Education is a field of study for many subjects, each of which deals with only one of its particular aspects, without being able to ensure its consistency with other approaches. Here the author considers the conditions that would allow a direct and scientific — i.e. experimental theoretical — study of the core of the teaching relationship in what he has specific knowledge to be transmitted. It thus tries to prolong the traditional, essentially normative, didactic system under the aegis of experimental epistemology and anthropology. The ultimate aim remains to produce, improve, reproduce, describe and understand the teaching situations that bring students to the activity and culture of mathematicians. Chapter 1 shows an example — teaching rationals and decimals — how general concepts allow the sources of questions to be classified, certain phenomena to be identified, explained, and specific problems to be drawn about the type of situations the pupil needs to take control of in order to acquire a correct knowledge. The information retrieved from the observation of the implementation of these situations provides the basis for a genuine experimental epistemology. Chapter 2 is a study of how educational situations work from the point of view of pupils who fail electively. Chapter 3 defines the basic concepts of the theory of didactic situations and presents a study of their consistency and a method of comparing them with experience or observation. Chapter 4 refers to a number of questions on the methodology of didactic research and a method of checking the factorial analysis of correspondence by analysing the explanatory space. Chapter 5 shows two possible applications of theory in the study of enumeration and geometry teaching. The annexes present detailed studies of certain aspects of this work, in particular the thesis of epistemological barriers in mathematics.