Graduate Thesis Or Dissertation
 

Characterizing and Understanding Semiotic Representations in Combinatorics Problems

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/jd473501k

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  • Combinatorics is a field of mathematics that concerns enumeration and existence, and its most notable applications are in computer science and statistics. Most students are introduced to combinatorics through counting problems, where they are tasked with determining the cardinality of a set of outcomes. Such problems are well-known for being simple to pose but deceptively difficult to solve (Tucker, 2004), in part because errors in solutions can be challenging to identify (Batanero et al., 1997) which can make solutions hard to verify (Eizenberg & Zaslavsky, 2004). This dissertation adds to the combinatorics education literature by using qualitative data to examine and characterize an approach to counting that focuses on external representations of the objects being counted in order to illuminate underlying structure. The set-oriented perspective to counting is one that attends to the set of outcomes being counted in order to identify and leverage structure in those outcomes (Lockwood, 2014). This contrasts a more traditional approach where counting is performed by determining a problem type and applying the correct formula. Research demonstrates that a set-oriented perspective can lead to robust ways of reasoning about fundamental counting principles and help students to avoid certain errors (e.g., Lockwood et al., 2015; Tillema, 2018; Wasserman & Galarza, 2019). Literature that investigates or presents how students reason about sets of outcomes often does so by presenting how students reason about representations of the sets of outcomes, and they argue that certain representations may provide desirable affordances. Yet, there is scant literature (Lockwood & Ellis, 2022; Montenegro et al., 2022) that investigates more broadly the role of representations in counting, and how representations satisfy those roles. There is thus is a gap in the literature that addresses the cognitive issues involved with reasoning about sets of outcomes, specifically the roles of semiotic representations in this reasoning. In this dissertation, I seek to investigate the numerous roles of semiotic representations in solutions to combinatorics. I report on qualitative interview data of undergraduate students as they worked through a series of counting problems written to foster the production of semiotic representations. I then employed inductive reflexive thematic analysis on this data to: 1) categorize the semiotic representations that appeared in the interviews, 2) understand how the participants came to produce those representations, 3) identify the roles of the representations, and 4) establish how those roles were satisfied. In my first paper, I build on prior literature for semiotic representations to develop a theoretical perspective that frames the use of representations to mediate solutions to counting problems. Mediation motivates the notion of ‘proximity’ of representations, which describes the cognitive load necessary to perform conversions between representations as well as the naturality of those conversions (i.e., likeliness to perform without additional scaffolding). I then use empirical data to illustrate the use of mediating representations in combinatorics, and why the use of representations in combinatorics is different than other areas of mathematics. In my second paper, I apply my theoretical perspective from Paper 1 to the empirical data collected for my dissertation in order to characterize the nature of combinatorial representations, and to understand their roles in solutions to counting problems. This paper thematically analyzes qualitative data of undergraduate students in clinical task-based interviews as they worked through a sequence of combinatorial tasks. This work seeks to capture the nature of the representations, address why the participants used representations in their solutions, and examine how the representations led to breakthroughs or clarification in the solution. One major result is a way to categorize representations as canonical, methodized, improvised, or blended. This categorization is separate from representational systems and registers, and it contributes to the literature by providing a means of characterizing non-institutional representations students use in solutions. I also address the cognitive mechanisms involved with identifying and creating representations of outcomes and sets of outcomes for problems. Then, I address the cognitive mechanisms involved with refining or transforming those representations, and producing a mathematical expression that determines cardinality. I conclude with a discussion of applications for researchers, and pedagogical implications. In my third paper, I focus on a case study of two undergraduate students as they developed and used two related systems of representation, which I call sequential and positional representations, across a number of different problems. These representation systems merit additional consideration apart from the rest of the study because of their ubiquity in the current literature, and their ability to illuminate connections among problems despite their relative simplicity. Additionally, this case study provides a deep-dive that chronicles how methodized representation systems can be developed over a series of tasks. I find that sequential and positional representations were first used to model sequences of physical events (e.g., coin flips) but the students adapted the representation systems to model numerous problem types. The representation systems were used to form connections among different problems, and they also afforded some flexibility in how certain problems were modeled. At the same time, the students leveraged structural elements of sequential and positional representations to convert between sets of outcomes and mathematical expressions. This paper addresses calls in the literature to identify representational systems for sets of outcomes that can be used both as a pedagogical tool to demonstrate reasoning about sets and as a computational tool to assist in the conversion between sets of outcomes and mathematical expressions for cardinalities of those sets.
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