### Describing knowledge in terms of praxeologies

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.

*Recommended readings*

*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.