Teacher guidance of algebraic formula building: functional grammatical analysis of a whole-class conversation

How does teacher-guided whole-class interaction contribute to expanding students’ potential for making and exchanging mathematical meanings? We address this question through an interpretative analysis of a whole-group conversation in a sixth grade class taught by an experienced teacher in a school in Southern Argentina. The extended interaction (circa 160 speaker turns) concerned a problem involving remainders that calls for building a direct formula and expressing it in symbolic algebraic language. Our functional-grammatical analysis of the selected conversation revealed that, in synch with the deployment and use of the function table, the manner in which the teacher worded her questions was instrumental to the simultaneous accomplishment of three goals. First, her use of the personal pronouns “I” and “we” with generalized referents helped shape a collective zone of proximal development for the class to think aloud together about the algebraizing task at hand. Second, her verb choices supported students in moving among material, mental, operative, and relational processes on the road toward the targeted formula. Finally, by means of repetition, conjunction, demonstrative pronouns, and pointing gestures, the teacher molded the conversation as a coherent and cohesive multi-semiotic text.