**Credits : **All modules are presented by Marianna Bosch (Universitat de Barcelona and Universitat Ramon Llull), except Module 0 by Jean-Luc Dorier (University of Geneva). All videos are produced by Aitor Ruiz Santiago (Tecnocampus Mataró, Universitat Pompeu Fabra, Barcelona), except the video of module 0 produced by Alexandre Bourquin (University of Geneva).

Welcome to the Anthropological Theory of the Didactic! This module presents some of its main methodological principles, following the introduction of the book including works from several researchers in France, Spain, Denmark, Japan, Belgium, Portugal and Mexico, edited by Marianna Bosch, Yves Chevallard, Francisco Javier Garcia and John Monhagan, published by Routledge in 2020 under the title Working with the anthropological theory of the didactic. A comprehensive casebook.

**Bibliography**

Bosch, M., Chevallard, Y., Garcia, F. J., & Monhagan, J. (2020). Working with the anthropological theory of the didactic. A comprehensive casebook. Routledge.

Introduction and Glossary

https://www.taylorfrancis.com/books/e/9780429198168

The theory of the didactic transposition, proposed by Yves Chevallard in the 1980s, corresponds to the first developments of the ATD. The focus is on the knowledge to be taught, a new entity that is usually taken for granted in educational research, and its relations with the scholarly knowledge and the knowledge actually taught. You will see in this module the kind of object of study defined by the ATD, which can initially appear as unconventional to the newcomer. The example of proportionality illustrates the type of analyses proposed.

**Bibliography**

A short and easy-to-read article about Didactic transposition in the Encyclopedia of Mathematics Education:

• Chevallard Y., Bosch M. (2014). Didactic transposition in mathematics education. In: Lerman S. (Ed.), Encyclopedia of Mathematics Education. Springer, Dordrecht.

An old text explaining the origins of the didactic transposition theory and pointing at some of its crucial notions, such as the notion of didactic phenomenon and the opposition between the school actor’s point of view and the didactician’s theoretical construction:

• Chevallard, Y. (1989). On didactic transposition theory: some introductory notes. In H. G. Steiner & M. Hejny (Eds.), Proceedings of the International Symposium on

Selected Domains of Research and Development in Mathematics Education (pp. 51−62). University of Bielefeld, Germany, and University of Bratislava, Slovakia.

The text of a conference given by Yves Chevallard in Osaka (Japan) in October 2016, presenting the main elements of the ATD. The theory of didactic transposition is presented in section 2:

• Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: an attempt at a principled approach. Hiroshima Journal of Mathematics Education, 12, 71-114.

The notion of praxeology is one of the key notions of the ATD. It provides a neutral and flexible tool to describe any kind of knowledge and also any kind of human activity. It overcomes the classical dichotomies of procedural and conceptual knowledge, of theory and practice, of thinking and doing. Praxeological analysis helps develop the example of proportionality introduced in Module 2.

**Bibliography**

In his plenary lecture at the 4th Congress of the European Society for Research in Mathematics Education (CERME 4), Yves Chevallard introduces the notion of praxeology for an international audience and presents the notion of study and research path (which is here named study and research programme or course):

• Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.) Proceedings of CERME 4 (pp. 21-30). Barcelona, Spain: Fundemi IQS.

One of the first papers that systematically uses praxeologies to analyse the teacher’s practice. It includes a description of the school mathematical organisation relating to limits of functions in terms of praxeologies:

• 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.

The text of a conference given by Yves Chevallard in Osaka (Japan) in October 2016, presenting the main elements of the ATD. The theory of praxeologies is presented in section 4:

• Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: An attempt at a principled approach. Hiroshima Journal of Mathematics Education, 12, 71-114.

The text of a lecture given by Carl Winsløw at the ICME 12 conference in Seoul (South Korea) using praxeological analysis to describe general features of university mathematics:

• Winsløw, C. (2015). Mathematics at University: The Anthropological Approach. In S. Cho (Eds.) Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 859-876). Springer, Cham.

This module develops the praxeology analysis previously introduced on the case of proportionality already used in the previous modules. Praxeologies of different “sizes” (pinpoint, local, regional and global) help consider this school mathematical content from a broad perspective and elaborate a research reference model about proportionality. The didactic transposition phenomena affecting the teaching and learning of proportionality can then be related to different mathematical domains, like the old theory of “ratios and proportions”, algebra, functions and, especially, the neglection of quantities and units in school mathematical work.

**Bibliography**

An article in the ICMI bulletin to celebrate the 25 years of the first publication of the book on didactic transposition. It presents praxeological analysis as an extension of analysis in terms of didactic transposition and introduces the notion of reference epistemological (or praxeological) model:

• Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI Bulletin, 58, 51–65.

A chapter introducing the ATD as a research methodology in a book about networking theories. The praxeological analysis is particularly developed about both mathematical and didactic (teaching and learning) activities:

• Bosch, M., & Gascón, J. (2014). Introduction to the Anthropological Theory of the Didactic (ATD). In A. Bikner-Ahsbahs, S. Prediger (Eds.), Networking of Theories as a Research Practice in Mathematics Education (pp. 67-83). Dordrecht: Springer.

A book including contributions from several researchers from France, Spain, Denmark, Japan, Belgium, Portugal and Mexico working within the ATD (Chapters 3, 4, 5, 6):

• Bosch, M., Chevallard, Y., Garcia, F. J., & Monhagan, J. (2020). Working with the anthropological theory of the didactic. A comprehensive casebook. Routledge.

The ATD is one of the first frameworks to develop an ecological approach in didactics. The main question is to study what makes a certain teaching and learning process possible, what conditions promote the existence of certain school activities and what constraints, on the contrary, hinder their implementation. Before trying to change our educational reality, it is critical to better know the conditions and constraints that can explain the current state of things. This requires researchers to adopt a broad perspective about the didactic activities they want to describe and analyse, in order to go beyond the level of the mathematical activities or contents at stake.

**Bibliography**

This module is based on a communication presented by Marianna Bosch, Jordi Cuadros, Ignasi Florensa and Noemí Ruiz-Munzón at the 2nd congress of the International Network for Didactic Research in University Mathematics (INDRUM) in 2018:

• Florensa, I., Bosch, M., Cuadros, J., Gascón, J. (2018). Helping lecturers address and formulate teaching challenges: An exploratory study. INDRUM 2018, INDRUM Network, University of Agder, Kristiansand, Norway. ffhal-01849937f

The lecture given by Marianna Bosch at the ICME 12 conference in Seoul (North Korea) illustrates with the case of school algebra the praxeological and ecological analyses made possible by the ATD:

• Bosch, M. (2015). Doing research within the anthropological theory of the didactic: The case of school algebra. In S. Cho (Eds.) Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 51-70). Springer, Cham.

A paper using the scale of levels of didactic codeterminacy to analyse international comparative research on mathematics education:

Artigue, M., & Winsløw, C. (2010). International comparative studies on mathematics education: A viewpoint from the anthropological theory of the didactic. Recherches en Didactique des Mathématiques, 30(1), 48–82.

• Wozniak, F. (2007). Conditions and constraints in the teaching of statistics: the scale of levels of determination. In Pitta-Pantazi, D. & Philippou, G. (Eds.), Proceedings of the European Society for Research in Mathematics Education (pp. 1808-1818). University of Cyprius. http://www.mathematik.uni-dortmund.de/~erme/CERME5b/WG11.pdf

In the ATD, didactic systems model the relationship between a student or a group of students, a teacher or a group of teachers and a didactic stake to study, which can be a piece of knowledge or a question. There are different ways to analyse a didactic system, depending if the focus is put on the didactic stake, on the evolution of the didactic system or on its ecology. This module considers the case where the didactic stake is a piece of knowledge modelled in terms of praxeologies, within the pedagogical paradigm of visiting works.

**Bibliography**

The module presents some examples that can be found in:

• 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., & Gascón, J. (2013). The ecological dimension in the teaching of mathematical modelling at the university. Recherches en Didactique des Mathématiques, 33(3), 307–338.

• Ruiz-Munzón, N., Bosch, M. & Gascón, J. (2013). Comparing approaches through a reference epistemological model: The case of school algebra. In B. Ubuz, Ç. Haser, M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 2870-2879). Ankara: Middle East Technical University.

• Bosch, M. (2015). Doing research within the anthropological theory of the didactic: The case of school algebra. In S. Cho (Eds.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 51-70). Springer, Cham.

The idea that every research work defines its own unit of analysis is exploited in the following paper, with the case of mathematical modelling at the university level:

• Barquero, B., Bosch, M., & Gascón, J. (2019). The unit of analysis in the formulation of research problems: The case of mathematical modelling at the university level. Research in Mathematics Education, 21(3), 314-330.

More generally, this chapter relates the notion of didactic engineering born in the Theory of Didactic Situations with the research methodologies proposed by the ATD:

• 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). Zürich: Springer.

This module focuses on the case in which the didactic system is built around a question to answer instead of a piece of knowledge to learn. This change of focus requires a change of paradigm where the visit of works is embedded into the broader paradigm of questioning the world. Notions like the Herbartian schema and the study and research paths are used to analyse inquiry processes, their components and dynamics.

**Bibliography**

Modules 7 and 8 rely on a lecture given by Marianna Bosch at the International Congress of Mathematicians in 2018:

• Bosch, M. (2018). Study and research paths: A model for inquiry. Proceedings of the International Congress of Mathematicians. Rio de Janeiro, Vol. 3 (4001–4022).

There are many texts in English about the change of paradigm and the new epistemology based on the inquiry into questions. Here are some examples in chronological order:

• Chevallard, Y. (2007). Readjusting Didactics to a Changing Epistemology. European Educational Research Journal 6(2), 131-134.

• Chevallard, Y. (2015). Teaching Mathematics in tomorrow’s society: A case for an oncoming counter paradigm. In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education (pp. 173-187). Springer International Publishing.

• Bosch, M., & Winsløw, C. (2015). Linking problem solving and learning contents: The challenge of self-sustained study and research processes. Recherches En Didactique Des Mathématiques, 35(3), 357–399.

• Bosch, M., Chevallard, Y., Garcia, F. J., & Monhagan, J. (2020). Working with the anthropological theory of the didactic. A comprehensive casebook. Routledge. (Chapters 7, 8, 9).

This module develops the analysis in terms of inquiry process to approach the critical issue of the conditions needed for the paradigm of questioning the world to prevail in our educational systems.

**Bibliography**

Modules 7 and 8 rely on a lecture given by Marianna Bosch at the International Congress of Mathematicians in 2018:

• Bosch, M. (2018). Study and research paths: A model for inquiry. Proceedings of the International Congress of Mathematicians. Rio de Janeiro, Vol. 3 (4001–4022).

Here are two interesting papers about the ecology of study and research paths and the difficulties to integrate them into Spanish current university education:

• Barquero, B., Bosch, M., & Gascón, J. (2013). The ecological dimension in the teaching of mathematical modelling at the university. Recherches en Didactique des Mathématiques, 33(3), 307–338.

• Barquero, B., Monreal, N., Ruiz-Munzón, N., & Serrano, L. (2018). Linking transmission with inquiry at university level through study and research paths: The case of forecasting Facebook user growth. International Journal of Research in Undergraduate Mathematics Education, 4(1), 8-22.

This module presents how cognition is modelled in the ATD, what are the main elements of the model proposed and how they can be used in didactic analysis.

The ATD ICMI AMOR series of modules has not adopted an axiomatic perspective. Otherwise, this would have been the first module of the series because it includes the basic notions of the theory, its primitive terms. The video’s protagonists are the notions of object, person, institution, position, and the personal or institutional relations to an object. A simple written formalism is used to work with these notions. As mathematicians know well, symbols are sometimes useful and aseptic tools to represent entities that our language permeate with connotations and assumptions that are difficult to control.

**Bibliography**

The elements presented in this module are fully introduced in several papers from Yves Chevallard and other colleagues. We include here the oldest one: a lecture given by Yves Chevallard at the 6th Summer School of Didactics of Mathematics in Plestin-les-Grèves (Bretagne, France) in August 1991:

- Chevallard, Y. (1992). Fundamental concepts in didactics: perspectives provided by an anthropological approach. Research in Didactique of Mathematics (selected papers), 131-167.

Original paper in French:

- Chevallard, Y. (1992). Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique. Recherches En Didactique Des Mathématiques, 12(1), 73–112.

The following are the most recent publications where the model of cognition is presented in updated versions:

- Chevallard, Y. (2020). Some sensitive issues in the use and development of the anthropological theory of the didactic. Educação Matemática Pesquisa, 22(4), 013-053.
- Chevallard, Y., Barquero, B., Bosch, M., Florensa, I., Gascón, J., Nicolás, P., & Ruiz-Munzón, N. (Eds.). (2022). Advances in the Anthropological Theory of the Didactic. Springer International Publishing AG.
- Chevallard, Y., & Bosch, M. (2020). Anthropological theory of the didactic (ATD). Encyclopedia of mathematics education, 53-61.

FURTHER READING: THE ATD AND THE TEACHING PROFESSION

As many other approaches in mathematics education, the ATD has been involved since its first developments with the design, analysis and implementation of teacher professional development programmes. Here are some references relating to this research axis:

- Barquero, B., Bosch, M. & Romo, A. (2018). Mathematical modelling in teacher education: Dealing with institutional constraints. ZDM Mathematics Education 50, 31–43.
- Bosch, M., Chevallard, Y., Garcia, F. J., & Monhagan, J. (2020). Working with the anthropological theory of the didactic. A comprehensive casebook. Routledge. (Chapters 10, 11, 12, 13).

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