CONSTRUCTION OF THE SUM OF TWO COVARYING ORIENTED QUANTITIES by JIYOON CHUN (Under the Direction of Leslie P. Steffe) ABSTRACT This study investigates how students construct the sum of two co-varying oriented quantities (denoted by “x + y=a”, where a is a constant) by reorganizing their counting schemes and units-coordinating schemes. Two 9th grade students, one who reasoned with the two levels of units and one who reasoned with three levels of units, participated in a year-long teaching experiment. I found major differences in how the two students constructed sums and differences of signed quantities. Carl, the student who reasoned with two levels of units, did not construct a negatively oriented quantity as the inverse of a positively oriented quantity nor did he find the sum of two oppositely oriented quantities in a way that respected the orientation of the quantities. In contrast, Maggie, the student who reasoned with three levels of units, did construct sums and differences of oriented quantities in such a way that respected their orientations. In situations that involved two oriented quantities, denoted by “x” and “y,” that co-varied in such a way that x + y = a (a is a constant), Carl found a set of discrete points that were representative of x + y = a by experientially plotting a few points on a coordinate plane. Although he said that he could find infinitely many points, he did not envision them as belonging to a line nor did he construct the counterbalancing relation between changes in each quantity. In contrast, Maggie constructed the counterbalancing relation by additively coordinating changes in the two quantities. Her schemes were anticipatory, and she could envision a two-dimensional trace of the co-variation of x and y as a line. My findings suggest that reasoning with three levels of units and reversible reasoning are both essential in constructing graphs of two oriented quantities that co-vary in such a way that their sum is a constant. INDEX WORDS: Radical Constructivism, Teaching Experiment, Reversible Reasoning, Counterbalancing Relation, Additive Inverse, Additive Reasoning, Oriented Quantities, Linear Function, Levels of Units, Anticipation, Scheme, Covariation, Quantitative Reasoning CONSTRUCTION OF THE SUM OF TWO COVARYING ORIENTED QUANTITIES by JIYOON CHUN B.S. Korea University, Republic of Korea, 2005 M.Ed. The University of Georgia, 2012 A Dissertation Submitted to the Graduate Faculty of the University of Georgia in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ATHENS, GEORGIA 2017 © 2017 Jiyoon Chun All Rights Reserved CONSTRUCTION OF THE SUM OF TWO COVARYING ORIENTED QUANTITIES by JIYOON CHUN Major Professor: Leslie P. Steffe Committee: Sybilla Beckmann Kevin C. Moore Electronic Version Approved: Suzanne Barbour Dean of the Graduate School The University of Georgia December 2017 iv DEDICATION I dedicate this work to my parents, 강혜숙, Connie D. Thompson, Tommy Thompson & April L. Storm, and to my dearest friend, Fabrizio Marsilli, who patiently read all the chapters for free. v TABLE OF CONTENTS Page LIST OF TABLES ........................................................................................................... viii LIST OF FIGURES ........................................................................................................... ix CHAPTER 1 INTRODUCTION .................................................................................................. 1 Motivation ......................................................................................................... 1 Problem Statement and Rationale ..................................................................... 3 Research Questions ........................................................................................... 8 2 THEORETICAL CONSTRUCTS .......................................................................... 9 Inaccessible Ontological Reality....................................................................... 9 Construction of Experiential Reality .............................................................. 18 Hypothetical Model of the Construction of Experiential Reality ................... 24 The Principles that Guide My Study ............................................................... 38 3 LITERATURE REVIEW ..................................................................................... 40 Students’ Mathematics .................................................................................... 40 Reorganization Hypothesis ............................................................................. 41 Quantity........................................................................................................... 43 Number Sequences and the Units-Coordinating Schemes .............................. 50 vi Operations, Adding and Subtracting Strategies of the Counting Schemes and the Units-Coordinating Schemes .............................................................. 57 Extension of the Counting Schemes and the Units-Coordinating Schemes ... 60 The Hypotheses Concerning the Construction of x+y=a ............................... 66 4 METHODOLOGY ............................................................................................... 69 Constructivist Teaching Experiment............................................................... 69 Data Collection ............................................................................................... 73 5 INITIAL INTERVIEWS ...................................................................................... 75 Overview of the Initial Interviews .................................................................. 76 Analysis of Carl’s Initial Interviews ............................................................... 85 Analysis of Maggie’s Initial Interviews ........................................................ 101 The Follow-Up to the Initial Interview ......................................................... 113 Summary ....................................................................................................... 120 6 COORDINATING TWO ORIENTED QUANTITIES ...................................... 123 Adding Two Positively Oriented Quantities ................................................. 123 Adding Two Negatively Oriented Quantities ............................................... 126 Adding two Oppositely Oriented Quantities................................................. 131 Adding Two Oppositely Oriented but Unknown Quantities ........................ 137 Finding Missing Addends I ........................................................................... 144 Finding Missing Addends II ......................................................................... 147 vii Finding Differences between Two Oppositely Oriented Quantities ............. 150 Summary ....................................................................................................... 164 7 SCHEMES AND OPERATIONS WHEN CONSTRUCTING x+y=a .............. 168 Distinct Meanings of the “-” Sign ................................................................. 168 Anticipating the Relationship Between Two Covarying Quantities ............. 175 Graphing x+y=a ........................................................................................... 196 Exploring the Counterbalancing Relation between Two Quantities............. 205 Representing Covariation on a Coordinate Plane: a Bouncing Ball ............. 221 Representing a+b=0 on a Coordinate Plane ................................................. 224 Summary ....................................................................................................... 239 8 CONCLUSIONS................................................................................................. 243 Additive Reasoning with Different Levels of Units-Coordination ............... 243 Constructions of x+y=a with Different Levels of Units-Coordination ........ 247 Essential Schemes and Operation when Constructing x+y=a as a Continuous Line ......................................................................................................... 252 Implications for Teaching and Research....................................................... 253 REFERENCES ............................................................................................................... 257 APPENDIX ..................................................................................................................... 263 THE COMPLETE LIST OF THE SELECTED TASKS .............................. 263 viii LIST OF TABLES Page Table 1 The Overview of Carl’s and Maggie’s Initial Interviews .................................................83 ix LIST OF FIGURES Page Figure 1. 1 Jack’s line graph (a) and Jamie’s point graph (b) ..........................................................4 Figure 2. 1 Chinese letter “symbol” and its origin........................................................................ 16 Figure 5. 1 Carl’s splitting, reconstructed by the author. .............................................................. 96 Figure 5. 2 Maggie (a) and Carl (b) pulled out their shares. ....................................................... 114 Figure 5. 3 Maggie’s and Carl’s distinct ways of measuring the whole. .................................... 115 Figure 5. 4 Maggie’s inserting the mid-sized piece into one-fourth of the whole. ..................... 116 Figure 5. 5 Carl’s and Maggie’s repartitions of their share. ......................................................