Steering (dis)course between metaphors and rigor: Using focal analysis to investigate an emergence of mathematical objects

This study deals with students' construction of mathematical objects. The basic claim is that the need for communication - any attempt to evoke certain actions by others - is the primary driving force behind all human cognitive processes. Effectiveness of verbal communication is seen as a function of the quality of its focus. Material objects may serve as a basis for creation of such a focus, but in some discourses, focus-engendering objects must be created. Such discursive construction is observed in analysis of one classroom episode. Special attention is given to metaphor, which is the point of departure for the construction process, and to the subsequent dialectical process of closing the gap between the metaphor-induced expectations and the need for a well-defined construction procedure to ensure effective communication.