Constructing and Consolidating of Algebraic Knowledge within Dyadic Processes: A Case Study

Studies of knowledge constructing often focus on the analysis of a single episode, without considering enough the history of the learners, or the future learners' trajectories with regard to the concepts learned. This paper presents an example of knowledge constructing within the context of peer learning. We show how the design of the task and the tools available to the students afford the constructing of conceptual knowledge (the phenomenon of exponential growth and variation, as it is expressed in its numerical and graphical representations). We trace the constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment. We show that knowledge is constructed cumulatively, each activity allowing for the consolidating of previous constructs. This pattern indicates the nature of the processes involved: knowledge constructing and consolidating are dialectical processes, developing over time, when new constructs stem from old ones already consolidated, which gain consolidation through the new construction, creating a new abstract entity. We also discuss the potential of the tool the students used (a spreadsheet program) to such processes of learning mathematics.