A task to connect counting processes to lists of outcomes in combinatorics

Abstract

Research has shown that solving counting problems correctly can be difficult for students at all levels, and mathematics educators have sought to identify strategies and interventions to help students reason conceptually about combinatorial tasks. A set-oriented perspective (Lockwood, 2014) is a way of thinking about counting problems that emphasizes the importance of reasoning about the set of outcomes being counted. From a set-oriented perspective, one possible type of intervention is to have students focus on the sets of outcomes rather than formulas and expressions, and specifically to reason about the structure of the set of outcomes. Yet, reasoning about sets of outcomes is not sufficient for students to make connections between outcomes and counting processes. In this paper, we investigate tasks where students wrote computer code to enumerate the set of outcomes in a specific order by implementing listing processes, and they were then asked to determine a specific numbered outcome in their list by using the structure of their enumeration scheme. We clarify particular aspects of a set-oriented perspective that were productive for students, and we demonstrate that tasks that asked students to name a specific outcome in their list elicited meaningful connections between counting processes and sets of outcomes. Further, such tasks reinforce desirable mathematical practices such as leveraging structure and connecting representations.