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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. [ABSTRACT FROM AUTHOR]

This article is concerned with an average student's understanding of the notion of function. It presents an analysis of a dialogue held with Kasia, a sixteen-years-old pupil of a general academic secondary school. The conversation concentrates on a non-standard problem – a functional equation. The theoretical framework used here is the PROCEPT THEORY formulated by Gray and Tall (1993, 1994). The article diagnoses Kasia's procept of function revealed during the dialogue. Her difficulties with understanding this notion can be described as based on a very limited procept of function and a misinterpretation of the symbols used in the functional equation. Three kinds of sources of the difficulties are identified: firstly, the intrinsic ambiguities in the mathematical notation, secondly, the restricted context in which some symbols occur in teaching and a limited choice of mathematical tasks at school, and, thirdly, Kasia's idiosyncratic interpretation of mathematical tasks. Moreover, the paper describes several important changes in understanding of symbols and of the concept of function that took place in the course of the dialogue. This article shows that we can use functional equations as both a research and a didactic tool in mathematics education. [ABSTRACT FROM AUTHOR]

As a unifying feature of this Special Issue, we have asked proponents of each framework to analyse an empirical classroom account of one student's process of solving a mathematical problem. Here, for the case study of ‘Frank’, we give the main data that were available to all authors. [ABSTRACT FROM AUTHOR]