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WORKING WITH THE
ANTHROPOLOGICAL THEORY OF THE
DIDACTIC IN MATHEMATICS
EDUCATION
This book presents the main research veins developed within the framework of the anthro-
pological theory of the didactic (ATD), a paradigm that originated in French didactics of
mathematics. While a great number of publications on ATD are available in French and Spanish,
Working with the Anthropological Theory of the Didactic in Mathematics Education is the first directed at
English-speaking international audiences.
Written and edited by leading researchers in ATD, the book covers all aspects of ATD
theory and practice, including teaching applications. The chapters feature the most relevant
and recent investigations presented at the 6th international conference on the ATD, offering
a unique opportunity for an international audience interested in the study of mathematics
teaching and learning to keep in touch with advances in educational research. The book is
divided into four sections and the contributions explore key topics such as:
The core concept of ‘praxeology’, including its development and functionalities.
The need for new teaching praxeologies in the paradigm of questioning the world.
The impact of ATD on the teaching profession and the education of teachers.
This is the second volume in the New Perspectives on Research in Mathematics Education
series. This comprehensive casebook is an indispensable resource for researchers, teachers and
graduate students around the world.
Marianna Bosch is Professor at Universitat Ramon Llull, Spain. She is an experienced
researcher in ATD with a long involvement in the dissemination of research to the
international audience.
Yves Chevallard is Professor Emeritus at Aix-Marseille Université, France and the main
developer of the ATD.
Francisco Javier García is a researcher and lecturer at Universidad de Jaén, Spain.
John Monaghan is Emeritus Professor at the University of Leeds, UK, and Professor at the
University of Agder, Norway.
NEW PERSPECTIVES ON RESEARCH IN MATHEMATICS
EDUCATION –ERME SERIES
Editors of the ERME Series
Viviane Durand-Guerrier (France)
Konrad Krainer (Austria)
Susanne Prediger (Germany)
Naďa Vondrová (Czech Republic)
International Advisory Board of the ERME Series
Marcelo Borba (Brazil)
Fou-Lai Lin (Taiwan)
Merrilyn Goos (Australia and Ireland)
Barbara Jaworski (Europe, United Kingdom)
Chris Rasmussen (United States of America)
Anna Sierpinska (Canada)
ERME, the European Society for Research in Mathematics Education, is a growing society
of about 900 researchers from all over Europe and beyond. In the ERME community and
beyond, a growing body of substantial research on mathematics education raises which is
shaped by the ERME spirit of communication, cooperation, and collaboration.
The contributions in the ERME Series seek to understand and improve learning and
teaching of mathematics at schools, colleges and universities, as well as in informal settings
(e.g., related to street mathematics or to self-organized networks of teachers). The ERME
Series puts an emphasis on reflecting the institutional, societal, and cultural contexts of
learners, teachers and researchers and how this context shapes research and adopts a variety
of perspectives on its research field.
The volumes are written by and for European researchers, but also by and for researchers
from all over the world. An international advisory board guarantees that ERME stays
globally connected. A rigorous and constructive review procedure guarantees a high quality
of the series.
The series aims to provide a range of books –from individual research monographs and
edited collections to textbooks and supplemental reading for scholars, researchers, policy
analysts and students.
Developing Research in Mathematics Education
Twenty Years of Communication, Cooperation and Collaboration in Europe
Edited by Tommy Dreyfus, Michele Artigue, Despina Potari, Susanne Prediger, Kenneth Ruthven
Working with the Anthropological Theory of the Didactic in Mathematics
Education
A comprehensive casebook
Edited by Marianna Bosch, Yves Chevallard, Francisco Javier García, John Monaghan
For more information about this series, please visit: https://www.routledge.com/European-
Research-in-Mathematics-Education/book-series/ERME.
WORKING WITH THE
ANTHROPOLOGICAL
THEORY OF THE
DIDACTIC IN
MATHEMATICS
EDUCATION
A comprehensive casebook
Edited by Marianna Bosch, Yves Chevallard,
Francisco Javier García and John Monaghan
First published 2020
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
52 Vanderbilt Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2020 selection and editorial matter, Marianna Bosch, Yves Chevallard, F. Javier
García, John Monaghan; individual chapters, the contributors
The right of Marianna Bosch, Yves Chevallard, F. Javier García, John Monaghan
to be identified as the authors of the editorial material, and of the authors for
their individual chapters, has been asserted in accordance with sections 77 and 78
of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in any
information storage or retrieval system, without permission in writing from the
publishers.
Trademark notice: Product or corporate names may be trademarks or registered
trademarks, and are used only for identification and explanation without intent to
infringe.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Names: Bosch, Marianna, editor.
Title: Working with the anthropological theory of the didactic in mathematics
education : a comprehensive casebook / edited by Marianna Bosch, Yves
Chevallard, F. Javier García, John Monaghan.
Description: Abingdon, Oxon ; New York, NY : Routledge, 2020. | Series:
New perspectives on research in mathematics | Includes bibliographical
references and index.
Identifiers: LCCN 2019023790 (print) | LCCN 2019023791 (ebook) | ISBN
9780367187712 (hardback) | ISBN 9780367187705 (paperback) | ISBN
9780429198168 (ebook)
Subjects: LCSH: Mathematics–Study and teaching–Research–Congresses.
Classification: LCC QA11.2 .W675 2020 (print) | LCC QA11.2 (ebook) |
DDC 510.71–dc23
LC record available at https://lccn.loc.gov/2019023790
LC ebook record available at https://lccn.loc.gov/2019023791
ISBN: 978-0-367-18771-2 (hbk)
ISBN: 978-0-367-18770-5 (pbk)
ISBN: 978-0-429-19816-8 (ebk)
Typeset in Bembo
by Taylor & Francis Books
CONTENTS
List of illustrations vii
List of Contributors ix
Introduction: An invitation to the ATD xii
A short (and somewhat subjective) glossary of the ATD xviii
Yves Chevallard, with Marianna Bosch
Acknowledgements xxxviii
PART 1
Unity in Diversity: ATD at Work 1
1 What kind of results can be rationally justified in didactics? 3
Josep Gascón and Pedro Nicolás
2 Research on ATD outside mathematics 12
Caroline Ladage, Marianne Achiam and Martha Marandino
PART 2
Praxeological and Didactic Analysis 31
3 Praxeologies to be taught and praxeologies for teaching: A
delicate frontier 33
Michèle Artaud
4 Developments and functionalities in the praxeological model 41
Hamid Chaachoua, Annie Bessot, Avenilde Romo and Corine Castela
5 Research on negative numbers in school algebra 61
Eva Cid, José M. Muñoz-Escolano and Noemí Ruiz-Munzón
6 The phenomenotechnical potential of reference epistemological
models: The case of elementary differential calculus 77
Catarina Lucas, Cecilio Fonseca, Josep Gascón and Maggy Schneider
PART 3
Questioning the World 99
7 The Second Spanish Republic and the project method: A view
from the ATD 101
Encarna Sánchez-Jiménez, Dolores Carrillo Gallego, Yves Chevallard
and Marianna Bosch
8 The ecology of study and research paths in upper secondary
school: The cases of Denmark and Japan 118
Britta Jessen, Koji Otaki, Takeshi Miyakawa, Hiroaki Hamanaka,
Tatsuya Mizoguchi, Yusuke Shinno and Carl Winsløw
9 The need for new teaching praxeologies in the paradigm of
questioning the world 139
Jean-Pierre Bourgade, Karine Bernad and Yves Matheron
PART 4
ATD and the Teaching Profession 157
10 The study of teachers’mathematical and didactic praxeologies as
a tool for teacher education 159
Gisèle Cirade and Anne Crumière
11 The education of prospective early childhood teachers within
the paradigm of questioning the world 169
Francisco Javier García, Tomás Ángel Sierra, Mercedes Hidalgo and
Esther Rodríguez
12 The education of school and university teachers within the
paradigm of questioning the world 189
Berta Barquero, Ignasi Florensa and Alicia Ruiz-Olarría
13 On the contributions of the ATD to the teaching profession 213
Klaus Rasmussen, Kaj Østergaard and André Pressiat
Index 232
vi Contents
ILLUSTRATIONS
Figures
3.1 The scale of levels of didactic codeterminacy 39
4.1 Example of a hierarchic tree for a subset of the values of
a variable 48
4.2 Example of a hierarchic tree for the set of values of a variable 48
4.3 From I
r
to I
p
, the transpositive effects model 57
6.1 System given by a set of numerical data and a graph 83
6.2 Structure of the activity diagram that redefines
functional modelling 85
6.3 Example of a functional modelling process in the discrete field 86
6.4 Example of a functional modelling process in the
continuous field 87
6.5 Example of an MP that allows the step from discrete
to continuous 88
6.6 Components of the activity diagram of functional modelling
worked in the reference institution 90
8.1 The scale of levels of didactic codeterminacy 120
11.1 A model of a study process according to the questioning of the
world paradigm 172
11.2 Reformulation of the teacher education problem in the ATD:
teachers’praxeological equipment develops from exploring
professional questions, supported by specific education devices 173
11.3 Teaching situation 1 about numbers in ECE 178
11.4 Teaching situation 2 about numbers in ECE 179
12.1 Instructions to build the box and examples of the
resulting boxes 196
12.2 Pre-algebraic-geometrical models for the box and the lid 198
12.3 Work of one of the groups regarding the accumulative with
decreasing amount savings plans 203
12.4 Data used for the experienced SRP 207
12.5 Question–answer map of one of the groups 208
13.1 PISA 2003 item: “Bricks”215
13.2 Oblique parallel projection 216
13.3 Cavalier perspective of a cube with α= 45° 217
13.4 Parallel projections 219
13.5 Affine map transforming a parallelogram into a parallelogram 219
13.6 Transvections, two documents 220
13.7 Cavalier perspective: an experiment in a grade 6 class 221
Tables
5.1 Phase 1 of the didactic sequence 71
5.2 Phase 2 of the didactic sequence 72
5.3 Phase 3 of the didactic sequence 73
5.4 Phase 4 of the didactic sequence 74
11.1 Possible professional questions for ECE teachers 175
12.1 SRPs-TE implemented at different school levels 192
Boxes
11.1 Teaching situation 1 178
11.2 Teaching situation 2: “Laying the table”179
viii List of illustrations
CONTRIBUTORS
Marianne Achiam, Department of Science Education, University of Copenhagen,
Denmark
Michèle Artaud, ADEF, Aix-Marseille Université, France
Berta Barquero, Faculty of Education, Universitat de Barcelona, Spain
Karine Bernad, Doctor in Sciences of Education, Université d’Aix-Marseille,
France
Annie Bessot, Laboratoire Informatique de Grenoble, Université Grenoble Alpes,
France
Marianna Bosch, IQS School of Management, Universitat Ramon Llull, Spain
Jean-Pierre Bourgade, ADEF, Aix-Marseille Université, France
Dolores Carrillo Gallego, Dpto. de Didáctica de las Ciencias Matemáticas y
Sociales, Universidad de Murcia, Spain
Corine Castela, LDAR member Emeritus, Universités de Rouen, Paris Diderot,
Paris-Est Créteil, Artois et Cergy Pontoise, France
Hamid Chaachoua, Laboratoire Informatique de Grenoble, Université Grenoble
Alpes, France
Yves Chevallard, Professor Emeritus, Aix-Marseille Université, France
Eva Cid, Departamento de Matemáticas, Universidad de Zaragoza, Spain
Gisèle Cirade, UMR EFTS, ESPE Toulouse Midi-Pyrénées, Université Toulouse
- Jean Jaurès, Université de Toulouse, France
Anne Crumière, UMR EFTS, ESPE Toulouse Midi-Pyrénées, Université
Toulouse - Jean Jaurès, Université de Toulouse, France
Ignasi Florensa, Escola Universitària Salesiana de Sarrià, UAB, Universitat
Autònoma de Barcelona, Spain
Cecilio Fonseca, Departamento de Matemática Aplicada I, Universidad de
Vigo, Spain
Francisco Javier García, Departamento de Didáctica de las Ciencias, Universidada
de Jaén, Spain
Josep Gascón, Departament de Matemàtiques, Universitat Autònoma de Barcelona,
Spain
Hiroaki Hamanaka, Department of Education in Mathematics and Natural
Sciences, Hyogo University of Teacher Education, Japan
Mercedes Hidalgo, Departamento de Didáctica de Ciencias Experimentales,
Sociales y Matemáticas, Universidad Complutense de Madrid, Spain
Britta Jessen, Department of Science Education, University of Copenhagen,
Denmark
Caroline Ladage, EA4671 ADEF, Aix-Marseille Université, France
Catarina Lucas, Institute of Public Health (ISPUP), University of Porto, Portugal
Martha Marandino, Faculdade de Educação, Universidade de São Paulo, Brasil
Yves Matheron, Institut de Mathématiques de Marseille, UMR 7373, Institut
Français de l’Éducation, France
Takeshi Miyakawa, School of Education, Waseda University, Japan
Tatsuya Mizoguchi, Department of Education, Tottori University, Japan
John Monaghan, Leeds University and Agder University, United Kingdom and
Norway
xList of Contributors
José M. Muñoz-Escolano, Departamento de Matemáticas, Universidad de Zar-
agoza, Spain
Pedro Nicolás, Dpto. de Didáctica de las Ciencias Matemáticas y Sociales,
Universidad de Murcia, Spain
Kaj Østergaard, VIA University College, Denmark
Koji Otaki, Department of Teachers Training, Hokkaido University of Education,
Japan
André Pressiat, Professor Emeritus, Université Paris Diderot, France
Klaus Rasmussen, University of Copenhagen & University College Copenhagen,
Denmark
Esther Rodríguez, Departamento de Investigación y Psicología en Educación,
Universidad Complutense de Madrid, Spain
Avenilde Romo, CICATA, Instituto Politécnico Nacional, Mexico
Noemí Ruiz-Munzón, ESCSE Tecnocampus , Universitat Pompeu Fabra, Spain
Alicia Ruiz-Olarría, Departamento de Didácticas Específicas, Universidad
Autónoma de Madrid, Spain
Encarna Sánchez-Jiménez, Dpto. de Didáctica de las Ciencias Matemáticas y
Sociales, Universidad de Murcia, Spain
Maggy Schneider, Institut de mathématique, Université de Liège, Belgium
Yusuke Shinno, Department of Mathematics Education, Osaka Kyoiku University,
Japan
Tomás Ángel Sierra, Departamento de Didáctica de Ciencias Experimentales,
Sociales y Matemáticas, Universidad Complutense de Madrid, Spain
Carl Winsløw, Department of Science Education, University of Copenhagen,
Denmark
List of Contributors xi
INTRODUCTION: AN INVITATION TO
THE ATD
The anthropological theory of the didactic (ATD) was the name Yves Chevallard
gave to the research framework in the didactics of mathematics developed since the
1980s with the first investigations of didactic transposition (Chevallard, 1985/
1991). This framework was born in the project of a “science of the didactic”pro-
posed by Guy Brousseau and initiated by his theory of didactic situations (TDS)
(Brousseau, 1997). The filiation of the ATD towards the TDS is doubtless, as well
as the relations to the theory of conceptual fields due to Gérard Vergnaud (1990).
What these three research frames have in common is their explicit aim to experi-
mentally model the knowledge and know-how that is at the core of teaching and
learning processes. This is why Josep Gascón (1993) proposed speaking of the
“epistemological paradigm”in didactics of mathematics, to distinguish it from
what was the prevailing paradigm at this period, more focused on the study of
learners’—and lately teachers’—cognitive processes.
This concise chapter is an introduction—or better, an invitation, in the style of
Peter L. Berger (1963)—to the ATD, its methods and conceptualisations, the pro-
blems it raises and intends to approach, and also its main assumptions and founda-
tions. It is certainly wise to start with a warning, in case the reader is not aware of
it. The conceptual framework developed by the ATD is broad and not always
intuitive. This is the consequence of a main methodological principle the ATD
shares with all social sciences, the emancipatory principle, which consists in avoiding
taking for granted the elements of the social world we are studying, a world we
know very well because of our experience as students and citizens and sometimes
also as teachers, educators or parents. Trying to control one’s assumptions is a basic
gesture in scientific work. The strategy proposed by the ATD consists in setting
forth a wide set of basic notions used to model—or conceptually reconstruct—the
didactic world in a “fresh”perspective, to avoid being contaminated by the visions
of the persons and institutions that are part of this world.
Using a specific terminology and specific notations to put this terminology at
work might be initially disturbing for the neophytes. However, this very dis-
turbance is part of the implicit assumptions—or prejudices—one has when con-
sidering a research framework about education. Our culture tells us that problems
about education have to be solved in an uncomplicated way, mostly using
common sense and common means and perhaps some results from other research
areas such as neurology, psychology, sociology, etc. The emancipatory principle
mentioned above leads to a rejection of this assumption. Moreover, we will add a
second methodological principle, essential to scientific creation, that we will call
the Humpty Dumpty principle, in reference to Lewis Carroll’s well-known character
in chapter VI of Through the looking-glass, and what Alice found there (1871):
“I don’t know what you mean by ‘glory’,”Alice said.
Humpty Dumpty smiled contemptuously. “Of course you don’t—till I tell
you. I meant ‘there’s a nice knock-down argument for you!’”
“But ‘glory’doesn’t mean ‘a nice knock-down argument’,”Alice objected.
“When I use a word,”Humpty Dumpty said, in rather a scornful tone, “it
means just what Ichoose it to mean—neither more nor less.”
The Humpty Dumpty principle seems obvious when we consider well-established
disciplines. The theorisation of reality, essential to put the emancipatory principle at
work, needs new terms. This is done by inventing new words—like “hetero-
skedasticity”,“molecule”or “unconscious”—or by assigning new meanings to
common words. Mathematicians, sociologists and psychologists talk about “fields”to
refer to different things, none of which have grass or flowers …. Moreover, the use of
written symbols is sometimes of great help in describing complex situations and ima-
gining new cases or relations that would not easily appear through ordinary discourse.
Words, expressions and discourses, as well as symbols, graphs, figures and gestures, are
crucial cognitive tools. They are indispensable for describing and scrutinising the
objects and phenomena we wish to study, as well as imagining and creating new rea-
lities, new entities and new relations; talking about what exists but also about what
could exist, but does not, as well as about its (non-existent) conditions of possibility.
Research in the ATD has generated its own system of terms, principles and rela-
tions, as well as its own way of questioning reality and approaching it—its own
methodology. A third important principle in this respect is the one that gives its name
to the theory: the anthropological one. Why is the ATD an anthropological theory? Why
was this term chosen? What does it stand for and what does it imply? The anthro-
pological principle affects the level of generality and specificity that is assigned
to didactic phenomena. In the ATD, didactic phenomena are considered as
inherent to any group of human beings, as part of humanity. Being human
beings means co-creating and disseminating knowledge, and also failing to do
so. Moreover, the anthropological approach also proposes a common vision of
all kind of human exchange or activity, trying to exclude all kinds of value or
categorisation brought about by the society and institutions researchers are
Introduction: An invitation to the ATD xiii
immersed in—and subjected to. For instance, an important distinction when
analysing teaching and learning processes affects the “content”of the process,
what is taught and learnt. It can be just a basic gesture—“Show me how to
open this tap”—or a body of knowledge like mathematics, music, grammar,
history or economics. In between there are multiple other options, such as
sawing, playing football, investing one’s savings, preparing a couscous or writ-
ing a formal letter of complaint. To approach all these processes in a unitary
perspective and, especially, to prevent assuming the complex distinctions our
different cultures put on all these contents, the ATD proposes to initially con-
ceptualise all of them as a same entity by using an invented word: praxeology.
The anthropological principle states that any human activity can be described as
a praxeology or as an amalgamation of praxeologies of different “sizes”.
The term praxeology results from the union of two Greek words: praxis and logos.
The praxis refers to the practical part of the activity, the “doing”and its related
know-how. It is split into two elements: a type of tasks T
i
and a technique τ
i
to carry
out the tasks of type T
i
. In accordance with the Humpty Dumpty principle, tasks and
techniques must be understood in a broad sense: creating a new molecule, driving a
boat, encouraging a sports team or drinking water are types of tasks, and they have
different possible associated techniques, not necessarily algorithmic. The second
element of the praxeology, the logos, also contains two types of elements: a technology
and a theory. If the praxis corresponds to the “doing”, the logos corresponds to the
“telling”, the description, presentation, explanation, justification of the technique,
together with the organisation of the types of tasks and their relations to the techni-
ques. A first level of description and justification is the level of the technologies θ
i
.The
word is understood here according to its etymology: a discourse (logos) about the
technique (technè). Technologies rely on principles and draw on concepts and terms
that belong to the second level of justification, the theories Θ
i
.
The anthropological principle states that any human activity can be described in
terms of praxeologies. It contains a subprinciple derived from the very notion of
praxeology: any human activity includes a praxis component or know-how together
with a logos component of knowledge. In other words, any practice is described and
justified in one way or another, sometimes very poorly—“I drink water like this
because this is the way to drink water. Is there another one?”—, other times with
complex developments, as the example of the creation of new molecules lets us guess.
Some praxeologies can be considered as having an underdeveloped praxis because the
techniques are not “performant”enough or because new types of tasks have appeared
without a good way to carry them out. Other praxeologies can be considered as
having an underdeveloped logos: techniques almost without any justification, without
specific words to describe them, etc. And there are also praxeologies considered as
having an overdeveloped logos, with too much to say for such a little doing.
Praxeologies are entities that evolve over time. New types of tasks appear, asking for
new techniques to perform them; some new conditions can make a technique fail in
some cases and require new developments; the new praxis will then lead to the
expansion of the logos to describe, explain or justify the new ways of doing. Other
xiv Introduction: An invitation to the ATD
evolutions are also possible. A new theoretical element produces new technologies—
for instance, in mathematics, a new theorem derived from a new property or relation
between notions—which lead to new ways of doing and the appearance of new types
of tasks.
A last ATD principle should be mentioned about the consideration of praxeologies
as evolving entities: their institutional relativity. We will use a simple mathematical
example (taken from Bosch and Gascón, 2014) to illustrate it. Let us consider Pytha-
goras’theorem. At first sight, at least for somebody who is not far away from the
educational system, this property of the lengths of the three sides of a right triangle is
considered a technological element of a praxeology that exists in many lower sec-
ondary schools, linked to a praxis that consists in types of tasks like determining if a
triangle has a right angle; or to calculate one of the lengths of a right triangle given the
other two; etc. The development of this praxis and the corresponding logos can give
rise to a whole body of knowledge—a broader praxeology—called trigonometry.
However, in another context, say a topographic praxis of making measurements in a
field, the Pythagorean theorem can just be a simple technique of drawing a right
angle when other tools are not available. Its status and function change, as well as the
justification discourse that is built around it—Pythagoras’theorem may be “a well-
known property”and nothing else. Finally, Pythagoras’theorem can move from a
technological or technical element to a theoretical one, when the space or its geo-
metry is considered to be Euclidean or not. Therefore, a given object can be part of a
type of tasks, a technique, a technological or a theoretical element depending on the
praxeology we are considering and the way this praxeology exists and evolves in a
given institutional setting. Not only does a praxeology change when moving from
one institution to another (we do not drink water in the same way everywhere), the
very elements of a praxeology can have different functions depending on the type of
praxeology that is considered.
After this brief introduction to some basic principles of the ATD, we can invite
the reader to cross the threshold to the other side of the mirror. Let us just add a
final comment. As with other knowledge domains, when we talk about the
anthropological theory of the didactic, we do not refer to a theory in the ATD
sense, but to a set of research praxeologies. As is often the case in science, “theory”is
used metonymically. This introductory chapter follows a traditional pattern, start-
ing from the ATD basic principles, which properly belong to the theory (in the
ATD sense). Many other theoretical and technological elements—primary notions,
derived notions, relations, properties—are presented in the form of a glossary to be
found right after this presentation. This glossary includes a list of current ATD
terms with their definition and some short developments to relate them to other
entries and explain some of their uses. Even if the ATD cannot be reduced to a list
of terms, we hope it will help the reader to better approach fully fledged ATD
research praxeologies.
Many other aspects of the ATD logos and praxis are presented in the list of
chapters that make up this book. They include recent research studies that were
presented at the 6th international conference on the ATD held in Castro Urdiales
Introduction: An invitation to the ATD xv
(Spain) in 2016, which was recognised as an ERME Topic Conference. These
studies illustrate some of the main research questions that are raised in the ATD,
the way they are formulated and approached and some of the results obtained. In
brief, they bring to the fore some research tasks and research techniques and their
outcomes. Of course, this research praxis is just evoked, mainly through the tech-
nological discourses that describe, explain and justify the work done—sometimes in
a summarised way due to space restrictions. The results obtained, the product of
the praxis, are tested by the research community and, when they appear to be
relevant and robust enough, they become part of the research logos. They will then
become part of new research methodologies and yield important tools to formulate
new research questions, thus enlarging the research praxis, its techniques and types
of task. As any praxeological open system, the ATD evolves through complex
interactions between the praxis and logos elements. In a written presentation, the
logos elements may acquire a stronger presence. We hope that this limitation will
not restrain the reader from accessing the whole of the ATD as a praxeological
complex. This is the mission of the research studies presented in this book.
References
Berger, P. L. (1963). Invitation to sociology: A humanistic perspective. New York: Knopf
Doubleday Publishing Group.
Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques
1970–1990. Dordrecht, The Netherlands: Kluwer.
Chevallard, Y. (1985/1991). La transposition didactique: Du savoir savant au savoir enseigné.
Grenoble, France: La Pensée Sauvage (2nd edition 1991).
Gascón, J. (1993). Desarrollo del conocimiento matemático y análisis didáctico: del patrón
Análisis-Síntesis a la génesis del lenguaje algebraico. Recherches en Didactique des Mathématiques,
13(3), 295–332.
Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des
Mathématiques, 10(2), 133–170.
Short bibliography on the ATD
Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s
practice: The case of limits of functions in Spanish high schools. Educational Studies in
Mathematics, 59, 235–268.
Barquero, B., & Bosch, M. (2015). Didactic engineering as a research methodology: From
fundamental situations to study and research paths. In A. Watson & M. Ohtani (eds), Task
design in mathematics education (pp. 249–271). Cham, Switzerland: Springer.
Bosch, M. (2015). Doing research within the anthropological theory of the didactic: The
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Introduction: An invitation to the ATD xvii
A SHORT (AND SOMEWHAT
SUBJECTIVE) GLOSSARY OF THE ATD
Yves Chevallard, with Marianna Bosch
Hence it appears that definitions are very arbitrary, and that they are never subject to
contradiction; for nothing is more permissible than to give to a thing which has been
clearly designated, whatever name we choose. It is only necessary to take care not to
abuse the liberty that we possess of imposing names, by giving the same to two
different things.
Blaise Pascal, The Geometric Spirit and the Art of Persuasion, ca. 1658
Activity. The word applies to all stabilised or slowly evolving, not haphazard or
erratic, practices in which someone engages. The main tenet of the ATD in this
respect is that all human activity breaks down into a number of tasks of determined
types. (See also Types of tasks.)
Answer. Afirst type of answers responds to how questions: basically, these answers
are made up of a technique which indicates how to accomplish tasks of a certain type,to
which is added the necessary technological and theoretical block in order to get a complete
praxeology. A second type of answers responds to why questions: they usually consist in a
technological and theoretical block, which may generate new praxis blocks,orcometoplay
the logos part in a number of praxeologies in need of a logos part. (See also Questions.)
Antididactic. See Didactic gestures and the possibly didactic.
Break. A break is a change in a person’s relations to a number of objects such
that the person’s new relations do not conform to the corresponding relations of
any surrounding institution, thus laying the basis for a potential, conspicuously new
institution. Although such a change should be called generally a cognitive break,itis
traditionally labelled an epistemological break. A rupture from a given institution’s
relations to a number of objects is termed an institutional break.
Civilisation. In the ATD, this word is not taken to mean a process—the process
of civilisation. It applies to a set of societies that, up to a point, can be said to share—
through historical fashioning—a large set of objects and relations to these objects. The
concept is functional in pointing to conditions and constraints present in a whole
range of societies, albeit under a specific guise. It helps in avoiding false beliefs about
conditions at first regarded as local—and therefore “shallow”—when in fact they
are deeply rooted in a whole civilisation—and therefore not so easily removable.
(See also Scale of didactic codeterminacy levels.)
Cognition. Persons and institutions are objects.IntheATD,anobjectis
anything that exists for (at least) a person xor an institutional position p.Given
apersonxand an object o,thepersonal relation of xto o, denoted by R(x,o),
the set of links of any kind between xand o.ItmaybethatR(x,o)=∅.In
this case, we say that “odoes not exist for the person x”or that “xdoes not know the
object o”. If, on the contrary, R(x,o)≠∅, the object oexists for x,andxknows o,
even if in a minimalist way. The same definitions apply when replacing the person x
with an institutional position (I,p). The relation of pto ois traditionally denoted by R
I
(p,o). We have either R
I
(p,o)=∅or R
I
(p,o)≠∅. We say that the institutional
position pdoes not know or on the contrary, knows the object o, and as well that o
does not exist or exists for p.
Cognitive algebra. The word “algebra”is taken here slightly metaphorically.
When an author writes “We have”or “One has”,asin“We have either R
I
(p,o)=∅,
or R
I
(p,o)≠∅”, who is the “one”or the “we”? More specifically, when the author
writes “We have R(î,o)≠∅”,the“we”is necessarily some personal or institutional
instance ĵ(of course ĵmay be the author). This is denoted by
ĵ
⊦R(î,o)≠∅,and,more
generally, by ĵ⊦ϑ,whereϑis any sentence. The statement ĵ⊦ϑis to be read “the
instance ĵjudges that ϑ[is true]”. (The symbol ⊦used here is the Unicode “assertion
character”.) Another instance k
̂may judge that ϑis not true, i.e., k
̂⊦¬ϑ. There is not
generally agreement but disagreement between different instances. This is an aspect of
cognitive relativity—as a general rule, instances “do not think alike”.Allthiscanbeeasily
generalised. The statement k
̂⊦(ĵ⊦R(î,o)≠∅) means, for example, that, according to
the instance k
̂, the instance ĵthinks that the îknows the object o. Likewise, if a
person x
0
asserts “They say that the person x
1
knows the person x
2
”, we shall write:
x
0
⊦∃î(î⊦R(x
1
,x
2
)≠∅), where the instance îstands for “they”.Thefactthatĵ⊦R
(î,o)≠∅(or, as well, that ĵ⊦R(î,o)=∅) tells us something about the relation R(ĵ,
R(î,o)), i.e., about the relation of ĵto the relation of îto o.Thefactthatk
̂⊦(ĵ⊦R(î,
o)≠∅) informs us about R(k
̂,R(ĵ,R(î,o))), i.e., about the relation of k
̂to the
relation of ĵto the relation of îto o,etc.
Cognitive base. See Cognitive nucleus.
Cognitive break. See Break.
A Short (and somewhat subjective) Glossary of the ATD xix
Cognitive equipment. Given an instance î, the universe of objects or cognitive
universe of îis defined by: Ω(î)≝{o/R(î,o)≠∅}. The set Ω(î) tells us which objects
exist for î. The cognitive equipment of îis then defined by: Γ(î)≝{(o,R(î,o))/o∈Ω
(î)}. The set Γ(î) specifies how îknows the objects they know, i.e., the objects o∈
Ω(î). (See also Praxeologies and praxeological analysis.)
Cognitive nucleus. Let ŵbe the reference instance. Let ȶ
0
and ȶ
1
be two suc-
cessive times. How can we make sense of the following semi-formalised statement ŵ
⊦[R(î,o)atȶ
1
is “better”than R(î,o)atȶ
0
]? At the start, ŵhas in mind the ordered
pair n
̄
=(î,o), called the cognitive base. This base will be completed so that ŵcan
arrive at a judgment. To do so, we must assume that ŵalso envisages a positional
instance (I,p) whose relation to o,R
I
(p,o), is regarded by ŵas a reference point
against which R(î,o) can be evaluated: the more R(î,o) will be recognized as “close”
to R
I
(p,o) and the better it will appear. But this is not enough. The reference
instance ŵmust also consider an evaluating instance v
̂, which is supposed to judge the
“proximity”of R(î,o)toR
I
(p,o). The triple ṉ=(I,p,v
̂)isthecognitive reference frame.
If ŵ, i.e., the reference instance, is a teacher y, the relation R
I
(p,o)maybetheone
expected at the exam for which yis preparing the students x =î, the evaluating
instance v
̂being the examination board. The quintuple ñ=n
̄
⁀ṉ=(î,o,I,p,v
̂)isa
cognitive nucleus (or kernel). The instances (I,p), v
̂, and even î, are often merely ima-
gined by ŵand are therefore respectively denoted by
*
I,
*
p,
*
v
̂or
*
î. The corre-
sponding cognitive nucleus,
*
ñ, is written as the case may be, (î,o,
*
I,
*
p,
*
v
̂), (î,o,
*
I,
*
p,v
̂), (î,o,I,
*
p,
*
v
̂), etc. Cognitive nuclei are therefore frequently underdefined.
Note that, of course, we can have, for example, v
̂=ŵor else v
̂=î. The evaluating
instance v
̂is supposed to be able to pronounce judgments like “R(î,o)conforms
with R
I
(p,o)”or “does not conform to R
I
(p,o)”, which are to be denoted,
respectively, by R(î,o)≅R
I
(p,o)andR(î,o)≇R
I
(p,o). But it is also assumed that v
̂is
able to estimate a “degree of conformity”φ(R,R
̄
) between R=R(î,o)andR
̄
=R
I
(p,o). Given relations Rand R′to o, we usually have one of the three
following verdicts: v
̂⊦φ(R,R
̄
)<φ(R′,R
̄
); v
̂⊦φ(R,R
̄
)>φ(R′,R
̄
); v
̂⊦φ(R,R
̄
)
≈φ(R′,R
̄
). In particular, Rand R′can be respectively the relation R(î,o)at
times ȶand ȶ′,withȶ<ȶ′. The notion of cognitive nucleus is the notion on
which hinges the decisive concept of a possibly didactic situation.
Cognitive reference frame. See cognitive nucleus.
Cognitive relativity. See cognitive algebra.
Cognitive universe. See cognitive equipment.
Conditions and constraints. In the ATD, a condition is anything purported to
have influence over at least something. Essentially, a condition can thus be identi-
fied to the state of some system, and therefore to the values of a set of variables that
govern this system. A constraint is any condition which appears to be unmodifiable
xx A Short (and somewhat subjective) Glossary of the ATD
by occupants—acting as such—of a given institutional position. Very often, for any
condition, there exists at least one position in society from which this condition
can be modified. It is the universal aim of the natural as well as the social and
behavioural sciences to create means to turn ancient, lasting constraints into con-
ditions modifiable from at least some (in general newly created) institutional
positions.
Constraint. See Conditions and contraints.
Curricular path. See Curriculum.
Curricular trail. See Curriculum.
Curriculum. The word curriculum derives from Latin currere “to run”and has
come to mean a “course of study”at a college, university, or school. In the ATD it
refers to a general, nonnormative notion. We say that, as seen from the instance ŵ’s
vantage point, an institutional position p
s
is an antecedent of a position pif p
s
is a
position seen by ŵas preparing persons xto occupy the position p, which can be
denoted by ŵ⊦p
s
⇝p. The position p
s
may be regarded, or not, as a student
position by some “authorised”institution such as a university. We define a ŵ-cur-
ricular path from p
1
to p=p
n
to be a (finite) sequence of positions p
1
,p
2
,…,p
n
=p
so that ŵ⊦p
i
⇝p
i+1
for 1 ≤i≤n–1. To the path (p
1
,p
2
,…,p
n
) corresponds the
curricular trail (π(p
1
), π(p
2
), …,π(p
n
)), where π(p
i
) is the praxeological equipment of
position p
i
. For the sake of brevity, we leave implicit, for each position p
i
of the
positional path (p
1
,p
2
,…,p
n
), the potential position p
̄
i
of “teacher”(in an exten-
ded sense of the word) that may exist according to ŵ.Wethendefine a ŵ-curriculum
(π
1
,π
2
,…,π
n
), where π
i
=π(p
i
), for 1 ≤i≤n, as the curricular trail associated with
the ŵ-positional path (p
1
,p
2
,…,p
n
). Note that a curriculum is an existing social
reality (as seen by an instance ŵ) and must not be mistaken for a curricular project.
The consideration of the notion of curriculum reveals many key questions. Given a
ŵ-curricular path (p
1
,p
2
,…,p
n
), what percentage of people in position p=p
n
have
arrived at this position by following the path (p
1
,p
2
,…,p
n
)? More generally, what
percentage of those in position p
i
have arrived there by following the path (p
1
,p
2
,
…,p
i
)? For 1 ≤i≤n–1, what percentage of “students”in position p
i
gain access to
the position p
i+1
? How can a person accede to the initial position p
1
?Doesŵ
acknowledge other curricular paths (p′
1
,p′
2
,…,p′
m
), with p′
m
=p? There also exist a
whole array of questions about what goes on in each and every position p
i
and in
particular about the praxeological equipement π(p
i
) or about the links between π(p
i
)
and π(p
j
)withj>i, including the praxeological obstacles to the establishment of π(p
j
)
that π(p
i
) may generate. All this pertains to the crucial question of curriculum
development.
Curriculum development. See Curriculum.
A Short (and somewhat subjective) Glossary of the ATD xxi
Deranging conditions. See Didactic gestures and the possibly didactic.
Derived questions. See Herbartian schema.
Dialectic. Any praxeology that enables one to overcome two opposed types of
constraints by turning them into a new kind of conditions that supersede them. In this
context, one, therefore, speaks of supersession (French dépassement, German Aufhebung,
Spanish superación).
Dialectic of black boxes and clear boxes. Praxeology that allows one, when
confronted to some praxeological element, to manage one’s way between full
ignorance (black box) and supposedly complete knowledge (clear or white box) of
that element. To cast it in formulaic style: this dialectic helps one determine the right
“shade of grey”to work with—there is no such thing as a purely “white”box.
Dialectic of conjecture and proof (or dialectic of media and milieus). In
the course of an inquiry on a question Qby a didactic system S(X;Y;Q), Xis
confronted with statements expressed by what is generically called media,amedium
being any system that issues messages—a textbook, a teacher, a newspaper, the
Internet are all media. Of course, this list should also include Yas well as Xinsofar
as this group utters statements regarding the question Q. Notwithstanding their
plausibility, mostly all the statements “received”by X(including those coming
from Y) should be regarded as conjectures, i.e. as statements based on incomplete
evidence. Looking for evidence is thus the sinews of inquiry. Proof of a statement
ϑshould be looked for by questioning media which, with respect to ϑ, behave like
“adidactic”milieus. Such an adidactic milieu—or simply milieu, if no ambiguity is
to be feared—is a system deemed to be devoid of any intention to prove or dis-
prove σ, much like a part of the inanimate world. The dialectic of media and
milieus enables the pursuit of truth—even in cases where there is no decisive test.
Dialectic of dissemination and reception. Whatever the answer A
♥
to some
question Q, be sure that it will disseminate outside of S(X;Y;Q). For example, if
[X;Y] is a school class, A
♥
will be known, in essence, to other teachers, to parents,
etc. Bringing an answer to a question is a social act, the product of which cannot
be restricted to a single place—“leaks”are sure to happen. The dissemination that
takes place alters the ecology of A
♥
and may, therefore, diminish its viability, even
within [X;Y]. How A
♥
will be received is thus a crucial concern for its producers
and potential users. The dialectic of dissemination and reception is, therefore, a key tool
of inquiry.
Dialectic of media and milieus. See Dialectic of conjecture and proof.
Dialectic of on-topic and off-topic. At school, the course followed by an
inquiry is traditionally supposed to remain on-topic all the time, without
xxii A Short (and somewhat subjective) Glossary of the ATD
wandering off-topic even for a short detour that would seem promising in terms of
unexpected but hopefully relevant encounters. Proper mastery of the dialectic of
on-topic and off-topic makes it possible to overcome this institutional limitation
and go away at times from the apparent right path, in search of the unforeseen.
Dialectic of persons and institutions. The tension between people and insti-
tutions and its supersession is at the heart of the continued creation of a society. For a
person x,“being a member”of an institution Imeans occupying a certain position p:
we say that the person xis subjected to Iin position por is a subject of I in position p. For
example, a member of a school class may hold the position of student or the position of
teacher (or such other existing position); a member of a family may hold the position of
mother, father, or child; etc. A given person xis, and has been, subjected to a multi-
plicity of institutional positions (I,p). All persons are the results of the dynamical
system of their institutional subjections, which gives them their power of thought and
action. A person xcan occupy at the same time the positions of adult, female, mother,
teacher—a person is a plural reality (the word person originally meant a mask, that
Greek actors used to wear, and of which they could change). When we address a
person, we often question, in that person, the subject of this or that institutional
position: “You who are a man…”,“She is the form teacher”,“You are the mid-
wife?”, etc. A person xmay want to free themself from some of their institutional
subjections. Such desubjections are made at the cost of contracting new subjections,
which frees up old ones and gives new power of thought and action while creating
new constraints. Such trade-offs are constitutive of the person x’sfreedom.
Dialectic of questions and answers. See Dialectics of inquiry and Herbartian schema.
Dialectic of reading (or “excribing”) and writing (or inscribing). Most
information comes to us in texts, as happens with the answers A
◊
appearing in the
developed Herbartian schema. Texts are made of assertions that both follow from
and manifest praxeologies which, usually, remain hidden to the casual reader.
These praxeologies have been “inscribed”(and thus concealed) in the text, so to
speak; conversely, the serious reader, who feels concerned with the praxeologies
put to use to produce the assertions he reads, will have to “undo”the inscribing
by—to use a neologism—“excribing”them, i.e. by questioning the text about its
hidden content, so as to bring to the fore normally latent praxeologies. It follows
from all this that, reciprocally, in producing A
◊
,X(and therefore Y) has to devote
much effort to “inscribing”it into the text that will preserve it from oblivion and
make it known more widely. Altogether, all this necessitates considerable writing
and above all different kinds of writing (such as in a notebook, a progress report,
adraft,etc.)
Dialectic of the individual and the group (or dialectic of idionomy and
synnomy). In a school class [X;Y]whereY={theteachery}, it is usually supposed
that every student x∈Xinquires about a question Qon his or her own to
A Short (and somewhat subjective) Glossary of the ATD xxiii
produce an answer A
♥
x
. In general, the answer A
♥
x
supplied by xwill cease to
have any relevance from the very moment the teacher discloses his or her own
answer A
♥
y
, which will displace all answers A
♥
x
,x∈X, in accordance with the
degenerate Herbartian schema:S(X;Y;Q)➥A
♥
Y
. This leads students to develop
an individualistic relation to knowledge (and to ignorance), caught as they are
between their idionomy (from Greek idios “one’sown”and nomos “law”)and
the heteronomy imposed by the teacher. By contrast, in an inquiry as modeled
by the (non degenerate) Herbartian schema, answers A
♥
x
are no more but no
less than answers A
◊
that will constitute part of the milieu Mfrom which A
♥
is
to be produced according to the semi-developed Herbartian schema, [S(X;Y;
Q)➦M]➥A
♥
,withM={A
◊1
,A
◊2
,…,A
◊n
,W
n+1
,…,W
m
,…}. In such a
perspective, a student is no longer accountable only for his or her own answer A
♥
x
:
all students are collectively accountable for the answer A
♥
and its construction. Their
main need is, therefore, to establish in the class a common law, determined and applied
collectively, to which they will be accountable. Such a synnomy (from Greek syn
“together”) must counterbalance the idionomy that remains indispensable for each
student in his or her personal effort to investigate the question Qand bring his or her
share to the advancement of the inquiry.
Dialectic of the parachutist and the truffle hound. When looking for infor-
mation in the course of some inquiry, one has to sweep vast areas, thus acting as a
(military) parachutist, while knowing that the information searched for will be found
(in the way a truffle hound—or hog, or pig—does) only in some sporadic, unexpected
places. The capacity to do so is identical with the mastery of the dialectic of the parachutist
and the trufflehound.
Dialectics of inquiry. In the ATD, the action taken to provide an answer Ato
a question Qis called an inquiry into Q. An inquiry is minimally represented by the
so-called reduced Herbartian schema S(X;Y;Q)➥A
♥
, in which the exponent ♥
(heart) means that the answer A
♥
satisfies conditions specifictoS(X;Y;Q), i.e., is
an answer according to S(X;Y;Q)’s heart, so to speak. In carrying out an inquiry,
Xand Yshould put to use the dialectics of inquiry, namely the dialectics of on-topic
and off-topic,of the parachutist and the truffle hound,of black boxes and clear boxes,of
conjecture and proof (also called dialectic of media and milieus), of reading (or “excrib-
ing”)and writing (or inscribing), of dissemination and reception,of the individual and the
group (also called dialectic of idionomy and synnomy). All the work done generates
derived questions and therefore partial answers from which the answer A
♥
will be
produced. So that the dialectics mentioned above are carried out in constant
interaction with the key dialectic of questions and answers.
Didactic. See Didactic gestures and the possibly didactic.
Didactic analysis. A didactic analysis is an analysis of the didactic present in a
given social situation according to a given instance ŵ. Such analysis should include
xxiv A Short (and somewhat subjective) Glossary of the ATD
an account of every didactic system S(X;Y;♥) which must consist minimally in
more or less comprehensive answers to the following questions: What is X? What
is Y? What are the questions or praxeologies [T/τ/θ/Θ] that the didactic stake ♥is
made of? What are the didactic praxeologies put to use by Xand Yand what
didactic means have proved necessary to do so? How does S(X;Y;♥) evolve over
time? What praxeological equipment can be engendered in Xas a short-term and as a
long-term result of the functioning of S(X;Y;♥)? And lastly, what does Yas well
as some institutional environments of S(X;Y;♥) may have learnt in the process?
To answer these questions properly, it seems crucial to identify the main conditions
and constraints composing the ecology of the situation and their potential effects on
it. To do so, one should then scan the scale of levels of didactic codeterminacy to make
explicit conditions or constraints too often described—or even simply alluded to—
as “natural”and (“therefore”) inconsequential. Didactic analysis is carried out
through the elaboration of models (in the scientific sense) that are used to provide a
description of the elements of S(X;Y;♥) and their evolution. In the ATD, these
models pertain to the theory of didactic moments when ♥is a praxeology or an
amalgamation of praxeologies, and to the Herbartian schema when ♥is a question Q
formulated within a project Π.
Didactic gestures and the possibly didactic. The word gesture is used in an
extended and neutral sense. That being said, how then can we define a didactic
gesture? Strictly speaking, the answer is: there’s no way of doing it simply. We can
only speak of possibly didactic gestures. But in fact every gesture is “possibly didac-
tic”. Let ŵbe a reference instance and ñ=(î,o,I,p,v
ˆ)acognitive nucleus. Let us
suppose that some instance û“makes a gesture”δ. When will we say that the
gesture δis didactic for ŵ,orŵ-didactic, with respect to ñ? To do so, we have to
introduce a pivotal parameter, the set Cmade up of all the conditions that prevail
before the gesture δtakes place. The quadruple ς=(ñ,û,δ,C) then denotes a
possibly didactic situation. Let Rbe the relation R(î,o) before the gesture δis per-
formed and let R′be the “same”relation after the accomplishment of δ. We say
that the gesture δis didactic for ŵ(or ŵ-didactic) with respect to ñand Cif ŵcon-
jectures that v
̂will consider R′“closer”to R
̄
=R
I
(p,o) than was the case of R.Ifŵ
conjectures that v
̂will consider it further away from R
̄
=R
I
(p,o) than was the case
of R, we say that the gesture δis antididactic for ŵ(or ŵ-antididactic). The gesture δis
said to be isodidactic for ŵ(or ŵ-isodidactic) with respect to ñand Cif ŵconjectures
that v
̂will find Rand R′are almost equally compliant with R
̄
. To conjecture that
the gesture δ(performed by û)isdidactic or isodidactic or antididactic with respect to ñand
C,ŵrelies in particular on ŵ’s relations to R(î,o), (I,p), v
̂and C,i.e.,onR(ŵ,R(î,o)),
R(ŵ,(I,p)), R(ŵ,v
̂)andR(ŵ,C). The gesture δchanges the prevailing conditions C,
in particular, because it modifies R(î,o). Let us denote by C′the new set of conditions
created by δ.WecallC′aderangement of Cand write C′=C
δ
(which can be read “C
deranged by δ”), where the symbol used (λ) is the caret insertion point. We thus have
C′=C
0
∪D
δ
,withC
0
⊂Cand D
δ
∩C=∅. This may lead to rewrite the situation
ςas ς
̂=(ñ,û,D,C), where Dis the set of deranging conditions. Note that the instance ŵ
A Short (and somewhat subjective) Glossary of the ATD xxv
forms their judgment before the gesture δtakes place about the judgment that the
evaluating instance v
̂will issue after δhas been performed. In the case when ςand δare
(roughly speaking) reproducible, ŵmay have integrated into their relation to ςresults
observed in past occurrences of the situation. But ŵwill nevertheless issue an a priori
judgment relating to the a posteriori judgment of v
̂. One of the major difficulties of
forecasting in general (and not only in didactics) is our lack of knowledge about the set
Cof conditions and their effects on R(î,o). The commendable effort to neutralize
some of these conditions does not eliminate the fact that we are unaware of even more
of them. Although the possibly didactic is always unsure, it is a vital necessity to human
societies, who nonetheless tend to repress it as if it were an insuperable flaw. They
therefore hide it in selected, isolated places, e.g., schools and classrooms. Didacticians
should however look for the possibly didactic wherever it occurs in society—not only
where society pretends to maintain it.
Didactic milieu. See Dialectic of conjecture and proof and Herbartian schema.
Didactic moments. These are the moments discernible in study processes. In
the study of a how question relating to a type of tasks T, ATD recognizes six
such “study moments”: the moment of the first encounter with T;themoment of
exploration of Tand emergence of a technique τ; the moment to build the tech-
nological and theoretical block [θ/Θ]; the moment to work on the praxeology pro-
duced, [T/τ/θ/Θ] and particularly on the technique τ; the moment to
institutionalise it; the moment to evaluate the praxeology produced and one’s
relation to it. (See also Moment.)
Didactic organisations. Given an object oand an instance î, how can the
relation R(î,o) be formed? In the case where îis a person x, the basic answer is:
because xhas occupied (or occupies) the position of student in at least one formal or
informal didactic system S(X,Y,♥) whose functioning involves the object o,
notably, but not only, if ♥is the object oitself. In all such cases, ois the subject of
an inquiry (in an extended sense of this word). In many cases, this inquiry is still-
born, or quickly vanishes, or vegetates indefinitely. This is the basic regime of the
possibly didactic: individually or collectively, one quickly “forgets”to inquire into
a word, a use, an event, a phenomenon. There is atrophy and repression of the
(possibly) didactic. Didactics could stop here. But Homo sapiens, the “learned”man,
is just as much Homo discens, the “learning”man, and Homo docens, the “teaching”
man. For this reason, the (possibly) didactic is everywhere around us: the economy
of the didactic is flourishing and could be even much more, under well-known or
new forms, as revealed by the ecology of the didactic. To study the didactic, from an
economic and ecological point of view, we can start from the level of the didactic
system S(X,Y,♥). Such a system is formed, under the aegis of an institution
assuming a school function, by a social contract that brings together people x∈X,
possibly other people y∈Y(we can have Y=∅), and a didactic stake ♥.Wehaveto
examine what S(X,Y,♥) will do and not do to inquire into ♥. In the case of the
xxvi A Short (and somewhat subjective) Glossary of the ATD
paradigm of visiting works,Ycontains (and often is reduced to) a distinguished element
y, the teacher. The didactic stake ♥is a work oon which yhas investigated prior to
the formation of S(X,Y,♥) and on which the x∈Xare supposed not to have
investigated on their own account. Under the name of lecture,ythen presents to X a
“report of inquiry”μ
y
,inwhichthex∈Xhave no part and through which they are
supposed to “learn”o. Here, the topos of yand the topos of the x∈Xare essentially
disjointed. There are many variations: for example, ycan choose an inquiry report μ
z
of which yis not the author and then guide the x∈Xin their study of μ
z
. In recent
decades, in an increasing number of societies, the paradigm of visiting works has
increasingly been viewed as antididactic. At the same time, we observe the rise of a
paradigm called the paradigm of questioning the world, in which the work ois a question
qand the inquiry into qis carried out “in the classroom”by the x∈Xunder the
supervision of y. This opens up a sometimes dissonant variety of didactic organisa-
tions. The study of any work ois generally limited to the study of questions q
1
,q
2
,
…,..,q
n
relating to o(origin, structure, use of o, etc.). But, in general, the study of o
is not generated in the classroom by the study of a question q
0
,sothatXwill not
wonder whether ocould be a possible tool in the study of the generating question
q
0
. For this and other reasons, the pedagogies of inquiry are considered by some
instances as antididactic. For example, if we look at the work oas a tool to inquire
into the question q
0
, we will not necessarily ask ourselves all the questions asked in
older curriculums which, de facto, were held to “define”what it was like to “know
o”. We have here a huge field of questions, which has been worked on for many
years by a growing number of researchers in the context of the ATD.
Didactic stake. See Didactic organisations and Didactic system.
Didactic system. A didactic system S(X;Y;♥) results from the forming, around
what is called a didactic stake,♥, of a group made up of two functionally distinct
subgroups, Xand Y, the former being composed of students of ♥while the latter is
the team of “study assistants”(or “helpers”) among whom is usually at least one
“study director”who directs (and, up to a point, plans) the study underway. The
most eye-catching didactic systems are those formed at school, in classrooms; in
such a case, Xis the class, and Yis ordinarily a “singleton”(in mathematical parlance)
whose unique member is “the teacher”y.
Didactic transposition. See Praxeologies and praxeological analysis.
Didactician. A researcher in didactics.
Didactics. Didactics is defined as the science—still in infancy—whose object is
to study the conditions and constraints that govern the dissemination of praxeologies
in institutions across society. Dissemination is taken here in an extended sense: it
also includes non-dissemination and the most notably deliberate withholding of
praxeologies concerning specific institutions.
A Short (and somewhat subjective) Glossary of the ATD xxvii
Discipline. All institutional stabilised human activity is “disciplined”in the sense
that it draws on determined praxeological equipment: the discipline of the activity is
then tantamount to using that very equipment. Of course, the praxeological
changes that affect any institution entail periods in which the activity within the
institution is “dedisciplined”, before being “redisciplined”after a while. “Dis-
cipline”in ATD thus covers much more than the slowly evolving repertoire of
“school disciplines”: it applies equally well to institutional disciplines with a much
lesser pedigree. The conscientious reader should also note that the word discipline
has meant from the start the “instruction given to a disciple”, a meaning which by
far predates the notion of “order necessary for instruction”and that of “treatment
that corrects or punishes.”
Ecology and economy. Abstractly, an ecology is simply a set of conditions.The
ecology of a system of a given type, whether animate or inanimate, existing in the
natural or the social world is the set of prevailing conditions under which this system
lives in actual fact. It is said to be a hostile ecology if it causes the system to malfunc-
tion or, a fortiori, to cease to exist. A deliberately created condition (economy)
becomes a condition of the prevailing ecology. The economics of the (possibly)
didactic studies the conditions which, under the set of given initial conditions C,are
created by specificgesturesδ. Ecology, on the other hand, studies which conditions
are, under given initial conditions C, possible, difficult but possible or just impossible.
Economy. See Ecology and economy.
Epistemological break. See Break.
Evaluating instance. See Cognitive nucleus.
Evaluation. To evaluate something is to determine its value. However, this is
only half the story: there is no such thing as an intrinsic value. The value assigned
depends on the project in which the “something”is called to play a part. In the
reduced Herbartian schema S(X;Y;Q)➥A
♥
, the value of answer A
♥
depends on
what use will be made of it. The same is true of any praxeology or, for that matter,
of a person’s relation to some object: their value is dependent on the situations in
which the person will have to make use of that praxeology or to relate to that
object. (See also Cognition.)
Genre of tasks. See Types of tasks.
Gesture. See Didactic gestures and the possibly didactic.
Herbartian schema. Johann Friedrich Herbart (1776-1841) was a German
philosopher and the founder of pedagogy as an academic discipline. The saying
goes that, according to Boyer’s law, “mathematical formulas and theorems are
xxviii A Short (and somewhat subjective) Glossary of the ATD
usually not named after their original discoverers”. The Herbartian schema is no
counterexample to the extended law referred to Carl B. Boyer (1906-1976), though
its name is not a misnomer either, for it retains something of Herbart’spedagogical
views. Minimally, the Herbartian schema requires a question Qand a group of
persons X—which may be reduced to a singleton—with the project to study
question Q. This induces the formation of a didactic system S(X;Y;Q), where the
team of “study helpers”Ymay be empty (Y=∅). The functioning of S(X;Y;Q)
must lead to the production of an answer A
♥
to question Q, a process represented
thus: S(X;Y;Q)➥A
♥
.Thisisthereduced Herbartian schema.ToproduceA
♥
,
however, S(X;Y;Q)needs“materials”; these materials make up the didactic milieu
Mconstituted by S(X;Y;Q) and represented as follows in the semi-developed
Herbartian schema: [S(X;Y;Q)➦M]➥A
♥
. In the didactic milieu M,itis
customary to distinguish different categories. First, Maccommodates “readymade”
answers A
i◊
drawn from available “resources”(including members of X∪Y)and
derived questions Q
k
induced by the study of Qand A
i◊
. It also includes other works
W
j
specifically drawn upon to make sense of A
i◊
, analyze and “deconstruct”them,
bring appropriate answers to the questions Q
k
, and, last but not least, build up
A
♥
. Finally, Mincludes sets of data of all natures the D
l
gathered in the course
of the system’sinquiryonQ, on which the didactic system’s answers to the
questions under consideration partially rest. The didactic milieu is therefore represented
generically thus:
M¼A1;A
2;...;A
m;W
mþ1;W
mþ2;...;W
n;Q
nþ1;Q
nþ2;
f
...Qp;D
pþ1;D
pþ2;...;D
qg;
hence the developed Herbartian schema:
SX;Y;QðÞ➦A1;A
2;...;A
m;W
mþ1;W
mþ2;...;W
n;Q
nþ1;Q
nþ2;
f
½
...Qp;D
pþ1;D
pþ2;...;D
qg➥A♥:
How question. See Question and also Answer.
Humankind. See Scale of didactic codeterminacy levels.
Infrastructure and superstructure. The notion of infrastructure (or substructure)
is, in the ATD, a general concept: it refers to the underlying base needed to
develop any determined, superstructural activity. It should be clear, for example,
that the “superstructural”activity that consists in watching TV at home requires an
enromous infrastructural base. In a school system Σ, the infrastructure allows the
appropriate actors of Σto engage in the superstructural activities of creating and
managing the schools σthat the system Σwill consist of. In each of these schools σ
A Short (and somewhat subjective) Glossary of the ATD xxix
there are also infrastructural means to create and manage classes c, for example by
solving problems of time and place of operation. In each class, there are similarly
infrastructural devices that allow the superstructural activities that make up the class
to be carried out. In a mathematics class, there is a gradually built infrastructure
allowing the mathematical (superstructural) activities to be carried out by the stu-
dents. To be able to write that we have 141217/3215763 = 0.04391…≈4.39%,
for example, we need to have available the division operation and the system D≥
of nonnegative decimal numbers, without forgetting a sufficient calculation time by
hand or a calculator, together with the notions of “almost equality”and percentage
and their respective symbols (≈and %). It should be noted that, in many cases, at
least within the paradigm of visiting works, the time taken to build the mathematical
infrastructure leaves relatively little room for the (superstructural) mathematical
activities that this infrastructure is supposed to make possible. Things go differently
within the paradigm of questioning the world, insofar as the mathematical infrastructure
is built according to the needs of the superstructural mathematical activities that
one wishes to develop. In this perspective, it should be noted that the infrastructure
made available by the Internet and digital information technology offers a quite
favorable framework to the pedagogies of inquiry.
Inquiry. See Dialectics of inquiry and Herbartian schema.
Instance. In the ATD, the word instance refers to either a person xor an insti-
tutional position (I,p). In the former case we have a personal instance î =x, in the
latter a positional instance î =p. We denote by R(î,o) the relation of the instance îto
the object o.
Instance of reference. See Research in didactics and the plurality of instances.
Institution. See Persons, institutions, and institutional positions.
Institutional break. See break.
Institutional instance. See Instance.
Institutional relation. See Cognition.
Institutional transposition. See Praxeologies and praxeological analysis.
Isodidactic. See Didactic gestures and the possibly didactic.
Lecture. See Didactic organisations.
Logos block. See Praxeologies and praxeological analysis.
xxx A Short (and somewhat subjective) Glossary of the ATD
Media. See Dialectic of conjecture and proof.
Milieu. See Dialectic of conjecture and proof.
Moment. A moment is an invariant in the accomplishment of a task tof a given
type T; i.e. it is a type of tasks T
*
that will necessarily appear in carrying out the
task t, whatever the technique (within a certain family of techniques) used to do so.
The reason for the choice of the word moment is explained by the phrase “there
comes a moment when”: there comes a moment when a task t
*
∈T
*
has to be
carried out, whatever the way of accomplishing task t.
Moment of evaluation. See Didactic moments.
Moment of exploration. See Didactic moments.
Moment of institutionalisation. See Didactic moments.
Moment of the first encounter. See Didactic moments.
Moment of the technique. See Didactic moments.
Moment of the technological-theoretical block. See Didactic moments.
Noosphere. See Scale of didactic codeterminacy levels.
Objects. See Cognition.
Paradigm of questioning the world. See Didactic organisations.
Paradigm of visiting works. See Didactic organisations.
Pedagogy. See Scale of didactic codeterminacy levels.
Person. See Persons, institutions, and institutional positions. See also Dialectic of
persons and institutions.
Personal instance. See Instance.
Personal relation. See Cognition.
Persons, institutions, and institutional positions. Aperson is any human
being. In particular a newborn, an infant, or a toddler are persons. An institution is
any created reality of which people can be members (permanent or temporary).
For example, a class, a couple, a football club, a bar, a conference, a concert, etc.,
A Short (and somewhat subjective) Glossary of the ATD xxxi
are institutions. In all institutions, there is a number of positions, at least one. In a
classroom, there is at least the position of student and the position of teacher; in a
bar, the position of client and the position of waiter; in a football team, that of
goalkeeper, etc.
Position. See Cognition and Persons, institutions, and institutional positions.
Positional instance. See Instance.
Possibly didactic situation. See Didactic gestures and the possibly didactic. See also
Research in didactics and the plurality of instances.
Praxeological equipment. See Praxeologies and praxeological analysis.
Praxeological universe. See Praxeologies and praxeological analysis.
Praxeologies and praxeological analysis. A researcher in didactics ξexamines
which gestures δare held, by which instances ŵ, concerning which cognitive nuclei ñ
under which set of conditions C, to be didactic (or antididactic, or isodidactic), and
what changes do these gestures generate in C—including learning and unlearning
processes. The verdict pronounced by the instance ŵis based on their relation to ς,i.e.,
R(ŵ,ς), which depends on the relations R(ŵ,o), R(ŵ,î), R(ŵ,R(î,o)), R(ŵ,(I,p)), R
(ŵ,v
̂), R(ŵ,R(v
̂,(I,p))), R(ŵ,û), R(ŵ,δ), R(ŵ,C), etc. Where do these relations come
from? What are they born of? What is their genesis? These are the key questions, which
can be answered thanks to the notion of praxeology. The ATD contains a theory of
human action whose starting point is the notion of a type of tasks T (and the notion of
task t of type T, which is denoted by t∈T). As is often the case in the ATD, there are
no “size”criteria: “scratching your ear”,“calculating the difference of two integers”,
“writing a novel”,“reading a newspaper article”,“getting married”, are all equally
types of tasks. The realization of task t∈Trequires the use of a technique τ
T
relating to T.
This applies to all actions: walking, singing, eating require learned techniques. In
reverse, imitating a person “denaturalizes”his or her behaviour. The ordered pair
made up of a type of tasks Tand a technique τrelating to Tis denoted by [T/τ]and
is called the praxis block (or “know-how”). Any technique requires an explanatory
or supporting comment, called its technology, denoted by the letter θ.Thetech-
nology θis itself coupled with a justifying discourse at a higher level, which is the
theory, denoted by the capital letter ϴ,ofthetechniqueτ. The ordered pair made
up of the technology θand the theory ϴis denoted by [θ/ϴ]: it is the logos block (or
“knowledge”). Any praxis block [T/τ] supposes a logos block [θ/ϴ]withwhichit
forms a praxeology p=[T/τ/θ/ϴ]. Let us set Π=[T/τ]andΛ=[θ/ϴ], we can write:
p=[T/τ/θ/ϴ]=[T/τ]⊕[θ/ϴ]=Π⊕Λ. In an institution I, the blocks Πand Λof a
praxeology p=Π⊕Λcoming from an institution I
0
are generally modified by the
phenomenon of institutional transposition of praxeologies (or works), which we can write
thus: Φ:p=Π⊕Λ↦p
*
=Π
*
⊕Λ
*
.Ifδis a transpositive gesture and ς=(ñ,û,δ,C)
xxxii A Short (and somewhat subjective) Glossary of the ATD
is ŵ-didactic, we’ll speak of ŵ-didactic transposition. All of the above applies to any
action. If, thus, ŵjudges ςantididactic, this judgment results from a praxeology that
allows ŵto make a judgment based on their relations to the many relevant objects
o. Where do these relations come from? We define the praxeological universe of îby
Ω
✦
(î)≝{p/R(î,p)≠∅}andî’spraxeological equipment by Γ
✦
(î)≝{(p,R(î,p))/p∈
Ω
✦
(î)}. Note that we have Ω
✦
(î)⊂Ω(î), where Ω(î)={o/R(î,o)≠∅}isthecognitive
universe of î,andΓ
✦
(î)⊂Γ(î), where Γ(î)={(o,R(î,o))/o∈Ω(î)} is the cognitive
equipment of î. We then posit that, in the reverse direction, Γ
✦
(î) generates Γ(î)in
the following sense: whatever the object o,therelationR(î,o) results from all the
relations R(î,p), where p∈Ω
✦
(î), involves the object o, whether technically, techno-
logically or theoretically. The above applies to any object o. Thus the relation
R(x,x′)ofpersonxto a person x′, for example to their own mother (x′≠x),
or to themself (x′=x), arises from all the praxeologies to which xhas a non-empty
relation and which involve x′. The same remark applies to institutional instances îand î′.
Praxeology. See Praxeologies and praxeological analysis.
Praxis block. See Praxeologies and praxeological analysis.
Question. Questions are the starting point and the main incentive of didactic life:
to ask oneself—or to ask someone—a question is the basic act that will ultimately
cause praxeologies as yet unknown to be met. Proper managing of questions is
therefore crucial. Two types of questions are mainly considered in ATD. The first
type is made up of how questions: given a type of tasks T,thehow question relating
to T,Q
T
,reads:“How can one accomplish a task tof type T?”An answer to such a
question consists essentially in a technique, τ
T
, complemented by a technological and
theoretical block, [θ
T
/Θ
T
]. The second type of questions is that of why questions—
Why is it so that…? In such a case, the answer consists of an “explanation”,which
finally refers to some technological and theoretical block, of which it is part and
parcel. Note that a why question, in fact, conceals a how question: “Why is it so
that…?”is equivalent to “How can one explain that…?”,thatistoahow question
hinging on the genre of tasks “to explain”. (Answering such a question, however, will
necessitate specifying a type of tasks within the genre.) Indeed, most questions, if not
all, boil down to how questions. For instance, a question about the nature of some
entity—What is didactics? What are miscellanies? What is a pupa? Etc.—hides a
question about how can one define such entity (the genre of tasks being here “to
define”). The same occurs with true-or-false questions—Is it true (or false) that…?—
which disguise questions of the “How can one determine the truth value of…?”
subtype. Moreover, in ATD, all these types of questions are conditional on the
institution to which they are referred: the question “How can one do, explain,
define, etc.”in fact should be construed as meaning “How in this or that institution
do they do, explain, define, etc.”(See also Answers.)
Reduced Herbartian schema. See Herbartian schema.
A Short (and somewhat subjective) Glossary of the ATD xxxiii
Reference instance. See Instance.
Relation. See Cognition.
Research in didactics and the plurality of instances. The field of research in
didactics may be denoted by Δ
̂.InΔ
̂regarded as an institution, there exist different
researcher positions r
̂, among which is the position ρ
̂of the researchers in didactics
working in the framework of the ATD. Researchers in didactics are generically
denoted by the letter ξ. From the point of view of the ATD, the positions r
̂
(including ρ
̂) have neither prerogative nor privilege as concerns their relations to
objects o. Whenever r
̂is presented as a position of “specialists”of an object oin Δ
̂,
one may be tempted to think that the “absolute”statement”R(î,o)≠∅actually
means: r
̂⊦R(î,o)≠∅. The ATD posits that this statement has no special privileges
over other statements of the form ĵ⊦R(î,o)≠∅. Quite often, the position r
̂relies
on a position p
̂reputed to be that of “true specialists”of o, with R(p
̂,o) being the
ultimate criterion to judge R(î,o). This attitude of r
̂towards p
̂is a risky one, for it
can generate an illusion of mastery (relating to o) in the researchers ξin position r
̂,
which in turn may stifleξ’seffort to make sense of the reality observed, including
R(p
̂,o). Now what is of interest to the researcher ξas such? It is the conditions that
can make a relation R(î,o) evolve, in particular that can make the instance îcome
to know the object o“better”. To move forward, ξhas to take a step aside. To this
end, we consider an instance ŵcalled the instance of reference, which can be any
instance; the prefixr
̂⊦is henceforth replaced by the prefixŵ⊦, and we will
therefore ask, for example, whether ŵ⊦R(î,o)≠∅or ŵ⊦R(î,o)=∅. Of course,
we can refocus the analysis on r
̂by taking ŵ=r
̂. The introduction of the reference
instance ŵopens the way for the introduction of the key notions of cognitive nucleus
and possibly didactic situation.
Scale of didactic codeterminacy levels. The ATD seems to differ from many
other theorisations in didactics in that it does not intend to ignore any of the conditions
that possibly exist in a given society. All these conditions are arranged on a scale known
as the scale of levels of didactic codeterminacy. The highest level of the scale is that of
humanity or human species. The deepest level is that of didactic systems S(X,Y,♥),
where Xis the set of students,Yis the set of study aids (teacher, etc.), and where the
symbol ♥indicates the object which is the didactic stake, “the thing to learn”.Too
many didacticians are only interested in the conditions that originate at the level of
didactic systems proper. Even more restrictively, a teacher often tends to focus on
conditions that he or she can create or modify by himself or herself as a teacher.
However, we know that what happens in a didactic system cannot be fully explained
by the gestures of the y∈Yand the x∈X. A didactic system presupposes, first of all, an
institution that makes its existence possible, namely a school (which can be a family, a
sports club, etc.), which itself is included in a society. At this point, the scale has the
following structure: Humankind ⇄…⇄Societies ⇄…⇄Schools ⇄…⇄Didactic systems.
The gestures accomplished, and the conditions created, within a didactic system are
xxxiv A Short (and somewhat subjective) Glossary of the ATD
called (possibly) didactic in the strict sense. The others are (possibly) didactic in the
broadest sense. This is the case for “pedagogical”conditions and gestures, that originate
at the level of pedagogies located between the level of schools and the level of didactic
systems. We thus have: …⇄Schools ⇄Pedagogies ⇄Didactic systems.Togofurther,we
use the notion of work (generically denoted by the letter o). Works are all objects whose
existence within a society is due to human action, that is to say…all objects—
which are never purely “natural”, nor, moreover, created ex nihilo.Thepurposeof
a pedagogy is to lead students to the work oto study. For this reason, we see that a
pedagogy is by no means independent of ♥=o. The existence of a school is thus a
pedagogical condition. The grouping of students into classes is also a pedagogical
condition—there is nothing “natural”about it. The division of knowledge into
disciplines, domains, sectors, themes and subjects of study is another one. These
conditions are unequally specificto♥=o, but they all aim to create the situations
that will allow the study of ♥. The prior arrangement of knowledge in disciplines,
etc., is typical of the pedagogies of visiting works.Itissomewhatdifferent with the
pedagogies of questioning the world. Between schools and societies, we can then situate
the level of the noospheres, an institution that, within society, “manages”the orga-
nization and future of schools. We then have this: …⇄Societies ⇄Noospheres ⇄
Schools ⇄…Here we include the level of noospheres in the level of societies: …⇄
Societies ⇄Schools ⇄…It then remains to examine a level with a pompous name,
the level of civilizations:Humankind ⇄Civilisations ⇄Societies ⇄…Here, the word
“civilization”does not refer to a globality: it has a local meaning. So let us assume
societies Sand S′,institutionsIin Sand I′in S′, and positions pand p′in Iand I′
respectively. (Note that we can have S′=Sand even I′=I). We will say that, for the
instance ŵ,(I,p)and(I′,p′)belong to the same civilisation as concerns the object oif ŵ⊦R
I
(p,o)≈R
I′
(p′,o). (I,p)and(I′,p′)belong to different civilisations if ŵ⊦R
I
(p,o)≉R
I′
(p′,o).
This will be the case, for example, if ŵ⊦R
I
(p,o)≠∅and ŵ⊦R
I′
(p′,o)=∅.Givenan
institutional position (I,p) in a society S,ifŵ⊦R
I
(p,o)attimeȶ
2
≉R
I′
(p,o)attimeȶ
1
,
where ȶ
1
<ȶ
2
, we will say that, between ȶ
1
and ȶ
2
, there has been a civilisational change
in the eyes of ŵas concerns o—achangein civilisation, not a change of civilisation. A
school, a class are institutions where students are almost constantly confronted with
civilisational changes.
School. See Scale of didactic codeterminacy levels.
Semi-developed Herbartian schema. See Herbartian schema.
Specimen of a type of tasks. See Types of tasks.
Student. See Didactic system.
Subject. See Dialectic of persons and institutions.
Supersession. See Dialectic of.
A Short (and somewhat subjective) Glossary of the ATD xxxv
Tasks. See Type of tasks.
Teacher. See Didactic system.
Technique. See Praxeologies and praxeological analysis.
Technological and theoretical block. See Praxeologies and praxeological analysis.
Technology. See Praxeologies and praxeological analysis.
The didactic. See Didactic analysis. See also Type of tasks.
Theory. See Praxeologies and praxeological analysis.
Topos. See Didactic organisations.
Transposition. See Praxeologies and praxeological analysis.
Type of tasks. Every natural language provides an analysis of human activity:
when we say “He washes his hands”,“She solves the equation for x”,“They sing
an old lullaby”,“The man swims to the buoy”,“The mother teaches her three-
year-old to blow his nose”, we refer to some determined task of a certain type:to
wash one’s hands, to solve an equation for x, to sing an old lullaby, to swim to a
buoy, to teach one’s three-year-old to blow one’s nose. A task tof a given type T
is called a specimen of that type, which is usually written t∈T. Grammatically, all
these statements are generally made up of a verb of action (to wash, to solve, to
sing, to swim, to teach) and a direct object (one’s hands, an equation, a lullaby, [the
distance] to the buoy, one’s three-year-old to blow one’s nose). It must be
remarked that, in the last case, the “direct object”involves a nested type of tasks,
“to blow one’s nose”; in fact, the sentence displays the ternary structure typical of
the didactic: someone (Y, in this case, the mother) does something to help someone
(X, her three-year-old) learn something (♥, to blow one’s nose). It must also be
emphasized that a verb of action (to sing, to walk, to draw, etc.) refers not to a type
of tasks but to a genre of tasks, which will be narrowed into different types of tasks by
specifying the object to which the action applies. Although it is usual to say that one has
learnt “to sing”,or“to draw”,or“to cook”,orto“solve equations in one unknown”,
etc., only types of tasks can be objects of learning: genres of tasks are beyond reach since
any new type of objects can create a new type of tasks necessitating a brand-new
technique. Conversely, one cannot learn to accomplish a task regarded as unique: any
task has to be recognised as belonging to a certain type—i.e. as a specimen—so that an
appropriate technique can be applied to it (or can be built). One should, therefore,
beware not to speak of “the task”when indeed the type of tasks is meant.
Why question. See Question.
xxxvi A Short (and somewhat subjective) Glossary of the ATD
Work. A work (French œuvre) is any intentional product of human activity. Two
types of works are of special interest to ATD. The first one is that of questions
(which are indeed products of human activity); the second one is that of prax-
eologies. Both categories of works appear in the reduced Herbartian schema central to
ATD, viz. S(X;Y;Q)➦A
♥
, where Qis a question and A
♥
is a praxeology (or a
significant part of a praxeology).
A Short (and somewhat subjective) Glossary of the ATD xxxvii
ACKNOWLEDGEMENTS
The elaboration of this book has benefited from the support of projects RTI2018-
101153-B-C21 and RTI2018-101153-A-C22 from the Spanish Ministry of Science,
Innovation and Universities, the Agencia Estatal de Investigación (AEI) and the
European Regional Development Funds (MCIU/AEI/FEDER, UE).
1
WHAT KIND OF RESULTS CAN BE
RATIONALLY JUSTIFIED IN DIDACTICS?
Josep Gascón and Pedro Nicolás
1 The question of the presumed normativity in didactics
Let us start with a question about the alleged normative character of didactics:
To what extent, how and under what conditions can (or must) didactics issue
value judgements and normative prescriptions about how to organise and manage
study processes?
Most of the approaches and theories in didactics have an implicit position about
this question. However, explicit discussions about the legitimacy of didactics to
express normative prescriptions remain scarce, let alone specific deliberations
about the logical form of the results one should expect. At the same time,
discourses in didactics often contain value judgements concerning teaching and
learning processes, which sometimes lead to proposals for rules of action—the
well-known “implications for teaching”. Including value judgements and nor-
mative prescriptions in the discourse, often as the main conclusion, appears as a
tacit taking of position that has a strong effect on the sort of research questions
to be considered and the kind of endorsed answers. It may even lead one to
believe that research results in didactics can (or even must) be stated in terms of
values and rules.
How can we answer this question according to the ATD’s basic assumptions
and principles? First, we will explain the ATD perspective about the object of
study of didactics and the corresponding research results. This will make clear
that value judgements and normative prescriptions have no place in the realm
of research results in didactics. Another consequence will be the need to clarify
the type of assertions didactics can formulate on solid foundations (Gascón &
Nicolás, 2017).
2 Max Weber’s thesis on social sciences
The origin of one of the main points of this chapter is (Weber, 1917/2010). In this
work Max Weber stands up for the following thesis:
Social sciences can only state assertions about the rationally suitable means to
achieve ends previously fixed but whose validity cannot be rationally established.
The precept of avoiding value judgements is a consequence of the distinction
between two spheres of reality, the knowledge sphere and the values sphere, each
one containing questions of different nature.
In the knowledge sphere, the main questions raised relate to the behaviour of a
certain portion of the world. Why is this portion of the world the way it is? What
are the required conditions for this portion of the word to be modified in a certain
direction, and which are the obstacles to this modification?
In the values sphere, the main questions are about what we should do in a given
situation; how we value this situation; if we should do something to change this
situation in a given direction, and, if so, in which one.
In the knowledge sphere, it is possible to make a critique of a given value via an
analysis of the required means to attain this value, regarded as an end. Indeed, one can
conclude that, given a school institution I, a certain kind of teaching (regarded as valu-
able, as representing a positive value) is not achievable in Ibecause the required means to
putitintopracticecannotbeimplementedinI. As we can see, this critique cannot state
whether this value is estimable or worthy. It can only say that certain means are suitable
or not to achieve a certain goal, that certain conditions make it easier or more difficult to
attain. Therefore, the object of study of social sciences, according to Weber, concurs
with the object of study of didactics according to the ATD sense (see section 4).
The principle according to which science should not make value judgments has
a positive reverse: the ability of science to provide criteria to guide us on the steps
to follow. These criteria do not permit us to shun the responsibility to make
decisions, but they can help us to foresee the consequences of our actions.
Ultimately, the coercive aspect of Weber’s theses points out the issues and decisions
excluded from the scientific scope. Science cannot teach us what to do,onlywhat can be
done and the corresponding consequences. Thus, for instance, economy, as a science, is
not qualified to decide how material goods should be distributed, in the same way,
sociology cannot decide about the best social or political structure for societies or
physics about what should be done with atomic energy. But, in all these cases, and
many others, science can reveal some relevant consequences of each possible decision.
3 Reinterpretation of Weber’s thesis in epistemological terms
To prepare our description of the object of study of didactics, let us express Weber’s
theses in terms of explanations and scientific laws. Science is important to society to
the extent that it provides laws which support non-trivial and general explanations
4Josep Gascón and Pedro Nicolás
about the occurrence of objective facts. For a fact to be objective we require it to be
intersubjective—perceptible by everyone in standard circumstances—and substantive—its
existence does not depend upon someone’s perception or representation.
Ascientific explanation can be considered as an answer to a question of the type:
“Why the occurrence eis the case (instead of the occurrences d
1
,d
2
,…)?”Of
course, deductive arguments are always welcome, but sometimes something dif-
ferent can be accepted as a scientific explanation, for instance, inductive or abduc-
tive arguments. The reader is invited to consult (Andersen & Hepburn, 2016;
Woodward, 2017) for an overview of this topic.
Typically, a scientific explanation is a valid argument based on a so-called
scientific law. What a scientific law is, remains a controversial issue in the philosophy
of science. We will therefore only present a summary of the most commonly
accepted features of scientific laws. The logical structure of a scientific law is as
follows: “Every occurrence of type A is also of type B”. This logic structure makes
it clear that the basic parts of scientific laws are occurrences-type (defined as the
extension of a property, or giving an exhaustive list, or by recursion, etc.). There-
fore, scientific laws are general statements and not assertions about particular
occurrences localised in time-space. Another feature of scientific laws is that they
seem to express a necessary relationship between types of occurrences, and are not
merely accidental coincidences.
For instance, the law stating that metals are good conductors should not be
regarded as pointing out a fortuitous relationship between the occurrence-type
“being metal”and the occurrence-type “being conductive”. On the contrary, it
seems to say that, given what we know about the world, metals are good con-
ductors and it could not be otherwise. This is an important feature of scientific laws,
which tells the difference between them and incidental true generalities, as, for
example, the statement: “All the mountains on Earth have an altitude of fewer
than 9,000 meters.”Indeed, the property “being a mountain on Earth”does not
seem to imply the property “having an altitude of fewer than 9,000 meters”.
As we have seen, it is difficult to find the formal distinction between scientific
laws and incidental generalities. Nevertheless, there are other features that distin-
guish them. Unlike arbitrary general statements, scientific laws are part of a scien-
tific theory in which they are logically linked to other statements. Via its laws and
its premises, a scientific theory always describes a certain part of the world. Despite
being the compression of the complexity of reality, this description still allows us to
master a certain part of the world: answering questions; state predictions; etc.
As an example of a law in didactics, we can consider the statement about the existence
of the didactic contract (Brousseau, 1997; see also Nicolás, 2015). This law states that, under
certain conditions, a study community formed by students and a teacher, along a study
process (occurrence of type A), behaves according to specific (perhaps implicit) clauses
which rule, among other things, the conduct that students expect from the teacher and
vice versa, regarding the knowledge to be taught (occurrence of type B).
For the law of the didactic contract to be scientific, we should show, on the one
hand, that both types of occurrences are objective and, on the other hand, that the
Justifying results in didactics 5
law itself is objective. To claim this objectivity of the type of occurrences we
should provide a sharp, precise description of them. For instance, we should answer
the question: what are the precise clauses of this contract? If this is not clear
enough, how could we verify the law? To ensure the objectivity of the law we
could check its validity in a representative sample of the kind of study communities
under consideration, and then use statistical inference. Anyway, we could also
consider the law provisionally accepted if we show that, for the moment, it is the
best explanation for a certain mysterious or disturbing phenomenon.
The case of the age of the captain is an example of such a phenomenon. As
explained in Equipe Elémentaire IREM de Grenoble (1979), a research team
noticed that students at school tend to operate with the numbers of the formula-
tion of a problem in mathematics regardless their appropriateness for the solution.
For instance, when they face the problem “A captain of a ship owns 26 sheep and
10 goats. How old is the captain?”, the majority of students give as an answer a
number which results from operating with 26 and 10.
In Chevallard (1988) we find an explanation for this phenomenon: students do
not pay attention to the meaning of the numbers because their behaviour is gov-
erned by a didactic contract including a clause according to which school problems
always have a solution, reachable from the numbers appearing in the formulation.
Therefore, the law that states the existence of the didactic contract including the
clause above appears to be the best explanation for an intriguing phenomenon.
Hopefully, after this brief reflection about scientific laws we are persuaded that
the questions placed by Weber in the knowledge sphere admit scientific answers
(in terms of scientific laws and explanations), unlike the questions placed by Weber
in the values sphere. Indeed, the occurrences involved in the questions of the values
sphere are not objective.
4 The object of study of didactics
The positive part of Weber’s thesis is that we have interesting questions falling
under the knowledge sphere. Let us formulate these questions to link them to
didactics. According to the ATD, the theory of praxeologies is not only rich
enough but also especially useful to describe all the human behaviours relevant for
didactics. Therefore, in the questions included in the knowledge sphere, we will
change the expression “portion of the world”by the term “praxeology”(see
Glossary). Moreover, in our formulation, we will take into account one of the
basic assumptions in ATD according to which the primary object of study in
didactics are the processes of genesis and diffusion of institutional praxeologies.
After these considerations, the resulting questions are as follows. Which is the beha-
viour of a certain praxeology in a given institution? Why is this praxeology the way it is in
this institution?Whicharetherequiredinstitutional conditions for this praxeology to be
modified in a given direction, and what are the institutional obstacles to this modification?
All this is coherent with the object of study of didactics according to the ATD:
‘the ATD suggests the following definition of didactics: didactics is the science of
6Josep Gascón and Pedro Nicolás
conditions and constraints for the diffusion of praxeologies in the institutions of the
society’(Chevallard, 2011, p. 27, our translation).
To study the aforementioned conditions and constraints for the diffusion of
praxeologies and, more explicitly, to account for the didactic phenomena linked to
that diffusion and which appear in the different institutions of the society, didactics
formulates questions belonging to the following three fundamental dimensions of a
research problem: economic, ecological and epistemological (Gascón, 2011).
The economy of any system (of the body of an animal, of a plan, of a language, of
a discourse, of a book, etc.) is formed by the set of rules and principles which
govern the structure and the running of the system. In particular, given an insti-
tution, I, the praxeological economy of Iat a certain period is given by the set of rules
and principles which govern the institutional life of the praxeologies of Iduring
this period. Thus, when we tackle the economic dimension of a didactic problem in
which there are several praxeologies involved, we are concerned with those rules
and principles.
The ecology of a system is given by the set of conditions and constraints that have
an impact on this system. Those conditions and constraints allow an explanation, at
least in part, of the evolution of the system, its current behaviour (its economy) and
its possible future evolution. As a consequence, when we tackle the ecological
dimension of a problem in which there are several praxeologies involved, we deal
with questions concerning the conditions and constraints which:
explain the behaviour of the corresponding praxeologies in the considered
institution at a certain period (that is to say, the economy of those praxeologies
at that moment in that institution),
are required to promote or impede the life of certain kind of praxeologies in
the given institution,
facilitate or hinder the modification of certain kind of praxeologies in a given
direction.
In order to study the economy and the ecology of praxeologies involved in the
study of a certain didactic problem, researchers use as a reference both a model of
the praxeologies relative to the knowledge at stake and a model of the corresponding
didactic praxeologies—those related to the teaching, learning, study and dissemination
of the previous ones. The first model is said to be a reference epistemological model
(REM, see chapter 6), which accounts for what to study, and the second one is said
to be a reference didactic model (RDM), which accounts for how to study. In practice,
both models determine each other so we should speak of a reference epistemological-
didactic model (REDM). The epistemological dimension of a didactic problem includes the
questions devoted to getting criteria for the construction of a suitable REDM.
The three dimensions of a research problem are mutually determined. Still, in
practice there seems to be a certain hierarchy among them. Indeed, to deal with a
question included in the ecological dimension—e.g. how to change the conditions
for the study of a certain piece of knowledge?—one seems to need certain answers
Justifying results in didactics 7
to questions included in the economic dimension—e.g. what is the current state of
play with this piece of knowledge?—which, in turn, are based on the (perhaps
implicit) assumption of an epistemological point of view—e.g. a certain description
of the piece of knowledge at stake (Gascón, 2011, p. 206). Moreover, when we
wonder about the current behaviour of certain praxeologies (economic dimension)
we have to deal with certain derived questions that can be answered only by
researching what happens when we try to change those praxeologies in a given
direction (ecological dimension).
We can now reformulate the negative part of Weber’s theses in terms of prax-
eologies and institutions, to get the kind of questions which, being included in the
values sphere, cannot be addressed from a scientific point of view. They include
questions about how to value the presence of certain praxeologies (for instance,
describing a very algorithmic and rote-learning approach to teaching mathematics)
in a given institution; about changing the presence of these praxeologies in this
institution in a given direction; or about deciding which one of two didactic
praxeologies seems to be the best.
5 Nature of didactic results in the ATD
In the ATD, research results—that is, the (tentative) answers to research questions—
can be formulated in terms of didactic laws describing certain didactic phenomena.
The ATD provides different didactic laws whose formulation relies on the reference
epistemological-didactic model used in each case. Didactic laws addressing the economic
dimension relate to how certain praxeologies behave in a given institution. Let us give
two examples of statements that are good candidates to become didactic laws.
Example 1a: In secondary education in Spain—and, most likely, in similar school
institutions—mathematical praxeologies about proportionality have given rise to a gen-
eral pervasive epistemological-didactic model that isolates mathematical praxeologies
about proportionality from other mathematical praxeologies concerning functional
relations (occurrence of type A). In the corresponding didactic praxeologies one finds
certain phenomena like the avoidanceofalgebraand the characterisation of proportion-
ality as a purely arithmetical relationship (occurrence of type B) (García, 2005).
Example 2a: The mathematical praxeologies about elementary differential calculus in
secondary education in Spain—and, most likely, in similar school institutions—have
given rise to a general pervasive epistemological-didactic model in which praxeologies are
separated from the activity of functional modelling (occurrence of type A), which consists
of using elementary functions as models to provide answers to certain questions. Con-
sequently, the corresponding didactic praxeologies do not attribute to elementary differ-
ential calculus a raison d’être related to the construction, manipulation and interpretation
of different kinds of functional models (occurrence of type B) (Lucas, 2015).
Didactic laws addressing the ecological dimension give an account of the condi-
tions that made possible the current state of the praxeologies (the economy), the
conditions required to change those praxeologies in a certain direction and the
constraints that hinder this change. To state those laws, ATD uses, among other
8Josep Gascón and Pedro Nicolás
things, the theory of didactic transposition (see Glossary), which enlarges the field of
study and provides insights about the constraints which affected, and still affect, the
taught knowledge (Chevallard, 1985/1991). Another important tool is the scale of
levels of didactic codeterminacy (Chevallard, 2011, see Glossary), which enlarges the
kind of conditions and constraints researchers in didactics must consider.
An epistemological-didactic reference model is a conjecture (a scientific
hypothesis) that provides a (partial) answer to the questions appearing in the three
dimensions. In the epistemological dimension, this model redefines the piece of
knowledge at stake (in particular, it might attribute to it an alternative raison d’être).
In the economic dimension it serves as a reference to compare with and char-
acterise the pervasive epistemological-didactic model. Finally, in the ecological
dimension it conjectures that certain conditions will permit to avoid some phenomena
appearing in the mathematical-didactic organisation of the knowledge at stake.
In this sense, reference epistemological-didactic models are tools by which the
researcher emancipates her/himself from the prevailing epistemological and didactic
models (Gascón, 2014). Moreover, they also state implicitly a didactic law:ifina
given institution one follows a study process based on this reference epistemologi-
cal-didactic model (occurrence of type A), then along this study processes certain
didactic phenomena will (not) take place (occurrence of type B).
Let us present, in what follows, examples of possible didactic laws corresponding
to the ecological dimension.
Example 1b: If in the final courses of secondary education in Spain (14–16 years)
one carries out a study process about proportionality based on the epistemological-
didactic reference model presented in García (2005), in which the study of pro-
portionality is integrated into a general study of the possible functional relations
between two magnitudes (occurrence of type A), then the didactic phenomena
described in the example (1a) before will not take place (occurrence of type B).
Example 2b: If in the transition from secondary to tertiary education in Portugal,
France or Spain one carries out a study process about elementary differential calculus
based on reference epistemological-didactic model presented in Lucas (2015)
(occurrence of type A), then the didactic phenomena described in the example (2a)
before will not take place (occurrence of type B).
Why might one be interested in the elimination of certain didactic phenomena as
the ones presented in the two previous examples? Inevitably, as any other didactic
approach does, the ATD assumes certain principles concerning which are the ends of
education. This assumption already conditions the way ATD detects and values phe-
nomena. Of course, the fact that certain ends of education are prioritised among others
belongs to the values sphere, so cannot be rationally founded.
6 Enquiring into the relationship between didactic research
and teaching
As laws in sociology describe some features of the social world and laws in economy
describe some features of the economic world, laws in didactics describe the behaviour
Justifying results in didactics 9
of institutional praxeologies. Those laws are general statements that can be used to
sustain a good explanation for certain (social, economic, didactic) phenomena, but
they never claim value judgments or normative prescriptions or proscriptions.
However, some expressions appearing in works by people using the ATD—for
instance, “to turn the study of proportionality into something meaningful”,“true
raison d’être of the elementary algebra”,“undesirable consequences of the disin-
tegration of school mathematics”—can be disturbing and misleading, as they
assume value judgments which, in turn, induce normative prescriptions or proscriptions
concerning teaching.
Actually, these judgments and norms, although frequent in ATD work, cannot
be presented as research results, not even as consequences of them. Research results
belong to the knowledge sphere and teaching proposals belong to the values
sphere, to the extent that “proposing”entails a taking of positions concerning the
ends of education. Thus, when a theory in didactics fosters certain teaching
proposals, it is necessarily assuming certain ends.
The assumed ends of education are part of the principles embraced by a theory
in didactics. By “principles”we mean those unquestioned statements on which the
whole theory rests. Among these principles, we find not only the aims of education
but also an ontological description of the part of the world the theory deals with
together with methodological premises.
To raise those principles may help to not only illuminate the link between
research and teaching proposals but also increase the degree of self-awareness of the
theory. Only in this way, with our principles explicit, is it possible to distinguish
between research results and values and make clear to what extent we help to
enlarge our knowledge of the didactic dimension of human societies.
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NOTES
Chapter 4
1It should be noted that the technique is described as a set of types of tasks.
2Partial fractions: an integrationist perspective. Math 153 Section 55. Vipul Naik. University
of Chicago. https://vipulnaik.com/math-153-sequence.
3http://public.iutenligne.net/automatique-et-automatismes-industriels/verbeken.
4The Laplace transform is a linear integral operator that transforms f(t) with a real argument
t(t≥0) to a function F(p) with complex argument p. It transforms f’(t)top·F(p).
Chapter 8
1The two countries have rather different educational systems, which again differ from the
French system where the design tool emerged. Therefore the countries are considered
suitable for this analysis.
2A Computer Algebra System is a computer programme able to perform algebraic
manipulations and part of what is called ICT.
3European Credit Transfer and accumulation System introduced by the EU as part of the
ERASMUS programme. Sixty ECTS points corresponds to one year of full-time
university studies.
4The next version was published in 2018, but we use the term current for the 2009
version in this chapter. This is because the 2018 version was not implemented at the
time of writing.
5See for example the results of a national survey on ICT equipment in schools by MEXT:
www.mext.go.jp/a_menu/shotou/zyouhou/1287351.htm.
6See for example the website of T^3 Japan (Teachers Teaching with Technology): www.
t3japan.gr.jp/.
Chapter 9
1Institut de Recherche sur l’Enseignement des Mathématiques.
Chapter 10
1In France, the Certificat d’Aptitude au Professorat de l’Enseignement du Second degré
(CAPES) is the competition for the recruitment of certified teachers for general subjects.
The winners of this competition then follow a second-year training course at the École
Supérieure du Professorat et de l’Éducation (ESPE) as trainee teachers.
Chapter 12
1ECTS Stands for European Credit Transfer System. ECTS credits are a standard for
comparing the volume of learning for higher education. One academic year corresponds
to 60 ECTS credits.
Chapter 13
1We appreciate that some of the descriptions above may be difficult for some readers to
follow (full details are not possible given the space restrictions).
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