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vi LIST OF TABLES  xii LIST OF FIGURES  xiii LIST OF ABBREVIATIONS xv CHAPTER ONE: INTRODUCTION 1 Background 1 Problem Statement 3 Purpose Statement 4 Research Questions 4 Definitions of Terms 4 Significance of the Study 10 CHAPTER TWO: LITERATURE REVIEW 12 Theoretical Framework 12 Conceptual Frameworks 16 Using Mathematical Models 16 Modes of Representation 17 Overview of Proportional Reasoning18

The Influence of Tape Diagrams and Bar Models on Middle School Students' Proportional Reasoning