Our research is concerned with teacher's knowledge, and especially with teacher's processes of learning, in the classroom, from observing and interacting with students' work. In the first part of the paper, we outline the theoretical framework of our study and distinguish it from some other perspectives. We argue for the importance of distinguishing a kind of teacher's knowledge, which we call didactic knowledge. In this paper, we concentrate on a subcategory of this knowledge, namely observational didactic knowledge, which grows from teacher's observation and reflection upon students' mathematical activity in the classroom. In modeling the processes of evolution of this particular knowledge in teachers, we are inspired, among others, by some general aspects of the theory of didactic situations. In the second part of the paper, the model is applied in two case studies of teachers conducting ordinary lessons. In conclusion, we will discuss what seems to be taken into account by teachers as they observe students' activity, and how in-service teacher training can play a role in modifying their knowledge about students' ways of dealing with mathematical problems.KEY WORDS: case study, didactic knowledge, didactique of mathematics, ordinary mathematics lesson, theory of didactic situations, teacher's activity, teacher's knowledge Educational Studies in Mathematics (2005) 59: 205-234

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L’introduction de la notion de variable dans le modèle praxéologique rejoint en partie les raisons de son introduction dans la théorie des situations didactiques. Nous allons dans un premier temps rappeler ces raisons puis présenter l’intérêt d’introduire la notion de variable dans la théorie anthropologique du didactique.

On a building site, the operations carried out by the workman are the most visible part of his work. These actions, which have a professional purpose, are the outcome of a body of knowledge which is less visible to a building site observer (because it is unformulated and is often incapable of being expressed). This knowledge forms part of the device which makes it possible for the actions observed to take place. We describe a methodology which allows mathematical objects present in professional pratices to be made visible. An analysis a priori of possible techniques for creating the formwork of a wall on a sloping slab fed into an investigation on a construction site course which attests to the presence of mathematical knowledge within the structure of vocational training. The problem of assuming responsibility for this knowledge within vocational education forms part of the discussion of this paper.

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