An Introductory Set of Activities Designed to Facilitate Successful Combinatorial Enumeration for Undergraduate Students

Abstract

In this paper, we present a set of activities for an introduction to solving counting problems. These activities emerged from a teaching experiment with two university students, during which they reinvented four basic counting formulas. Here we present a three-phase set of activities: orienting counting activities; reinvention counting activities; and outcome-focused review activities. In presenting the activities, we include a list of sample tasks, suggestions for prompts for effective implementation, and data excerpts that provide evidence for the effectiveness of these activities.

Keywords:

• Combinatorics
• discrete mathematics
• counting problems
• reinvention

Notes

¹ We have reported on this data set previously [12], in which we describe the overall reinvention in detail and emphasize insights about student reasoning. Specifically, we highlight the students’ tendency to rely heavily on empirical patterning and frame this phenomenon within Lockwood’s model of combinatorial thinking [9]. This current paper is novel in that we focus on the instructional implications of this study, providing a practical set of tasks and instructional interventions that may support students in solving counting problems by making informed, strategic decisions.

² We acknowledge that there cannot be a direct analog between what we did in our teaching experiment and what can be done in the classroom – after all, we worked with two above-average students across ten 90-minute sessions. However, we feel that the sequence of tasks proposed here might be effectively used as a set of introductory activities to facilitate meaningful counting for undergraduate students.

³ The Language Books problem is adapted from Tucker [18], and the Apples & Oranges problem is from Martin [13].

⁴ We developed these problems but were generally influenced by problems in Tucker [19] and Martin [13].

Additional information

Notes on contributors

Elise Lockwood

Elise Lockwood is an assistant professor in the Mathematics Department at Oregon State University. She completed her Ph.D. in Mathematics Education from Portland State University and was a postdoctoral fellow at the University of Wisconsin – Madison. Her research focuses on undergraduate students’ combinatorial thinking, and she is passionate about better understanding how to help students solve counting problems more effectively. Since June 2014 she has been a contributing editor for the AMS blog “On Teaching and Learning Mathematics.” In her free time, Elise enjoys cooking, cheering for the Portland Trail Blazers, running, and playing with her cats.

Craig A. Swinyard

Craig A. Swinyard is an associate professor in the Mathematics Department at the University of Portland and recently became the Director of Alumni Relations. He received his Ph.D. from Portland State University. His research interests include students’ understanding of the formal definition of limit, as well as the teaching and learning of discrete mathematics. In addition to teaching, he also loves running, watching sports, and spending time with friends and family.