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The Role of Intuitive Arithmetic in Developing Mathematical Skill University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 The Role Of Intuitive Arithmetic In Developing Mathematical Skill Emily Maja Szkudlarek University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Cognitive Psychology Commons, and the Developmental Psychology Commons Recommended Citation Szkudlarek, Emily Maja, "The Role Of Intuitive Arithmetic In Developing Mathematical Skill" (2019). Publicly Accessible Penn Dissertations. 3498. https://repository.upenn.edu/edissertations/3498 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3498 For more information, please contact [email protected]. The Role Of Intuitive Arithmetic In Developing Mathematical Skill Abstract Symbolic mathematics allows humans to represent and describe the logic of the world around us. Although we typically think about math symbolically, humans across the lifespan and a wide variety of animal species spontaneously exhibit numerical competence without reference to formal mathematics. This intuitive ability to approximately compare, estimate, and manipulate large non-symbolic numerical quantities without language or symbols is called the Approximate Number System. The four chapters of this dissertation explore whether non-symbolic, approximate calculation can function as a bridge between our Approximate Number System and symbolic mathematics for children at the beginning of formal math education and university undergraduates. Chapter 1 explores how non-symbolic and symbolic ratio reasoning relates to general math skill and Approximate Number System acuity in elementary school children. Chapter 2 examines whether children and adults can perform a non-symbolic, approximate division computation, and how this ability relates to non-symbolic and symbolic mathematical skill. Chapter 3 tests the robustness and mechanism of a non-symbolic, approximate addition and subtraction training paradigm designed to improve arithmetic fluency in university undergraduates. Chapter 4 investigates whether the negative relation between math anxiety and symbolic math performance extends to approximate, non-symbolic calculation. Together, Chapters 1 and 2 provide evidence that non- symbolic calculation ability functions as a mechanism of the relation between Approximate Number System acuity and symbolic math. Chapters 3 and 4 identify populations of students for whom practice with non-symbolic calculation may or may not be beneficial. In sum, this dissertation describes how non- symbolic, approximate calculation allows students harness their intuitive sense of number in a mathematical context. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Psychology First Advisor Elizabeth M. Brannon Keywords Approximate Number System, Arithmetic, Cognitive Development, Numerical Cognition Subject Categories Cognitive Psychology | Developmental Psychology This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/3498 THE ROLE OF INTUITIVE ARITHMETIC IN DEVELOPING MATHEMATICAL SKILL Emily Maja Szkudlarek A DISSERTATION in Psychology Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2019 Supervisor of Dissertation __________________________ Elizabeth M. Brannon Professor of Psychology Graduate Group Chairperson __________________________ John Trueswell, Professor of Psychology Dissertation Committee Daniel Swingley, Professor of Psychology Allyson Mackey, Assistant Professor of Psychology THE ROLE OF INTUITIVE ARITHMETIC IN DEVELOPING MATHEMATICAL SKILL COPYRIGHT 2019 Emily Maja Szkudlarek iii ACKNOWLEDGMENT First, I would like to thank my advisor, Dr. Elizabeth Brannon, for her guidance and support throughout my time in graduate school. I am forever grateful for the opportunity to work in the Brannon lab. I would also like to thank the members of my committee, Dr. Daniel Swingley and Dr. Allyson Mackey for their advice and feedback regarding this research. I am further thankful to Dr. Whitney Tabor and Dr. Etan Markus for their endless support of my research interests as an undergraduate at the University of Connecticut. Without their mentorship, this dissertation would not be possible. Many past and present members of the Brannon lab have offered help and advice during my time in graduate school, and to all of them I am thankful. Two members of the lab in particular, Nick DeWind and Stephanie Bugden, have contributed to the current work in innumerable ways. I cannot imagine graduate school without either of them. Our near daily chats about work and life (whether in the office or not) have impacted this research and beyond. I am immensely grateful for our friendship. I would also like to thank Eleni Kursten and Emily Kibbler for their consistent belief that I am moving in the right direction. I am also thankful to my family for their continual interest in what I’ve been working on, especially Kim Szkudlarek, Celeste Truskolawski, Małgorzata Podłuska, Alicja Szkudlarek, Marek Szyndel, and my grandmother, Valerie Truskolawski. Though my grandmother is not here to see the end of my time in graduate school, I know she was already excited for the next things to come. To my parents, Justine and Lech Szkudlarek, thank you for everything. There are truly too many things to list, but a few include helping me to move (three) times, timely reminders that my car is due for an oil change, and sparking my interest in science. Finally, thanks to my parakeets Pickle, Parsnip, and Ollie for their happy chirps each morning that became the soundtrack of my graduate career. iv ABSTRACT THE ROLE OF INTUITIVE ARITHMETIC IN DEVELOPING MATHEMATICAL SKILL Emily Maja Szkudlarek Elizabeth M. Brannon Symbolic mathematics allows humans to represent and describe the logic of the world around us. Although we typically think about math symbolically, humans across the lifespan and a wide variety of animal species spontaneously exhibit numerical competence without reference to formal mathematiCs. This intuitive ability to approximately compare, estimate, and manipulate large non-symbolic numerical quantities without language or symbols is called the Approximate Number System. The four chapters of this dissertation explore whether non-symbolic, approximate calculation can function as a bridge between our Approximate Number System and symbolic mathematics for children at the beginning of formal math education and university undergraduates. Chapter 1 explores how non-symbolic and symbolic ratio reasoning relates to general math skill and Approximate Number System aCuity in elementary school children. Chapter 2 examines whether children and adults can perform a non-symbolic, approximate division computation, and how this ability relates to non-symbolic and symbolic mathematical skill. Chapter 3 tests the robustness and mechanism of a non-symbolic, approximate addition and subtraction training paradigm designed to improve arithmetic fluency in university undergraduates. Chapter 4 investigates whether the negative relation between math anxiety and symbolic math performance extends to approximate, non-symbolic calculation. Together, Chapters 1 and 2 provide evidence that non-symbolic Calculation ability functions as a mechanism of the relation between Approximate Number System acuity and symbolic math. Chapters 3 and 4 identify populations of students for whom practice with non-symbolic calculation may or may not be beneficial. In sum, this dissertation describes how non-symbolic, approximate calculation allows students harness their intuitive sense of number in a mathematical context. v TABLE OF CONTENTS ACKNOWLEDGMENT ........................................................................................III ABSTRACT ........................................................................................................ IV LIST OF TABLES ............................................................................................. VIII LIST OF ILLUSTRATIONS ................................................................................. IX GENERAL INTRODUCTION ................................................................................1 Introduction to approximate number representations ............................................................ 1 Potential mechanisms of the link between the ANS and mathematics................................... 4 Non-symbolic, approximate calculation................................................................................... 5 The relation between non-symbolic calculation and symbolic math ...................................... 7 Dissertation approach............................................................................................................... 9 CHAPTER 1: FIRST AND SECOND GRADERS SUCCESSFULLY REASON ABOUT RATIOS WITH BOTH DOT ARRAYS AND ARABIC NUMERALS ......13 Introduction ............................................................................................................................. 13 Materials and Methods ............................................................................................................ 19 Subjects ................................................................................................................................ 19 Procedure ............................................................................................................................
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