Defining and demonstrating an equivalence way of thinking in enumerative combinatorics

https://doi.org/10.1016/j.jmathb.2020.100780Get rights and content

Highlights

  • Equivalence is a fundamental idea within combinatorics (and mathematics more broadly).

  • We draw on Harel’s duality principle and describe and demonstrate an equivalence way of thinking in combinatorics.

  • This equivalence way of thinking can be valuable for students in a variety of combinatorial situations.

Abstract

Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning 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.

References (35)

  • L.D. English

    Children’s strategies for solving two- and three-dimensional combinatorial problems

    The Journal of Mathematical Behavior

    (1993)
  • N. Hadar et al.

    The road to solving a combinatorial problem is strewn with pitfalls

    Educational Studies in Mathematics

    (1981)
  • A. Halani

    Students’ ways of thinking about enumerative combinatorics solution sets: The odometer category

  • G. Harel

    DNR perspective on mathematics curriculum and instruction, part I: Focus on proving

    ZDM Mathematics Education

    (2008)
  • G. Harel

    A DNR perspective on mathematics curriculum and instruction. Part II: With reference to teacher’s knowledge base

    ZDM Mathematics Education

    (2008)
  • G. Harel

    What is mathematics? A pedagogical answer to a philosophical question

  • S. Kavousian

    Enquiries into undergraduate students’ understanding of combinatorial structures

    (2008)
  • Cited by (4)

    • An initial framework for analyzing students’ reasoning with equivalence across mathematical domains

      2022, Journal of Mathematical Behavior
      Citation 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.

    • Teaching and learning discrete mathematics

      2022, ZDM - Mathematics Education
    View full text