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Problem Formatting, Domain Specificity, and Arithmetic Processing: the Promise of a Factor Analytic Framework

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Problem Formatting, Domain Specificity, and Arithmetic Processing: the Promise of a Factor Analytic Framework Georgia State University ScholarWorks @ Georgia State University Psychology Dissertations Department of Psychology 8-11-2015 Problem Formatting, Domain Specificity, and Arithmetic Processing: The Promise of a Factor Analytic Framework Katherine Rhodes Follow this and additional works at: https://scholarworks.gsu.edu/psych_diss Recommended Citation Rhodes, Katherine, "Problem Formatting, Domain Specificity, and Arithmetic Processing: The Promise of a Factor Analytic Framework." Dissertation, Georgia State University, 2015. https://scholarworks.gsu.edu/psych_diss/138 This Dissertation is brought to you for free and open access by the Department of Psychology at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Psychology Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. PROBLEM FORMATTING, DOMAIN SPECIFICITY, AND ARITHMETIC PROCESSING: THE PROMISE OF A FACTOR ANALYTIC FRAMEWORK by KATHERINE T. RHODES Under the direction of Julie A. Washington, Ph.D. and Lee Branum-Martin, Ph.D. ABSTRACT Leading theories of arithmetic cognition take a variety of positions regarding item formatting and its possible effects on encoding, retrieval, and calculation. The extent to which formats might require processing from domains other than mathematics (e.g., a language domain and/or an executive functioning domain) is unclear and an area in need of additional research. The purpose of the current study is to evaluate several leading theories of arithmetic cognition with attention to possible systematic measurement error associated with instrument formatting (method effects) and possible contributions of cognitive domains other than a quantitative domain that is specialized for numeric processing (trait effects). In order to simultaneously examine measurement methods and cognitive abilities, this research is approached from a multi- trait, multi-method factor analytic framework. A sample of 1959 3rd grade students (age M=103.24 months, SD=5.41 months) were selected for the current study from the baseline time points of a larger, longitudinal study conducted in southeastern metropolitan school districts. Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, and the Exact versus Approximate Calculations Hypothesis (a specification of Triple Code Theory) were evaluated with confirmatory factor analysis, using 11 measures of arithmetic with symbolic problem formats (e.g., Arabic numeral and language-based formats) and various problem demands (e.g., requiring both exact and approximate calculations). In general, results from this study provided support for both Triple Code Theory and Encoding Complex Theory, and to some extent, Exact Versus Approximate Calculations Theory is also supported. As predicted by Triple Code Theory, arithmetic outcomes with language formatting, Arabic numeral formatting, and estimation demands across formats were related but distinct from one another. The relationship between problems that required exact calculations (across formats) also provided support for Exact Versus Approximate Calculations Theory’s stipulation that exact calculation problems may draw from the same cognitive processes. As predicted by Encoding Complex Theory, executive function was a direct predictor of all arithmetic outcomes. Language was not a direct predictor of arithmetic outcomes; however, the relationship between language and executive function suggested that language may play a facilitative role in reasoning during numeric processing, particularly for language- formatted problems. INDEX WORDS: Arithmetic cognition, Numeric processing, Format effects, Common method variance, Language, Executive functioning, Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, Exact versus approximate calculations PROBLEM FORMATTING, DOMAIN SPECIFICITY, AND ARITHMETIC PROCESSING: THE PROMISE OF A FACTOR ANALYTIC FRAMEWORK by KATHERINE T. RHODES A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the College of Arts and Sciences Georgia State University 2015 Copyright by Katherine T. Rhodes 2015 PROBLEM FORMATTING, DOMAIN SPECIFICITY, AND ARITHMETIC PROCESSING: THE PROMISE OF A FACTOR ANALYTIC FRAMEWORK by KATHERINE T. RHODES Committee Chairs: Julie A. Washington Lee Branum-Martin Committee: Lynn Fuchs Rebecca Williamson Electronic Version Approved: Office of Graduate Studies College of Arts and Sciences Georgia State University August 2015 vi ACKNOWLEDGMENTS I would like to thank all of the children, families, teachers, and schools who participated in this study. Without their willingness to participate, this research would not be possible. I would also like to thank my dissertation chairs and committee members for their willingness to guide me and answer questions as I completed this research. Finally, I would like to thank my family, colleagues, and mentors for their unwavering support throughout my years of graduate school. vii TABLE OF CONTENTS ACKNOWLEDGMENTS .......................................................................................................... VI LIST OF TABLES ....................................................................................................................... X LIST OF FIGURES .................................................................................................................. XII 1 CHAPTER 1: ARITHMETIC MASTERY, COGNITIVE DOMAINS, AND MEASUREMENT FORMATTING............................................................................................ 1 1.1 Introduction ................................................................................................................ 1 1.2 Arithmetic Mastery and Mathematics Achievement Difficulties ........................... 3 1.3 Interpreting Test Results: Math Content Versus Test Formatting ....................... 5 1.4 The Quantitative Domain and the Development of Arithmetic Cognition ........... 7 1.4.1 Neural bases of arithmetic cognition. .................................................................. 7 1.4.2 Other neural circuits in arithmetic cognition. .................................................... 9 1.5 Problem Formatting and Common Method Variance .......................................... 11 1.6 Cognitive Theories of Arithmetic ............................................................................ 13 1.6.1 Abstract code theory. .......................................................................................... 13 1.6.2 Encoding complex theory. .................................................................................. 16 1.6.3 Triple code theory. .............................................................................................. 19 1.6.4 Exact versus approximate calculations: An extension of triple code theory. .. 22 2 CHAPTER 2: MODELING ARITHMETIC PERFORMANCE AND THE ROLE OF PROBLEM FORMATTING ..................................................................................................... 25 2.1 Modeling Leading Theories of Arithmetic Cognition ........................................... 26 2.1.1 Abstract code theory. .......................................................................................... 27 2.1.2 Encoding complex theory. .................................................................................. 32 viii 2.1.3 Triple code theory. .............................................................................................. 39 2.1.4 Exact versus approximate calculations specification of triple code theory. .... 45 2.2 Hypotheses ................................................................................................................. 49 3 CHAPTER 3: METHODS ................................................................................................... 50 3.1 Participants ............................................................................................................... 50 3.2 Procedures ................................................................................................................. 54 3.3 Measures .................................................................................................................... 54 3.3.1 Mathematics achievement measures with language formatting. ..................... 54 3.3.2 Mathematics achievement measures with Arabic numeral formatting. .......... 56 3.3.3 Mathematics achievement measures involving estimation or analog magnitude. ...................................................................................................................... 59 3.3.4 Language measures. ........................................................................................... 60 3.3.5 Executive functioning measures. ....................................................................... 61 3.4 Design ......................................................................................................................... 64 4 CHAPTER 4: RESULTS ..................................................................................................... 70 4.1 Proposed Analyses Overview ................................................................................... 70 4.2 Phase 1: Measurement Models for Arithmetic, Language, and Executive Functioning ............................................................................................................................ 70 4.2.1 Abstract code model...........................................................................................
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