Undergraduate students’ initial conceptions of factorials

ABSTRACT

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics.

KEYWORDS:

• Combinatorics
• discrete mathematics
• counting problems
• factorials

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Statements of multiplication principle vary widely and can contain subtle mathematical details (see [28] for a detailed discussion). Our preferred definition is Tucker's (2002) [29]: ‘Suppose a procedure can be broken down into m successive (ordered) stages, with r₁ different outcomes in the first stage, r₂ different outcomes in the second stage, …, and r_m different outcomes in the mth stage. If the number of outcomes at each stage is independent of the choices in the previous stage, and if the composite outcomes are all distinct, then the total procedure has r₁ × r₂ ×…× r_m different composite outcomes’ (p. 170).

2. In this institution, discrete mathematics is a junior-level transition-to-proof course that is required of all mathematics majors. It includes topics such as logic, set theory, proof techniques, functions and relations, and introductory combinatorics.