A Model of Students’ Combinatorial Thinking: the Role of Sets of Outcomes

Combinatorial topics are prevalent in undergraduate curricula, and research indicates that students face difficulties when solving counting problems. The literature has not sufficiently addressed students' ways of thinking about combinatorial concepts at a level that enables researchers to understand how students conceptualize counting problems. In this paper, a model of students’ combinatorial thinking is presented that emphasizes relationships between formulas/expressions, counting processes, and sets of outcomes; additionally, the model is used to frame several examples of students’ reasoning about counting problems. The model serves as a conceptual analysis of students' thinking and activity related to counting, providing language to describe and explain aspects of students' counting activity. In this way, the model has practical implications, both for researchers (providing a lens through which to examine data on combinatorics education) and for teachers (providing an aid to instructional design based on student thinking).