The Question of Method in a Vygotskian Semiotic Approach

Abstract

This paper presents two methods of analysis of interaction processes in mathematics classes—the analysis of argumentation and the analysis of participa-tion –, and it furthermore explores the relationship between these methods and their resulting impact on the development of elements of an interaction theory of math-ematics learning. The main theoretical assumption of this article is that learning mathematics depends on the student's participation in processes of collective argumentation. On the empirical level such processes will be analyzed with meth-ods that are based on Toulmin's theory of argumentation (Toulmin, SE. (1969). The uses of argument. Cambridge, UK: Cambridge University Press) and Goffman's idea of decomposition of the speaker's role (Goffman, E. (1981). Footing. Forms of talk. ders. Philadelphia: University of Philadelphia Press). Different statuses of par-ticipation in processes of argumentation will be considered, which allow a theoreti-cal description of different stages in the process of learning mathematics from the perspective of an interaction theory of mathematics learning. Who did it a different way? 1 This question of a second grade mathematics teacher might serve as a catchy utterance that highlights my interest in the analysis of argumentation and participa-tion. The teacher was a member of an American-German project funded by the Spencer Foundation (Cobb and Bauersfeld 1995). The research team analyzed videotaped classroom situations, which had been accomplished in the following way: First, there were group work sessions, in which groups of children were sup-posed to solve the problems of given work sheets. The instructions for the children were that they should solve as many problems as they want and that they have to explain to each other their ideas about the solution. After a group work session, the teacher initiated a whole class discussion, in which the students were asked to pres-ent the results of their group work. After the presentation of a few solutions, the teacher generally asked: " Who did it a different way " . The intention of this question was that the second graders should present a variety of different ways of solving the given mathematical problems. 2 While analyzing parts of the video sequences, it was striking to me, that the presented ways of solving the problems mainly serve in the interaction as an explanation and justifi cation for the found result. That means they have an argumentative function in the sense that the students attempt to demonstrate to the class what they did and to convince the class that their way of solving the problem is 'ok'. Moreover, in the emerging sequence of presenting different solutions for the same problem, the question arose for me, how inventive and independent were the presented solutions at the end of such a round of classroom discussion. Methodologically, this insight led to the search for procedures that allow the accomplished argumentation of the participants of a mathematics class to be ana-lyzed and that allow a classifi cation of the originality of their contributions in the series of ongoing presentations of different solving methods. In the following, I introduce an analysis of argumentation and an analysis of participation, each method exemplifi ed by the same two primary mathematics classroom interactions.