Appropriating geometric series as a cultural tool:a study of student collaborative learning

摘要

The aim of this article is to illustrate how students, through collaborative small-group problem solving, appropriate the concept of geometric series. Student appropriationof cultural tools is dependent on five sociocultural aspects: involvement in joint activity,shared focus of attention, shared meanings for utterances, transforming actions andutterances and use of pre-existing cultural knowledge from the classroom in small-groupproblem solving. As an analytical point of departure, four mathematical theoreticalcomponents are identified when appropriating the cultural tool of geometric series: (1)estimating of parameters, (2) establishing of the general term, (3) composing of the sum and(4) deciding on convergence. Analyses of five excerpts focused on the students' socialprocesses of knowledge objectification and the corresponding semiotic means, i.e., lecturenotes, linguistic devices, gestures, head movements and gaze, to obtain shared foci andmeanings. The investigation of these processes unveils the manner in which the studentsestablished links to pre-existing mathematical knowledge in the classroom and how theysimultaneously combined the various mathematical theoretical components that go intoappropriating the cultural tool of geometric series. From the excerpts, it is evident that thestudents' participation changes throughout their involvement in the problem-solvingprocess. The students are gaining mathematical knowing through a process of transformingand by establishing shared meanings for the concept and its theoretical components.