Elise Lockwood

Set Partitions and the Multiplication Principle

Abstract To further understand student thinking in the context of combinatorial enumeration, we examine student work on a problem involving set partitions. In this context, we note some key features of the multiplication principle that were often not attended to by students. We also share a productive way of thinking that emerged for several students who were then able to resolve the issues in question. We conclude with pedagogical implications.

Reinforcing Mathematical Concepts and Developing Mathematical Practices Through Combinatorial Activity

As a branch of discrete mathematics, combinatorics is an area of mathematics that offers students chances to engage with accessible yet complex mathematical ideas and to develop important mathematical practices. In this chapter, we focus on a combinatorial task involving counting passwords, and we provide examples of affordances that undergraduate students gained by engaging with the task. We highlight two kinds of affordances—those that strengthened understanding about fundamental combinatorial ideas, and those that fostered meaningful mathematical practices. We hope that these examples of rich and sophisticated student work will contribute to an overall goal of elevating the status of combinatorics specifically, and discrete mathematics more broadly, in the K–16 curriculum. We conclude with a handful of pedagogical implications.

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.

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.