Mathematical tasks and the collective memory: How do teachers manage students' prior knowledge when teaching geometry with problems

This is a study of how teachers manage students' prior knowledge in problem-based mathematics teaching. I propose that geometry teachers manufacture a collective memory which they use to hold students accountable for what they should remember and what they should forget when they work on problems. This hypothesis is put to work in analyzing two corpuses of data. I inspect a corpus of video records of a problem-based unit on quadrilaterals, where a teacher made changes to usual practices in two ways, by asking students to call forth knowledge from prior mathematics classes and by having students anticipate a theorem that the teacher had not stated yet. The second corpus consists of proceeds of five focus group sessions in which experienced geometry teachers viewed and discussed records of problem-based teaching in geometry and where they designed tasks in which they would engage their students. The analysis uncovered teachers' assumptions and normative stances on how to manage students' prior knowledge. In addition, from the analysis I describe a catalogue of teaching actions that teachers accept they might avail themselves for shaping the collective memory of a class. Methodologically, the study shows how to investigate teachers' management of prior knowledge by applying tools from Systemic Functional Linguistics to transcripts of mathematics classroom talk and to transcripts of conversations among practitioners. This work is a contribution to the study of teaching by describing the kinds of resources that teachers could use in teaching with problems, and the underlying rationality for teaching actions to manage students' prior knowledge.