A cognition-based analysis of undergraduate students' reasoning about the enumeration of permutations

ABSTRACT

Counting is an intellectual activity that is usually identified with the assignment of positive whole numbers ("1, 2, 3, 4, ...") in one-to-one correspondence with a collection of items in one's attentional field. Taking into account a broader, combinatorial meaning of the term, counting extends far beyond this description. Enumerative combinatorics, the field of mathematics concerned with problems and techniques of counting, typically involves determining the number of different "ways" a given set of objects can be arranged, in relation to each other, into particular kinds of composite structures. One basic but fundamental type of structure is a linear arrangement, or permutation. Most often, permutations consist of sets of spatial objects placed in a line, where one permutation differs from another if there is a difference in how objects within the arrangements are placed in relation to each other. However, permutations can also take the form of sequences of actions or events that occur in a temporal order in relation to each other.In general, the number of permutations of n distinct items can be computed using a multiplicative expression of the form n x (n-1) x ... x 2 x 1, which is usually denoted n! (and read as "n factorial"). A vast range of combinatorial structures - including (but not limited to) partial permutations, permutations with repetition, combinations, and combinations with repetition - can be conceptualized in terms of permutations, and they can be enumerated using operations on factorial expressions. Thus, permutations constitute a fundamental concept for conceptualizing and reasoning about many additional combinatorial structures. Yet many important questions about student reasoning and learning of permutations remain unanswered, questions which constitute the focus of this work. What mental actions/operations, concepts, and strategies do non-STEM college students, without prior combinatorics instruction, use to count permutations? Can a progression of levels of sophistication be identified in students' reasoning? How can students' knowledge of counting permutations serve as a constructive resource for their initial enumerations of partial permutations? How can a theory of levels of ion and of spatial-temporal-enactive structuring be used to capture students' reasoning about permutations? Lastly, what instructional approaches were productive toward supporting student learning?To address these questions, I conducted one-on-one teaching experiments with undergraduate students enrolled at The Ohio State University. In this dissertation, I report on data from two of these students' case studies. These two students, Ashley and Mary (pseudonyms), were chosen because they had not received prior instruction in combinatorics or probability, and because I found their development throughout the study to likely be representative of a broader case of undergraduate students' reasoning in relation to students who participated in a prior research study. Because of the COVID-19 pandemic, each teaching experiment was conducted remotely using the video conferencing system Zoom. Each student's teaching experiment consisted of a pre-assessment, a set of 9 to 10 teaching episodes, and a post-assessment. Pre- and post-assessments were conducted as semi-structured interviews, while teaching episodes consisted of tasks and follow-up questions intended to elicit perturbations and/or new insights. Multiple tasks involved digital learning environments, developed in Geometer's Sketchpad, within which students were able to use perceptual materials to reason about combinatorial structures. Through a careful and detailed analysis of the two case-study students' actions, concepts, and strategies that emerged throughout their teaching experiments, I address the overarching questions that guided this work. (PsycInfo Database Record (c) 2023 APA, all rights reserved)