Leveraging a Categorization Activity to Facilitate Productive Generalizing Activity and Combinatorial Thinking

Abstract In this paper, we present data from two iterative teaching experiments involving students’ constructions of four basic counting problems. The teaching experiments were designed to leverage the generalizing activities of relating and extending to provide students with opportunities to reflect on initial combinatorial activity when constructing these formulas. We discuss three combinatorial ways of thinking that emerged from their work on this task and provide commentary on the interplay between the students’ generalizing activities and engagement in the emergent ways of thinking. We address the practical goals of identifying ways that students productively reason in combinatorics, and we provide a theoretical commentary on students’ reflection on activity as it occurs in the context of particular generalizing actions.