### Didactic systems and units of analysis

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.

*Recommended readings*

*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.