In this research, we are interested in a didactic study of the arithmetical contents, where arithmetic refers to elementary theory of numbers. We aimed to study choices of the teaching of arithmetic in France from the early XXth century and to identify institutional constraints for the reintroduction of arithmetic in the secondary education that occurred in the early XXIth and their effects on teaching practices and students’ experiences. First, we lead an epistemological analysis to describe the different mathematical organizations, and definitions that should be chosen for the teaching of arithmetic that we have completed with a review of previous researches in the Anglo-Saxon world on one hand, and in the French works on the other hand. We lead then an institutional analysis of the arithmetic in an ecological perspective to reveal different systems of constraints and conditions that should have an influence on the evolutions of this knowledge during the process of internal didactic transposition, by analyzing the programs and the textbooks in two institutions: Middle school and the fifth year of High school, from the reform of 1902 till 2010, tracking the mathematical organizations and the definitions. Second, we lead a study of the personal relationships of teachers and students regarding the arithmetical concepts involved in fifth year of high school through two questionnaires, including a comparison between teachers’ answers and the answers of their own pupils. A main result of our research is the great instability of the arithmetical content in the French curriculum at Middle school and at the transition from Middle school into High school.
