Units Coordination, Combinatorial Reasoning, and the Multiplication Principle: The Case of Ashley, an Advanced Stage 2 College Student

ABSTRACT

The multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research literature, however, has not captured, generally, how students establish these unit relationships. We provide a second order model of an advanced stage 2 college student, Ashley, who had no prior combinatorics instruction, as she engaged in solving combinatorics problems that we considered to involve the MP. Our findings suggest that Ashley began by interpreting combinatorics problems using her whole number iterative units coordination scheme. Through engagement with the teacher-researcher, Ashley constructed combinatorial composites using a pairing operation, units coordination, and units simplification. We also found that Ashley was able to create a three-level-of-unit structure in activity, and to use notation that she produced to re-instantiate the reasoning that produced this unit structure. Doing so provides novel insights into how advanced stage 2 students, especially those at the college level, can use notation to manage complex unit relationships.

KEYWORDS:

• Combinatorics
• multiplication principle
• scheme theory
• teaching experiment
• units coordination

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Thompson’s definition of figurative material generalizes Piaget’s (1970) definition where Piaget defined figurative material as pertaining to kinesthetic or sensory-motor experience and operative as pertaining to mental operations.

² Hackenberg and colleagues use the stages of units coordination to refer to a subset of Steffe’s number sequences (see Ulrich, 2015, 2016a, for a discussion of the relationship).

³ Discrete units of one are units that are not contingent upon sensory-motor material that is constituted in perception nor to this material re-generated in visualized imagination (see, Steffe & Cobb, 1988, arithmetical unit).

⁴ Author 1’s use of the term units coordination differs from Steffe’s (1992) original definition of units coordination in that it entailed a units coordination between a composite unit and a composite number of pairs.

⁵ Elsewhere, Author 2 (date1) has called this type of solution an objects-times-positions solution.

⁶ We use a comma between numbers to represent a pairing and no comma to represent a composite unit. The numbers represent the following, 1 = red, 2 = blue, 3 = green, 4 = black, and 5 = pink. This equivalence between numbers and colors is a researcher introduced equivalence, but it is based on our inference that Ashley had interiorized an order for the units of her composite units. That is, we infer her solution of systematically moving a color to each position in the composite unit to be a result of having interiorized a unit of ordered units.