"I don't know—I'm just genius!": Distinguishing Between the Process and the Product of Student Algebraic Reasoning

Abstract

This paper reports on a semiotic-cultural analysis of two high-school minority seniors' participation in a mathematics instructional intervention conducted at a school for academically at- risk students. Focusing on the students' speech acts, gestures, and artifact production during their successful collaborative engagement in an algebraic pattern-finding task, I adopt and adapt Radford's (2003) framework to evaluate the quality of each student's actual learning as implicated by their discursive contributions. By analyzing each student's utterances as either "to" a semiotic mode (generalization) or just "in" the mode, and then overlaying these coded utterances, I show how one student's participation, which appears to mark a developing algebraic command, in fact blinds the teachers to underlying discontinuity in the student's meaning construction. The study illuminates critical tradeoffs in the design and facilitation of collaborative problem solving.