Defining and demonstrating an equivalence way of thinking in enumerative combinatorics
Section snippets
Introduction and motivation
Equivalence is a fundamental aspect of mathematics, and it arises in a wide variety of mathematical topics in settings ranging from elementary classrooms to graduate school. In combinatorics, equivalence lays the foundation for several central ideas – it is an integral aspect of justifying certain counting formulas (such as the formula for combinations), and it can offer unique insights into approaches to particular kinds of problems (such as counting arrangements with repetition). In multiple
Literature review
Combinatorics education is a rich and growing field, and there has been considerable work done to investigate students’ combinatorial thinking and activity (e.g., English, 1991, 1993; Maher & Martino, 1996; Maher, Powell, & Uptegrove, 2011; Sriraman & English, 2004; Tillema, 2013, 2018; Tillema & Gatza, 2016). While much of this work shows students reasoning robustly about counting problems, there is plentiful evidence that students struggle with solving counting problems correctly, which
Theoretical perspective
In this section, we outline two theoretical perspectives that frame the results presented in this paper – Harel’s (2008c) duality principle and Lockwood’s (2014) set-oriented perspective. We also include a brief mathematical discussion, although the primary mathematical discussion in the paper occurs within the results. Then, we conclude the setion by defining an equivalence way of thinking.
Methods
In this paper, we present results from a paired teaching experiment (Steffe & Thompson, 2000) that we conducted with two undergraduate students. The data described in this paper were collected as part of a broader study in which our research team interviewed students to learn more about their generalizing activity within the domain of combinatorics. These data offer instances of equivalence ways of understanding and ways of thinking that emerged during their combinatorial problem solving.
Results
In this Results section, we discuss three different mathematical contexts – permutations, combinations, and arrangements with repetition, in which students leveraged an equivalence way of thinking. For each of these settings, in addition to the mathematical discussion, we offer examples that show the students developing and leveraging equivalence as they solved problems and justified formulas. By showing the students’ use of equivalence on a variety of problems and in a variety of situations,
Discussion, conclusion, and implications
In this paper we have attempted to articulate ways in which equivalence can arise in counting problems, and we characterized an equivalence way of thinking as an approach that specifically involves first recognizing that, in a given set of outcomes, there are certain outcomes that can be considered equivalent, and then using the operation of division to account for the occurrence of such equivalent outcomes. We have argued that this way of thinking can apply in a variety of problem types within
Author statement
Descriptions in this manuscript are accurate and agreed upon by all authors.
Acknowledgement
This material is based upon work supported by the National Science Foundation under Grant No. 1419973.
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An initial framework for analyzing students’ reasoning with equivalence across mathematical domains
2022, Journal of Mathematical BehaviorCitation Excerpt :The examples in Sections 4.2.1 and 4.2.2 come from a study (Lockwood & Reed, 2020) of the mathematical activity of novice counters8 as they engaged in combinatorial tasks. The particular episodes presented were from sessions wherein two undergraduate students (pseudonyms Rose and Sanjeev) developed understandings of four basic counting formulas as well as what Lockwood and Reed (2020) called an equivalence way of thinking in combinatorics. Our focus here, however, is on using the examples of students’ activity from previous studies to illustrate aspects of our framework and its utility for illuminating key aspects of students’ reasoning about equivalence.
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