Conceptualizing and Justifying Sets of Outcomes with Combination Problems

Combination problems are a cornerstone of combinatorics courses, but little research has been done examining the ways that students perceive and differentiate among different combination problems. In this article, we investigate how mathematics education students (n = 18) in a discrete mathematics course view two categorically different combination problems (Category I and II combination problems). In particular, we look at how participants conceptualized each problem's sets of outcomes, counting processes, and formulas, while also exploring the means by which students justified their relationships. Review of the data collected suggests that students tend to be less consistent and have more trouble utilizing and justifying combinations with a collection of ordered indistinguishable objects (Category II) than they do with a collection of unordered distinguishable objects (Category I). Based on these findings, we provide recommendations for the teaching and learning of combinations in combinatorics education.