designed tests concerning their conception of mathematical topics they have encountered, in education or in their diverse experiences in everyday life. The didactical project is something else entirely. Il consists in turning this perspective upside down and is concerned with the problems and situations themselves, for the way in which they inform us of the knowledge bring into play and that they activate. Thus, one no longer studies the subject “in abstracto” but instead the situation for the potential it must offer the student, whether this be in his mathematical activity or in the context of learning teaching process as a subject of a didactical institution.
- The second point of view takes as a starting point the consideration of the non didactical situation, in other words the context of employing mathematics whether that be in the work of mathematicians or a everyday user in an environment of specific practice. Indeed, the knowledge of mathematics could never be reduced to just the knowledge of theorems and algorithms, but requires the ability to recognize when necessary the conditions of their use. The meaning of a mathematical concept does not depend on an interplay of external obligations linked, for example, to the use of a piece of knowledge, a requirement that exists in all didactical commands. Based on this analysis, the main theoretical approach then consists in studying the conditions of setting up a didactical system of situations that involve the student as in non didactical ones. Guy Brousseau refers to these situations as “adidactical”. For him, it is a matter of showing that it is possible to set up adidactical situations and to be aware of their function. Both on a theoretical level (the rule required relation to the knowledge in question) and in the contingency (by examining through observation the conditions of “didactical viability”, in other words their creation within the constraints of the mathematical classroom.
Guy Brousseau puts in evidence that the success of establishing these conditions involves two aspects, which he decides to study more closely.
The first aspect concerns the setting up of the situation itself. This led him to propose a new concept, that of “devolution”: if knowledge pre-exists to students, their understanding requires a common practice obviously expected by the teacher, but that cannot be imposed by him to the students,; that is the paradox of devolution : “If the teacher says what he wants from the student, he can no longer obtain it !” (Brousseau, 1998). Brousseau initially endeavoured to study this paradox in the sixties by looking at the conditions in which this paradox can be overtaken by the devolution to the student of adidactical situations(Which basic strategies can students develop in this situation? Which retroactions will he get from it? What didactical variables are likely to keep the meaning of the target knowledge? The teacher attempts to ensure that the student’s actions are carried out and justified purely by the demands of the milieu and not by the interpretation of the teacher’s didactical behavior nor its expectation.
The second aspect is closely linked to the first since it concerns the conditions for maintaining the students commitment to the situation. Based on a clinical case today well known amongst the community of mathematical didacticians, “The case of Gaël”, Brousseau studied the set of mutual obligations that each partner in the didactical situation imposes or believes to be imposed on the others, and those that are imposed on him or that he believes to be imposed on him, concerning the knowledge in question: this is the concept of “didactical contract”. It corresponds to the outcome of an often implicit negociation of the setting up of the relationship between a student, a certain milieu, and an educational system. This is not a real contract : it is neither explicit nor consented to freely, since it relies upon knowledge necessarily unknown to the students. It positions the teacher and the student to face a truly paradoxical demand : if the teacher explains what he wants the student to do, he can only obtain it as the carrying out of an order and not through using knowledge and judgment. The reverse is also true; if the student accepts that the teacher shows him the solutions and the answers, he will not discover them by himself and therefore will not be able to appropriate it. Learning requires therefore the refusal of the contract in order to tackle the problem independently (devolution). Learning will therefore depend not only on the correct functioning of the contract, but also on the ruptures of it, hence the importance of studying the actual conditions of these ruptures more closely.
Moreover, as an actor in the situation the subject is aware of the knowledge, but this is not enough for it to be learned, because if the students experience is a necessary condition, the knowledge activated must also be recognized as such, then classified and incorporated into socially accepted knowledge. Guy Brousseau thus highlighted the need for “institutionalization” and paved the way for a new field of theorization of educational phenomena.
The theory put to the test by facts : the methods and the COREM^{[9]}
A major concern of Guy Brousseau was to carry out experimental study of the phenomena of mathematical education, a scientific project that was born out of a general schema based on the interaction between the topic studied, understood within the frame work of an adapted theoretical paradigm. In this case, the theory cannot determine what it must be. He provides a model of the facts, summons and brings the phenomena to light in order for it to be analyzed and interpreted in a paper published in 1978, entitled “The observation of didactical facts”, Guy Brousseau provides a solid foundation for the method that was at the heart of his work. Il is constructed around observation applied in the field of didactics: it is then a matter of putting together a collection of facts and constructing them as didactical phenomena, studying their reproducibility and their degree of generalization and consistency.
The COREM, the principle of which had been defined by Guy Brousseau at the end of the sixties and that he was able to be realized with the support of the authorities from 1972, enabled him to conduct this study. These research facilities, unfortunately unique in their kind, continued to function until the end of the nineties. The COREM was the product of joining together a primary school with facilities, welcoming the research and observation of classroom’s situations proposed by the researcher. These situations were designed and constructed, using the theory of didactical situations, upon the questions and hypotheses particular to the research undertaken and on the expertise of the teachers that took the responsibility of the class. The theoretical notion and practice of “didactical engineering” takes into account the workings of a system that depends on a close collaboration between teachers and researchers.
Moreover, in order to back up this scientific research, Guy Brousseau contributed to the development of the use of statistics in research in mathematics education from a heuristics point of view (multidimensional analysis for example) and theoretical hypothesis testing (inferential statistics, descriptive statistics and the investigation of the facts). He contributed, in particular to the creation and use of implicative analysis in didactics. (Gras and Lerman)
The main notions developed in the field of didactics
- The fundamental notion is that of situation; it can be modeled as a formal game. The possibility of isolating in the specially constructed situations, like “the race to twenty (20)^{[10]}” for example, moments of action, moments of formulation, moments oriented towards validation and the tools involved at each of these moments, and finally, moments for institutionalization constituted a major part of the work carried out for more than thirty years on various mathematical topics. This shows both the significance and heuristic value of this theorization and demonstrates the success of Guy Brousseau’s scientific research project.
- The didactical transposition is a concept that was originally developed by Yves Chevallard to explain the transformations that mathematical subjects undergo when made to enter a didactical system. In the paradigm of the theory of situations, this concept is defined and activated by the notion of the fundamental situation for a knowledge, that constitutes a privileged study tool of phenomena involving transposition by defining the conditions for preserving the meaning of knowledge at the moment of transposition.
- The concept of didactical contract, central to the analysis of the workings of the didactical systems, was recently taken up again by Guy Brousseau himself, from the perspective of modeling different types of contracts. Other researchers have studied, from a different perspective, the didactical situations likely to explain why certain students prove to be more sensitive than others to the implicit factors raised by the contract, as well as the links this phenomenon of sensitivity to the contract has with the traditional question of educational differences. (B. Sarrazy)
- The concept of obstacle, taken from the work of the French epistemologist Gaston Bachelard, enabled original approaches to be developed concerning conceptual difficulties and analysis of students’ errors. This concept has been particularly productive in the analysis of the difficulties experienced when moving from whole numbers to decimals.
The proposed distinction between the knowledge actuated in action (C-knowledge), the product of the subject’s activity in his relationship with the milieu and the knowledge acquired in the institutions (S-knowledge) has opened up a new field of study related to the role of enumeration in the construction of numbers (J. Briand) and another concerning the treatment of relationships between spatial knowledge and Euclidean geometry (R. Berthelot, M.-H. Salin).
- The concept of milieu for action and its organization enables one to create a model of the necessary ruptures implemented in the subject’s change of references in a didactical context (distinction between learning situation and didactical situation). This concept, introduced right at the beginning of the theorization of didactical facts, was taken up again and developed by C. Margolinas, in particular to analyze the teacher’s action in ordinary classes.
- Didactical memory is a fundamental concept that enables one to explain phenomena linked to didactical time and it’s progression: conversion of knowledge through the action of institutionalization (J. Centeno).
- The position and the role of institutionalization that consists in laying down components taken from knowledge developed in adidactical situations, contribute to the construction and explicit location of knowledge and thus ensures the establishment of consistency between learning and the teaching objectives set by the institution (A. Rouchier).
- The notion of didactical assortment is more recent. It enables one to study the structuring of the groups of activities and exercises brought together for teaching purposes (F. Genestoux).
The mathematical fields studied
Whether it be directly, through his own work or that of his students or even through work conducted in the paradigm that he set up, Guy Brousseau was interested in all areas of mathematics, notably those covering the curriculum of compulsory education.
The difficulties of learning some standard algorithms of multiplication and division, the aspects of other algorithms, from the point of view of both facilitating the learning process and their use, the early stages of teaching them : the meaning of the operation and construction of the algorithm (G. Brousseau).
The first lessons on numbers and numeration. The fundamental situation of numbers, averages in order to make a set “equipotent” to a given set combined with the use of didactic variables enables one to generate a large number of situations concentrating on action or communication, allowing one to successfully structure the early stages of learning.
The creation of a code of designation in a group context at kindergarten level.
Probably at the end of elementary school: meeting situations in which the early notions of probability are means of decision making (G. Brousseau).
Rational numbers and decimals: fundamental situations and complete early progression constructed following a program that lasts several years (G. Brousseau, N. Brousseau).
The required diversity of contexts and situations in which mathematical reasoning is specified: solving classroom arithmetic problems, situation of multiple choices, etc (P. Gibel, P. Orus, B. Mopondi).
Fixing the position of prior knowledge that has not been formalized and its effective treatment in education: the case of geometry (R. Berthelot, D. Fregona, M-H Salin) , enumeration (J. Briand) and that of reasoning (P. Orus).
The teaching of subtraction and the group of situations set out in the box game (G. Brousseau).
The study of the conditions of the transition from classroom arithmetic to algebra (D. Broin).
The notion of function and the role of graphical representation (P. Alson, I. Bloch, E. Lacasta)
The early stages of proportionality: a fundamental situation based on the notion of equitable share (E. Comin).
An active role in the commitments of a generation
Guy Brousseau’s commitment to mathematical education, and the study of the questions it raises wasn’t only noticeable in the realm of research.
On a national level, he played an extremely important role, notably as part of the Association of Mathematics Teachers, which enabled him to actively contribute to the creation and setting up of the IREM. These are institutions unique in the French institutional context, from which crossed collaboration was developed to serve mathematical education by supporting three areas : research , innovation and teachiers training. It was on his initiative that a national work group was created, which has united trainers of elementary school teachers for thirty years: the COPIRELEM (the Permanent Commission of the IREM for Elementary School).
He also took very active role in the creation of numerous instruments for collective scientific activity, dedicated to the training of young researchers; with the support of Professor Jean Colmez, Guy Brousseau created the first postgraduate course in Didactics in Mathematics in France, to the debate and circulation of ideas : amongst them, one must mention the scientific journal (RDM)^{[11]}, the association of scientists (ARDM)^{[12]}, the Summer School and the National Seminar on Didactics of Mathematics.
One can also notice his commitment on an international level, Guy Brousseau, developing on the work of Caleb Gattegno, Jean Piaget, Willy Servais, Zofia Krygowska, Lucienne Félix, Hans Freudenthal, Ephraïm Fishbein and many other major researchers, was the tireless driving force behind the CIEAEM which he was in charge of for several years and that he kept regular contact with during his summer trips ranging from Switzerland to Mexico, Hungary to Great Britain from 1960 to the beginning of the nineties. Moreover the term of driving force doesn’t fully express the diversity and depth of the work that had to be carried out in a structure that was subject as little as possible to institutional constraints, as was the CIEAEM during the sixties, seventies and eighties. Guy Brousseau equally played a central role in the initial launch of the international group “Psychology of Mathematical Education” at the International Conference of the ICME in 1976 in Karlsruhe. He has been and continues to be regularly invited to contribute to collective works and international scientific conferences concerning mathematical education. Guy Brousseau was awarded an Honorary Doctorate from the University of Montreal in June 1997.
Instruments for teacher activity and training and for the research
Guy Brousseau’s influence goes beyond the realm of research. In the seventies for example in the INRP (National Institute of Educational Research) and in the IREM, numerous teams were formed to develop experimental products for teaching aiming to generalize through books for teachers and student text books.
These products focused mainly, on the one hand on the theoretical setting provided by the theory of didactical situations, and on the other, on numerous suggested situations and problems constructed and studied at the COREM. The recognition of the role and position of the mathematical exercise task as a likely driving force behind learning for the student, the consideration of epistemological and didactical obstacles, methodical focus, emphasis on fundamental situations, attention given to formulation are as much acquired as they are absorbed into the curriculum and practices of French teachers.
Teacher training has always been a concern for Guy Brousseau. The concepts he developed, proven in their ability to further the understanding of didactical actions, have strongly influenced the current syllabus for training elementary school teachers. One also finds this influence in the recruitment process. Indeed, students wishing to become teachers learn to analyze student output and educational documents through concentrating on categories of analysis taken from the theory of didactical situations. One also finds this influence in other moments of the course, moments when the young student teachers learn about other components of their profession: the construction of teaching and learning situations. Finally, through his contribution to the creation of COPIRELEM whose work he closely followed right from the start, he has enabled elementary school mathematics to have at its disposal a unique tool for national organization of teacher training, linked to the IREM and IUFM.
Official citation of the ICMI
The first Felix Klein Award of the Internal Commission on Mathematical Instruction (ICMI) is awarded to Professor Guy Brousseau. This distinction recognises the essential contribution Guy Brousseau has given to the development of mathematics education as a scientific field of research, through his theoretical and experimental work over four decades, and to the sustained effort he has made throughout his professional life to apply the fruits of his research to the mathematics education of both students and teachers.
Born in 1933, Guy Brousseau began his career as an elementary teacher in 1953. In the late sixties, after graduating in mathematics, he entered the University of Bordeaux. In 1986 he earned a 'doctorat d'état,' and in 1991 became a full professor at the newly created University Institute for Teacher Education (IUFM) in Bordeaux, where he worked until 1998. He is now Professor Emeritus at the IUFM of Aquitaine. He is also Doctor Honoris Causa of the University of Montréal.
From the early seventies, Guy Brousseau emerged as one of the leading and most original researchers in the new field of mathematics education, convinced on the one hand that this field must be developed as a genuine field of research, with both fundamental and applied dimensions, and on the other hand that it must remain close to the discipline of mathematics. His notable theoretical achievement was the elaboration of the theory of didactic situations, a theory he initiated in the early seventies, and which he has continued to develop with unfailing energy and creativity. At a time when the dominant vision was cognitive, strongly influenced by the Piagetian epistemology, he stressed that what the field needed for its development was not a purely cognitive theory but one allowing us also to understand the social interactions between students, teachers and knowledge that take place in the classroom and condition what is learned by students and how it can be learned. This is the aim of the theory of didactic situations, which has progressively matured, becoming the impressive and complex theory that it is today. To be sure, this was a collective work, but each time there were substantial advances, the critical source was Guy Brousseau.
This theory, visionary in its integration of epistemological, cognitive and social dimensions, has been a constant source of inspiration for many researchers throughout the world. Its main constructs, such as the concepts of adidactic and didactic situations, of didactic contract, of devolution and institutionalization have been made widely accessible through the translation of Guy Brousseau's principal texts into many different languages and, more recently, the publication by Kluwer in 1997 of the book, 'Theory of didactical situations in mathematics - 1970-1990'.
Although the research Guy Brousseau has inspired currently embraces the entire range of mathematics education from elementary to post-secondary, his major contributions deal with the elementary level, where they cover all mathematical domains from numbers and geometry to probability. Their production owes much to a specific structure – the COREM (Center for Observation and Research in Mathematics Education) – that he created in 1972 and directed until 1997. COREM provided an original organisation of the relationships between theoretical and experimental work.
Guy Brousseau is not only an exceptional and inspired researcher in the field, he is also a scholar who has dedicated his life to mathematics education, tirelessly supporting the development of the field, not only in France but in many countries, supporting new doctoral programs, helping and supervising young international researchers (he supervised more than 50 doctoral theses), contributing in a vital way to the development of mathematical and didactic knowledge of students and teachers. He has been until the nineties intensely involved in the activities of the CIEAEM (Commission Internationale pour l'Etude et l'Amélioration de l'Enseignement des Mathématiques) and he was its secretary from 1981 to 1984. At a national level, he was deeply involved in the experience of the IREMs (Research Institutes in Mathematics Education), from their foundation in the late sixties. He had a decisive influence on the activities and resources these institutes have developed for promoting high quality mathematics training of elementary teachers for more than 30 years.
THE ANTHROPOLOGICAL THEORY OF THE DIDACTIC Floriane Wozniak^{[1]}, Marianna Bosch^{[2]}, Michèle Artaud^{[3]} |
Yves Chevallard’s importance in the field of the didactics of mathematics comes from the singular way he considers teaching and learning of mathematics as much as from the types of empirical data he offers to study. By enlarging the analysis’ scope of didactical phenomena he managed to point out the main constraints that bear on the educational system and built fruitful theoretical and methodological tools. Furthermore, one of his great achievements is to have shown the necessity to associate the analysis of mathematical knowledge with the study of institutional practices, in which these elements of knowledge are created, developed, used, spread, taught and learned.
Trained as a logician, Yves Chevallard started his career as a mathematics researcher in this field in the beginning of the 70s. However, he rapidly focused his interest on questions about the teaching of mathematics, a field of investigation that he discovered while attending a conference by Guy Brousseau in 1976. Inspired by his reading of Michel Foucault, Pierre Bourdieu and Louis Althusser – whom he discovered while attending his lecture at the École Normale Supérieure in Paris – Yves Chevallard chose right from the beginning of his work, to built a didactic theory clearly in the line of the Theory of Didactic Situations, that Guy Brousseau was developing at that time. His originality is to try to take into account the institutional relativity of knowledge, on which he bases his analysis of didactical phenomena. His work in the 80s bears on phenomena that he interprets in the light of the didactic transposition, that will be enlarged from the 90s into the Anthropological Theory of the Didactic (ATD).
At the origins: an emancipative theory
The creative power of Yves Chevallard’s research work lies at first in his epistemological and institutional emancipative positioning with regard to the institutions, in which the elements of knowledge studied by the didactics of mathematics “live”. For him, indeed, things cannot be considered as “here since or for ever”, elements of knowledge are products of human constructions, their place and function vary according to places, societies, and periods of time. An engineer modelling activity on a production chain, a journalist commenting on recent pools, an architect calculating the resistance of some hardware, a teacher teaching addition… all participate socially in the diffusion of mathematical knowledge or know-how, among different groups. In this context, mathematics is made of human activities, produced, spread, managed, taught, among a large variety of social institutions.
Objects studied by researchers in the didactics of mathematics live within institutions, of which they are themselves subjects. This situation necessitates a critical position in order not to assume as given what indeed needs to be questioned. In this sense, the emancipation offered by the ATD lies in its rejection to validate intellectual products naturalised in common culture, in its attention to the relativity of contents and forms of knowledge and its claim that researcher in the didactics of mathematics necessarily need to make a step aside, in order to analyse institutions, of which they are themselves subjects. The ATD offers modelling and analysis tools for these human activities, which allow a control of the implicit constraints that any institution imposes on any practice that it shelters. This explicit search for an epistemological break is what allowed to point out phenomena that could be interpreted in terms of didactic transposition. “Where does knowledge present in different didactical systems originate?” was the first question, whose study gave birth in the 80s to the theory of didactical transposition, for which Yves Chevallard’s name is now famous all around the world (Chevallard, 1985a).
At the origins: the theory of didactic transposition
The theory of didactic transposition questions what seems obvious, about knowledge present in didactical systems (and therefore breaks a certain illusion of transparency), about the fact that identical objects could live under different names, or more generally about the inclination to see only what institutions point out as worth of interest. Looking form a certain distance is the only way to see the effects of the institutions accurately. Mathematical knowledge is most often produced outside school and is subject to a series of adaptations before being accepted for teaching: mathematical objects created by mathematicians are not the ones taught in school. The object of the theory of didactic transposition is precisely to describe and explain the phenomena of transformation of knowledge from its production up to its teaching (Bosch & Gascón, 2005).
This is how the theory of didactic transposition allows the distinction between academic knowledge produced, for instance, by mathematicians, knowledge to be taught defined by the educational system, knowledge taught by the professor and finally knowledge learnt by students. This work of transposition is a social construction made by lots of different persons within various institutions: political authorities, mathematicians, teachers and their associations define the issues of teaching and choose what should be taught, as well as under which form. This level of institutional organisation is what Chevallard calls the “noosphere”, it sets up the limits, redefines and reorganises the knowledge in socially, historically or culturally determined contexts, which make possible or not certain choices. Beside the reference book that Chevallard published for the first time in 1985, La transposition didactique – Du savoir savant au savoir enseigné, which has been reedited and translated in Spanish, several works have studied phenomena of didactic transposition about various mathematical domains: elementary algebra (Chevallard 1985b, Kang 1990, Coulange 2001), proportionality (Bolea et al. 2001, Comin 2002, Hersant 2005), volume (Menotti 2001), geometry (Tavignot 1991, Chevallard & Jullien 1991, Matheron 1993, Bolea 1995), irrational numbers (Assude 1992, Bronner 1997), functions and calculus (Artigue 1993, 1998 ; Ruiz Higueras 1994, 1998 ; Chauvat 1999 ; Amra 2004 ; Barbé et al. 2005), linear algebra (Ahmed and Arsac 1998, Dorier 2000, Gueudet 2000), arithmetic (Ravel 2002), proof (Arsac 1989, Cabassut 2005), modelling (García 2005), statistics (Wozniak 2005), mathematics in economy (Artaud 1993, 1995) ; but also in other disciplines as different as physics (e.g. Johsua 1994), music (e.g. Beaugé 2004), or sport (e.g. Barbot 1998).
Furthermore, the wish to fight against the illusion of transparency has motivated, from the mid 80s, the introduction of the “ecology approach” in didactique of mathematics (Rajoson, 1988). This approach is based upon a set of persistent questions: What does or does not exist? What should exist? What could exist? What are the conditions, which favour, allow or on the contrary make difficult or even prevent the existence of such object? (Artaud, 1997). The answers given to such questions bring to light conditions of existence of mathematics in the educational system, which bear on mathematics itself, as well as on the systems in which they live. Bringing in the notion of ecosystem makes it possible for the researcher in didactique of mathematics to consider in relation with mathematics several new objects outside mathematics. The ecology viewpoint is today an essential positioning in the use of techniques of analysis with tools from the ATD. Its field of intervention has been enlarged and enriched. The various works consisting in determining ecological conditions of existence of mathematical objects have finally led to a structuring schema in nine levels, called levels of didactical codetermination going from the most specific (subject, theme, sector, domain, discipline) to the most generic (pedagogy, school, society, civilisation). This structuring schema has proven to be most productive recently, while bringing into light the most determining elements constraining the didactical systems (Wozniak 2007).
A didactic anthropological theory
The type of questioning generating the theory of didactic transposition calls for a more accurate distinction between objects which seem to be the same, but do not live in the same manner from one institution to another, since they are not used to do the same thing. Moreover, to describe and analyse the genesis and evolution of elements of knowledge in a given institution, as well as personal and institutional relations to these elements, it is necessary to design a model of these elements of knowledge or know-how. The difficulty is that no elements of knowledge can be totally isolated, but is rather always part of an aggregate. Within the ATD, a significant breakthrough came with the modelling of such aggregates in terms of praxeologies made of the two components: praxis and logos. This model came initially from an attempt to describe the mathematical activity in relation with the concept of institutional relations and with use of the notion of ostensif (Chevallard 1994, Bosch & Chevallard 1999).
The notion of praxeology insists on the techniques, which allows to accomplish certain types of tasks, bringing to light the plurality of techniques for one type of task, hidden within the subjection to a didactical system. On the other hand, it insists on the technological function of knowledge (for producing, justifying and making techniques comprehensible). This points out a system of conditions and constraints bearing on the existence or absence of such technique, in such institution. An element of knowledge is before all a discourse making possible to justify, produce, make comprehensible techniques and not only what the culture designates as obvious under the label “knowledge”. In this sense, the praxis refers to the practice, the know-how in some ways, while the logos refers to the theory, the discourse describing, legitimising, explaining, the praxis. Therefore, a praxeology does not encompass the study of human practice, but the “science”, personal or institutional, of a certain practice. It is thus relative to the person using it or to the institution in which it can live. The use of the notion of praxeology gives a fundamental model in order to apprehend the elements of knowledge, to study their transformations, and to give account of what is done with them in any particular institution. It makes explicit the epistemological model of reference, which nourishes the analyses of transposition phenomena.
From the profession of teacher to the epistemological refoundation
Chevallard’s first works, centred on the study of didactical transposition phenomena and the use of the ecology viewpoint, immediately produced elements of knowledge on didactical systems and contents for mathematics teachers’ training. Yves Chevallard developed these contents during in service training sessions, in the context of the IREM^{[4]} of Aix-Marseille, with a constant care for answering the needs of the profession of mathematics teacher. This attention to what is now called the problems of the profession (Cirade, 2006) leads, along with the constitution of a clinic of didactical phenomena, to a development of the theory as well as its practical realization.
As soon as he got a position as professor in the IUFM^{[5]} of Aix-Marseille – into the creation and development of which he has been strongly involved – he set up most of his work in the context of mathematics teachers’ pre-service training, that he has been supervising for the last 15 years. The various didactical designs that he has set up have allowed, through the years, the constitution of a text of professionally orientated knowledge, in terms of “archives for training”.
The research design that he conjointly sets up is one of the main originalities of Yves Chevallard’s activity in research. It is common for a researcher in the didactics of mathematics to use the classroom as a “laboratory” for the study of didactic engineering, by experimenting the didactic situations that he elaborates. Based on his experience as teachers’ trainer in a IUFM, Yves Chevallard sets up, more than a laboratory, a clinic for mathematics classes, their teachers and their students. Innovative training designs are created (Chevallard, 2006), such as one called “the questions of the week”: each trainee-teacher is invited to raise an issue in relation with his own teaching practice. Some of these issues are then debated and studied within the whole group of trainees. These questions of the week, adding up to around a thousand every year, reveal the major problems of a profession in mutation, especially for these repeated year after year.
The whole set of data produced by teachers-trainees and Yves Chevallard’s seminar, – about 450-500 pages every year – constitute the “archives for training” and give to researchers clinical data, which have allowed recently the development of what has become the “clinic for training” (Chevallard 2007, Cirade 2007) in relation with a new approach known as the dialectic of medias and milieux (Chevallard, 2006). One plays against a system, which is not free of didactical intention. The goal is to point out among the “responses” of the system, the elements, which have some chance of not sustaining any intentional strategy, but are only here, like any symptom which is not commanded.
This position of trainer, open to problems of the profession, led Yves Chevallard in the second half of the 90s, to introducing the model of the didactic moments, as a means of analysis of the didactic praxeologies. This means studying and analysing the difficulties of teachers while implementing a new teaching design (called modules) imposed by the French institution (Ministry of Education). Indeed, how can one describe the diffusion and in particular the difficulties of diffusion of didacitc praxeologies in an institution, especially school? How can one explain that a didactic situation cannot ‘live’ in school, that the conditions and constraints on the teacher or school prevent that such didactic situation can ‘live’ in the class? An essential condition is that the elements of knowledge be apprehended from the viewpoint of the raison d’être. Why, for instance, should one teach the properties of triangles? What are the questions that this subject allows to study? In order for school to be able to let these questions live as generative of knowledge, one must act in two directions: epistemological and didactical. Yves Chevallard’s indefectible care for answering the needs of the profession of teacher and of the society led him to the exploration of these two ways (Chevallard, 2002a, 2002b). The first way consists in developing a functional way to approach an element of knowledge, that Yves Chevallard structures in terms of Study and Research Activities (SRA) and more recently in terms of Study and Research Path (SRP). In doing so, he meets one of the central aspect of the Theory of Didactic Situations developed by Guy Brousseau, precisely the conception of fundamental situations. Moreover, the study of didactical systems, leads to the emergence of the notion of moments of the study, each of them corresponding to one specific didactical function in the process of the study. The didactical moments then appear as some types of task for the study. The model of the mathematical organisations in terms of praxéologies and of didactical organisations in terms of moments of the study allows the study of the didactical systems, from the viewpoint of knowledge as well as its activation. Today, the study of the didactical praxéologies constitutes one of the most promising vehicles of development for the ADT, especially in the specific context of the use of new technologies.
In conclusion, one can say that three different ingredients are therefore essential in the theorisation that Yves Chevallard has been conducting in the last thirty years:
a deep anchorage within mathematics,
a willingness for breaking the illusion of transparency (not trusting what the institution put into light and pointing out the conditions explaining what exists or not),
a clinical approach to didactical phenomena, in articulation with their theorisation which complete the experimental approach like most research programs in mathematics education.
Devotion to the community of research in didactique of mathematics
Yves Chevallard has always cared to create the conditions for production and diffusion of research in the didactics of mathematics to the widest audience. In this sense, he has been dircetor of the IREM of Aix-Marseilles between 1984 and 1991. He also took a great part in the creation of the IUFM of Aix-Marseilles, in 1991, being a member of the administrative board from the beginning, as well as director of the scientific and pedagogic council from 1991 until 1999 and director of research and development from 1991 until 1997. He also created and directed the scientific journal Skholê, and has been head of the mathematics department since 1991. Recently he enlarged his audience in the context of the department of education science in the university, in order to claim that “the didactic care is an eminent social duty”.
Yves Chevallard has also been chief editor of the international journal Recherches en didactique des mathématiques from 2000 until 2002, member of the scientific board of the collection Raisons éducatives published by the Faculty of Psychology and Education Science of Geneva University, member of the editorial board of the journal Éducation et didactique recently created. His care for the diffusion of the theoretical framework he has created takes shape in his important participation to juries for doctorates and his electronic publications through Internet (http://yves.chevallard.free.fr/). Yves Chevallard is indeed a prolific researcher, whose list of publication covers over 13 pages: 3 books in French, one translated in Spanish, 1 book in Spanish, also translated in Portuguese, 15 participations to collective books, 36 articles in international journals more than 60 communications in international congress, and many in various seminars.
Outside French speaking countries, Yves Chevallard has close cooperation with Spanish and latino-american countries. The Spanish translation of his book about didactic transposition in Argentina in 1997 has widely contributed to the diffusion of this approach in all parts of education. His book in Spanish (Chevallard, Bosch et Gascón 1997) is about to be diffused by the Ministry of Education in all Mexican schools in pocket edition. The ATD today represents a spreading field of research regrouping about 200 French or Spanish speaking researchers over four continents, Europe, America, Asia and Africa. The two international congresses on the ADT (Baeza, Spain, 2005 and Uzès, France, 2007) are proofs of the dynamics and importance of the projects around which a community of research of ADT is being built. A teachers’ training program set up in Marseilles (since 1990); a project of curricular development encouraged by the Ministry of Education in Chile that mobilises since 2002 a whole team of researchers working with teachers and students of 300 primary schools; a research team about renovation of secondary and tertiary education using mathematical modelling in Spain; and research teams working on different subjects in Latin America, Canada, Vietnam, North Africa, South Africa, and Europe (Belgium, Denmark, France, Italy, Sweden, Switzerland).
Je ne saurais pas dire tout ce que la collaboration avec Yves m’a apporté comme idées et comme plaisir. Sa culture, la précision de sa pensée, son écoute aussi m’ont vraiment « éduqué » sans jamais infléchir mes propres démarches.
These words of recognition addressed by Guy Brousseau to Yves Chevallard during the first international congress on the ADT in Baeza reveal, beyond the friendship of these two exceptional didacticians, the close and original relation that bounds their two theories and therefore the essential place of each of them in French Didactique of mathematics but also in the world of research in Mathematics education.
References
Ahmed, B., Arsac, G. (1998). La conception d’un cours d’algèbre linéaire, Recherches en Didactique des Mathématiques 18/3, 333-386.
Amra, N. (2004). La transposition didactique du concept de function. Comparaison entre les systèmes d’enseignement français et palestinien. Doctoral dissertation, Université Paris 7.
Arsac, G. (1989). Les recherches actuelles sur l’apprentissage de la démonstration et les phénomènes de validation en France, Recherches en Didactique des Mathématiques 9/3, 247-280.
Artaud, M. (1993). La mathématisation en économie comme problème didactique. Une étude de cas. Doctoral dissertation, Université d’Aix-Marseille II.
Artaud, M. (1995). La mathématisation en économie comme problème didactique. La communauté des producteurs de sciences économiques : une communauté d'étude. In Margolinas, C. (eds.) Les débats de didactique des mathématiques. La Pensée Sauvage, Grenoble, 113-129.
Artaud, M. (1997). Introduction à l’approche écologique du didactique. L’écologie des organisations mathématiques et didactiques. In actes de la IXe école d’été de didactique des mathématiques, 101-139.
Artigue M. (1993). Enseignement de l'analyse et fonctions de référence, Repères, 11, 115-139.
Artigue, M. (1998). L'évolution des problématiques en didactique de l'analyse, Recherches en didactique des mathématiques 18/2, 231-261.
Assude, T. (1992). Un phénomène d’arrêt de la transposition didactique. Ecologie de l’objet “racine carrée” et analyse du curriculum. Doctoral dissertation, Université de Grenoble I.
Barbé, Q., Bosch, M., Espinoza, L., Gascón, J. (2005. Didactic restrictions on the teacher’s practice. The case of limits of functions. Educational Studies in Mathematics 59, 235-268.
Barbot, A. (1998). Conception et évaluation d’un projet d’enseignement des sports de combat de préhension et de combat en éducation physique et sportive. Science et motricité 32/33, 88-101.
Beaugé, P. (2004). Transposition didactique de la notion musicale de hauteur. Revue d’Éducation Musicale 22, 1-32.
Bolea, P. (1995). La transposición didáctica de la geometría elemental. In Educación abierta: aspectos didácticos de matemáticas 5, I.C.E. de la Universidad de Zaragoza, 89-126.
Bosch, M., Chevallard, Y. (1999). La sensibilité de l'activité mathématique aux ostensifs. Objet d'étude et problématique, Recherches en didactique des mathématiques 19/1, 77-123.
Bosch, M., Gascón, J. (2005). La praxéologie comme unité d’analyse des processus didactiques’. In Mercier, A., Margolinas, C. (eds.), Balises pour la didactique des mathématiques. La Pensée Sauvage, Grenoble, 107-122.
Bosch, M., Gascón, J. (2006) 25 years of didactic transposition. ICMI Bulletin, 58, 1-2-3, 51-64.
Bronner, A. (1997). Étude didactique des nombres réels. Idécimalité et racine carrée. Doctoral dissertation, Université de Grenoble I.
Cabassut, R. (2005). Démonstration, raisonnement et validation dans l'enseignement secondaire des mathématiques en France et en Allemagne. Doctoral dissertation, Université de Paris VII.
Chauvat, A. (1999). Courbes et fonctions au Collège. Petit x 51, 23-44.
Chevallard Y. (1985a), La transposition didactique - Du savoir savant au savoir enseigné, La Pensée sauvage, Grenoble (126 p.). Deuxième édition augmentée 1991.
Chevallard, Y. (1985b) Le passage de l’arithmétique à l’algébrique dans l’enseignement des mathématiques au collège – Première partie : l’évolution de la transposition didactique, Petit x 5, 51-94.
Chevallard, Y. (1994). Ostensifs et non-ostensifs dans l’activité mathématique, Séminaire de l’Associazione Mathesis, Turin, 3 février 1994, in Actes du Séminaire 1993-1994, 190-200.
Chevallard Y. (2002a), « Organiser l’étude. 1. Structures & fonctions », Actes de la XIe école d’été de didactique des mathématiques (Corps, 21-30 août 2001), La Pensée Sauvage, Grenoble, p. 3-32.
Chevallard Y. (2002b), « Organiser l’étude. 3. Écologie & régulation », Actes de la XIe école d’été de didactique des mathématiques (Corps, 21-30 août 2001), La Pensée Sauvage, Grenoble, p. 41-56.
Chevallard, Y. (2006). Organisation et techniques de formation des enseignants de mathématiques, In Actes du XIIIe colloque CORFEM.
Chevallard, Y, Bosch, M. et Gascon, J. (1997), Estudiar matemáticas. El eslabón perdido entre la enseñanza y el aprendizaje, ICE/Horsori, Barcelone.
Chevallard, Y., Jullien, M. (1991). Autour de l’enseignement de la géométrie, Petit x, 27, 41-76.
Cirade, G. (2006). Devenir professeur de mathématiques : entre problèmes de la profession
et formation en IUFM. Les mathématiques comme problème professionnel. Doctoral dissertation, Université de Provence.
Cirade, G. (2007). Devenir professeur de mathématiques : entre problèmes de la profession et formation en IUFM. Les mathématiques comme problème professionnel. In Actes du séminaire national de didactique des mathématiques (à paraître)
Comin, E. (2002). L’enseignement de la proportionnalité à l’école et au collège, Recherches en Didactique des Mathématiques 22/2.3, 135-182.
Coulange, L. (2001). Enseigner les systèmes d’équations en Troisième. Une étude économique et écologique, Recherches en Didactique des Mathématiques 21/3, 305-353.
Dorier, J. -L. (2000). Recherche en histoire et en didactique des mathématiques sur l'algèbre linéaire. Les cahiers du laboratoire Leibniz 12.
García, J. (2005). La modelización como herramienta de articulación de la matemática escolar. De la proporcionalidad a las relaciones funcionales. Doctoral dissertation, Universidad de Jaén.
Gueudet, G. (2000). Rôle du géométrique dans l’enseignement et l’apprentissage de l’algèbre linéaire. Doctoral dissertation, Université Joseph Fourier, Grenoble.
Hersant, M. (2005). La proportionnalité dans l’enseignement obligatoire en France, d’hier à aujourd’hui, Repères 59, 5-41.
Kang, W. (1990). Didactic Transposition of Mathematical Knowledge in Textbooks. Doctoral dissertation, University of Georgia.
Matheron, Y. (1993). Un étude de la transposition didactique du Théorème de Thalès entre 1964 et 1989. Doctoral dissertation, Université d’Aix-Marseille.
Menotti, G. (2001). De l'influence de la constitution de classes de niveau sur la pratique du professeur de mathématiques : le cas de l'enseignement du volume. In actes de la XIe école d’été de didactique des mathématiques.
Ravel, L. (2002). Arithmétique en Terminale S spécialité mathématiques: Quel(s) enseignement(s)?. Repères 49, 96-116.
Rajoson, L. (1988). L’analyse écologique des conditions et des contraintes dans l’étude des phénomènes de transposition didactique : trois études de cas. Doctoral dissertation, Université Aix-Marseille II.
Ruiz Higueras, L. (1994). Concepciones de los alumnos de Secundaria sobre la noción de función: Análisis epistemológico y didáctico. Doctoral dissertation, Universidad de Jaén.
Ruiz Higueras, L. (1998). La noción de función: Análisis epistemológico y didáctico. Servicio de Publicaciones. Universidad de Jaén.
Tavignot, P. (1991). L’analyse du processus de transposition didactique. Exemple de la symétrie orthogonale au collège. Doctoral dissertation, Université Paris V.
Wozniak, F. (2005). Conditions et contraintes de l’enseignement de la statistique en classe de seconde générale. Un repérage didactique. Doctoral dissertation, Université Claude Bernard Lyon 1.
Wozniak, F. (2007). Conditions and constraints in the teaching of statistics: the scale of levels of determination. In actes de CERME 5, (à paraitre).
François Conne |
French didactics of mathematics is hugely indebted to Gérard Vergnaud, the French “ critical thinker ” who dedicated himself, internally, to counterbalancing the weight of “utopian thinkers, and externally, did much to indicate the possible links between these new theories and those which were current elsewhere.
Utopian theory, critical theory in the field of didactics
Work on theory is necessary if only to put some order into the wealth of ideas, research works, experiments and observations, and there are at least two ways in which a researcher can contribute to it. He can either set up a personal framework in the aim of reconsidering the field in its whole, hoping in this way that a new order will emerge. This research mode has a formalising aim, and mathematicians are particularly at ease in this mode. For the sake of simplification, let us give it the name of utopian theory. It consists in developing a relatively detached construction which can be used as a framework for the study and transformation of practices of the dissemination of knowledge. According to the other mode of theory, which we will qualify as critical theory, the researcher intervenes directly on the state of the field and its multiple ramifications, taking an interest in the interplay between the numerous links built up in didactic systems, and how these systems develop and are renewed. The critical researcher dedicates himself to questioning existing theories in order to redirect them to a constantly changing reality whose transformations owe little to our speculations.
These two theories are in contrast. They are unfortunately only too often placed in sterile competition. However, each is necessary to the other. On one hand, holders of utopian theories will never be able to burn all their bridges. Their ideas and concepts will always be deformed by their combination with ideas and concepts stemming from other theoretical horizons. Moreover, any order which emerges will always be shown to have had a precursor. A new theory can claim to represent a rupture, it will however inevitably either disappear or become part of the tradition. In short, no theory can be self-sufficient and its assimilation into its field requires a theory of the second order. On the other hand, just as trees grow both through their branches and their roots, a scientific construction can only expand if it reworks its own foundations. Hence the search for new systematic natures, however utopian they may appear, offer new perspectives to critical theory.
A third mode of theory is more directed towards experimentation. In psychology, as far as questions of development and learning are concerned, Piagetian research is impressive. Unfortunately the practices developed by this school, the Piagetian clinical interview (sometimes also called critical), is no longer in vogue. Such a regrettable fate demonstrates that this perspective cannot, no more than the other two, be self-sufficient. Moreover, such fragility in experimental work in psychology is nothing compared to that which characterises research in didactics. For instance, it is far more difficult to set up real laboratories in didactics, as they must be set up in training structures. However, these centres are already monopolised by many other missions than the possibility of proposing ground for research. On top of that, a researcher cannot totally avoid the strong time pressure to which any teaching is subjected. The fact that the experimental Jules Michelet school and the COREM directed by Guy Brousseau remained an exception and were not able to survive is witness to this fact.
Gérard Vergnaud, a critical thinker, psychologist and didactician
Two facts probably explain the position of Gérard Vergnaud in the field of mathematics education and more particularly in that of didactics. First of all, Gérard Vergnaud is a psychologist. As a result, as a psychologist benefiting from peer recognition, his theoretical work has been able to become a critique of existing psychological theories which address questions of epistemology and teaching. Secondly, for Gérard Vergnaud, the adequacy of theory is what is required for any effective action on reality. Consequently, although psychology aims to contribute to questions of didactics, its theories must leave themselves open to questioning by didactic reality. In order to allow themselves to be convinced of this, readers may refer to articles published by Gérard Vergnaud and note the frequency of the interrogative in their titles. The relevance of psychological theories can be measured more through their propensity to evolve and revise themselves in relation to didactic realities than through the benefits that pedagogues, teachers, educators or parents may obtain by referring to them. Hence, for Gérard Vergnaud, on one hand didactic reality provide him with information and push him to adapt his theories incessantly, and on the other hand the bridges and links he sets up between researchers in contact with different realities allow his critical work to gain the general nature required by his theoretical sight.
One might assume that questions of didactics are marginal in relation to psychology as such, and that any researcher in psychology would be better off focusing on more purely psychological realities. Two arguments may be presented to counter this notion. The first is pragmatic: an exclusive focusing of this nature would only put off the study of crucial questions without any justification, without anyone being able to state that psychology really cannot understand or benefit from the study of didactic questions. The second is methodological: theory must be in adequacy with reality in a general way, otherwise it will only be in adequacy with its “own” reality, a convenient reality. By rendering itself in this way almost impossible to qualify, it will soon become insignificant. From this ensues strong sensitivity of any critical theory towards marginal questions.
For Jean Piaget, seeking answers to epistemological questions through research in psychology already consisted in taking position on its edges. The remarkable thing about Gérard Vergnaud’s work is not only that he took the opposite approach, but that this approach was taken following two different dimensions: 1/ developing psychological theories in response to questions raised by didactic realities, and 2/ undertaking such development along the lines of a theory which is general enough to respond to both academic and professional questions, which may be of interest to schools as well as businesses. Hence Gérard Vergnaud’s critical theory addresses both psychology and didactics, both academic and professional education, both on the development of children and adults.
Gérard Vergnaud’s career is also distinguished by his terrific entrepreneurial spirit: he is the initiator of numerous movements and gatherings of researchers on the international scene. We shall only quote the following here: The International Group for the Psychology of Mathematical Education – PME – of which he is a co-founder (ICME3, 1976) and of which he was president from 1977 and 1982, or from 1977, the Séminaire National de Didactique des Mathématiques (French National Mathematics Didactics Seminar) in Paris, then from 1980, l’Ecole d’Eté de Didactique des Mathématiques (Mathematics Didactics Summer School) as well as the Recherches en Didactique des Mathématiques – RDM – journal. His influence is great in the French-speaking sphere (he is for example Dr Honoris causa of the University of Geneva), but he also maintains numerous collaborations both in the west, in North and South America, and in the east (he is for example a member of the Russian Academy of Psychological Sciences). For his numerous other contributions we refer the reader to the Curriculum Vitae in the appendix.
Development of psychology brought about by the problems raised in rendering it operational
Gérard Vergnaud’s psychology explains to what extent action and its organisation are at the heart of of conceptualisation. The idea of a continuity between the most elementary actions of a subject and the most highly developed conceptualisations of science was proposed and firmly supported before Vergnaud by Jean Piaget. Gérard Vergnaud has brought this notion up to date by obliging his psychological theory to set up a dialectic between on one hand its operational contributions, and on the other hand conceptual contributions. (Why can’t research in psychology do without didactics and epistemology? Article published 2001). G. Vergnaud’s work offers proof of the pertinence of his point of view, as, over the years, it has demonstrated its ability to articulate highly varied disciplinary approaches, with incomparable ease and elegance.
Gérard Vergnaud is a developmental psychologist. Inspired by Jean Piaget, he has inflected the latter’s theoretical framework, highlighting the importance of learning contents in development and the role of mediation. This led him to Lev Vygotski, but with the aim of synthesis rather than confrontation. What strikes one most about his work is that it shows, through acts, how developmental psychology contributes to the development of psychology itself. What greater homage could a pupil pay to the Genevan master?
At the start of his career, Gérard Vergnaud approached questions about the teaching of mathematics in the manner of a specification of the results of genetic epistemology to the school context and particular mathematical contents. His perspective remained developmental. However, there was no longer question of major structures of intelligence linked to the most general logical-mathematical concepts such as number, space, function etc, but instead an effort aimed at defining the general and distant questions of school teaching and learning, in order to render it usable by teachers. The order he considered was no longer Jean Piaget’s over-rigid theory of states, but was designed as a partial order in development. This idea allowed questions relative to cognitive development to be opened up to the case of adults. This was to be defined in research applied to the classification of learning situations (Essai de classification des situations d’apprentissage, article 1964), then to the idea of psychogenetic complexity compared to additive structures (Structures additives et complexité psychogénétique, article 1976),or to the relationship between psychogenesis and hierarchies of the difficulty of school tasks (Psychogenèse et programmes d’enseignement : différents aspects de la notion de hiérarchie, article 1976-77). Gérard Vergnaud has remained faithful to the spirit of Piaget as he continuously makes parallels between the structure of mathematical contents and a pupil’s learning progress and development of knowledge. However, in contrast to Genevan researchers close to Bärbel Inhelder, he did not undertake his work of definition on the study of details which would have pushed his research towards microgenetic phenomena. He limited himself to categories of knowledge calibrated on school practice. This is what marks his commitment as a didactician. In this regard it is significant that Gérard Vergnaud persistently defended the fact that progress in learning at school should be considered in the long term, (ex. Le Long terme et le court terme dans l’apprentissage de l’algèbre, article 1988 ; Algebra, Additive and Multiplicative Structures. Is there any coherence at early secondary level?, article 1997), hence his insistence on longitudinal research both in psychology and didactics. Gérard Vergnaud’s psychology remained epistemological. His major work on this type of research is an article which he co-signed with Mme C. Durand, which we cited above: Structures additives et complexité psychogénétique.
Gérard Vergnaud has never renounced the strong hypothesis which considers the link between the genesis of acquired knowledge and the structure of mathematical knowledge to be central. This led him to take an interest more precisely in a relational logic, in the psychological concept of representation, and the mathematical concept of homomorphism, putting forward that what renders representation operational is precisely the fact that it is homomorphic, allowing subjects to act on its comparisons themselves. This prevented him from falling into the familiar trap of representation considered as the mental reflection of reality, or conversely as a formatting of reality based on models implemented in the mind. Before Vergnaud, Jean Piaget leant heavily on the idea of invariant structure which mathematicians had highlighted (in particular Erlangen’s programme), and which he transposed to his theory of the development of intelligence. Gérard Vergnaud has updated these notions in light of developments in Piagetian research, once again through a reversal of perspective: whereas Jean Piaget aimed to qualify the structure of intelligence, and to explain the stability of a subject’s acquired knowledge beyond the apparent fluctuations of reality, Gérard Vergnaud has focused on the description of how the acquisition of knowledge allows the learner to order and stabilise reality himself, first and foremost the effect of his acts on reality. At the same time he turned his attention to the structural pair: operational invariant/theorem in action, and on the functional pair: scheme/algorithm. This gave rise to numerous works on calculus and the learning of algebra. We may quote here the following articles: Calcul relationnel et representation calculable (article 1974-75); Invariants quantitatifs, qualitatifs et relationnels (article 1976-77) ; Homomorphisme reel-représentation et signifié-signifiant: exemples en mathématiques (paper published in Russian 1995) ; A comprehensive Theory of Representation for Mathematics Education, (paper published 1999) ; as well as Concept et scheme dans une théorie opératoire de la representation (article 1985). This last article is the most important of all.
His commitment to the emerging movement of French didactics of mathematics was to turn his work towards new categories of reality and its knowledge: situations and concepts. This caused him to insist on the importance of conceptualisation in learning. (Au fond de l’apprentissage, la conceptualisation, paper published 1996; Qu’apportent les systèmes de signes à la conceptualisation? paper published 2002 ; Conceptualisation , clé de voûte des rapports entre pratique et théorie, paper published 2003), and was to provide the corner stone of his Theorie des Champs Conceptuels (Theory of Conceptual Fields) which considers any concept to be a triple of three sets, and I quote (La Théorie des Champs Conceptuels, article 1991):
The idea of thinking in terms of conceptual fields considers that a concept never concerns a single type of situation but several, and that reciprocally, a situation always presents different interlinked conceptual facets. This is of capital importance as far as the teaching of mathematics is concerned, as on one hand it raises doubts about the operationality of any didactics which may divide up its objects too finely, and on the other hand it pleads that the long term processes of conceptualisation be seriously taken into account in the programming of school learning. Apart from the key article, quoted above, mention should be made of research which formed the primary support of his theory: that led with his team on the notion of volume (Didactique et acquisitions de la notion de volume, Représentation du volume et arithmétisation : entretiens individuels avec des élèves de 11 à 15 ans, & Une expérience didactique sur le concept de volume en classe de 5ème, 12-13 ans – article 1983) as well as less well-known but no less capital research on the comparison of the representation of data on temporal or spatial scales (Les fonctions de l’action et de la symbolisation dans la formation des connaissances chez l’enfant, collective work 1987).
After this time Gérard Vergnaud’s career turned towards questions of development and training of adults (La didactique a-t-elle un sens pour la formation des personnes peu qualifiées et peu motivées ?, paper published 1995), in which he launched the setting up of professional didactics (creation of a coordinated research group in didactics, Greco: professional groupe didactique). He also set up exchanges with the business world (Crin club "Evolution du travail et développement des compétences" (Evolution at work and skill development), whichbrings together business people, consultants and researchers, in order to construct research objects. See for example La forme opératoire de la connaissance : un beau sujet de recherche fondamentale et appliquées ? – paper published 1999 – or the writing of Volume 3 Les conditions de mise en oeuvre de la démarche compétence, journées internationales de la formation MEDEF, France). It was the impulse of the most urgent problems in this field, and following the profound changes that the world of work is undergoing, with on one side serious problems of professional mobility and redeployment of workers, and on the other side the pressure to specify skills in a company which led him to consider the question of skills as central (creation of the Association pour la recherche sur le développement des compétences, ARDéCO). Particular note should be taken of Compétence et connaissance théorique (paper published 1998) ; Les conditions de mise en œuvre de la démarche compétence (paper published 1998) ; Compétence, conceptualisation et représentation (paper published 1999).
Gérard Vergnaud’s major contributions to the development of a psychology useful to didactics
According to Gérard Vergnaud, 1/ there are two forms of acquired knowledge : predicative and operative. The terms used are important: predicative, not discursive (as there is predication in the knowledge in action); operative, not pragmatic, as pragmatism, except in its Peirician acceptation known as “pragmaticism”, tends to subject acquired knowledge to its usefulness. 2/ Of these two forms of knowledge, the operative form is the first. (Forme opératoire et forme prédicative de la connaissance, article 2001) His aim is to see how these two forms of knowledge exchange and interact in development and learning. It is probably here that he found inspiration in the work of Lev Vygotski. His position in relation to psychology led him to revise Piagetian concepts principally on the following points :
a) The question of representation and symbolic representations. Piaget studied the question of symbolic formation, but not the place of the symbol in the construction of acquired knowledge. Vergnaud’s synthesis with Vygotsky’s approach was established on this basis, subsequently giving rise to his pupils’ developments on the question of instruments, instrumentation and instrumentalisation in work.
b) The substitution of subject/situation and perceptive/gestural pairs for respectively subject/object and sensorial/motor pairs. This point is essential as far as didactics is concerned, where the importance of situations as a basis for learning is no longer in any doubt.
c) The question of operational invariants and their translation into theorems in action and concepts in action. His approach links these questions to those of anticipation and awareness, both of which closely affect didactics whether disciplinary or professional and on which psychology alone can shed light.
d) Gérard Vergnaud remains however strongly attached to the principle that research in psychology should not completely abandon an epistemological perspective in the broadest sense, as it represents a condition of its relevance to didactic questions. He restores the relationship between scheme and conceptualisation to its position in the paradigm of conceptualisation in action. The Theory of Conceptual Fields provides him with a framework to do this and also allows a bridge to be set up between academic (subject-based) and professional fields, which, from this point of view give rise to the same questions, problems and challenges. Indeed, the notion of conceptual fields, in the area of school learning, is opposed to subject-based fields, just as, in the professional area it is opposed to professional fields structured around academic or professional practice built on the basis of transposed knowledge. The theory of concept, as formulated by Gérard Vergnaud, allows this difficulty to be overcome. For him, a concept is a triple made up of operational invariants, situations and systems of signifiers. Imagining the concept in its relationship to situations (taken in a broader sense than that which G. Brousseau uses in his theory of situations) allows a link to be made between professional and conceptual fields, in that what arises from a frequently unorganised variety becomes, through a transformation into a conceptual field, an ordered variation concerning situations.
Unlike most didacticians in mathematics, on one hand he has approached this discipline from the point of view of a developmental psychologist, which led him to be careful of those who stressed strongly the specific nature of a single discipline. On the other hand, unlike occasional research into psychology or didactics, Gérard Vergnaud has tried to keep the developmental approach by setting himself “large” research objects: conceptual fields, which allow a longitudinal approach. The concept of conceptual field has allowed him to continue to maintain this approach in his empirical research. It has also allowed him to design, outside didactics of mathematics, a theoretical framework which could be applied to a wide variety of areas, including work. As a result the idea of development is enlarged, and henceforth applied as much to adults, in particular in their professional life, as to children, in particular in their school life.
Annexe
Curriculum Vitae of Gérard Vergnaud and list of his publications (2003)
Diplomas
Career in CNRS
Main responsibilities
Participation to institutional councils and comities
In CNRS
In other institutions
Présidences
Comités éditoriaux ou scientifiques
Organisation de Conférences, Colloques et Ecoles d'Eté (responsable ou co-responsable)
Direction of Theses
LIST OF PUBLICATIONS
This list only takes into account books and contributions to collective books, articles in international journals and published communications.
1. BOOKS
2. CONTRIBUTIONS TO COLLECTIVE BOOKS
3. ARTICLES IN INTERNATIONAL JOURNALS
4. PUBLISHED COMMUNICATIONS