Seeking Coherence in the Multiplicative Conceptual Field: A Knowledge-in-Pieces Account

Abstract The present study is motivated by a significant body of research documenting teachers’ perennial difficulties with a critical swath of topics related to multiplication. In response, we track how Nina, a future middle grades mathematics teacher, made progress constructing explanations across topics by reasoning with measurement-based definitions of multiplication and of fractions and by coordinating symbolic representations with math drawings. The dataset spans 1 semester of Nina’s in-class work during a content course—explicitly designed to foster coherence within the multiplicative conceptual field—as well as her written assignments for the course and her moment-to-moment reasoning during three interviews conducted near the beginning, middle, and end of the semester. A main result is that constructs from coordination class theory, a strand of theory within the knowledge-in-pieces epistemological perspective, were particularly useful for tracking and explaining Nina’s piecemeal progress. The broad contribution of the article is two-fold—(a) a shift in focus from research on reasoning about one or two topics toward reasoning across a wider range of topics related to multiplication and (b) highlighting refinement and coordination of knowledge resources as basic processes by which future teachers can progress toward coherent understandings of critical school mathematics content.