Construction of the Sum of Two Covarying Oriented Quantities

CONSTRUCTION OF THE SUM OF TWO COVARYING ORIENTED QUANTITIES

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

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

Jiyoon Chun

CONSTRUCTION OF THE SUM OF TWO COVARYING ORIENTED QUANTITIES

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

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.

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

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

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...... 119

Figure 6. 1 The measures of temperature that were drawn by the students on the thermometer

strips (reconstructed by the author)...... 126

Figure 6. 2 Carl’s directed segments when solving -12+-32, reconstructed by the author...... 129

Figure 6. 3 Maggie’s directed segments when solving -12+(-32), reconstructed by the author. 130

Figure 6. 4 Maggie’s and Carl’s coordination of the sum of the measure of the temperature at 5

a.m. and the change in temperature for 12 hours...... 134

Figure 6. 5 Maggie and Carl’s distinct ways of arranging the given quantities, reconstructed by

the author...... 140

Figure 6. 6 Carl’s vertical computation (a) and Maggie’s vertical computation (b) when finding

the missing addend...... 145

Figure 6. 7 Maggie's solution for Carl's problem...... 148

Figure 6. 8 Maggie’s problem for Carl...... 149

Figure 6. 9 GSP screen capture of the directed segments and Carl’s and Maggie’s equations. . 153

Figure 6. 10 Carl's equation and Maggie's directed segments ...... 158

Figure 6. 11 Carl’s equation (a) and Maggie’s directed segments (b) on GSP. In Screencapture

(a), Carl’s correction of his answer can be observed...... 159

Figure 6. 12 Carl's modeling the given situation ...... 161

Figure 7. 1 The screen captures of Carl’s and Maggie’s directed segments...... 170

Figure 7. 2 Maggie's solution to Carl's problem ...... 173

Figure 7. 3 The model of two quantities of a+b=0 ...... 176

Figure 7. 4 Maggie's dragging b to 0 ...... 180

Figure 7. 5 The screen capture that was taken when Maggie dragged the lower directed segment,

b, to -6 and moved its origin to 15...... 181

Figure 7. 6 Carl’s screen when finding a relationship between a and b that can be modeled as a=b

...... 186

Figure 7. 7 Carl's screen when finding m=-c ...... 187

Figure 7. 8 Carl’s screen when finding a relationship between a and b that can be modeled as a=b

...... 190

Figure 7. 9 Carl’s memos for the relation between a and b that can be modeled as a+b=8 ...... 194

Figure 7. 10 Carl's and Maggie's graphs ...... 201

Figure 7. 11 Carl’s (a) and Maggie’s (b) notes when finding changes in c ...... 207

Figure 7. 12 Maggie’s way of coordinating two consecutive changes in the measures of the

temperature of city C and city M...... 210

Figure 7. 13 Carl's way of finding the second trip to station 6 ...... 212

Figure 7. 14 Carl’s solution to 30+ x =6...... 215

Figure 7. 15 The comparison of Carl’s estimations of the length of the directed segment of 3 and

the directed segment of 30...... 216

Figure 7. 16 The two other task that Carl solved...... 217

Figure 7. 17 Carl’s attempt to find the second trip whenever the first trip is given in order ...... 219

Figure 7. 18 A screen capture of the ball bouncing video...... 222

Figure 7. 19 Carl’s graph (a) and Maggie’s graph (b)...... 223

Figure 7. 20 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first

number line) when A changes 0 to 10 (the second number line)...... 226

Figure 7. 21 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first

number line) when A changes 0 to 10 (the second number line)...... 228

Figure 7. 22 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first

number line) when A changes 0 to 10 (the second number line)...... 233

Figure 7. 23 The locus of (a,-a)...... 234

Figure 7. 24 Carl's point wise graphs ...... 237

CHAPTER 1

INTRODUCTION

Motivation

As a mathematics teacher in secondary schools, I faced many challenges when teaching functions. I had to witness that many middle school students lost their interest in mathematics after the linear functions. Many high school students gave up mathematics because they could not make sense of trigonometry, differentiation and Riemann sums. I worked hard to help my students improve their performances and preserve their interest in mathematics, but I faced the wall; no matter how hard I tried, I could not make them understand the concepts and love mathematics. Therefore, I decided to study more about mathematics education after four years of teaching.

In the spring of 2011, I took the course, Mathematics Curriculum in the Secondary

Schools, with emphasis on current issues and trends (EMAT 7080, taught by Dr. Leslie Steffe) at the University of Georgia. In the course, I reconstructed my concept of function from the relationship between inputs and outputs to the coordination of measurements of quantities. I found that thinking in terms of measurements of quantities instead of numbers was crucial in my construction of the concept of functions. Consequently, I became aware of the importance of measuring quantities, for it was the basis for my concept of function. While experiencing the joy of reconstructing my own concept, I could not help but asking a critical question to myself as a former teacher; “How much did I know about my students’ concepts of function?” To my shame,

2 my answer was “not much”. I realized that I was a performer of teaching acts, but not a good observer of my students.

Shifting Perspectives

The next question that I asked to myself was, “How can I be a good observer?” I learned that shifting my perspective as an adult to that of students was necessary in order to be a good observer. From an adult’s point of view, it was easy to refer to certain students’ ways of reasoning that were not compatible with that of adults’ as misconceptions: a conception that needs to be corrected. I found this approach was problematic because it prevented me from understanding students’ ways of reasoning by observing their language and actions. To understand students’ ways of reasoning, admitting that student's mathematics concept is reasonable and legitimate from their perspective (Steffe & Thompson, 2000) was essential. Thus, it was important for me to respect students’ ways of reasoning by considering them as intelligent beings who were capable of organizing their own concepts and knowledge. This is why I chose radical constructivism for my theoretical constructs for this study.

Radical Constructivism

From the radical constructivist perspective, knowledge is actively constructed rather than passively conceived (von Glasersfeld, 1990). A teacher cannot make students understand certain things. Students themselves construct their own concepts and knowledge that fit in their existing concepts. Therefore, the overarching hypothesis of this study is the reorganization hypothesis

(Steffe & Olive, 2010) that one constructs a new way of reasoning by reorganizing one’s existing one. My goal for this study is to formulate my own understanding of how students construct a constant sum of two co-varying quantities (hereinafter, x + y = a, a is constant) by reorganizing their existing schemes. So, it was important for me to understand students’ existing schemes such

3 as counting schemes, measuring schemes, the units-coordinating schemes as well as their mental operations involving the construction of object concepts. I elaborate on radical constructivism and essential mental operations when constructing experiential reality (including object concepts) In Chapter 2. Then, in Chapter 3, I review the studies of students’ schemes and operations (such as counting, measuring, and the units-coordinating schemes) that were essential for me to construct a hypothetical model of how students construct x+y=a. In Chapter 4, I introduce the constructivist teaching experiment (Steffe & Thompson, 2000) as a methodology that I chose for making a hypothetical explanatory model of my students’ construction of x+y=a.

In Chapter 5, I elaborate on the initial interviews that I conducted for the two students (Carl and

Maggie, pseudonyms) to find their existing schemes and operations. In Chapter 6, I explain my students’ schemes and operations involving modeling the sum of two oriented quantities. In

Chapter 7, I explain how the students reorganized their schemes and operations that they used to model the sum of two oriented quantities when constructing x+y=a.

Problem Statement and Rationale

Different Representations for a Linear Function

One of the most common ways to teach linear functions is to plot a few points that satisfy the function and connect them to make a line. From an adult’s point of view, it is apparent that the set of plotted points are collinear. However, is this also apparent to children? In the spring of

2014, I took part in a research project titled "Pathways to Algebra," during which I taught linear functions to two 10th grade students (Jamie and Jack, pseudonyms). When asked to represent the speed of an inchworm on the coordinate plane, provided that the inchworm crawls 3 inches for every five seconds, Jack was able to draw a line, but Jaime only plotted one point. Later, Jamie plotted several points, but he never understood those points as belonging to a line. It is clear that

4 being able to plot a few points on a coordinate plane did not necessarily mean that Jamie understood that the points belonged to a line that was the result of two variables co-varying in a constant ratio.

(a) (b)

Figure 1. 1 Jack’s line graph (a) and Jamie’s point graph (b)

I was very curious about what mental operations could be used to explain why Jack represented the speed of the inchworm as a line while Jamie did not. I believe, based on the teaching episodes, that Jack’s more sophisticated units coordination was one of the reasons why he could represent the speed by drawing a line. The tasks that I posed for Jack and Jamie were heavily based on rate reasoning and proportional reasoning. Although both ways of reasoning are instrumental in students’ construction of linear functions that involve multiplicative reasoning, many studies that investigated students’ construction of linear functions that involve rate

5 reasoning or proportional reasoning were only focused on positive quantities (Harel et al., 1994;

Karplus et al., 1983; Noelting, 1980; Simon & Blume, 1994). Furthermore, according to

Common Core State Standards for mathematics (CCSSI, 2012) and Mathematics Georgia

Standards of Excellence (Woods, 2016c), linear functions are first introduced as representations of proportional relationships with positive quantities in the 8th grades. This focus in both research and teaching raises some questions; why is there almost total emphasis on multiplicative reasoning when introducing linear functions, while additive reasoning might be less cognitively demanding than multiplicative reasoning? How do students extend their concept of linear function beyond the first quadrant on a coordinate plane?

The research on linear functions in which I am interested involves additive reasoning involving oriented quantities rather than multiplicative reasoning. For example, could Jamie have graphed x + y = a, in such a way that he could engage in covarying x and y? Will their different units-coordinating schemes engender different constructions of x + y = a? It was important for me to learn how students engage in additive reasoning involving negative quantities before I began my investigation. Fortunately, I had many opportunities to do so as a teaching assistant and an instructor.

The "Take-Away" Strategy

One of the interesting things that I observed in students’ additive reasoning when I was assisting a course “Teaching Algebra in the Middle School (EMAT 5320)” in the fall of 2012 and 2014 was that some students always used the “take-away” technique whenever they need to add a negative quantity to a positive quantity, while other students saw addition in terms of the relationship between the two counterbalancing quantities, regardless of their signs. When adding a negative quantity to a positive quantity, it seems natural to take away the magnitude of the

6 negative quantity from the positive quantity. Consider bank account activity as an example. If you deposit a $5 check, but then later spend $3 with your debit card, you can find the account balance by counting five up from your previous balance, then counting three back down. This counting down can be easily understood as taking 3 away from 5. However, this "take-away" strategy could be considered to be problematic in two ways. First, the "take-away" method might not make sense to children in certain situations. Consider a case where the expenditure is bigger than the deposit. If a child uses the "take-away" method, he or she might experience a difficulty when he or she sees there is more to take away than the amount of the deposit, as Bishop et al. pointed out (2014). Second, the "take-away" method might cause inconsistency in mental operations when constructing a linear functions such as x+y=a. Using the "take-away" method, one might not consider x+y=a as a sum of x and y, especially in the case of x and y are oppositely oriented because one needs to alter the operation from addition to subtraction when using the "take-away" method. I presupposed that this alternation of the operations would serve as a constraint on the construction of x+y=a as a continuous line as opposed to a set of discrete points.

Absence of Measurements of Quantities

When I was instructing the course of Teaching Secondary School Mathematics II (emat

4850) in 2016, I had many opportunities to observe how teachers taught integers using number lines in a middle school of a Southeastern State of the United States. Teachers plotted dots on a number line when teaching integers and taught rules for adding negative numbers to 6th grade students. For example, when adding 5 and -3, teachers showed the students to start at 5 and move

3 units to the left. It seems as if teachers were teaching the addition of integers as computation with numbers rather than quantities. In fact, the standards of the number system for 6th and 7th

7 grade of Mathematics Georgia Standards of Excellence are as follows:

MGSE6.NS.6 Understand a rational number as a point on the number line; MGSE6.NS.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; MGSE6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram (Woods, 2016a, p. 6). MGSE7.NS.1b Understand p + q as the number located a distance from p, in the positive or negative direction depending on whether q is positive or negative (Woods, 2016b, p. 4).

Based on my observation and the standards that I listed above, it seems as though many students have been taught to use points for representing numbers and then encouraged to find the sum of two oriented quantities, p + q, as a number located q units away from p. I observed many students (the middle school students and the juniors and seniors that I taught and observed at the

University of Georgia) used the "take-away" method treating quantities as numbers. From my perspective, this focus could be construed as a shortcoming of the construction of the concept of linear function because the measurements of quantities was fundamental in my reconstruction of the concept of function. Furthermore, thinking in measurement of quantities has been strongly supported from many researchers because it is fundamental when coordinating changes in a function (Castillo-Garsow, 2012; Johnson, 2012; Moore, 2012; Thompson, 1994, 2011, 2014;

Thompson & Carlson, 2017). Although inefficient, from my perspective, I believe the "take- away" method and reasoning with numbers instead of quantities should be considered to be legitimate mathematics from the students’ perspective. As a researcher, it is important to investigate how they developed their strategies based on their existing schemes and how these strategies impact their construction of x+y=a. Therefore, I chose counting schemes and the units- coordinating schemes as an explanatory tool in my study.

Research Questions

The overarching research question of this study concerns investigating the necessary schemes and operations that undergird constructing x+y=a. Because the reasoning involved when finding a sum of the measurements of two oriented quantities might affect the construction of x+y=a, and different schemes and mental operations might engender different constructions of x+y=a, the following questions were asked to refine and support the research question:

1. Does reasoning with different levels of units engender different constructions of additive

reasoning involving the measurements of two oriented quantities?

a. Are students with two levels of units-coordination able to construct the sum of

the measurements of two oriented quantities regardless of their orientation?

b. Are students with three levels of units-coordination able to construct the sum of

the measurements of two oriented quantities regardless of their orientation?

c. Is reversible reasoning necessary for students to construct the sum of the

measurements of two oriented quantities with opposite orientation?

2. How do students, who reason with different levels of units, construct x+y=a?

a. How do students reorganize the schemes and operations that they used when

finding the sum of the measurements of two oriented quantities when constructing

x+y=a?

b. What mental operations are required when constructing x+y=a?

CHAPTER 2

THEORETICAL CONSTRUCTS

Following radical constructivist epistemology, I believe that knowledge is actively constructed rather than passively conceived (von Glasersfeld, 1990). I consider ontological reality, if it exists, to be inaccessible to cognizing subjects, i.e., one cannot perceive or conceive ontological reality as thing-in-itself (Kim, 1999b, 2000a, 2000b; von Glasersfeld, 1974, 1995). I regard the world around me as my perception or interpretation of an active on-going construction of experiential reality that has been recursively organized by myself (von Glasersfeld, 1995) to make it fit within constraints rather than matching to ontological reality. Hence, the goal of this study is to explore and investigate how the students construct x+y=a based on my observation of their actions and operations and to develop an explanatory model of their constructions. In this chapter, I discuss inaccessibility to ontological reality and the importance of the construction of experiential reality. Then I introduce the principles of radical constructivism and elaborate on the hypothetical model of the construction of experiential reality.

Inaccessible Ontological Reality

I reject the metaphysical realist’s perspective that one can access ontological reality as thing-in-itself, regardless of its existence. Inaccessibility to ontological reality was suggested in

Western philosophy, in East Asian philosophies, as wells as based on recent observations that were made in distinct situations.

In Western Philosophy

According to von Glasersfeld (1984), the doubts — whether or not one’s knowledge depicts ontological reality — have been challenging metaphysical realism, which considers something is “true” only if it corresponds to an independent, “objective” reality” (p. 2), and which has dominated Western philosophy for about 2000 years (von Glasersfeld, 1984). One of the pre-Socratics, Xenophanes, claimed the impossibility of seeing and saying what is completely true because the actions of one’s seeing and saying inevitably entails one’s perceptions. According to von Glasersfeld (1984), this metaphysical perspective produces the essential question about “what is to know” (p. 6) and how we know in Western epistemology.

Later, Sextus Empiricus used an apple to question whether or not we can access the “real” apple; the properties that we know about an apple through our senses, such as its hue, color, and taste, does not directly imply the characteristics of the “real” apple. In addition, one cannot be sure whether or not the “real” apple possesses other characteristics that one could not perceive through one’s senses (von Glasersfeld, 1984). Similarly, von Glasersfeld (1990, 1995) and von

Glasersfeld and Cobb (1983) also reject the accessibility to ontological reality and metaphysics by elaborating on questions of the skeptics and Kant’s extension to the question: “could we ever tell whether or not the pictures our senses “convey” are accurate and true, if the only way they can be checked is again through our sense” (von Glasersfeld & Cobb, 1983, p. 4)? This question raises another important question; if one’s experience is not depicting or conveying any natural law as a thing-in-self, “how, then, can we explain that we nevertheless experience a world that is in many respects quite stable and reliable?” (von Glasersfeld, 1984, p. 7) von Glasersfeld (1984) quotes Giambattista Vico to answer this question:

As God’s truth is what God comes to know as he creates and assemble it, so human truth is what man comes to know as he builds it shaping it by his actions.

Therefore, science (scientia) is the knowledge (cognito) of origins, of the ways and the manner how things are made. (p. 7) von Glasersfeld continued, “God knows his creation, but we cannot; we can know only what we ourselves construct” (p.7) and emphasizes the importance of construction of reliable and stable experiential reality.

In East Asian Philosophies

The three major East Asian philosophies, such as Confucianism, Buddhism, and Taoism, uniformly address the inaccessibility to ontological reality.

In Confucianism. Interestingly, Confucius suggests an almost identical perspective to

Vico in regard of the importance of the construction of experiential reality. In the anecdotes of

Confucius, it is explicitly written that “子不語怪力亂神,” meaning that Confucius does/did not argue anything regarding strange mysticism or gods. In addition, Zichan (子產), philosopher and politician during the Spring and Autumn period in ancient China who lived around 500 BC, addressed that Heaven’s way can be hardly guessed or investigated; hence, whatever is known as

Heaven’s way is just arbitrary and random guesses. On the contrary, a human’s way can be investigated and explored by human’s reasoning ability. He argues the absurdity of deducting a human’s way based on vague and uncertain guesses about ontological reality that were given by

God and vice versa (surprisingly, Zichan’s argument is identical to that of Vico’s!). One of the salient characteristics in China in terms of their philosophy and traditions lies in the realization of the roles of the cognition of human beings (Park, 2005).

In Buddhism. In the case of Buddhism, some might argue that there exists some mysticism embedded in it as a religion, including fantastic stories of the creator Gods and reincarnation. However, the stories of gods and reincarnation were added later (from Hinduism) when King Ashoka adopted Buddhism as a doctrine of the ruling Maurya Dynasty. In fact, in the

Diamond sutra, the text known as the backbone of Buddhism in Korea (Kim, 1999a), as well as in the Heart sutra, which is known to convey the heart of Buddhism (Conze, 2001), the idea of reincarnation or the creator gods does not exist. The reason why Buddhism could not involve the idea of creator gods lies in its core belief of emptiness. The notion of emptiness could be confusing and misleading because emptiness does not necessarily imply nothingness. In the

Heart sutra that is translated by Conze (1991), the Buddhist notion of emptiness is thoroughly explained:

In emptiness there is no form, nor feeling, nor perception, nor impulse, nor consciousness; No eye, ear, nose, tongue, body, mind; No forms, sounds, smells, tastes, touchables, or objects of mind; no sight-organ element, and so forth, until we come to: No-mind consciousness element; There is no ignorance, no extinction of ignorance, and so forth, until we come to: there is no decay and death, no extinction of decay and death. There is no suffering, no origination, no stopping, no path. There is no cognition, no attainment and nonattainment. (p. 97)

The Buddhists consider that what we think is driven from what is perceived through our senses which is also the result of our cognition. Thus, the world around us is formed and shaped by the filtered data through our senses. In other words, whatever we feel, think, touch, see, hear, and taste are our own constructions. Hence, all the concepts are not objective but subjective, are driven from cognition, and can be considered as illusory images. They do not exist “until we come to,” hence, empty. So, what is real? Buddhists do not deny there is something out there, but they teach us the inaccessibility to “reality” that is yet to be experienced by our senses and mind.

This inaccessibility is explained as emptiness, which neither can be seized, nor felt or thought.

Therefore, the notion of emptiness implies possibility or potentiality before one’s experience instead of nothingness. However, when it is accessed by human beings through their senses and mind, this emptiness is colored by and formed with building blocks of the human mind and senses. Thus, the Heart sutra concludes that emptiness is form and form is emptiness. To

Buddhists, forms that were constructed by human minds are understood as a dynamic being involving constant changes in accordance with changes in our cognitions, which inherently exclude permanent eternity. Some might argue that the Buddhist notion of emptiness falls into solipsism, “the view that this world is only in my imagination and the only reality is the imagining “I”” (von Foerster, 1984, p. 59), as they argue with radical constructivists. However, in Buddhism, solipsism was not even considered to be an argument because the notion of emptiness denies selfness1; myself is also a construction of my mind. In the Diamond sutra, the emptiness of selfness is more clearly articulated. The Buddha said, “all these I must lead to

Nirvana, into the Realm of Nirvana which leaves nothing behind. And yet, although innumerable beings have thus been led to Nirvana, no being at all had led to Nirvana” (Conze, 1991, p. 16).

This sentence could be understood in two ways. First, the Buddha himself claims that no being at all had led to Nirvana by him. Since he does not have self as it is, he claims that it was not him but themselves who reached Nirvana. Secondly, the innumerable beings are not ontologically real. The innumerable beings including himself are also a constructed concept. This is the salient distinction between a metaphysical realist and a Buddhist. There is nothing a buddhist can do to lead someone to the Nirvana since all the conceptions, including selfness and the Nirvana itself, are constructed by a cognizing subject, and they are nothing but mere images of your own construction. This perspective is similar to Kant’s doubt about the existence of objects that are apart from one’s experience (von Glasersfeld, 1984).

Wonhyo, who lived from 617 to 681, which is the end of the Three Kingdoms of Korea and the Beginning of Unified Silla, is one of the most well-known Buddhist monks in Korean

1 In sanskrit, “diamond” was actually “thunderbolt”. Kim (1999a) explicates that the realization of selflessness is so powerful and shocking just as thunderbolt striking upon one’s head.

14 history. In 661, Wonhyo and his friend, Uisang decided to go to the Tang dynasty seeking the essence of Buddhism. One night, they were caught in a heavy rain in the middle of nowhere and decided to stay overnight in a place they believed to be an earthen shelter. Wonhyo woke up thirsty at night and looked for water to drink. In the pitch darkness, he felt a bowl filled with water. Assuming it was rainwater in a wooden bowl, he drank it and quenched his thirst. Next morning he woke up and found that the place in which they slept was an ancient tomb, the wooden bowl was a human skull, and the water was extremely filthy and nasty. This made him question what reality is. At this point, he hit the moment of realization: 일체유심조(一切唯心

造), which means “everything is nothing but a construction of one’s mind,” which became the core lesson in Hwaeom, a branch of Buddhism. He decided not to proceed to the Tang dynasty for he already realized that a way of Buddhism (佛道); the core teaching of Buddhism does not exist somewhere to be discovered but he, himself, is the one who needs to construct it. This is one of the most well-known episodes in regards to the epistemology and axiology of Buddhism in Korea, which is taught in middle school and high school as part of Korean history.

In Taoism. Although it is controversial when exactly Taoism was established (some say it is around 2700 BCE), one can guess that it had established at least 2,300 years ago, because the book “Tao de Quing” was discovered in an ancient grave that was formed around 300 BCE.

There are two chapters that explicitly address its epistemology and axiology (chapters 1 and 2) in

Taoism (Kim, 1999b, 2000b). The first chapter in which epistemology is discussed addresses the incapability of defining Tao, which implies (including but not limited to) all the natural laws and phenomena, as “Tao” as it is, i.e., Tao as thing-in-itself, because “Tao as it is” includes endless changes, such that the changes are continual, differ, but are inseparable, and a static word “Tao” cannot capture all the characteristics of “Tao as it is.” The reader’s understanding of this very

15 word, for example, will change with time, with the development of the reader’s ideas, with the reader’s understanding of Tao, and so on. However, Tao de Quing did not downgrade or underestimate one’s construction of the experiential reality. The very next sentence explicitly expressed the importance of nominality, by saying that naming things is the mother of everything. Hence, the first chapter addresses that inaccessibility to ontological reality and the importance of the construction of our own reality.

In the second chapter, in which axiology of Taoism is discussed, the author(s) claims that there is no ultimate reality but multiple realities with which some number of people agree. (Kim,

1999b, 2000a, 2000b) For example, in the second chapter, the author used “dislike” for the opposite word of “beautiful” to address that there was no ultimate or universal concept of beauty.

The author also used “not good,” to describe the concept of “opposite of good,” which is different from the concept of “evil.” Hence, Laozi teaches that what is known to be good can be not good as well, depending on the perspectives of the cognizing subjects. Thus, there only exists preferences, not the dichotomy of good and evil. He later elaborated all the other concepts such as long and short, high and low, difficult and easy, which were all constructed in comparisons.

Taoism avoided the dichotomy of ultimate good and evil by claiming that the concepts are relatively constructed by the subjects who name (hence, construct) the concepts themselves, which was based on the idea of inaccessibility to ontological reality.

In Chinese Etymology. The Chinese etymology of 象, meaning symbol, image, or things shows how Far Eastern people considered the world as their own construction of their experience, not as an icon depicting ontological reality. According to 韓非子 (Han Feizi), an ancient Chinese text that is considered to be attributed to the master Han Feizi, the Chinese word

象 originated from the skeleton of an elephant. In the days when the Chinese characters, 象 was

16 created, it was almost impossible to see a real elephant. Most people could only guess about this gigantic animal by its skeleton that they found buried in the ground. Thus, the Chinese word for symbol or image does not imply the pictorial and iconic copy of the real elephants. Rather, it suggests a hypothetical model, which serves as a pointer to the construction of the elephant concept of the cognizing subject.

Figure 2. 1 Chinese letter “symbol” and its origin

The Chinese philologists categorized chinese letters into six categories, and one of them is imitative drafts, which are “rough sketches representing the object” (Wieger, 1965. p. 10), such as 山 for mountain and 川 for stream. 象 also falls into this category. Their usage of rough sketches as letters might have consistently and implicitly suggested the inaccessibility to ontological reality and the importance of the construction of one’s own understanding of the letters and the meanings.

Taoism and Confucianism resonated with Buddhism for sharing the same premise: admitting the inaccessibility to ontological reality, which leads to the realization of selflessness and emptiness, as well as considering ourselves as active constructors of our own experiential world. Then, what are the implications of Confucianism, Buddhism, and Taoism? These East

Asian philosophies focused on how to live life from perspectives of human beings. In Buddhism, one of the objectives is to cut off causal-effect links of life (karma). Life is suffering (again, in

Buddhism, ‘suffer’ does not solely mean distress, agony or pain, because human’s feelings are

17 originated from our conceptions. Hence, ‘suffer’ should be understood as a word encompassing all the events and feelings in life) because of our obsession to images that we have created, such as happiness, fame, money, and so on. In the Diamond sutra, Buddha teaches that the realization of selflessness strikes one’s head like thunderbolt and removes these endless life-long causal- effect links by admitting the inaccessibility to the ontic world, because what one thinks as a cause of a certain effect is also nothing but one’s own construction!

It is interesting that the idea of inaccessibility to ontological reality was considered to be a very shocking and striking work in the Diamond sutra; Being free from causal-effect links of life, i.e., Karma, by realizing there exists no concept of self as a thing-in-itself that is yet to be experienced, hence, seeking causes for one’s experience outside of one’s cognition means little.

However, radical constructivism suggests that surrendering seeking causal-effect links does not imply the end of epistemology. Rather, it gives rise to a new model of epistemology. von

Glasersfeld’s (1990) principles of radical constructivism elucidate how this new idea that admits inaccessibility to the ontic world shifts the notion of knowledge from discovering Reality to constructing a viable explanation that fits within the constraints of the previously constructed experiential reality. von Glasersfeld’s Frogs & Hoffman’s Australian Jewel Beetles

von Glasersfeld (1974) drew an interesting example of the frogs’ visual conception and reaction. From single fibers of their optic nerve, the frogs have a visual system that includes four specialized detectors: one for sustained light-dark contrast; a second for small dark convex shapes; a third for a moving edge; and the fourth for sudden dimming of illusion. In other words, the detector is optimized to capture small dark moving items, such as flies. Thus, a frog will catch any small dark moving item, such as black beads or air-gun pellets, as though they were

18 flies. Similarly, Hoffman (TED, 2015) reported the case of Australian jewel beetles, which were in danger of extinction, because many male beetles mated with beer bottles due to the resemblance of their appearances to the female beetle’s bodies. Like frogs catch any small dark moving item as thought they were flies, the Australian jewel beetles mated with beer bottles that were dark, brown, shiny, and round. One of the implications of these two observations is that frogs’ or Australian jewel beetles’ certain actions were not associated with what was real from the observer’s perspective. Their actions were not caused by real flies or female beetles but were triggered by what they perceived of real, based on their sensory motor inputs and previous experience.

Construction of Experiential Reality

von Glasersfeld (1974) found the epistemological implications in the frog story to be that,

“whatever we perceive is basically composed of signals within our sphere of experience,” (p.16) and used the implication to emphasize the importance of cognizing subjects’ construction of experiential reality. Therefore, it is impossible to conceive the ontological reality that exists independently of one’s experience, and one constructs one's own reality based on one’s experience. He argues that one might consider the original signals to be the effect of some outside causes, but “there is no way of approaching or “observing” theses hypothetical causes except through their effects” (p. 16). von Glasersfeld explained how one constructs one’s reality using the analogy of a “black box”.

We may observe and record the “output” from the black box (in this case the “sense data,” the signals on our side of the interface), and we may observe and record the “input” to the black box (in this case “proprioceptive data” and “feedback signals,” again on our side of the interface); both are neuronal signals - but once we have imposed a differentiation of “input” and “output,” we can establish recurrent coordinations and more or less reliable dependencies between the two. Having done this, we can construct an “external world” and our “selves” on the basis of input-output relations. (p. 16)

von Glasersfeld continues that, in case of frogs, the concept of fly is “defined only in terms of the neuronal signals that concur with the experience and never in terms of the inaccessible hypothetical outside “causes” of these signals,” (p. 17) and claims that the ways of how cognizing subjects construct concepts are not different from how frogs construct the concept of a fly; One constructs the world based on the data that were driven from the black box in which they were processed and constructed by an existing interface. Similarly, von Foerster (1984) argues that cognition is nothing but recursive computations in terms of neurophysiology where there are at least two levels of computations: “(1) the operations actually performed and (2) the organization of these operations represented here by the structure of the nerve net” (p. 57). He elaborated that when there are stimuli, the nerve synopsis takes them as the inputs of electronic pulses or signals that are qualitatively identical regardless of the stimuli. It is the self-regulated system that computes the input signals and reorganizes them to send output signals to react to the given stimuli. Hence, what we perceive itself is an input that is already processed in a “black box.”

Another example that the knowledge or concepts were not given but constructed by cognizing subjects can be found in an interesting experiment that Rosenhan (1984) conducted in order to investigate whether “the salient characteristics that lead to diagnoses reside in the patients themselves or in the environments and contexts in which observers find them” (p. 118).

They hypothesized that if the characteristics of insanity are real and they are embedded in the patients, then one can match them to the diagnoses of insanity. The eight pseudopatients, who were considered to be normal and sane in varied groups, went to the admissions office of mental hospitals saying that they were hearing voices. Except for using pseudonyms and false

20 symptoms, they provided their own life stories and medical histories and acted normal, i.e., as they were, when they were sent to the ward. Although the patients were falsely diagnosed as insane, none of them were successfully recognized as sane while they acted as they were in the wards, and they were hospitalized an average of 19 days (ranged from 9 to 52 days).

Once labeled as insane, Rosenhan articulated that further diagnoses were geared toward to explicate why the patients were insane. In addition, all the pseudopatients reported that the staff in the wards treated them with depersonalization and invisibility, which eliminated possibilities to recognize their sanity. Rosenhan therefore concluded that diagnoses are constructed by the observers based on the environments and contexts in which they find them.

His finding is instructive in two ways: (1) The concept of sanity and insanity does not carry any a priori characteristics to which one can match certain diagnoses, and (2) observers construct the patients’ realities regardless of patient's’ own realities by fitting the given information in the observers’ own experiential realities.

Principles of Radical Constructivism

With the premise that the ontological reality is not accessible and that knowledge is actively constructed rather than passively perceived, von Glasersfeld (1990) stated the principles of radical constructivism as follows:

1. Knowledge is not passively received either through the senses or by the way of communication. Knowledge is actively built up by the cognizing subject. 2. a. The function of cognition is adaptive, in the biological sense of the term, tending towards fit or viability. b. Cognition serves the subject’s organization of the experiential world, not the

discovery of an objective ontological reality. (pp. 22-23)

von Glasersfeld’s (1990) radical constructivism is radical because even the input that one

21 takes in one’s neuronal system is a piece of data that is already processed by the cognizing subject’s perception and cognition; A cognizing subject is the one who imposes the inputs and outs of the processed data (von Glasersfeld, 1974). However, radical constructivism does not deny the existence of raw material that is to be experienced; the ontological reality as thing-in- itself is simply inaccessible, hence, unknowable, because once perceived, it is already a processed data that was imposed to be the output in the “black box”. von Glasersfeld (1990,

1995) and von Glasersfeld & Cobb (1983) suggest radical constructivism as an alternative scenario which can answer the skeptics questions as well as provide a viable explanatory model of epistemology. They claim that the ontological reality is inaccessible, the world one lives is an experiential world that is constructed by the cognizing subject, and knowledge is not discovered but constructed;

Traditional epistemology has always taken it [knowledge] as a matter of course that there is a knowable ontic world and that it is the knower’s task to get to know and describe it. The activity of “knowing,” thus, was always seen as the acquisition of something that was already there. Our theory, instead, focuses on the activity of “knowing” as a constructivist activity whose results are not merely compilations of material which the knower passively receives through senses or through some other experiential conduit, but rather coordination of elements which originate, within the knower, as products of knower’s own activities of generating and coordinating (von Glasersfeld & Cobb, 1983, p. 6).

Introducing radical constructivism as unpopular philosophy in Western countries, von

Glasersfeld (1990) explains that the radical constructivism is considered to be radical because the basic premise of western philosophy, i.e., the ontic world can be discovered, is disregarded.

Then, how one can guarantee the reliability of one’s own experiential world (von Glasersfeld,

1984)? von Glasersfeld (1984) suggests an exploratory model of this important “epistemological problem-how we acquire knowledge of reality, and how reliable and “true” the knowledge might be” (p. 2) by carefully introducing the notion of “fitting” instead of “matching”.

Fitting Instead of Matching

von Glasersfeld (1990, 1995) drew an analogy between keys and biological selection to elucidate the principles of radical constructivism. The key that comes with a lock is not the only thing that unlocks the lock. Anything such as pins or paper clips also can open the lock as long as it fits within the constraints of a lock. Given this notion of fit, Radical constructivism is based on

“the principle of adaptation to constraints rather than principle of causation” (von Glasersfeld &

Cobb, 1983, p. 8). In the context of evolution, the organisms that are adapted to (or fit in) the given environments despite the constraints within the environment survive. However, this viability does not depict the environment itself, in the constructivist perspective, such as knowledge cannot depict the ontic world but models the experiential world. While the viability of the organisms is fatal in survival, in the cognitive domain, errors or failures are not terminal but feed back into the cognizing subject’s operations creating adaptations so that one can solve the problem better later on, i.e., one can learn (von Glasersfeld & Cobb, 1983). In other words,

“knowing” and “knowledge” generate a dynamic self-regulating system.

Piaget (1968) conducted an interesting experiment showing how our memories entail adaptations as dynamic beings. One might think memories are iconic representations of certain things that can be either remembered as exactly as they were memorized. Piaget argues that memory itself evolves in accordance with the development of the subject’s mental operations

(Piaget, 1968). In his experiment, Piaget showed sticks that were arranged in order of increasing length from 9 cm. to 15 cm. to the participants (children ranging from 3 to 8 years old). He then asked them to draw or describe what they saw from memory a week later and then six months later. The participants’ memory of a week later differed in accordance with their development of their mental operations. The participants, who did not have ordinal sense, recalled a certain

23 number of equal length sticks. More advanced participants described sticks of two or three distinct lengths. Only the participants who had the most advanced operations available described a series of about 10 sticks of distinct lengths. Six months later, when asked the same questions, the majority of the participants exhibited better recollection. For example, some participants who previously described a certain number of sticks of equal lengths recalled sticks of two different lengths. The participants who described a number of sticks with three different lengths later recalled a series of sticks of distinctive lengths. Based on this experiment, Piaget (1968) argues that the children’s memory is not an iconic copy of their observations, but it “reflect(s) subjects’ assimilation” (p. 5) and “thinking means structuring reality by means of operation” (p. 35). von

Glasersfeld summarized Piaget’s theory of knowledge as, “The mind organises the world by organising itself” (von Glasersfeld, 1995), which entails recursive process in a self-regulated system. Interestingly, the radical constructivist notion of thinghood (including memory) coincide with the Taoist epistemological perspective that permanence entails continuum of changes.

Theory of Knowing in Radical Constructivism

With the principles of radical constructivism, von Glasersfeld (1995) explicated Piaget’s constructivist theory of knowing including the notion of learning. He explained (1) how one constructs one’s experiential reality by constructing object concepts and object permanence, (2) how one expands one’s experiential reality through assimilation and accommodation which leads to an equilibration, and (3) the notion of knowing in a constructivist perspective. von Glasersfeld and Cobb (1983) suggest the word “model” in cybernetical sense, i.e., “a conceivable arrangements and operations which, under similar circumstances and in similar situation, would produce results similar to those of which we ourselves, as cognizing subjects, become aware,” (p.

7) and introduce the notion of viability to explicate how a cognizing subject constructs one’s

24 experiential world that is not unlike the others’ experiential reality. They also emphasize that the model of one’s experiential reality is always hypothetical, because one cannot access the ontic world. von Glasersfeld (1995) developed a hypothetical model of the construction of experiential reality, which is employed in my study.

Before I address von Glasersfeld’s hypothetical model of how one constructs one’s experiential reality, I want to reemphasize the cyclic and recursive nature in the ways one constructs one’s experiential reality. von Glasersfeld frequently used the analogy of black box

(1974; 1984; 1990; and 1995) to emphasize the inputs that one uses when constructing that one’s experiential reality is not a raw material that is yet to be experienced but is the output of one’s cognizing processes. Although he explicates how one constructs one’s experiential reality in a certain order for the sake of efficiency, one must be aware that he considered that the processes were recursive and cyclic, forming a self-regulated system rather than a process in a linear order.

Hypothetical Model of the Construction of Experiential Reality

von Glasersfeld and Cobb (1983) introduced Maturana’s perspective of living organisms as conservative inductive systems which repeat only what brings desirable outcomes. They propose three premises that a cognitive subject is capable of when constructing the conservative inductive system. First, cognizing subjects can record or retract their experiences. Second, cognizing subjects can discriminate the desirables and undesirables for the sake of viability.

Lastly, cognizing subjects act in response to cognitive perturbation to produce desirable outcomes.

Recognition & Re-presentation

Piaget (1954) investigated how infants construct their experiential reality by observing their behaviors and suggested a model of infants’ construction of their experiential reality.

According to Piaget, investigating how one can construct object concepts is indispensable when explaining how one constructs his or her experiential world. von Glasersfeld (1995) also claims that Piaget’s intensive study of infants’ construction of their reality suggests, “how the basic concepts that constitute the essential structure of any individual’s reality can be built up without the assumption that such a structure exists in its own right” (p. 58). One might be curious whether infants’ concept of nipples is identical to that of adults’. Piaget explains that infants develop their object concept while coordinating and associating it with sensory signals. For example, infants who are a couple of weeks old would turn their head and try to suck the nipples when the nipples hit their cheeks. When a thumb accidentally hits their cheeks, infants assimilate the tactile sensation of the thumb as that of nipples and try to suck it. At this stage, Piaget explicated that infants recognize nipples by assimilating their tactile sensation to their previous successful experience. In terms of visual perception, Piaget observed that these infants did not look for objects when the objects disappeared in their immediate visual field. The infants around three to five months old started to look for hidden items when the observer hid the item in the infant’s visual field. However, they were yet to seek for the items that were hidden out of their visual field. The infants around four to six months old would not look for completely hidden items, but they could recognize the hidden items when particular part of the hidden items was shown. For example, one of the infants did fetch a baby bottle when the nipple part was exposed and the rest was hidden while she did not grab the baby bottle when the bottom part of the baby bottle was exposed. Infants around five to seven months old could find the hidden items but they were limited to finding the hidden items only when the items were located where they were previously located. When the observer changed the location of the hidden item from under one cushion to the other, the infants failed to find the hidden items. Only after the infants could find

26 the hidden items that were hidden out of their visual field, did Piaget infer that the infants constructed object permanence.

Infants can recognize objects when they establish objects concepts that were coordinated or associated with sensory signals and experientially abstracted. This empirical abstraction creates templates or prototypes of the objects, but it is triggered by the sensory motor input.

Then, infants progress to the second developmental phase in which they can “run through a sequence of physical actions” (von Glasersfeld, 1995, p. 59) in the absence of sensory motor data in their immediate experiential field. von Glasersfeld called this ability re-presentation of an object concept instead of “representation” to avoid possible confusion that might arise from the conventional meaning of representation as an iconic copy of the ontic world and to illuminate the recursive mental process present in one’s construction in one’s mind.

Object Permanence: Equivalence & Identity

von Glasersfeld continues that the ability to re-present an object is one of the two crucial mental operations for constructing the concept of object permanence. The other one is the notion of identity. Before constructing the concept of object permanence, one engages in comparing activities between the re-presentation and the present experience of objects to judge their sameness. He explains that there are two kinds of sameness:

On the one hand, there is the sameness of two experiential objects that are considered the same in all respects that have been examined (as in assimilation); This can be called ‘equivalence’. On the other hand, there is now the sameness of two experiences that are taken to be two experiences of one individual object. (...) In the first case, the set of characteristics which, as a group, differentiate a particular experiential item from all other constructs, is abstracted and maintained (given perdurance) for future use. It constitutes the template or prototype to which experiences can be assimilated as members of the class. This procedure is the basis of all classification and categorization. The concept of ‘object permanence’, on the other hand, is an abstraction from the second type of sameness. It characterized the situation where a child considers the object it is perceptually constructing at the moment, to be the identical (self-same) individual it experienced at some prior time. Perdurance is now attributed to the object whether or not

it is actually being experienced. (pp. 60-61)

Piaget (1954) differentiated these two types of sameness in terms of whether the concept of the sameness is associated with or bounded by one’s actions and perception. When the object concept is externalized in such a way that one can conjure up its re-presentation independently of present actions and perception, he considered the object concept to be permanent. Steffe (2004) found the distinction between these two ways of considering sameness in an account of explaining the difference between the equi-partitioning scheme and the splitting operation.

Consider the following two tasks:

(1) Share the given licorice equally among five people. Mark off your share. (2) The given licorice is five times longer than your licorice. Show your licorice.

When solving the first task, one might mentally re-present the given whole as the input for one’s partitioning activity. After partitioning the given whole into five equal pieces, one might disembed one piece and mark off the length on the given whole. When asked to check whether the marked-off share is a fair share, one might iterate the marked-off part five times producing the partitioned whole while assimilating the partitioned whole to the given whole.

This partitioned whole is equivalent to the given whole that is yet to be partitioned. On the other hand, when solving the second task, one must produce and posit the hypothetical part by simultaneously coordinating partitioning the whole and iterating the part. Thus, the input is the conceptual given whole that is already partitioned and iterated. If one mentally compares the partitioned and iterated whole to the given that has already been mentally partitioned and iterated, one might infer that he or she constructed the whole that is identical to the given whole.

To construct object permanence, von Glasersfeld claims that the construction of the second kind of sameness is required. This concept of object permanence gives rise to the idea of

28 the existence of objects. This is the crucial step for a child to construct his or her reality including the concept of time, space, etc., because the object permanence enables a child to think and re-present the objects that are not in their experiential fields. von Glasersfeld called this conceptual repository ‘proto-space,’ where the object concepts were stored. He continues that a child constructs proto-time in the same way, both of which are the building blocks of their experiential reality. He further explains that the construction of object permanence enables the cognizing subject to detach his or her perception and actions from themselves so that they externalize themselves in active experience. Then how does one utilize these building blocks in order to construct and expand his or her experiential reality? Based on von Glasersfeld and

Cobb’s (1983) three premises that a cognitive subject is capable of when constructing the conservative inductive system (one can retract one’s experiences and repeat the actions that bring forth the desirable outcomes of one’s actions), von Glasersfeld (1991; 1995) elucidates Piaget’s notion of assimilation, schemes, accommodation, equilibration, reflection, and abstraction as important operations in the construction of one’s reality.

Assimilation

One might observe, as Piaget did, that a newborn baby sucks not only a nipple but also her thumb, possibly thinking that sucking whatever touches her cheek would nurse her. Piaget

(1954) explicates this baby’s behavior using the notion of assimilation; Assimilation is a term that Piaget borrowed from biology to explain the mental operation when a cognizing subject uses to fit a new experience into an existing conceptual structure. According to von Glasersfeld

(1995), “assimilation always reduces new experiences to already existing sensorimotor or conceptual structures,” (p. 63) by disregarding whatever does not fit in the existing structures. In the case of the baby’s sucking thumb action, she assimilates the tactile sensation of her thumb to

29 that of a nipple by disregarding other characteristics that do not fit her nipple experience.

However, assimilation is not enough to explain how one expands their experiential reality by learning new things. von Glasersfeld introduces Piaget’s notion of scheme to better explain assimilation and what learning is.

Scheme

According to Piaget (1968, p. 17), scheme is a, “part of an action or operation which is repeatable and generalizable in another action or operation.” von Glasersfeld (1995) explains

Piaget’s notion of scheme using an infant’s sucking scheme. When an infant feels something touching her cheek, she turns her head toward the sensation and finds something to suckle because the infant anticipates that the result of her actions will provide her nutrition that is essential to survive. The infant will repeat this action because it works. von Glasersfeld continues that the way we organize our knowledge and concepts are not unlike the way an infant constructs the sucking scheme. We perceive the given situation using our perception and recognize it in a certain way, and we carry out certain actions or operations expecting results that we previously experienced. Thus, a scheme is composed of the following three parts:

(1) recognition of a certain situation; (2) a specific activity associated with that situation; (3) the expectation or the results that the activity produces a certain previously experienced result (von Glasersfeld, 1995, p. 65)

von Glasersfeld continued to explain that recognizing a situation as well as the result of one’s actions are the products of assimilation, and assimilation triggers goal directed activities.

Learning as Accommodation

Assimilation is not enough to explain how one can expand one’s experiential world. von

Glasersfeld (1995) introduces the notion of accommodation. In the biological sense, organisms

30 perish if they fail to modify their structures to fit within the constraints in their environment. In the cognitive sense, a cognitive subject makes adaptations by modifying the previous schemes so that the new scheme can restore equilibration, by which von Glasersfeld (1995) means, “an increase in the range of perturbations the organism is able to eliminate” (p. 67). While convenient, von Glasersfeld was concerned about the possibility that one might misunderstand the concept of cognitive adaptation when one uses the analogy of biological adaptation (the modification of the structure of the organisms) in the case of explaining the cognitive sense of adaptation. In the biological sense, one might consider the environment to be external to the organisms, and adaptation implies modifying the structure of the organisms so that it can fit in what is externally given. From the radical constructivist perspective, “adaptation does not mean adequation to an external world of existing things-in-themselves, but rather improving the organism’s equilibrium, i.e., its fit, relative to experienced constraints” (von Glasersfeld, 1995, p.

63). For example, what if one day, the baby sucked what was touching her cheek, but her goal, getting milk, was not achieved? More generally, what if the goal of the scheme is not achieved and one gets an unexpected result to one’s dismay or surprise? In this case, the equilibration will be perturbed, and the cognizing subject will review what was disregarded when one assimilated a new experience if the previous assimilation is retrievable. Then the cognizing subject would refine, extend, or vary the activity patterns that encompass the disregarded factors of the inputs and the unexpected results by making new schemes, which results in equilibration. This operation is called accommodation, which is learning in the constructivist perspective.

Equilibration

While equilibration can be generally understood as, “a generic term for the elimination of perturbation” (von Glasersfeld, 1995, p. 66), from the constructivist perspective, the concept of

31 equilibration entails active and dynamic equilibrium that makes a system stable by neutralizing perturbations. von Glasersfeld (1980) considered counteracting perturbation to be a characteristic of living organisms and homeostatic devices. He explained the notion of “control” in the cybernetic sense as a counteraction to perturbation of a system. von Glasersfeld (1980) stated:

Control always involves at least three things:

a) a possible state of equilibrium defined as a constant relation between an input variable and a reference variable; b) perturbations that tend to modify the relation between the input and the reference values; and c) an activity by means of which the system can, at least potentially, restore the equilibrium. (p. 75)

Based on this notion of “control,” one can consider organisms to be the agents of acting in response to perturbation. This control system works, i.e., is viable, when the activity brings forth a change in relation between an input and a reference whenever it detects perturbation that upsets the equilibrium. Unlike an artificial homeostatic device into which a reference variable is inserted by its creator, in terms of cognitive development, the references are constructed by cognitive subjects themselves. A cognizing subject then constructs inferential connections between the activity and its result and recursively uses them in a self-regulating controlling system. In this sense, von Glasersfeld (1980) considered equilibration to be a goal toward which the system acts.

In order to detect perturbation, it is indispensable to reflect one’s experience on a previous experience. von Glasersfeld elaborated (1991; 1995) on reflection as a crucial mental operation in the construction of one’s experiential reality.

Reflection

When asked to compare the taste of apples, one might recall how apples taste to compare them. von Glasersfeld (1991) explained that the mental operation enabling this comparison is reflection and utilized Wilhelm von Humboldt’s aphorisms in his explanation:

1. The essence of thinking consists in reflecting, i.e., in distinguishing what thinks from what is being thought. 2. In order to reflect, the mind must stand still for a moment in its progressive activity, must grasp as a unit what was just presented, and posit it as object against itself. 3. The mind then compares the units, of which several can be created in that way, and separates and connects them according to its needs. (p. 2)

He continued his explanation by stating that when one abstracts one’s experience, the individual’s “focused attention picks a chunk of experience, isolate[s] it from what came from before and from what follows, and treat[s] it as a closed entity” (p. 2). This is the simplest kind of abstraction, called the unitizing operation,2 which is used to construct an object concept. This simplest kind of abstraction is crucial in the construction of counting schemes (Steffe & Cobb,

1988). Once abstracted, one can re-present the abstracted experience, i.e., “posit it as object against itself.”

Abstraction and Generalization

Once abstracted, the isolated chunk of one’s experience can be compared, connected, and separated, which leads to a further step of abstraction. von Glasersfeld (1991) referred to this abstraction as generalization, which is crucial when categorizing. For example, if a child grows up in a region where all the apples are red, the child will likely associate redness with the

2 In the section titled in the literature review “The Unitizing Operation,” I will elaborate on the unitizing operation.

33 concept of apple. Later, when the child experiences green apples, he or she first will be surprised, but perhaps due to social pressure, he or she will accommodate the concept of apple so that it includes green apples, forming a generalization of apples. According to von Glasersfeld (1991), one can abstract a general idea of one’s experience by substituting “a kind of place-holder” (p. 4) for some of the particular sensory properties that are also abstracted from our experience. He continues that these “general ideas are not “figurative” but “operative”” (1995, p. 92), and this operational structure functions like a program, which can be called a concept.

Four Kinds of Abstraction

The ability to abstract items that can serve to recognize them does not necessarily imply the ability to re-present them without perceptual inputs (von Glasersfeld 1995). For example, one might recognize a person’s face that he or she met at a cocktail party while he or she cannot quite recall the person’s face. von Glasersfeld addressed Piaget’s four kinds of abstraction: empirical, reflective, reflected, and pseudo-empirical abstraction, which are relevant to the distinction that I just mentioned. Empirical abstraction means isolating certain common sensory properties from two more conceptual items so that it can serve to assimilate new items to an existing object concept. Pseudo-empirical abstraction refers to an ability to re-present certain things only when suitable sensory-motor materials are available. Hence, the empirical abstraction and pseudo- empirical abstraction are not fully operative, because both require some perceptual inputs to trigger a scheme. Reflective abstraction is an internalized mental operation or activity that enables one to cognitively reconstruct and reorganize what has been processed. Once reflectively abstracted, experience can be projected onto different situations. Applying multiplying schemes when finding a commensurate fraction is an example. Reflected abstraction is an interiorized mental operation including subjects’ awareness of what has been abstracted. Reflected

34 abstraction enables monitoring one’s own mental operations while one is operating. von

Glasersfeld referred this reflected abstraction to “reflecting on reflection” (p. 12). He differentiated reflective abstraction and reflected abstraction by citing Piaget’s work, “thought as a process of construction” and “ thought as a process of retroactive thematization.” (p. 12) He continued that reflective and reflected abstractions enable a child to “step out of the experiential flow” and reorganize the existing cognitive structure “in function of new givens” (p. 12).

Anticipation and Reversible Reasoning

Recall that a scheme is composed of the following three parts (von Glasersfeld, 1995, p.65): (1) recognition of a certain situation, (2) a specific activity associated with that situation,

(3) the expectation that the activity produces a certain previously experienced result. von

Glasersfeld (1990; 1997) stated that the expectation can be understood as anticipation of a scheme. Therefore, a scheme is a goal-directed activity accompanied by the anticipation of its own scheme. I will explain the relation between a goal-directed activity and anticipation of a scheme.

To explicate the notion of anticipation, von Glasersfeld (1990) drew an operational analysis of the constructs of cause. According to von Glasersfeld, when a cognizing subject asks oneself why he or she experiences a certain effect, he or she can engage in abstracting the cause of the effect by isolating the factors that produce the effects while paying attention to the changes that the selected cause may bring. von Glasersfeld (1998) continued that “cause-effect link [...] is based on the belief that, since the cause has produced its effect in the past, it will produce in the future” (p. 6). von Glasersfeld claimed that the process, re-presenting (or projecting) the experientially formulated cause-effect links into the future experience that is yet to be experienced, involves anticipation. Thus, when one formulated the link that if A happens then B

35 follows, one will try to generate A when his or her goals of a certain activity are to bring about

B. In other words, anticipation triggers goal-directed activities.

von Glasersfeld (1988) continued that anticipation involves the ability “to reflect on past experiences, to abstract specific regularities from them, and to project these as predictions into the future” (p. 8). He then differentiated three different kinds of anticipation. The first kind of anticipation is in the form of implicit expectation such as preparing to control motions of a body when stepping down in the dark. This kind of anticipation does not require cause-effect rules, but it requires some familiarity of the conditions. The second kind of anticipation is “the expectation of a specific future event, based on the observation of a present situation” (p. 9). With this second kind of anticipation, one can predict the result of his or her actions based on the consequences he or she had in the past. The third kind is the “anticipation of a desired event, situation, or goal, and the attempt to attain it by generating its cause” (p. 9). The third kind is also based on the consequences of the past experience as well, but the difference lies in that the third kind of anticipation involves a conceptual model that explains the cause-effect connections. Note that the second kind of anticipation is based on the observation of a present situation whereas the third kind of anticipation includes not only the prediction of the desired goal of an activity but also the attempt to attain the goal by generating its cause. It seems that the second kind of anticipation is related to inductive reasoning, and the third kind of anticipation requires not only inductive reasoning but also deductive reasoning to construct a conceptual model that explicates the cause-effect links of how the cause can be generated rather than observed. Moreover, the abstracted past experience that is associated with a particular cause-effect link feeds back one’s thinking in order to generate a particular cause. Then, how can one generate the cause that attains the goal of his or her activity?

In the analysis of the structural integration of formal thought, Inhelder and Piaget (1972) elaborated the explicit distinction between the concrete thought and formal thought in terms of their forms and contents of the operations. Inhelder and Piaget stated:

From the standpoint of form, concrete operations consist of nothing more than a direct organization of immediately given data… From the standpoint of content, concrete thought has the limiting characteristic that it cannot be immediately generalized to all physical properties. Instead, it proceeds from factor to another… (p. 249)

They continued that concrete thought, therefore, is essentially bounded by empirical reality. The anticipation that concrete thought produces seems related to von Glasersfeld’s

(1988) second kind of anticipation. Inhelder and Piaget claimed that formal thought includes:

(...) a reversal of the direction of thinking between reality and possibility in the subjects’ method of approach.(…) The most distinctive property of formal thought is this reversal of direction between reality and possibility; instead of deriving a rudimentary type of theory from the empirical data as is done in concrete inferences, formal thought begins with a theoretical synthesis implying that certain relations are necessary and this proceeds in the opposite direction. Hence, conclusions are rigorously deduced from premises whose truth status is regarded only as hypothetical at first; only later are they empirically verified. This type of thinking proceeds from what is possible to what is empirically real. (p. 251)

According to Inhelder and Piaget, formal thought is a two-way process: (1) establishing hypothetical anticipation based on the hypothetical cause of the effect that brings about the anticipation, and (2) the empirical verification of the hypothetical anticipation by recursively using the scheme that one used to form the anticipation. I argue that von Glasersfeld’s (1998) third kind of anticipation involves reversible reasoning, because it includes the attempts to attain the predicted goal by generating its causes, which requires reversible reasoning, according to

Inhelder and Piaget.

My hypothesis is that one engages in reversible reasoning when recursively using the result of the scheme to produce a cause. To project a particular cause-effect link onto a new

37 experience, one must reflectively abstract the cause and the effect. To isolate the cause in order to produce it, one must disembed the cause. To disembed, one must have reconstituted the given by reflectively abstract the cause and effect, and use the reconstituted link to isolate the cause.

The Principles that Guide My Study

As a person who had lived in the country of the epistemology, cosmology, and axiology of Buddhism and Taoism as well as a practitioner of these philosophies, I believe in the inaccessibility to ontological reality. Thus, I reject the premise that one can discover the ontological truth that was embedded in a thing-in-itself, because as long as one is a human being, what we perceive is distorted, colored, and formed by our senses and cognition. Similarly, I refuse to believe that one can find the exact and every cause for a certain phenomenon or effect because the creator God created things for a reason and everything has its beginning. I agree with

Russell’s (1925) claim that “the idea that things must have a beginning is really due to the poverty of our imagination” (p. 7). However, I do not deny that there might exist something that

I cannot capture with my senses and cognition. Rather, I regard things that are yet to be experienced as possibilities and potentialities. In addition, I admit that the world that I perceive is constructed based on my senses and cognition. If there exists an ontic world, it might be distorted and colored by my senses and cognition as soon as it is thought, and there exists no way for me to notice such distortions. Hence, I do not seek for any linear cause-effect links for the ontic truth. Rather, I look for viable explanations of my observations that fit my current understanding.

Radical constructivism provides the same premise of the inaccessibility to ontological reality and suggests detailed explanations of how one constructs one’s own experiential world including time and space. Therefore, I chose radical constructivism as the frameworks that guides my study, and furthermore, as the philosophy that guides me as a lifelong learner, researcher, and a teacher.

One of the crucial points in my study is admitting and explicitly being aware that I am an observer to the participants of this study. There does not exist any way that I can find the

39 mysterious function of their cognition and depict it as it is. As von Glasersfeld (1990) suggested, they are black boxes of which the internal functions are unknowable. I conduct my study to explore and investigate ways of constructing x+y=a, and to construct a viable and compatible explanation of how the participants construct it. In other words, I seek a way to understand their construction of x+y=a based on my understanding of x+y=a, my observation of how the participants construct x+y=a, as well as other researcher’s studies that are viable to my understanding and observations.

CHAPTER 3

LITERATURE REVIEW

The purposes of the literature review are (1) to introduce and explain the reorganization hypothesis (Steffe & Olive, 2010) as an overarching hypothesis of this study, (2) to explicate the units-coordinating schemes while providing rationales for which I chose them as an explanatory tool to model students’ construction of x+y=a, and (3) to revisit my research questions in connection with the literature review. Based on radical constructivism, I focus on students’ operations and actions when constructing x+y=a rather than conventional mathematical definitions or concepts from an adult's’ perspective.

Students’ Mathematics

Steffe & Thompson (2000) used “students’ mathematics” to refer to students’ mathematical reality that is constructed by their ways of understanding in mathematics (Steffe &

Thompson, 2000). If we consider students as autonomous and self-governing entities who can organize their knowledge and concepts as we, adults, do, and if we regard students’ mathematics as “a product of functioning their intelligence”, students’ mathematics becomes legitimate mathematics, which can be explained and justified in adults’ perspectives. However, observers cannot directly know students’ ways of operating. Students’ mathematics can only be inferred and interpreted based on their actions, languages and interactions (Steffe, 2007) while they are engaging in goal-directed mathematical activities and reasoning. Hence, it is important to “look behind” students’ language and actions to understand the ways of their operating. The process

41 involved in investigating and analyzing the ways of students’ operating is called conceptual analysis of students’ mathematics (von Glasersfeld, 1995).

Modeling Students’ Mathematics: The Second Order Model

If we admit that students construct knowledge in accordance with their means and ways operating, then students are reasonable in their own perspectives. Therefore, the errors and mistakes that students make should be understood as results of their ways of operating. In the radical constructivist point of view, students’ mathematics is considered as coherent and internally consistent mathematics. It is an observer’s role to construct an explanatory model of students’ mathematics based on analysis of their actions, language and interactions (Steffe,

2007). To capture what radical constructivism suggests as a theory of knowledge, von

Glasersfeld & Cobb (1983) used “modeling” to suggest a conceivable arrangement of operations that produces the similar result under the similar situation and environment. Following von

Glasersfeld and Cobb, what I mean by “modeling students’ mathematics” is constructing observers’ knowledge of students’ mathematics based on conceptual analysis in order to suggest viable explanation of students mathematics within the observers’ ways of operating. Steffe

(2007) refers to modeling students’ mathematics as a second order model to emphasize the observer’s perspective while referring to the construction of one’s own explanatory model as a first order model. Following Steffe, the goal of this study is constructing the second order models of how students construct x+y=a.

Reorganization Hypothesis

The overarching hypothesis of this study is the reorganization hypothesis (Steffe & Olive,

2010): One constructs new schemes by reorganizing one’s existing schemes. According to Steffe

& Olive (2010), there are two ways of understanding the reorganization of prior schemes. The

42 first is to construct the new schemes by “operating on novel material in situations that are not a part of the situation of the preceding schemes” (p.1). This new scheme is used as the preceding scheme in prior situations, but it does not supersede the prior schemes. Steffe & Olive (2010) claimed that children construct their fractional knowledge by reorganizing their counting schemes in a way explained by this hypothesis. The second way is to construct new schemes that supersede the prior schemes that not only can solve the new situations, but also solve previous situations better. Children’s developing more complicated number sequences based on their preceding number sequences falls into the second kind of reorganization. This study employed the latter reorganization hypothesis; I hypothesize that the students construct x+y=a by reorganizing their units-coordinating schemes. For example, a child who reasons with two levels of units might find a missing addend by either counting up-to or counting down-to. Later, the child may reflectively abstract his or her counting acts, then review the sum as a unit containing three levels of units constructing an inclusion relation among the addends and the sum.

Furthermore, when asked to explain how x and y co-vary where x+y=a, the child might take the sum as an input, where the sum is a unit containing two quantities, and apply changes in each quantity by recursively using his or her three levels of units-coordinating scheme.

When a child constructs a superseding scheme, one might think the child will immediately use the new scheme in all relevant contexts. According to von Glasersfeld (1995), this is not the case. Spreading a novel way of operating takes time, and a child occasionally uses the old scheme in a certain context while using the new scheme in another.

Following the reorganization hypothesis and radical constructivism, my study focuses on students’ operations that are involved in the construction of x+y=a. I chose the units- coordinating schemes as an explanatory tool to model my students’ construction of x+y=a

43 because the units coordinating schemes are based on children’s counting schemes and the unitizing operation that generates extensive quantity (Steffe, 1991). In addition, observing how the students engaged in quantitative reasoning and utilize it in order to construct x+y=a was pivotal in my study. Thus, I explain the notion of quantity and the operation that generates quantity.

Quantity

Although essential, Steffe (1991) considered mathematical definitions or concepts that exist independently of children’s experience are not sufficient to specify the operations that children use when generating quantity. While I agree that the notion of extensive quantity is derived from counting or measuring activities (Schwartz’s, 1988), I focus on the operations that generate the concept of quantity and how the students utilize them in order to construct x+y=a rather than focusing on the definition of quantity, which is defined based on adults’ perspectives.

From the radical constructivist point of view, quantity is a scheme that is abstracted as a conceptual structure (Steffe, 1991; Thompson, 2011). Steffe (2012) emphasizes the importance of the concept of units using Thompson’s notion of quantity—perceiving a quantity includes perceiving appropriate measuring units. Thompson (1994) explains that a quantity is a scheme that is comprised of (1) an object, (2) the quality of the object, (3) an appropriate unit or dimension, and (4) quantification, a process by which to assign a numerical value to the quality.

He further states that a “quantitative operation is a mental operation by which one conceives a new quantity in relation to one or more already-conceived quantities” (p. 9). Thompson’s notion of quantitative operation is employed in my study when I observed and analyzed how the students found the sum in relation to two oriented quantities and how they constructed x+y=a

44 based on their units-coordinating schemes.

In the following sections, I first explain von Glasersfeld’s (1995) unitizing operation that is used in the construction of an object that has a quantitative property, and how one can utilize the unitizing operation when constructing the concept of units. Then, I introduce Steffe and

Cobb’s (1988) distinct kinds of units that were produced by children and elaborate on the number sequences and the units-coordinating schemes.

The Unitizing Operation

In the extension of Piaget’s work, Steffe (1991) elaborated extensive quantity as the result of unitizing operation. The unitizing operation is a mental operation that is involved in the construction of object concepts, which is essential in the construction of numerical units, according to von Glasersfeld’s (1995) hypothetical model of construction of concepts. He suggested that the construction of object concepts starts from perceptual elements, and the object concepts are the results of an abstracting activity carried by a cognizing subject. Following Silvio

Ceccato’s claim that the structure of certain abstract concepts could be interpreted as patterns of attention, von Glasersfeld explained how the construction of abstract concepts could be applied to the idea of a numerical concept. Applying his attentional model, von Glasersfeld outlined how sensory-motor items, unitary items, and abstract unit items are developed. He elaborates in the following:

Attention, in this model, is conceived as a pulse-like activity that picks out, for further processing, some of the signals from the more or less continuous multitude of signals which the organism’s nervous system supplies. That is to say, a single pulse or moment of attention can be, not need to be, focused on a particular signal. When it is unfocused it does not pick out particular signals, but this does not mean that there are no signals that could have been picked out. The unfocused moment merely creates a break in the process of composition. On the other hand, attention can focus on items that are not present at active sensorimotor signals, but as re-presentation of signals (or composites of them) that have been picked out at some prior occasion. (pp. 167-168)

Von Glasersfeld’s attentional model explicates the mental operation that is involved in the construction of object concepts. According to this model, a discrete perceptual item consists of the attentional focused moments that are bounded by unfocused moments. For example, the concept of an apple contains many different characteristics including its hue, taste, shape, and color. By smelling, looking, feeling, and tasting an apple quite frequently, one could extract some sensorimotor signals that are common to all occasions, which leads to the construction of a perceptual apple concept. He emphasized that “the concept is the result of a reflective abstraction that separated and retained an operational pattern from the sensorimotor material that provided the occasion for its constitution.” (p. 169). When a cognizing subject reapplies the unitizing operation to results of the unitizing operation (a perceptual item, i.e., an apple), the perceptual apple concept becomes further abstracted in such a way that it becomes the generic attentional pattern of a unit in which attentional pulses become fused (reprocessing a perceptual item). This reprocessing of perceptual items as “a means of taking them together opens the way to focus attention on the unitary wholeness” (Steffe & Olive, 2010, p. 31). This process is a unitizing operation of the sensory-motor items. However, the repetition of the focused attentional patterns that create unitary items is not counting but merely establishes a plurality (von Glasersfeld,

1995).

Counting Schemes and Types of Units

In counting, schemes are composed of the items as follows: (1) The child’s indication of plurality, i.e., recognition of countable items that was constructed by the unitizing operation and her intention to count, (2) the child’s counting of countable items, and (3) the result or the anticipation of counting. Steffe (1991) considered counting to be a quantitative scheme, because it measures quantity, which can lead to the development of numerosity. In addition, Steffe (1991)

46 introduced different types of units that children produce by using the unitizing operation: perceptive, figurative and abstract unit items. Only when children constructed abstract units did

Steffe refer to them as being numerical; children who were limited to counting perceptive and figurative items were referred to as being pre-numerical. Steffe (2012) emphasizes the significance of the construction of units, describing it is crucial in the construction of measuring schemes as well as in the process which “begins in infanthood the first time a baby recognize an object that it has experienced before, and it continues on throughout childhood and beyond” (p.

13).

Bounded Plurality and Countable Items. According to Steffe and Cobb (1988), pluralities are constructed by the repeated use of an object concept. Suppose there is a multiple number of cups in a child’s visual field and the child has constructed a “cup” concept as a permanent object. After he made an association of his perceptual signals to the word “cup,” he might look around and find other items in his visual field. Using his recognition template, the child might assimilate another cup to the “cup” concept. This repeated use of object concept can lead to an indication of plurality if the child keeps track of these cups in repetition. However, the indication of plurality does not automatically lead to coordinating number words with empirically abstracted items. Consider walking. When walking, one might know that one moves his leg more than once, but one does not know how many times one moves his legs unless one intentionally focuses on when to start and to end counting his steps. Thus, plurality must be bounded to be countable, and the counting action is triggered when a cognizing subject is aware of plurality, or of more than one item and the goal is to make definite what is indefinite. When a child uses an object concept repeatedly in either recognition or re-presentation, then the child creates perceptual or figurative plurality. When the perceptual or figurative plurality is bounded

(when the items become countable items to the child), the child might want to count them if she is aware of the bounded plurality and wants to specify it.

Perceptual Unit Items. A child can merely say “cup” four times when she sees four cups. To count, one must be able to coordinate the number words and counting actions, such as pointing actions, and must have the intention to count. When a child makes a pseudo-empirical abstraction of the unitary wholeness of perceptual items using the unitizing operations rather than focusing on particular sensory-motor records in the recognition template, Steffe (2012) regards the child as constructing perceptual unit items. If a child uses this perceptual unit item to re-process the collection of perceptual items, she generates a collection of perceptual unit items that Steffe (2012) and Steffe and Olive (2010) call a perceptual LOT. When a perceptual LOT is activated, a child can be aware of more than one perceptual unit item. Steffe refers this awareness to an awareness of perceptual plurality and considers it to be a quantitative property of a perceptual LOT.

I regard an awareness of perceptual plurality as a “quantitative” property of a perceptual LOT. It is what permits children who are restricted to establishing the perceptual unit items of a perceptual LOT as countable items to engage in purposeful counting activity in that it is their goal to make definite their sense of indefiniteness induced by their awareness of more than one perceptual unit item. (p. 18)

As the child counts, the assumption is that the child creates perceptual unit items and coordinates each with pointing acts with number words, which is an act of counting. However, the ability to count perceptual unit items does not necessarily imply the ability to re-present the unit items in their visualized imagination. The child who did not further internalize her perceptual unit items at the level of re-presentation cannot count hidden items. In fact, Brenda, one of the participants in Steffe and Cobb’s (1988) teaching experiments, could not count hidden marbles. She tried to uncover the screen and counted exposed items only. Steffe (2012) considers

48 that these children’s perceptual LOTS are yet to be externalized, i.e., not permanent objects.

Then, what operations are involved when re-presenting a perceptual LOT?

Figurative Unit Items. A single perceptual item can be a permanent object when a child can visualize the hidden item in her imagination, which Steffe (2012) calls a figurative item. By repeatedly re-presenting a figurative item, a child produces a collection of figurative items. Then the child may focus on the recurrence of the figurative item, which results in the construction of a figurative unit item. Sequentially re-presenting figurative items and unitizing these figurative items results in the construction of a figurative LOT. When it is activated, a child is aware of figurative plurality, which triggers counting actions when a child wants to specify the indefiniteness of the bounded plurality. Steffe calls this an awareness of the quantitative property of a figurative LOT. For example, Susan, one of the participants of Steffe and von Glasersfeld’s teaching experiments, provides a good example of her figurative unit items. When asked to count

12 checkers (the last seven checkers were hidden by a screen) Susan counted five visible counters in a row. She then continued to count 6, 7, 8, 9, 10, 11, fixing her gaze on the screen and pointing to the probable locations of each checker successively. She stopped counting when her imaginary checkers filled up the screen. Susan re-presented the checkers on the screen and counted the figurative unit items until her counting acts exhausted the space of the screen. A stronger indication of a figurative LOT can be found when a child put up fingers as substitutes for the hidden checkers as in what immediately follows. Hence, the abstraction that generates figurative LOTs is pseudo-empirical, because children who were limited to counting figurative items require perceptual inputs, such as pointing actions, and they need to start counting from one.

Motor unit items. When a child substitutes motor acts for figurative items, one can say

49 that the child has counted motor unit items. According to Steffe and von Glasersfeld (1988), the

“substitution provides the child, for the first time, with complete independence from its immediate perceptual world when he or she counts, because motor acts can be produced at will”

(p. 5). The motor unit items could be exemplified by Tyrone, one of the participants of the teaching experiments. When asked to count checkers where eight checkers and three checkers were hidden underneath clothes separately, Tyrone counted 1, 2, 3, 4, 5, 6, 7, and 8 pointing over the first cloth. He then continued 9, 10, 11 accompanied by his pointing activity. While counting,

Tyrone frequently looked away from the cloth, and he completed his counting with his rhythmic patterns. Thus, unlike Susan who counted figurative unit items that were bounded by the screen, one can infer that Tyrone counted his motor acts.

Verbal unit items. If a child internalized his sensory-motor activity in such a way that the utterance itself signifies unit items, then one can infer that the child counts verbal unit items.

Brenda, one of the participants in Steffe’s (1988) teaching experiments, was asked to "do fifteen take away three" (p. 104). She uttered 1 through 15 and opened both hands. Then she uttered 1,

2, 3 folding three fingers and said “twelve” right after she uttered 3. Steffe inferred that Brenda’s utterances referred to her abbreviated finger pattern for 15, which can serve as an indication that her utterance was the substitute for the figurative items.

Perceptual, figurative, motor, and verbal unit items are not abstract unit items, because they all require some sort of sensory-motor input or some substitute for sensory-motor input to count. Steffe and Cobb (1988) claim these four unit items are pre-numerical because of the dependency on sensory-motor inputs. To construct the abstract unit item, which does not require sensory-motor input while counting, one needs to recursively apply the unitizing operation and reflectively abstract the unitary wholeness so that sensory contents from the focused attentional

50 patterns are stripped away, i.e., the concept of one becomes a place holder of unitary oneness constructing an abstract LOT (Steffe & Olive, 2010). When a child can count-on without substituting sensory-motor material for countable items, Steffe and Cobb inferred the child constructed abstract units.

Number Sequences and the Units-Coordinating Schemes

Abstract Units and an Initial Number Sequence

Abstract units are “interiorized attentional patterns that contain records of counting acts”

(Steffe & Olive, 2010). A child with abstract units can count on, because the number word contains the record of counting and the child does not need to activate counting. Steffe (2012) explains how Tyrone, one of the participants of Steffe and Cobb’s (1988) teaching experiments, constructed abstract units. According to Steffe (2012), Tyrone did establish figurative number patterns for the number words two, three and four, but he did not have a figurative number pattern for five to keep track of counting. After showing him two sets of counters (seven counters and five counters) and then covering them with cloths, Tyrone was asked to find how many counters were under the cloths. Tyrone first counted seven counters, pointing seven times to the cloth covering the seven counters. He then counted from eight to thirteen with his pointing actions. Then, he self-corrected himself and recounted from eight to twelve. Steffe explains that

Tyrone’s self-correction (self-regulation) and monitoring of his counting actions serve as an indication of the operations that generate numbers.

To monitor one’s counting acts, Steffe and Cobb (1988) explain that Tyrone might have interiorized his counting acts by re-proessing his internalized counting acts (mental representation of his figurative units), which is an act of reflective abstraction (von Glasersfeld,

1995; Steffe & Cobb, 1988; Olive, 2000). Olive (2000), Steffe and Cobb (1988) considered that a counting activity can be internalized when a child mentally re-presents his or her counting acts, e.g., Tyrone’s counting from eight to thirteen. Steffe and Cobb (1988) further explain that

Tyrone’s re-processing his counting acts led him to the interiorization of his counting acts so that the contextual details of his internalized counting acts were stripped off. Unitizing each represented counter strips the sensory quality of the counter, which results in producing abstract units containing the sensory records of the counters (Steffe, 1992). Steffe (1992) called the sequence of abstract slots as numerical patterns. In addition, when a child takes the counted blocks to be counted and “the abstract unit items also contain record of counting” (p. 266), he called the sequence of the abstract unit items ‘a numerical composite’” (p. 266) and considered the construction of a numerical composite as the beginning of the construction of the Initial

Number Sequence (hereinafter, INS): “a sequence of arithmetical units containing records of counting acts” (Steffe & Olive, 2010, pp. 36-37).

Steffe further explains that Tyrone generated numbers by reorganizing his previous counting scheme: “to monitor counting activity, there must be an explicit awareness of the material in re-presentation, an awareness that is indicated by ‘reflection on Internalized Counting

Acts’” (Steffe & Cobb, p. 310). In other words, Tyrone’s monitoring his counting acts suggests that his counting scheme was “a self-referencing scheme” (Steffe, 2012, p. 22), which allows him to use the scheme recursively so that “he could take records of counting acts as countable items” (Steffe, 2012, p. 12). Furthermore, Tyrone’s self correction indicates that his scheme was an auto-regulated process, which generates INS (Steffe & Olive, 2010). When a child assimilates a situation using his or her INS, he or she can “regenerate an experience of the included number word sequence” (Steffe, 1992, p. 263). For example, the number word “five” contains the record

52 of counting, and the child can count on without activating counting from one to five. However, the number word is yet to be symbolized as an entity for a child who is limited to INS.

Children counting with different units produce distinct results, which can be categorized into two types of numerosity: (1) proto numerosity and (2) numerosity. Steffe and von

Glasersfeld (1988) referred to the result of the counting acts of four pre-numerical units as proto numerosity. A child who has not constructed abstract units can count the collection to form its proto numerosity. For example, one can count one’s unit item with repeated instantiation of the act of counting until he or she exhausts the given units while keeping track of the repeated action. On the other hand, the result of counting abstract unit items contains numerical structure; the number word “eight” means a collection of eight unit items that involves interiorized counting actions (the number words are operative). Numerosity of a collection is finally established as a result of counting abstract unit items; extensive quantity is generated by counting abstract unit items. With INS, a child can count-on because number words point to the record of counting actions, i.e., INS is permanent. In addition, the counting actions result in the construction of numerical composites.

Composite Units and A Tacitly Nested Number Sequence

By uniting abstract units, a child with a Tacitly Nested Number Sequence (hereinafter,

TNS) can create a composite unit and take the counted items as countable items: “the TNS is recursive in that the results of counting can be reconstituted as a “situation” of counting, whereas the INS is a “one-way” scheme” (Steffe, 1992, p. 278). Thus, a child with TNS can double count or counts their counting acts. Another distinction between the INS and the TNS resides in the units-coordinating schemes. A child with INS can attempt to coordinate numerical composites to unit items in activity by sequentially uniting figurative composite units. However, the child who

53 is limited to INS cannot take the counted items as an input. For example, an INS child may count by twos, but he or she cannot keep track of how many twos they counted because the child cannot consider the “two” as a countable item (Olive, 2000). On the contrary, a child with TNS can coordinate composite units to unit items in activity. When asked how many strings of four beads they could make out of 24 beads, a TNS child might count “1, 2, 3, 4; one; 5, 6, 7, 8; two;

… 21,22, 23, 24; six,” while keeping track of counting fours. Olive (2000) explains that this monitoring of composite units involves the two levels of units coordinating scheme, which produces a numerical composite of composite units.

The Two Levels of Units-Coordinating Scheme

According to Steffe and Olive (2010), a units-coordinating scheme is “a multiplication scheme that gets its name from the coordination of, to the observer, two composite units of units where one composite unit is inserted into each unit item of the other composite unit” (p. 91).

Coordinating two levels of units refers to coordinating “unit items and composite units” (Olive,

2000, p. 7). For example, to an observer, children who have a two levels of units-coordinating scheme can review their number concept of five as five ones as well as one five and can project it into the elements of another unit. When children project a composite unit into another unit, they can treat a composite unit as a single item and iterate it as if they were iterating a unit item in order to produce a numerical composite. However, composite units are not yet iterable for children who are limited to coordinating two levels of units. From an observer’s point of view, a child who can coordinate two levels of units can produce a composite unit containing composite units (i.e., three levels of units) in activity. However, children with a two levels of units- coordinating scheme do not necessarily have three levels of units as given before they carry out iterating operations (Steffe & Olive, 2010). For example, children who can iterate their

54 composite unit of five three times to produce fifteen do not necessarily need to know that fifteen is composed of three composite units, each of which contains five single units before their iterative actions. In other words, to be regarded as being able to coordinate two levels of units from an observer's perspective, producing three levels of units in activity (Hackenberg, 2012) is required. However, taking three levels of units as given and using this for further operations is not necessary.

An Iterable Unit of One and an Explicitly Nested Number Sequence

Although a child with TNS can produce a numerical composite of composite units, the coordination of composite units only takes place in activity. When asked how many strings of four beads they could make out of 24 beads, a TNS child will count by 4, keeping track of counting fours. However, if a child was asked to find how many beads were in each of six strings, he or she may not be able to find the answer because he cannot take a composite unit, six, as given and project it into 24 (Olive, 2000). Steffe (1992) explains that one of the characteristics of this non-reversible many-to-one scheme was the “lack of inversion between

“situation” and result” (p. 287). The counting-by actions were not interiorized in such a way that the result of counting could be taken as an input to review it. Steffe and Olive (2010) further explain that, for a child who is limited to TNS, a composite unit is yet to be abstracted in such a way that the units in the composite unit can be withdrawn leaving empty slots that can be filled by iterating any one of the units in the composite unit. In other words, an iterable unit of one is yet to be constructed.

A child who constructed an iterable unit of one knows that “five” is not only five “ones” but also any one of those five “ones” can be iterated to form a composite unit of “five” (Steffe &

Olive, 2010). Steffe and Olive (2010) further explain that disembedding a part from a whole

55 without destroying the whole is fundamental in the construction of an iterable unit of one. For example, when asked to find the missing addend in the case of 27 + ⬜= 36, Jason, one of the participants of their teaching experiments, answered nine: “‘twenty-seven plus seven—it’s nine more!’” (p. 41). They analyze that Jason’s ability to estimate the missing addend and to adjust it can serve as an indication that he could treat the missing addend not only as a part contained in the whole, 36, but also as “a unit structure apart from 36” (p. 41). They further analyze that the construction of an iterable unit of one enabled Jason to disembed parts from the whole and to treat the composite units as if they were single units. They call the number sequence with which

Jason assimilated to 36 an Explicitly-Nested Number Sequence (hereinafter, ENS).

A child with ENS can construct a multiplicative scheme involving two levels of units

(Olive, 2000) and produce a composite unit of composite units. For example, a child with ENS can consider 30 as a composite unit containing six groups of five, i.e., three levels of units, but they may not take the three levels of unit structure as an input for further operations (Olive,

2000). Steffe and Olive (2010) explain that a child who is limited to ENS treats composite units as a TNS child treats abstract units, and iterable composite units are yet to be constructed.

Iterable Composite Units and a Generalized Number Sequence

When a composite unit is further re-interiorized and it becomes iterative, it allows for (1) coordinating composite units side by side, and (2) taking a unit of units of units as given and unpacking it into the composite units. Steffe and Olive (2010) claim that the operations are available to a child with GNS.

When asked to make 24 toys using the copies of 3-string and 4-string, Nathan, one of the participants of Olive and Steffe’s teaching experiment (2010), found that four threes and three fours made up 24. Nathan explained that three and four were seven, and three sevens made up

21. Instead of counting threes and fours separately, Nathan united two composite units of four and three into a unit of seven and established a unit of seven as a unit of composite units of four and three. Not only could Nathan take the result of his operations and decompose it into the composite units but also he could coordinate two number sequences (four and three composite units) side by side. Olive and Steffe (2010) infer that Nathan had constructed a generalized number sequence.

The Three Levels of Units-Coordinating Scheme

Coordinating three levels of units refers to coordinations that involve different levels of the elements of a unit of units of units or a composite unit of composite units (Steffe & Olive,

2010). When children coordinate two levels of units, they can produce three levels of units by iterating a composite unit. However, the ability to produce three levels of units by iterating a composite unit does not necessarily imply an ability to take three levels of units as given and to recursively use them as inputs for further operations. Following Steffe and Olive’s (2010) notion of units-coordination, I regard coordinating three levels of units as a scheme involving iterable composite units.

Iterable composite units require the re-interiorization of composite units (Olive, 2000).

When re-interiorized, a composite unit is abstracted in such a way that children can take and use it as if it is an iterable unit of one. When children construct iterable composite units, they can construct a composite unit of composite units as well, similarly to the way that they constructed a composite unit with the iterable unit of one. The construction of a composite unit of composite units enable children to treat it as one thing with awareness of its three levels of units. In other words, the children can take their three levels of units as given prior to carrying out iterating actions. Hence, children with a three levels of units-coordinating scheme can insert composite

57 units into each unit of a composite unit establishing a composite unit of composite units (three levels of units). They can also take their construction of three-level units as given and recursively use it for further operations.

Operations, Adding and Subtracting Strategies of the Counting Schemes and the

Units-Coordinating Schemes

In the following sections, I explain the strategies that the children, the participants in

Steffe and Cobb’s (1988) teaching experiments, used when finding sums and differences and mental operations that were available to them. Then I discuss the notion of reversibility and the reversibility of the disembedding operation.

Strategies of Children with INS: Increasing or Decreasing by One

Children with INS can add/subtract by increasing/decreasing by one (Steffe & Cobb,

1988). What they cannot do is to find the sum/difference by increasing/decreasing addends/minuends by more than one, because they cannot reprocess the numerical composite as a given and proceed with further operations. In addition, the children develop their subtraction strategy independently of their addition strategy because they do not view subtraction as the inversion of addition (Steffe & Cobb, 1988). To indicate that 13-6=7 implies 7+6=13, a child needs to treat six and seven as two entities composing the whole, 13.

Strategies of Children using the Two Levels of Units-Coordinating Scheme

When children interiorized their numerical composite and can take it as an input, i.e,. as a composite unit, they can start to coordinate composite units with numerical composite. The children can now add or subtract by recursively increasing or decreasing one unit of its addends, a minuend or a subtrahend. For example, knowing 31+6=37, children can effortlessly find the sums of 31+7=38, 31+9=40, etc. (Steffe & Cobb, 1988).

Compensation Strategy of Children with TNS

A child in this stage can use a compensation strategy to find the sum or difference of two numbers. For example, he or she can use 12+8=20 to solve 13+7 explaining that he or she has to take one from eight and give it to the 12. Transferring one unit from one to the other indicates that individual unit items were “the objects of reflection” (Cobb & Steffe, 1988, p. 261). In other words, the number word 12 implies a number sequence that could be unpacked into twelve individual abstract units. Thus, a child can see the increase of one and the decrease of one compensate each other.

Part-Whole Operation of Children with ENS: Addition and Subtractions as Inverse

Operations

According to Steffe and Cobb (1988), when asked to find what 22-17 was, Tyrone found

5 by counting 5 down to 17 from 22. However, when asked whether he can solve the given problem by counting forward, Tyrone shook his head. To be able to know that counting 5 down to 17 is equal to counting 5 up from 17, Tyrone should have treated the result of his counting- down-to operation, 5, as given and considered the given minuend, 22, as the result of his operation. Tyrone’s mental operation was not reversible, i.e. he could not proceed to further operations to find the given (the minuend, i.e. 22) with the result of his operation, 5, as an input.

Children who assimilate with part-whole operation in their counting can use composite units as given and proceed to further counting. To take composite units as given, one must re- interiorize composite units. A crucial result of re-interiorization of composite units is the construction of iterable units, which is fundamental when disembedding numerical parts from a numerical whole (Steffe & Cobb, 1988). With the part-whole operation, “both numerical parts and the numerical whole were ‘out there’ for the children” (Steffe & Cobb, 1988, p. 271);

59 children can anticipate the sums or differences and can consider addition and subtraction as inverse operations.

Operative Addend/Minuend-Increasing/Decreasing Strategy

With the part-whole operation, a child can find sums and differences by adjusting the addends/minuend and the sums/differences simultaneously. Tyrone could find sums and differences without carrying out any counting activity but by using the result of his prior operations in the later part of Steffe and Cobb’s (1988) teaching experiments. When asked to find 21+23, Tyrone immediately found the sum, 44, saying that 23+23=46 and he took two away. Tyrone did not carry out his counting activity, and he adjusted the addend and the sum simultaneously. Steffe and Cobb (1988) considered Tyrone’s adjustment of the addend and the sum to be operative, because Tyrone’s way did not require the carrying out of any counting activity.

The Notion of Reversibility and Reversible Reasoning

Inhelder and Piaget (1958) define reversibility as “the permanent possibility of returning to the starting point of the operation in question” (p. 272). Steffe (1994) explained the reversibility of the units-coordinating schemes as “the process of constructing a system of operations that would feed the results of using units-coordinating scheme back into its

“situation”” (p. 28). Extending Inhelder and Piaget’s (1958) and Steffe’s (1994) notion of reversibility, I claim that a child is engaged in reversible reasoning when she uses either the result of a scheme or the anticipation of the scheme as an input and recursively uses the scheme to return to its starting point by reconstituting the given situation.

Reversibility of the Disembedding Operation

The disembedding operation refers to a mental operation involving an ability to disembed a part from a whole without destroying the whole (Steffe & Olive, 2010). For example, when adding 5 and 17, children with the disembedding operation can envision 22 as a composite unit including two composite units, 5 and 17. By being explicitly aware of the inclusion relations among the composite units, one can disembed 5 while focusing on the remainder part, 17. This powerful operation enables a child to keep track of the given parts that are explicitly nested in the whole and to engage in reversible reasoning. This is so because in order to disembed a part from a whole, one must reconstitute the whole with the parts that can be disembedded and must take the whole as an input. This also enables a child to construct subtraction as an inverse operation of addition. Constructing additive structures by reflectively abstracting the addends enables one to reconstitute the given situation using composite units and review it as if it were out there, i.e. reconstituting the re-presentation of the given.

Extension of the Counting Schemes and the Units-Coordinating Schemes

Although essential, the units-coordinating schemes and the counting schemes alone are not sufficient to make a hypothetical model of how one constructs x+y=a, because, from my perspective, the construction of x+y=a requires (1) the ability to find one quantity in relation to the other quantity, (2) the construction of the relationship between x and y, (3) reasoning with signed quantities, and (4) the ability to reorganize one’s counting schemes in the contexts of continuous quantity. In the following sections, I first introduce the operations that generate continuous quantity (Steffe, 1991). Then I elaborate how one reasons with signed quantities using one’s units-coordinating schemes and counting schemes (Ulrich, 2012). Finally, I introduce the notion of quantitative unknowns and variables (Steffe et al., 2014) and how one specifies a quantitative unknown in relation to other quantities as well as how one finds the

61 relationship between two quantitative unknowns (Hackenberg, 2017).

Operations that Generate Continuous Quantity

While the involved experiences may be different, Steffe (1991), from a perspective that

“counting is a special form of measuring,” claims that “the operations that generate a measurable quantity are not different from the operations that generate a countable quantity” (p. 77). He used telephone poles as an example of how a child might construct a continuous but segmented quantity. After a child constructs an object concept of telephone poles and takes the set of the telephone poles as a figurative collection, he or she might focus on scanning action between the telephone poles and construct perceptual rows. At this point, the child’s focused attention shifts from telephone poles to the interval between two telephone poles. The child constructs “a figurative row of intervals” (p. 78) when he or she re-presents one unfocused moment (a telephone pole) between the two focused moments (two intervals). By re-processing this figurative row of intervals, Steffe considers that the child may construct “an abstract row of intervals” (p. 78), which enables him or her to re-present “a continuous but segmented unit” (p.

78). Steffe further explains that these abstract rows of intervals can serve as “a conceptual ruler”

(p. 78) when a child regenerates the scanning experience and uses it to make an indefinite length definite. For example, Martin, a 5-year-old participant in Steffe and von Glasersfeld’s teaching experiment, used his counting strategy when he was asked to measure the total length of two objects, a 6-inch-long ribbon and a 12-inch-long straw: He first measured each object using a ruler marked in inches and counted six and twelve while putting up his fingers. He then counted twelve and afterwards counted from thirteen to eighteen (Steffe, 1991). Steffe analyzes that

Martin’s self-initiated measuring acts were an indication that he assimilated the given objects

“using an abstract concept of a row of intervals—a ‘ruler’” (p. 79). Steffe continues to explain that Martin’s counting acts suggest that he reorganized the length of the given items as countable items. Steffe and Olive (2010) also explain how children develop fraction schemes by reorganizing their counting schemes. Following Steffe (1991), I hypothesize that one can construct a sum of two continuous quantities by reorganizing their counting schemes and the units-coordinating schemes.

Reasoning with Signed Quantities and Units-Coordinating Schemes

Bishop et al. (2014) explain how the development of the concept of integers was challenging in the history of mathematics, and they analyzed difficulties and affordances when reasoning with signed quantities. Due to the lack of understanding 0 as a reference but not as an absolute zero, even De Morgan in the 20th century disregarded the case when the magnitude of a subtrahend is bigger than that of a minuend, according to Bishop et al. (2014). They continued that studies regarding students’ integer understanding is sparse while the concept of integers has been challenging to many mathematicians in history as well as to students; they found similar difficulties in the students’ understanding of integers that are not unlike those of past mathematicians, such as addition cannot make the sum smaller, a subtrahend cannot have bigger magnitude than that of a minuend, and understanding zero as an absolute zero.

Thompson and Dreyfus (1988) reported a way to reason with integers as transformation.

Although there are some textbooks using arrows for integer addition, they considered the arrows to be pointers of specific locations of the numeric values, not as directed segments entailing transformations. Based on Vergnaud’s and Janvier’s studies about integer reasoning, they claimed that understanding integers as transformation is beneficial for children when solving

63 more complicated integer problems. In addition, the transformation approach opens a path for children to reason with continuous quantities when engaging in adding and subtracting integers.

Ulrich (2012) analyzed four students’ (Adam, Michelle, Justin, and Lily) ways of reasoning with signed quantities in accordance with their units-coordinating schemes and number sequences. She found that establishing inclusion relations among positive quantities was essential when constructing additive relationships with signed quantities. Adam, one of the participants of her study with TNS, took the first addend as a starting point of his counting actions and assimilated the second addend with his counting actions. It was possible for him to take the second addend as a transformation, but his counting was experientially bounded. He could not reflectively abstract his counting action to review the sum as a combination of two composite units. For example, he did not recognize the two missing addend situations that can be modeled as 35+__=51 and 16+__=51 to be the same (Ulrich, 2012). Another indication was found when he was asked to find the missing addends that can be modeled as “+52+K=-12” (p.

250). Adam answered that there would be a decrease of 40 because “52 minus 40 is 12” (p. 252).

Ulrich’s finding is consistent with Steffe and Cobb’s (1988) analysis: ENS is essential in order to consider subtraction as an inverse function to addition.

Ulrich continued to explain that a child with ENS could use an additive inverse when adding two oppositely oriented quantities in activity. Michelle, who could coordinate two levels of units, could find the sum of two positive quantities by simultaneously adjusting the addends and the sum, which is available for a child with ENS. Given the elevation situation where she descended 137m and then ascended 90 m, she was asked to find her location in relation to where she started to move. She found that it would be below 47 from the reference and explained that she kept adding 10 nine times until she reached 90 and then found 40 and seven more. However,

Ulrich hypothesized that Michelle was yet to reflectively abstract -137 as a combination of the inverse of 90 and -47, because Michelle was looking at the diagram when she answered the question. In a later teaching episode, Michelle found (+10)+(-24)=-34 while conflating the addend and the sum, which can serve as a corroboration to her hypothesis. Ulrich considered that

Michelle’s solution did not stem from her misunderstanding of the given situation because

Michelle was engaging in solving several similar problems that day. Rather, she analyzed that

Michelle was lacking “reasoning though 0,” i.e. “using additive inverse of the first addend to construct the sum as an explicit additive comparison with the second addend” (p. 255). She continued that the disembedding operation is essential in counting beyond 0, and coordinating with three levels of units is essential when modeling an additive situation with signed quantities.

For example, when solving a problem that can be modeled as 9+(-13), one might need to construct -13 as a composite unit containing the additive inverse of 9 and the resulting quantity,

-4, while assimilating 13 to a unit containing three levels of units. She compared the operations involving the construction of improper fractions; they are similar to the operations when counting down from zero, because in both cases, one needs to go beyond the original whole while monitoring it, which involves assimilating to a unit containing three levels of units.

Quantitative Unknowns and Variables

Steffe et al. (2014) define quantitative unknowns as “the potential result of measuring a fixed but unknown extensive quantity before actually measuring it” (p. 50), and quantitative variable as “the potential result of measuring a varying but unknown extensive quantity at any but no particular time” (p. 51). Because the units-coordinating schemes and the counting schemes are triggered by the goal of making indefinite numerosity (or magnitude) definite, I consider the recognition of a certain situation with the goal of specifying its magnitude as the

65 awareness of quantitative unknowns.

While one can find a quantitative unknown as a result of one’s activity such as counting or measuring, there are some situations where one must take a quantitative unknown as given and carry out further operations. Consider the following situation: When the given ribbon is your friend’s ribbon that is five times as long as yours, cut off the length of your ribbon (Steffe &

Olive, 2010). To cut off the length of the ribbon, one must posit a hypothetical ribbon in one’s mind, iterate it five times, and disembed it. In other words, one must take a quantitative unknown as given and carry out further operations. Steffe and Olive explain that the ability to posit a hypothetical stick requires the awareness of the multiplicative relationship between the part and the whole, which requires coordinating two levels of units. This is consistent with the claims of

Hackenberg et al. (2017) that taking two levels of units is essential when unitizing quantitative unknowns.

In addition, Hackenberg et al. (2017) studied 21 middle school students (nine students could take three levels of units as given, and 12 students could take two levels of units as given) and found that taking three levels of units is essential when constructing a multiplicative relationship between two quantitative unknowns. For example, when asked to find the relationship between the height of two crops, where the height of one crop is five times the height of the other, Tim, one of their participants, said he could only get the approximation, because he did not have the exact height of the shorter crop. Other participants who are also limited to taking two levels of units as given were restricted to finding the height of the other crops by either experientially iterating the height of the shorter crop or substituting specific numeric values for the height of the shorter crop.

The Hypotheses Concerning the Construction of x+y=a

I hypothesize that (1) children construct x+y=a by reorganizing their units-coordinating schemes, (2) reasoning with different levels of units engenders different constructions of x+y=a, and (3) the reversible reasoning is essential when constructing x+y=a in order to construct the counterbalancing relations between two signed quantities and to make a continuous line as opposed to a set of points. In my perspective, the units-coordinating schemes are essential in the construction of the sum of two extensive quantities, because the units-coordinating schemes are used to find the sum or missing addends. Based on Steffe and Olive’s (2010) reorganization hypothesis, I will analyze how my students used their units-coordinating scheme in the context of continuous quantity. I also hypothesize that reasoning with different levels of units engenders different constructions of x+y=a, based on Ulrich’s (2012) studies. According to Ulrich, taking a unit containing three levels of units as given is essential when reviewing the sum of oppositely signed quantities as a unit containing two composite units. When the magnitude of the second addend is greater than the first addend, children who were limited to coordinating two levels of units were not able to review 0 as a unit containing the first addend and the part of the second addend that acts like the inverse of the first addend. Thus, I hypothesize that a child who reasons with the two levels of units-coordinating scheme will find y by keeping track of their counting acts or using the “take-away” method. As a result, I anticipate that the child cannot reflectively abstract the counterbalancing relation between x and y. In contrast, I hypothesize that taking a unit containing three levels of units as given is essential when reviewing the sum a as a unit containing two composite units as well as reflectively abstracting the counterbalancing relation between x and y. The study by Hackenberg et al. (2017) that showed that taking two levels of units as given is required when taking quantitative unknowns as material for further operations

67 and taking three levels of units as given is essential when constructing the multiplicative relationship between two quantitative unknowns supports my hypothesis.

Finally, I hypothesize that reversible reasoning as well as the third kind of anticipation is necessary to construct x+y=a as a line as opposed to a set of points. I believe that the reversible reasoning is involved in (1) disembedding the addends in a sum, (2) unpacking a quantity into a combination of a previous quantity and the amount of change to find the counterbalancing relationship, and (2) verifying the hypothesis (counterbalancing relationship) that the child generated.

Counterbalancing Relations in x+y=a

I define a counterbalancing relation between x and y in x+y=a as a relation between two quantities where, if one quantity increases, the other decreases by the same amount (and vice versa) in order to maintain a constant sum. If one knows that the sum a is constant, he or she may have a sense of the relationship between the two quantities (i.e. he or she will be able to intuitively predict the result of the operation, with or without particular given quantities). Then, one might notice that two quantities vary in tandem (i.e. one quantity must counterbalance the other quantity to add up to a). To construct the counterbalancing relation of x and y in x+y=a, from my perspective, one must have re-presentation of x+y=a as a unit containing three levels of units and be able to disembed quantities to compare an increase in one quantity and a decrease in the other. To be able to understand the amount of increase compensating the amount of decrease, my hypothesis is that a child needs to (1) vary one variable, say from x to x+t, (2) find the amount of changes in y in accordance with the change in x, i.e. y+(-t) (3) unpack the changed variables x and y in such a way that the changed values of the variables comprise the previous values and their changes, and (4) have the image of the amount of changes that compensate each

68 other. I argue that disembedding the addends from the sums as well as unpacking the changed values of the variables into two parts, the previous value of the variable and its change, requires reversible reasoning.

In addition, I believe that the third kind of anticipation involves reversible reasoning when constructing x+y=a as a continuous line as opposed to a set of discrete points. If a child can only engage in the second kind of anticipation, he or she may find the number pairs that satisfy the given condition by discretely varying values of x and y. However, I argue that the child can not indicate that the set of points composes a line, because the result of his or her operations is bounded by his or her immediate empirical experience. The child might predict several points based on the observation of the sequential changes in the values of each variable, but he or she will not predict all possible cases. If a child could engage in the third kind of anticipation, I believe that the child can establish the counterbalancing relation between two quantities by unpacking the sum into a combination of two quantities including the changes in each quantity. The child then formulates the hypothetical anticipation that no matter how the quantities vary, the counterbalance relationships between two quantities maintain the sum of two quantities constant. This is consistent with the finding of Inhelder and Piaget (1958) that making possibilities an empirical reality involves reversible reasoning.

CHAPTER 4

METHODOLOGY

Constructivist Teaching Experiment

I chose the constructivist teaching experiment to explore and explain my research questions based on my theoretical constructs. The constructivist teaching experiment enables a teacher-researcher to explore and explain students’ mathematical activity over extended time periods (Steffe & Thompson, 2000). It is a methodology derived from Piaget’s clinical interview method; both methods are designed to explore and explain students’ mathematical thinking.

While Piaget’s clinical interviews focus on students’ current knowledge and ways of thinking, the constructivist teaching experiment is designed to trace the progress of students’ understanding over an extended period of time. In addition, teaching actually occurs in the form of responsive, intuitive and analytic interactions that are bounded and constrained by students’ language and actions (Steffe & Thompson, 2000).

Some might argue the constructivist teaching experiment is not a scientific methodology by claiming that it neither guarantees replicability nor social aspects (Lerman 1996, 2000).

However, in radical constructivism, interactions are considered to be essential when constructing experiential reality that is viable with others’ experiential reality (von Glasersfeld, 1995). Steffe and Thompson (2000) emphasize that the constraints, such as students’ language, actions, and essential mistakes that teacher-researchers experience, are “a basis for understanding students’ mathematical construction” (p. 267). Social pressure and different concepts of others feed back to the existing scheme causing perturbations, and one makes adaptations to one’s own schemes

70 by reorganizing the existing schemes. The goal of the constructivist teaching experiments is, therefore, constructing viable explanations about one’s actions and mental operations from an observer’s perspective (Steffe & Thompson, 2000).

In the Merriam-Webster Dictionary, “scientific method” is defined as “principles and procedures for the systematic pursuit of knowledge involving the recognition and formulation of a problem, the collection of data through observation and experiment, and the formulation and testing of hypotheses” (2017). To make a viable explanation of one’s ways of constructing certain concepts, the constructivist teaching experiment allows a teacher-researcher to form hypotheses and test one’s hypotheses through observation and teaching. Thus, I conclude that constructivist teaching experiments are a scientific method. In addition, replicability is not realistic, because it is impossible to replicate the same conditions, such as the same students with the same ability, identical interactions and environments. The constructivist teaching experiment, therefore, suggests the reconstitution of previous teaching experiments in a way that fits the given conditions.

The Goal of the Constructivist Teaching Experiment of This Study

The goal of the constructivist teaching experiment is to “bring forth the schemes that students have constructed thought spontaneous development” (Steffe & Thompson, 2000) and to use the model of students’ mathematics to form research hypotheses of the teaching experiment.

According to Steffe and Thompson (2000), spontaneous development consists of the following four factors: (1) ontogeny, (2) physical interactions (3) social interactions and (4) self-regulation.

In the radical constructivist point of view, a cognitive subject constructs and organizes his or her experiential reality in the frame of his or her ways of thinking. Thus, children’s cognitive development is ontogenic, i.e., a result of their cognitive functions. Physical interactions imply

71 any physical activity of children that is observable to the observer, and it is regarded as a reenactment of their re-presentation. Social interactions are important in spontaneous development, because children experience constraints and facilitation of their thinking through social interactions. It is also crucial for a teacher-researcher in modeling children’s mathematics, because exploring children’s interactions exposes how they modify and adjust their ways of thinking through the negotiation (social interaction does not exclude the interaction between the teacher-researcher and the students). Physical interactions and social interaction engender self- regulation. Through physical and social interactions, children modify their scheme and operation to fit in the frame they created as the product of physical and social interactions. Hence, it becomes ontogenetic. Steffe and Thompson (2000) used “spontaneous” to refer to nontraditional teaching and learning actions based on mathematical tasks, children’s self-regulation in interactions and its unpredictability rather than stimuli and reaction.

Through children’s activities and interactions between peers and a teacher-researcher and the students, a teacher-researcher can explore students’ ways of understanding and make and test her hypotheses on the spot. In a teaching experiment, a teacher-researcher actually teaches and researches at the same time. A teacher-researcher teaches the students by enhancing and facilitating the ways students think with prepared semi-structured mathematical tasks, and responsive interactions while making and testing hypotheses either on the spot or retrospectively.

Conceptual analysis

According to Thompson (2000, p. 59), conceptual analysis is “the conjoining of a theory of mathematical understanding and radical constructivism as an epistemology”. Von Glasersfeld

(1995, p. 88) emphasized that, “conceptual analysis pertains to conceptual structure”, which is knowledge of a cognizing subject. When one regards students’ mathematics as legitimate

72 mathematics not unlike ours, analyzing “what mental operation must be carried out to see the presented situation in the particular way one seeing it” (von Glasersfeld, 1995, p.78) becomes important to understand students’ rational ground of their reaction to goal directed mathematical activities. Thus, conceptual analysis of students’ mathematics provides explanatory models of students’ mathematics. I employ this conceptual analysis in this study to model students’ mathematics in the construction of x+y=a for the following purposes: (1) To understand what mental operations and schemes the students utilize, (2) to build up explanatory models of why the students can or cannot make assimilations or accommodations and (3) to form and test hypotheses based on their ways and means of operating when constructing x+y=a.

Exploratory Teaching

Exploratory teaching is an important factor in the constructivist teaching experiments for teacher-researchers to decentralize themselves and focus on ontogenesis of students’ thinking at an experiential level (Steffe & Thompson, 2000). Decentralization implies stepping out of the flow of one’s own thinking and reflecting a model of students’ mathematics on the frame of one’s own ways of thinking. Thus, decentralization requires the ability to reflectively abstract one’s own thinking, re-present it on abstracted students schemes and activities to assimilate students’ mental operations. A teacher-researcher herself also experiences functional and metamorphic accommodations by doing so. Moreover, a teacher-researcher experiences unexpected and surprising responses in the interactions to which she needs to be able to be flexibly responsive. In other words, a teacher-researcher also needs to learn the followings: (1) abstracting her own understanding of mathematics, (2) modeling students’ mathematics, (3) learning strategies to be responsive and flexible in the interactions.

I was engaged in the teaching experiment from the fall of 2012 as a part of the project

“The Pathways to Algebra”. From the fall 2012 to the end of 2013, I served as an observer and participated in data analysis. Through this opportunity, I learned how to construct explanatory model of students’ mathematics and how to plan the tasks to test hypothesis. In the spring of

2014, I became a teacher-researcher of my pilot study and had 16 teaching episodes with Jamie, and 4 episodes with Jack. This exploratory teaching opportunities provided me a more concrete idea about the teaching experiment along with my research questions.

Data Collection

For one school year (from fall semester of 2014 to spring semester of 2015), the constructivist teaching experiment was conducted at a rural high school in a southern eastern states of the United States. 9th grade students who were in on-level mathematics classes and who were interested in working with the research team were invited to the presentation informing the purpose of the study including time and location. Among volunteered students, I conducted initial interviews to choose two students reasoning with different levels of units. The teaching experiments were conducted during the advisement time of the high school, which is 30 minutes after the first period on Mondays and Fridays to avoid a conflict with the students’ schedule. 33 teaching episodes each of which lasted approximately 25 minutes were collected.

Carl, who reasoned with two levels of units, and Maggie, who reasoned with three levels of units, were selected based on the initial interviews. For the retrospective analysis, I videotaped each teaching episode and initial interviews using multiple cameras depending on the tasks. For initial interviews, I videotaped them using two cameras: one for the interaction between a student and the teacher-researcher and the other for focusing on the student’s actions including hand- writing. For the teaching episodes, one camera recorded the interaction between the teacher- researcher and two students. When the tasks required using computer programs, the screen of

74 each computer was recorded in addition to the camera for the interactions. When the task is composed of hands-on activities, two additional cameras recorded students’ actions, in addition to the camera for the interactions. Pens and paper were provided for the students’ convenience, and students’ memos were collected.

Meetings (one professor and four graduate students) were held every week to analyze the previous data and determine preceding tasks. The tasks were semi-structured so that I, the teacher-researcher, could make hypotheses on the spot and test them if necessary.

Initial Interviews

Nine students volunteered and two initial interviews, each of which lasted about 25 minutes, were conducted for each student in order to find two students reasoning with different levels of units: two levels of units and three levels of units. The tasks of the initial interview are composed of hands-on activities to determine students’ levels of units. The initial interviews were semi-structured so that the teacher-researcher can modify questions in accordance with the students’ different ways of operating. For example, if a student is considered to reason with two levels of units and could not engage a three levels of units task, the teacher-researcher asks more questions related to the ways he or she operates rather than proceeding to other tasks that are related to three levels of units. The students were encouraged to anticipate the result before they actually carry out certain activities and explain the reason of certain activities. Using hands-on materials will be encouraged to confirm or modify their anticipation.

CHAPTER 5

INITIAL INTERVIEWS

The goal of the initial interviews was to make hypotheses of my students’ units- coordinating schemes. Two initial interviews, each of which lasted approximately 25 to 30 minutes, were conducted with each student, Carl and Maggie. These interviews were semi structured (Steffe & Thompson, 2000) so that I could be more responsive to the students’ language and actions and to provide the students with the tasks to test my hypothesis of their schemes and operations on the spot. It is crucial for a teacher-researcher to understand that initial interviews might not provide solid indications of the students’ schemes, because the teacher- researcher’s interaction with the children may not be enough to fully understand what their actions and language imply. Through initial interviews, a teacher/researcher makes hypotheses of their students’ schemes and operations, and they need to seek corroborations of their hypotheses in later teaching episodes.

Dr. Steffe and four graduate students participated in making tasks for the initial interviews. To investigate the students’ units-coordinating schemes, strips in various lengths, such as wikistiks, fur sticks, and long paper strips were provided, where they were referred to candies or cakes in the given situations. Scissors and pens were also provided to the students for them to freely mark or cut the given strings, sticks, or paper strips. I planed the second interview based on the analysis that I had made in the first interview to effectively prepare for their distinct units-coordinating schemes as well as their unique responses.

Overview of the Initial Interviews

The initial interviews were composed of the following tasks: (1) An equi-partitioning task, (2) a missing addends task, (3) a coordinating two levels of units task, (4) a splitting task,

(5) a recursive partitioning task, and (6) a three levels of units-coordinating task. The tasks and the operations that were involved in each task are as follows:

The Equi-Partitioning Task

● Preparation: Show a wikistik that is approximately 18 cm to the student, and tell him or

her that it is his or her candy. Pens and scissors are available for the student to use

whenever it is necessary.

● Question: Let’s say this (the wikistik) is your candy. You want to equally share this

candy among five people. Can you mark off your share? How can you show me if your

share is fair?

This equi-partitioning task is designed to investigate children’s two levels of units- coordinating scheme. To be able to mark off their share, children must envision the given stick comprising five of their shares and disembed their share (Steffe & Olive, 2010). Some children may make uncanny estimations, and others may mark off their share with a bit of inaccuracy, depending on their previous sharing experiences. The important part of this task is asking the children to check to see if their share is a fair share. When a child folds or cuts off their marked off share, iterates it five times and adjusts their marked off share in order to make five of their share fit into the given string, I infer that the children can envision the given stick comprising five iterations of their share and disembed their share to mentally check if five of their adjusted share fit into the given whole. Thus, I hypothesize their actions, iterating and mentally adjusting their share, as indications of their two levels of units-coordinating scheme.

Depending on their previous sharing experiences, the task might not serve in triggering their schemes on the spot. In case of the children cutting the given string into five unequal pieces, an interviewer can allow the children to retry by showing them that two cut pieces are not the same length. If children cut off a share that is too small, it is important for an interviewer to remind them to exhaust the whole.

The Missing Addend Task

● Preparation: Provide the student with two fur strings; one is 29 cm long, and the other is

14 cm long.

● Question: This 29 cm long string represents my candy. The 14 cm long one represents

your candy. How much more candy do you need to make your candy as long as mine?

I located this task right after the equi-partitioning task to test my hypothesis of Carl’s and

Maggie’s levels of units-coordinating scheme as well as to investigate how they utilize the given strings when coordinating extensive quantities. One might solve this problem by counting up to

29 starting from 14, or iterate their candy onto 29 cm to establish 29 cm as composed of two of their candies and one more centimeter. According to Steffe and Cobb (1988), a child with ENS can find a missing addend by constructing an explicit inclusion relation between the addends and the sum. The construction of an explicit inclusion relation entails “the ability to put two number sequences side by side and understand that one of them can be included in the other” (p. 313).

Hence, a child with ENS can disembed the included number sequence, say 14, from 29 without destroying its explicit inclusion relation.

Following their analysis, I hypothesize that the two levels of units-coordinating scheme and ENS is essential when finding a missing addend by envisioning the explicit inclusion relation between the sum and the addend. However, I will not impute ENS to the students if they

78 find the missing addends by counting up to 29 while keeping track of their counting actions. In this case, I will hypothesize that they can assimilate with TNS and test my hypothesis in the coordinating two levels of units task.

The Coordinating Two Levels of Units Task

● Preparation: Use the fur strings that the student used when solving the missing addend

task. Cut the 14 cm long string in about half and cut about 5 cm off the 29 cm long string.

● Question: I cut off each candy a little bit. Now, we do not know exactly how long they

are. Can you show me how many of your candies fit into my candy?

This task is designed as a sequel of the missing addend task. When the students use their numerical computation algorithm, such as vertical subtraction, to solve the immediately preceding task, the interview can ask them if there is another way to find the missing addend. If the students say that there is no other way but vertical subtraction, it is hard to observe how they coordinated composite units or how they engaged in quantitative reasoning. By cutting off a little bit of each string, the students will not be able to use their vertical subtraction because they do not have specific numerical values any more. Thus, the interview can observe whether or not the students can iterate the composite unit (their candy) and find how many of their candies fit into the longer one.

The Splitting Task

● Preparation: A wikistik, representing my candy, is given to the student. Additional

wikistiks in various lengths, pens and scissors are available for the student to use.

● Question: This is my candy. Let’s say that my candy is five times as long as yours. Can

you make your candy? You can use these wikistiks (showing wikistiks in various lengths)

to make yours.

From an observer's perspective, the splitting task might seem to be identical to the equi- partitioning task because both of the tasks require disembedding and iterating operations.

However, according to Steffe and Olive (2010), the splitting task demands a simultaneous coordination of disembedding and iterating operations, while a child can successfully find his or her share by sequentially coordinating disembedding and iterating operations for the equi- partitioning task. For example, Jason, a third grade student who was one of the participants in their teaching experiment, could not make his share in a splitting task, while he could successfully complete the equi-partitioning task. Steffe and Olive explained that Jason needed to

“not only posit a hypothetical stick, but also posit the hypothetical stick as one of the five equal parts (of the teacher's stick) that had been already iterated five times and see the result of iterating as constituting the teacher's stick" (p. 99). They further described children’s ability to posit the hypothetical stick as the “awareness of a multiplicative relation between a whole and one of its hypothetical parts [that] is produced by the composition of partitioning and iterating

(pp. 111-112). In addition, the students who are limited to TNS will not be able to solve this task with an immediate success (Hackenberg, 2012; Steffe & Olive, 2010), due to the lack of an iterable unit of one, i.e.,the lack of the ability to interiorize composite units so that each unit of a composite unit can be freely withdrawn and substituted for another unit (Steffe & Olive, 2010).

With an iterable unit, a child can treat a composite unit as a single unit and use it for further operations. Thus, in order to iterate the hypothetical stick and to take the result of the iteration of it as an input for further operation, the hypothetical stick must be iterable.

Following Steffe and Olive (2010), I used this splitting task when my students successfully completed the equi-partitioning task (1) to infer the corroboration of their two levels

80 of units-coordinating scheme, (2) to investigate how they coordinate partitioning and iterating, and (3) to find whether or not their hypothetical stick is iterable.

The Three Levels of Units-Coordinating Task

● Preparation: Give the student three pieces of paper strips, which are 1 inch, 4 inches and

28 inches long.

● This smallest one is your candy. The mid-sized one is mine, and the largest one is the

cameraman’s candy. My candy is four times as long as yours, and the cameraman’s

candy is seven times as long as mine. Can you tell me how long his candy is compared to

yours?

This task is designed to test whether the students can enact the three levels of units- coordinating scheme. To successfully find the length of my candy compared to the length of theirs, the students need to mentally iterate their candy four times and project them onto my candy. Then, they need to iterate the composite unit (my candy that contains four of their candies) and project them onto the cameraman’s candy producing a composite unit containing composite units (i.e., three levels of units). However, the task can be successfully solved by enacting three levels of units coordinations. In other words, taking three levels of units as given is not necessary to complete the task because children can produce 28 cm by monitoring their counting, say, four is one, eight is two, etc…, by keeping track of seven, then by using part- whole comparison. To impute the three levels of units coordination scheme to a student, the indication of his or her construction of 28 as a composite unit containing four composite units of seven are necessary. In other words, they need to “see” these three levels of units from a distance and be explicitly aware of the inclusion relations. Therefore, the interviewer planned to proceed

81 to this task after making hypotheses about the students’ units-coordinating schemes based on the implications that were made in the equi-partitioning task and the splitting task.

The Recursive Partitioning Task

● Preparation: A yard long tape, representing a long piece of candy, is given to the student.

Pens and scissors are available for later use.

● Imagine that this is a long piece of candy. You want to share this equally among five

people. Take your portion out (mentally). You want to share your portion equally among

three people. What fraction is your share of the whole?

This recursive partitioning task is designed to investigate how the students coordinate three levels of units along with splitting operations. Hence, the task is planned to be presented to the students after the interviewer made a hypothetical model of the students’ levels of units- coordinating schemes based on the equi-partitioning and the splitting scheme. According to

Steffe and Olive (2010), the splitting operation and the three levels of units-coordinating scheme were essential in the construction of the recursive partitioning scheme. They were also very careful when imputing the recursive partitioning scheme to their students because from an observer’s perspective without a thorough analysis of their students’ hypothetical models of operations, coordinating three levels of units in activity or using multiplying facts might seem like the construction of the recursive partitioning scheme.

For example, Laura, a fifth grade student who was one of the participants in their teaching experiment (she was paired with Jason, another participant of their study), showed how coordinating her units-coordinating scheme and splitting operations made recursive partitioning possible. When asked to make a 12/12 bar without dialing 12 in JavaBars, Laura first made a

11/11 bar, pulled out one part of it, and added it to the 11/11 bar to make twelve equal pieces.

After being reminded that she could not change the whole, Jason partitioned the whole into three equal pieces and then partitioned each one-third part into four equal pieces. Laura agreed that

Jason made one-twelfth, but Steffe and Olive did not infer that she had constructed the recursive partitioning scheme due to the lack of her initiation of the partitioning actions. She frequently used multiplying facts when making a 15/15 bar (such as three times five makes fifteen), and they did not impute the recursive partitioning scheme to her as well because they could not find a strong indication that she was not limited to part-whole comparison. Their analysis was convincing when they asked Laura to find another name for 5/15 after she partitioned the whole into three equal pieces and repartitioned each piece into five equal pieces, based on the multiplying fact that three times five is fifteen. Laura said it was one-third, not because five of one-fifteenth constituted ⅓ of the whole, but because she recalled her partitioning action— partitioning the whole into three equal pieces. Only after they observe that Laura independently executed partitioning actions when constructing 1/27 using 1/9 and explained three of 1/27 were contained in 1/9, did they infer that she had constructed the recursive partitioning scheme.

Following Steffe and Olive (2010), I would not consider that the students operated with three levels of units when they partition the whole into five equal pieces, and repartition each piece into three equal pieces to make fifteen equal pieces. The students must independently initiate the partitioning actions and be able to explain how one-third comprises five of their shares and one whole is composed of the three one-thirds, each of which contains five of their shares, which make the given whole three composite units containing five unit items. In other words, the students’ recursive use of partitioning actions and their explanation reconstituting the whole as a unit containing three levels of units would convince me that they had constructed the

83 recursive partitioning scheme; hence, they could take a three levels of units-coordinating scheme as given.

The use of the recursive partitioning in the initial interview is twofold. First, when an interviewer found strong indication that the students operated with a three levels of units- coordination, the recursive partitioning task could serve as a way to corroborate her analysis because three levels of units-coordination is essential when recursively partitioning the whole. If an interviewer hypothesized that the students were limited to a two levels of units-coordinating scheme, the recursive partitioning task would provide a corroboration of their hypothesis that the students did/could not coordinate three levels of units. Second, recursive partitioning provides good opportunities for a teacher-researcher to investigate how students coordinate disembedding and iterating operations that are essential when coordinating changes in each quantity of x+y=a.

The Plans for Carl and Maggie

To test whether the students had the two levels of units-coordinating scheme, the first three tasks (i.e., the equi-partitioning task, the missing addend task, and the coordinating two levels of units task) were given to each student on the first day of their initial interview.

According to their actions and language, I planned to make hypotheses of their units- coordinating schemes and to proceed to different tasks if necessary. If the implications of the their actions and language were not strong enough to infer whether or not they coordinate two levels of units, I planned to proceed by giving her the splitting task to further investigate their disembedding, iterating, and partitioning schemes. When the implications of the students’ two levels of units-coordinating schemes were strong, I decided to provide the three levels of units- coordinating task to identify whether or not they could operate with a unit containing three levels of units. In fact, while interviewing Carl with the first three tasks, I decided to give him the

84 splitting task to further investigate his two levels of units-coordinating scheme. In the case of

Maggie, I proceeded by providing her with the three levels of units-coordinating scheme to explore how she coordinated three levels of units. The following is an overview of Carl’s and

Maggie’s initial interviews (see Table I.).

Table 1 The Overview of Carl’s and Maggie’s Initial Interviews

Carl Maggie

1st September 5, 2014 September 19, 2014 interview

Tasks 1. The equi-partitioning task 1. The equi-partitioning task 2. The missing addend task 2. The missing addend task 3. The coordinating two levels of units 3. The coordinating two levels of units task task 4. The splitting task 4. The three levels of units- coordinating task

2nd September 8, 2014 September 22, 2014 interview

Tasks 5. The recursive partitioning task 5. The splitting task 6. The three levels of units- 6. The recursive partitioning task coordinating task 7. The three levels of units- coordinating task II

Analysis of Carl’s Initial Interviews

The Equi-Partitioning Task

In the initial interview that was conducted on September 5, 2014, Carl was asked to mark off his share when sharing a licorice stick equally among five people. Carl estimated his share, immediately saying that he needed to split it into five pieces.

Protocol 5.1. Carl’s equi-partitioning scheme

I: (Puts a string on the table) Let’s say that this is a licorice. You want to share this fairly among five people. Using this pen, can you mark off your share? Carl: I mark… I: Mark off your share. Carl: So like… hm, so like, you can split it up like five, right? I: Yes. Carl: Then, there are five marks? I: Just mark off yours. Your share. Carl: (He tries to mark off his share, looking at the string intensively. He moves his pen up and down showing a little bit of hesitation.) I: Before you do that, what are you thinking about? Carl: Since there are five people, the measure is smaller? (He estimates his share using his thumb and an index finger.) I: Mm-hmm. Carl: So, it’s about… here? (pointing to a spot near 1/8 of the length of the string) I: Okay, mark it. Carl: (Marking on the string) I: (Pointing to his mark with a thumb and an index finger) So, that’s your share? Carl: Yes, ma’am. I: So, how could you convince me that that’s the fair share? Carl: Huh… So, there are five people, and it needs to be smaller so that it can divide it out between five people. I: Uh-huh. Carl: So, if you get like that (using his thumb and index finger, he pinches the mark that he made), then you can get that, that, and that (pinching it three more times in increments along the wikistik. The locations at which he points seems to divide the wikistik into five unequal pieces). And, they are about equal, I guess?

I: Okay. Can you show me? If you want to use scissors, you can use the scissors (there was a pair of scissors on the table). You can also (taking an additional wikistik out of a bundle)... This is a wikistik so that you can fold it if you want. Carl: Alright (he grabs his wikistik and folds it five times, beginning at his mark). I guess it is a bit smaller, isn’t it? (keeps folding it)...four, five (making five equal pieces plus one piece left over). I: (looking at the leftover) Oh… I want you to use up all of the candy. Carl: All of it? I: Yes. All the licorice. Use it all, and share it equally among five people. Carl: With five people? Can I use these scissors? I: Sure. Carl: Should I cut right there (his mark), or can I cut somewhere else now? I: Well, if you changed your answer, you can… Carl: Is that okay? I: Yeah. Carl: Alright. I don’t know… (aims scissors at a little further to the right of his previous mark in order to cut off a bigger piece than the previous one. It is about one- fourth of a string.) here? (he cuts it off.) I: Okay. So, how can you show me that’s the fair share? Carl: Huh...hm… (he put the cut-off piece on top of the longer piece and picked up the cut-off piece), because when I marked like this (pointing to the mark), it was way too small. And, then, when I kinda folded it up, I kinda saw how much was left. So, I increased it by a little… and guess that it just how I…. ‘Cause there was about this much (waving his cut-off piece) left. So, I figured if I did like this much (pointing to the increase in length), then that much wouldn’t—like the little bit left— that wouldn’t be left there (implying the leftover string). Then, that will be all divided out equally (sweeping his hand from left to right). I: Oh… Now you cut off this part (touching his cut-off part), and you said that this is your share. So, can you show me that this is a fair share—if that is a fair share? Carl: (Sighs.) I: I mean, I understood your explanation, and that was really good. How can you convince me and show me that this is a fair share? Carl: You want me to cut it? Is that what you need? I: Whatever you need to do it, do it. Carl: As long as I can show you this is a fair share? I: Exactly. Anything you want. Carl: (He puts the cut-off piece on top of the remaining string and cut off another equal- sized piece. He repeats his action one more time (three times of cutting actions in total) and makes four equal pieces.). Oh, there are only four. I: Oh, what would you say about that?

Carl: Uh… I guess, I didn’t do something right? I don’t know. I: Hm… so, if you think that you didn’t do something right, can you fix it? (Putting a new wikistik in front of him.) Carl: (Sighs.) I: How would you fix it? Before you do that, can you tell me, what would you do? Carl: Uh… make these (touching four pieces) a tad bit smaller? I: A tad bit smaller? Carl: I need one more of these (arranging four pieces in a row). So, I guess, I cut this this much less (indicating about one-third of his cut-off string). Can I use this (the cut off piece with the mark on it)? I: Sure. Carl: (Putting the cut-off piece on a new wikistik and cutting the new string between the mark and the end of the cut-off part). I: So, how can you show me that this is a fair share? Carl: (Putting the smaller cut-off part below the bigger cut-off part) so, this is two, three, four, five (repeatedly moving his index finger to mark the next location)... (He found that there are six instead of five. He laughs.) I: So, what do you think about it? Carl: So… I still did not cut… it’s too less? I: So, next time, if you try again, what would you do? Carl: Then, I will make it a little bit bigger.

Even though Carl said “split up like five", and “it needs to be smaller so that it can divide it out between five people", it was unclear to me whether he was explicitly aware that all shares must be in equal sizes. At most, I could infer that he wanted to make smaller pieces for his share while making five pieces so that his share became one out of five pieces. His four pinching motions while synchronously saying “that” and producing five unequal pieces exhausting the whole suggests that his “split up” might imply making five pieces out of all the candies.

However, when asked to show me if his share is a fair share, he independently chose to fold his share and iterate it five times in order to find whether five of his shares fit into the whole. Before he finished iterating his share five times, he already knew that his share was too small to exhaust the whole. At this point, I infer that the task triggered his equi-partitioning scheme.

Knowing that the first piece he cut was too small (approximately one-eighth of the whole), he next cut off a bigger piece (approximately one-fourth of the whole). His statements— that he found the leftover after folding four times and that he could compensate for it by marking off a piece a little bit to the left—indicate that he might have envisioned that five of the bigger pieces would make up the whole. He tried a second time, cutting off a bigger piece that was about one-fourth of the wikistik. Knowing that his piece was too big, he cut a smaller piece the third time, which was approximately 1/6 of the wikistik. After these three tries, it was clear that he was trying to split the wikistik into five equal pieces. His folding actions, cutting off a piece and iterating the piece on the wikistik indicate that he was able to iterate his share. His estimations, which were approaching one-fifth of the whole on each try, along with his independent adjustments imply that he could envision the given stick as comprising five iterations of his share and disembed his share in order to check whether five of his adjusted shares fit into the given whole. Thus, I infer that Carl used his number concept five as a template and projected it onto the given whole by coordinating two levels of units.

The Missing Addend Task

When I hypothesized that Carl had a two levels of units-coordinating scheme, I wanted to explore how he utilizes his units-coordinating scheme when finding a missing addend. Two fur strings, one 29 cm long and the other 14 cm long, were given to Carl, and he was asked to find how much more candy he would need to make a 14 cm long candy string as long as 29 cm one.

Protocol 5.2. Carl’s numerical computations

I : (Putting the 29 cm long stick in front of me, and the 14 cm long one in front of Carl) Let’s say that your stick is 14 cm and mine is 29 cm. Okay? How much of a string do you need to add on yours to make your string as long as mine? Carl : Um … 15 cm? I : Why do you say that?

Carl : Cause this is 29 cm and this is 15. I subtracted 29 from 15. And added that much more. I : How did you subtract them? Carl : Uh, did you say why or how? I : How. Carl : Uh, I guess 29 minus, you said 14? I : Yes. Carl : 29 minus 14 is 15. So … I : Can you check it if that is a right answer? Carl : Um … I mean, I can in my head. I : What happened in your head? How could you check it? Carl : I did, first I did 9 minus 4, which is 5, and then you said that was 29, so 2 minus 1 is one. So it’s 15.

The primary purpose of the missing addend task was to explore how he used his units- coordinating scheme and to determine whether he constructed the explicit inclusion relations between the addends and the sum. However, there was no basis to infer that he used his units- coordinating scheme when finding the missing addend in this initial interview because Carl did not display any actions related to units-coordination or iteration. He did not use his string to compare his to mine; instead, he immediately answered that it was 15. It may seem as if he was aware of the place values of 14 and 29 when he said, “first I did 9 minus 4, which is 5, and then you said that was 29, so 2 minus 1 is one. So it’s 15”, but there was no clear indication that he combined 1 and 5 while considering their place values. I refer to his engagement in traditional computation as Carl’s procedural scheme.

The Two Levels of Units-Coordinating Task

It was very challenging to find implications of his units-coordinating schemes when he used his procedural scheme. However, I decided not to question his method because his schemes and actions were legitimate to him without containing conflicts within each other. Considering this was the first time I interacted with Carl, it was better not to further question his correct

90 answers and procedures from his perspective in order to establish a rapport with him. To further investigate how he would use composite units in the context of extensive quantities, I chose to proceed to the two levels of units-coordinating task.

I cut off each of the strings so that Carl did not know how long each string was in order to trigger his iterating operation rather than the procedural scheme. Then, Carl was asked to determine how much of his string would fit into the longer one. In fact, approximated three and three-fourths of his string fit in the longer one.

Protocol 5.3. Carl’s use of composite units

I: So, now, we don’t know how long they are exactly. Can you somehow find how many or much of your string go into this long string? Carl: Uh, do like that (putting his string underneath the long one). Going back and forth (referring to iterating his string)? I: Oh, yeah. Go ahead. Carl: (Marking each iteration on the longer string.) I: Okay. What can you tell me about this? Carl: Uh, this (the longer string) is three and half times bigger than this (his string)? I: Three and half times? Does it (the last part of the longer string that is approximately 3/4 of his string) look like half of it (his string)? Carl: (Comparing the both) no, it doesn’t. I: Can you be more… Carl: Specific? I: Uh huh. Carl: Um… Maybe three and thirds? I: Three and one third? Carl: Yeah, three and one third. I: Is one third bigger than one half? Which one is bigger? Carl: (Smiling)...s…smaller, right? I: Yeah… Okay.

The task was successful in triggering Carl’s iterating actions. His first estimation that the given string was three and a half times longer than his string and his later agreement that it would be more than three and half times longer than his string suggest that he knew that three lengths of

91 his string plus slightly more would fit into the long string. He also knew that one-third is smaller than one-half. In retrospect, it seems as if he wanted to say it was three and two-thirds rather than one-third (he first said thirds before I asked him whether it was one-third). It was also possible that he later paid attention to the left over part of his string (he did not feel any conflict when answering that one-third was smaller than one-half). Had I been a better interviewer, I would have asked him why he said “thirds” to investigate what exactly he meant instead of asking him if he meant by three and one-thirds. However, I was satisfied with the fact that the task triggered his iterating operation and decided to further investigate how he would use his iterating operation to solve the problems. I was confident that he would not use his procedural scheme once his iterating operation was activated.

Protocol 5.4. (Cont.)

I: Let’s say that this (his string) is 7 cm and this (the longer one) is 24 cm. So, the same situation and the specific numbers are given. Can you tell me how many of this (his string) go into the longer string? Carl: Uh… (pointing at each part of the long string) seven, fourteen, twenty-one, and, then, it’s twenty-eight centimeters, right? I: Um, twenty-four. Carl: Twenty-four centimeters? That means… What’s the questions again? I: So, this is seven centimeters and this is twenty-four centimeters. Can you figure out how many of these (pointing at his string) go into this (pointing at the longer string)? Carl: By centimeters, or how many? I: How many times. Carl: This is four (picking up his string)? I: No, Seven. Carl: (Iterating his piece) so that means, one, two, three, and you get to here again. You said twenty-four, right? I: Uh huh. Carl: So, that means, this (his string) is seven. That means, three and one-third! I don’t… I: Do you have one-third? What is the left over here? What centimeters... Carl: (pointing at each part of the long string) this is 24, right?

I: Right. Carl: Six. I: Is it six? Can you count one more time? This (picking up his string) is seven, right? Carl: Oh! (laughing) Cause you said it was twenty-four, I was… I: Yeah, the whole thing is twenty-four. Can you count it loud for me? Carl: (Iterating) Seven, fourteen, twenty-one, and then twenty-eight? But this isn’t small enough to fit inside of that. I: Right. So approximately, if you think a fraction… Carl: You need a fraction? I: Uh huh…(looking at Carl’s confused face) Okay. How long is this (pointing at the left over part)? Carl: Six. I: Six? Why is it six? Carl: Cause you said that the whole thing was 24. So… Instead of… Cause it is divided into this many parts (tapping each piece of the string, total four times)...cause this (his string) is seven… So that means… twenty...this is eighteen right here (pointing at the third piece, looking confused)... I: This is seven. You counted really well. Carl: Seven, fourteen, twenty-one… I: Okay, how long is this (pointing at the left over part) Carl: Oh, I know what I’m doing wrong. Cause I am saying that this is six, but, it’s seven. I: Right. Carl: And you are asking how much is left? I: Uh huh. How many centimeters are left when you count… Carl: Six. I: Six? Why do you say that? Carl: Um, cause, I keep on thinking these parts are like actually parts. I said six cause these are all six. I: These are six? I thought these are sevens. Carl: They are seven, aren’t they? It’s hard. Then, it is 24, I guess. I: Yeah, the whole thing is 24. How long is this part? Carl: (Putting his string underneath the left over part of the given string) I say, five? I: Why do you say that? Carl: Cause this (his string) is seven, and it’s two centimeters shorter. So, seven minus five is two.

Carl’s statement that he “keep[s] on thinking these parts are like actually parts” suggests that he was envisioning the given string as comprised of four equal parts. Hence, he first counted by sevens and found the given string was 28 cm long. After being reminded that the string was

24 cm long, he thought his string was 6 cm long instead of 7 cm long. This is another indication that he considered the given string to be comprised of four equal parts. I inferred that he saw that the given string was divided into four pieces, and using this visual information, he reconstituted the given string as four times his string, which was seven centimeters long, when he was monitoring the length of his string, or as four times six when he was paying attention to the length of the given string. His struggle when monitoring the length of the iterating string suggests that he is with TNS. I wanted him to pay attention to the leftover part and asked him to find how long the last part was. He put his string underneath the last part of the given string to make a visual comparison. He estimated that the last part was two centimeters shorter than his string and concluded that it was five centimeters long. This local comparison serves as another indication that Carl was with TNS.

Had he reconstituted the given string as three of his string and three centimeters more, this would have been an indication that he could monitor his iterating actions while being explicitly aware of the composite unit of seven. His local comparison of his string and the last piece of the given string suggests that he did not reconstitute the given string as three levels of units as a unit containing three sevens and three centimeters more. Thus, I hypothesize that Carl could iterate a composite unit in activity, but he could neither take the result of his schemes and actions as given nor reconstitute the given string as a composite unit containing other composite units (i.e., three sevens and three more). I could have asked him to count by sevens and add five

94 more to check his answer, but I saw that he became frustrated. Considering that this was his first experience working with me, I decided to proceed to the next task to minimize his frustration.

The Splitting Task

To further explore Carl’s iterating and disembedding operations, I presented the splitting task for him.

Protocol 5.5. Carl’s attempt to split

I: (Showing a wikistik) Let’s say that this is my licorice. My licorice is five times longer than yours. Can you make your licorice? (putting wikistiks in various lengths in front of him) You can use any of these to make yours. Carl: (Trying to take out a much longer wikistik than the given one.) I: Uh, can you tell me what you are thinking? Carl: Uh, trying to get the biggest one of these. Cause this is five times longer. That means one five times of yours. I: Uh, so, mine is five times longer than yours. Carl: (Surprising) Oh, is longer than mine? I: Yeah… Carl: Oh, O.K.. I: Tell me what would you do. Carl: I would get the smallest one, ‘cause yours is five times bigger. And, then, split it in half? I: Can you show me? Carl: (Cut the piece in half that was approximately ⅓ of the given wikistik.) I: So, which one is your licorice? Carl: (Lifting the one in his right hand, which is slightly smaller than the left one.) I: Can you show me? Carl: (Putting his piece right beside the given wikistik and starting to count how many of his pieces would fit into the given one) One, two, three times bigger…. I: O.K.. So can you fix that? Carl: Maybe, I cut this in half (cutting it in half)? I: O.K. Carl: (Iterating the smaller piece) One, two, three, four, five. I: (Pointing to the left over that is approximately ⅛ of his piece) There is a little bit left. But, you think you did it, or you have to fix it? Or… Carl: Uh… It is about to be equal, but maybe, I need to be that (pointing to the left over) bit longer. (Then, he grabs the other piece that he held in his left hand when he made a first cut and finds that the length of his new piece is equal to his first

piece and the left over. He puts three pieces side by side. See Figure 1.). This part (looking at the third piece with the given wikistik). Do you want me to count it again? I: Yes. Carl: One, two three, four, five. I: Yes, you got it!

To make his licorice, Carl needs to (1) understand his licorice is one-fifth of mine, (2) posit a hypothetical licorice stick that is his, (3) mentally iterate his hypothetical licorice five times, (4) project five of his licorice onto mine, and (5) disembed one-fifth of my licorice.

According to Steffe and Olive (2010), the splitting task requires simultaneous coordination of iterating and disembedding operations as well as an anticipation of these operations. I hypothesized that Carl would experience difficulties in simultaneously coordinating iterating and disembedding operations because he was with TNS. He was limited to coordinate two levels of units and lacked reconstituting the given situation by recursively using his scheme.

Carl first attempted to make his licorice five times longer than mine. Being reminded that mine was five times longer than his, he picked up a small wikistik, cut it approximately in half, and chose the smaller piece, which was one-third of my string. There was no indication of anticipating the result; he needed to iterate the cut-off piece in order to see whether it fit. He then cut his piece in half again. According to Inhelder and Piaget (1958), the children, who could not anticipate the relationships between a number of parts and a given whole, kept cutting the whole in half in order to make more pieces. Carl’s cutting a piece into halves twice suggests that he did not anticipate the result, and he wanted to make smaller pieces to try whether five of his piece can fit in my string. When he found that there was a little left after he iterated his piece five times, he said he needed to make his piece a little bit longer. When he saw the other half piece was slightly longer, he iterated it five times (see Figure 5.1) and was satisfied with the result.

I do not think his last try was anything but fortuitous. Based on the equi-partitioning task,

I found a strong indication that he could adjust the length of pieces envisioning five of his pieces were contained in the given string. What he was lacking was anticipating the result of the task by

(1) positing a hypothetical string in his mind, and (2) simultaneously coordinating iterating his hypothetical string. His first try to make a smaller piece that was one-third of the given string and his repeatedly cutting the piece in order to make smaller ones corroborates my hypothesis. His trial and errors also suggest that he did/could not reconstitute the given string by mentally disembedding and iterating his part and anticipate the result.

Figure 5. 1 Carl’s splitting, reconstructed by the author.

Note. The first diagram (a) shows Carl’s first cut off piece (the read bar), his second try

(the right part of the first cut off piece) and his third try (the left part of the first cut off piece).

The second diagram (b) shows the difference in length of his second and third cut off piece.

Based on his ways of solving the two levels of units-coordinating task and the splitting task, I hypothesized that Carl was limited to coordinating two levels of units. He was lacking (1) monitoring the composite units while iterating, (2) disembedding the composite unit that he was iterating to make a comparison, (3) taking a result of his operation as given for further operations, and (4) simultaneously coordinating iterating and disembedding operations.

I planned the second interview to further explore Carl’s units-coordinating schemes.

Although I had identified plausible indications to infer that he was limited to coordinating two levels of units, I wanted to find corroborations to my hypothesis. In addition, Carl addressed his difficulties when solving the tasks several times. Considering that this was the first time that he was interacting with me and he was not used to solving the kinds of tasks that were given, the second interview was necessary for testing the hypothesis that I had formulated in the first initial interview. The second initial interview was held on the 8th of September, and I started the interview with a recursive partitioning task.

The Recursive Partitioning Task

I chose the recursive partitioning task to investigate if Carl could coordinate three levels of units. My hypothesis was that he was limited to coordinating two levels of units. Before I started, I asked him what he thought about the tasks of the first initial interview. He said the previous tasks were very challenging for him, especially lining up strings together. From his statement, I inferred that he was not used to engaging in hands-on experiences, and this new experience might have frustrated him. In later teaching episodes, I could also observe that he used his procedural scheme most frequently whenever it was possible. When he needed to quantitatively compare two quantities, he depended heavily on visual comparisons. To explore how he could coordinate units without visual aids, I showed him a long strip of paper and

98 covered it so that he could not see it. Carl then was asked to find one-fifth of one-third of the whole.

Protocol 5.6. Carl’s repartitioning the whole

I: So, you have this long piece, and ... Carl: Cut this into three pieces, and cut that one piece for five people. I: Yes. Then, one little piece is gonna be your share. Do you have the image in your head? Carl: (Nods.) I: What fraction is your share of the whole cake? Carl: One-third. I: One third? (uncovering the piece of the paper strip) So, this is the long cake … Carl: One-fifth? I: One-fifth? Can you show me? (Taking out scissors) You can use this or use the pen to mark off … Carl: (Cutting off about one-third of the paper strip, he put it above the strip to copy the length, and cut off the second piece. The last piece was longer than the other two pieces, and he wanted to cut the last piece off to make all the pieces the same lengths.) I: Can you hold it right there? I know that you wanted to make three equal pieces (Carl nods his head), and let’s just assume that they are all equal. Carl: (Cuts off about one-fifth of the last piece) I: So, what fraction is that of the whole cake? Carl: One-fifth? I: Can you show me why it is one-fifth? Carl: Oh, you are talking about the long one! I: Yes, the long cake (providing another long strip of paper). I am talking about this long cake. What fraction is that (pointing at the small cut-off piece in his hand) of the whole cake? Carl: Huh! Um … I guess … Maybe … (putting the small piece onto the longest strip of paper and iterating it three times) see that’s how many of this fit into that? I: Yes. Carl: (Putting the smallest piece on top of the long one and starting to iterate it.) I: Before you actually do it, can you somehow do it in your head? Carl: You mean guess? I: Not just a guess. It’s reasoning. Thinking about what fraction is that of the whole cake. Carl: Um … One-twentieth? I: One-twentieth? Why do you say that?

Carl: Because this (the small piece) is small, and twenty of these pieces fit into that (the longest strip of paper).

From Carl’s description of how to cut the given piece and his actions of cutting off his share, I infer that he understood the task required him to partition the given into three equal pieces and repartition each of the pieces that were one-third into five equal pieces. When asked to find what fraction his share was of the whole cake, he answered that it was one-fifth. He realized that he needed to compare his share to the whole cake only after being requested to show why his share was one-fifth of the whole. His surprise suggests that he was not aware that the partitioned pieces composed a whole as a unit containing three levels of units. His estimation, one-twentieth, based on visual comparison corroborates my hypothesis.

I later intervened and asked him to consider the relationship between the pieces; five of his shares fit into one piece that he previously cut off, and three of those pieces fit into the whole.

With strong guidance, he later said that there would be fifteen of his pieces: five of his shares fit into the three pieces that he previously cut. However, I could not find any indications that he independently inserted five of his units into the three pieces that he previously cut while monitoring that the three of those comprised the whole. Rather, he was reminded that the previously partitioned piece was composed of five of his shares, and he counted five three times using the visual aids (the three strips that were lying in front of him). According to Steffe and

Olive (2010), recursive partitioning requires operational awareness: being aware of the level of units on which one is operating while operating. In other words, I could not impute the recursive partitioning scheme to Carl, because he did not show any indication that he intentionally partitioned his share with being explicitly aware that each of his previously partitioned pieces comprised five of his share, and three of those composed the whole.

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The Three Levels of units-coordinating Scheme

From the recursive partitioning task, I identified corroborations of my hypothesis that (1)

Carl was limited to coordinating two levels of units, (2) he was lacking operational awareness when coordinating three levels of units, and (3) he could/did not take three levels of units as given for further operations. I presented the three levels of units-coordinating task to investigate whether he could enact coordinating three levels of units.

Protocol 5.7. Carl’s producing three levels of units

I: (Putting three strips of paper in front of Carl) This (the smallest piece that was an inch long) is your cake, and this (the mid-sized one that was four inches long) is mine, and that (the longest piece that was twenty-eight inches long) is Hamilton's (the cameraman’s) cake. Mine is four times as long as yours, and his is seven times as long as mine. Can you tell me how big your cake is in terms of his cake? Carl: (Picking up his cake and looking intently at the longest piece) Um … Maybe … One … Maybe … One-seventieth? I: One-seventieth? Why do you say that is one-seventieth? Carl: Maybe it’s not one-seventieth. I don’t know. I just said that cause that one (the longest piece) is really big, right? So, and this is small, so if you put these (his piece) in it (the longest one), it’s gonna take a pretty long time. I: I definitely agree with that. Carl: But, I don’t know about seventy, but … I don’t know. I: What about my cake? My cake is four times longer than yours, and his cake is seven times longer than mine. Knowing the relationship between your cake and my cake will help figure it out how many of this (his cake) go into there (pointing at the longest strip)? Carl: Wait, you said this (the longest piece) is seven times bigger than yours, right? I: Right. And, mine is how many times … Carl: Four. So, that means … Four of these (his piece) will fit in that (pointing at my piece), and if that (the longest piece) is seven times more, that means there are 28 of these. This piece (the longest one) is twenty-eight times bigger than this one (his piece).

When Carl compared his cake to the longest piece of cake, he did not even glance at my cake. His solution, one-seventieth, without confidence suggests that he was estimating the lengths of the two pieces without considering my cake, the mid-sized piece. Only after being

101 reminded did he identify the relations between his, mine and the cameraman’s cake. It is possible that he projected four of his share onto mine, mentally iterated it seven times and found the relationship between his cake to the longest piece of cake by using the facts of multiplication.

Thus, I infer that he could re-apply his two levels of units coordinating scheme. However, I could not find any indications that he could review the longest cake as a unit containing three levels of units.

Analysis of Maggie’s Initial Interviews

The Equipartitioning Task

I conducted the initial interview for Maggie on September 19. While solving the equipartitioning task, she tried seven times to mark off her share. It was impressive that she persevered in solving the task using different strategies in each try. Her perseverance in solving the given task using a variety of methods suggested that she might have plenty of experience with measuring and sharing activities.

When asked to mark off her share when sharing the given string among five students equally, she first marked the given string in half and marked the half pieces into another half, producing four pieces. She then realized that there were five people, not four, and marked the half of one of the quarter pieces. When asked if that was a fair share, she replied that it was not a fair share and said she just wanted to make a smaller piece. She said it was impossible to make a fair share without measuring the string. I wanted to explore more about her iterating and disembedding operations, so I asked her if there were any other ways to make her fair share without measuring the string. She folded the given string in half, stopped folding and unfolded the string. The protocol below illustrates why she stopped folding the string during her third attempt.

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Protocol 5.8. Maggie’s third try in the equipartitioning task

Mag: No, I don’t think it’s possible to make it accurate, cause it’s five. I: You are not a machine, so you cannot not make it 100% accurate. But, there must be some strategies that you are thinking of. Can you tell me what you are thinking? Mag: Yeah, I can measure it like using a thumb. (Using her second phalanx of her thumb as a ruler and iterating it five times while synchronously counting) One, two three, four, five. I: Can you mark off your share? Mag: (Marks off the first tick mark using her thumb, and repeats it four more times.)

Her iterating actions suggest that she could experientially iterate her hypothetical share five times. I hypothesize that she had the image that the given string comprises five lengths of the phalanx of her thumbs. I asked her to make marks on the string so that she could review her reasoning. After she made her marks, she realized that measuring the given string with her thumb did not produce accurate measure as well, for she found that the last piece was a little bit longer than the others. Unsatisfied, she marked off her share a little bit longer on her fourth try.

Although each of the pieces were all slightly different lengths, it looked as though she produced approximately equal-sized pieces. At my request to show whether her share was fair, she cut off her share and put it on top of the string in order to copy the length of her share while cutting more pieces. When she was cutting off other pieces, she ignored the mark that she previously made and produced four equal pieces plus a much smaller piece. In her fifth try, she said she needed to make a smaller piece. She put her share on top of the new string, marked it off so that she could produce a smaller piece than the previous one. She repeated her actions twice more while iterating her previous string twice. She stopped iterating the piece and asked me if she needed to exhaust the whole. After realizing that she needed to exhaust the whole, she stopped marking off the string and made a new mark so that the first piece became slightly longer than her fifth try. This was an indication that she could anticipate the result of her iterating actions

103 prior to completing her actions. Unfortunately, her sixth try resulted in the pieces being too small. It was surprising to observe her failure when she checked the length of her string while she made fairly good estimates on each of her tries. However, I found solid indications that she could envision the string as five of her shares, based on her iterating and adjusting her share.

Therefore, I hypothesized that she could coordinate two levels of units.

I wanted to explore how she would use a ruler to find the accurate length of her share.

Thus, I allowed her to use a ruler toward the end of the task. The string was 6 inches long, and she made inch marks on the string using a ruler. Then, she cut off each piece a little bit longer than an inch, saying, “because it [the given whole] is longer than five [inches]". Her estimation suggested that she could envision the given string as five inches plus one inch more and reconstituted the given string as five of her shares that were slightly longer than one inch in order to compensate for the extra inch. Although the estimation was not accurate, I found solid indications that she could coordinate two levels of units, so I decided to proceed to the next task.

The Missing Addend Task

Protocol 5.9. Maggie’s missing addend task

I: Um, let’s say that your candy is 14 cm, and my candy is 29 cm. How many more or much more candy do you want to add on your candy to make it as large as mine? Mag: Um, one more and then one centimeter? I: So, that is, how many centimeters… Mag: That will be…it will equal to 29. That’s 28 (iterating her string once) and 1 centimeter. I: Okay, I like that.

Maggie’s iteration of her string while keeping track of its numerosity indicates that she could envision the 29 cm long string as two 14 cm-long strings plus one centimeter more. Her clear explanation, “it will be equal to 29. That’s 28 and 1 centimeter” suggests that she could

104 produce three levels of units by combining composite units and a unit, and she was explicitly aware that 29 was composed of two fourteens and one more. This suggested that she envisioned the explicit inclusion relationship between 29 and two of her shares plus one by producing three levels of units. However, it was not clear whether she envisioned 29 as three levels of units prior to the activity. To further investigate how she coordinated units, I preceded to the next task.

The Two Levels of Units-Coordinating Task

I cut off some of her string and some of mine and asked Maggie how many times longer was my string to hers. At first, my string became almost three times longer than hers. She iterated her string three times and said mine was three times longer than hers. To explore how she coordinated her string and its fractional part, I cut some more of my string. Now, my string was about two and two-thirds times longer than hers. Maggie iterated her string twice on top of my string and saw there was some left over. She said, “it is one and a half times bigger, because you add one and a little bit more".

It was unfortunate that the following situation was almost identical to the previous missing addend task. Although she first understood the questions in a way that I intended and answered correctly, the similarity between the following task and the missing addend task, along with her successful experience in the missing addend task, suggests that her previous experience might have induced her to answer that my string was one and half times more while explaining that it was one and a little bit more. Hence, I could not find further corroboration to my hypothesis in this task. I decided to proceed with the three levels of units-coordinating task for further investigation.

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The Three Levels of Units-Coordinating Task

Three strips of paper of differing lengths were presented to Maggie. The shortest one

(one inch long) was hers, the mid-sized one (six inches long) that was six times longer than hers was mine, and the longest one (18 inches long) that was three times longer than mine was the cameraperson’s. I asked Maggie to express the amount of her piece in terms of the length of the cameraperson’s piece.

Protocol 5.10. Maggie’s coordinating three levels of units

Mag: What is yours to hers (the cameraperson’s)? I: Hers is three times longer than mine and… Mag: Mine to yours is six. I: Yes. Mag: So, I can use yours so that’s … three times longer? (in deep concentration.) Can I move it around ... I: Before you do it, can you tell me what you are thinking? Mag: If yours is six times longer than mine and this is three times longer than yours, I can see how … um… I don’t know. I: Okay, what do you want to do? Mag: (putting her piece and my piece side by side) I am trying to think how to put it. (Whispering) six times longer … Is that what really is? Or, you just made … I: They are in actual sizes. Mag: (Putting my piece near the longest one) Cause I could figure it out, like, cause it’s three times longer, I am trying to figure out how many this (her piece) is for that (the longest piece). So, I can figure … six and three … it’s like twenty-seven. I: Twenty-seven? Why do you say that? Mag: Cause I timed it. This (my piece) is six; I mean six. It will be eighteen. I: Eighteen? Mag: Cause this is three … times (inaudible whispering), and this is six. So, I think if I (sliding her piece right next to the longest piece and tracing it along the longest piece) … I am not sure. I: You are doing a good job. What did you want to do? So, you said that you multiplied, right? Why did you want to multiply? Mag: Cause … Now it makes sense. It’s eighteen. Maybe more, but … probably eighteen. I: Can you tell me more about why you said that?

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Mag: Okay, this (my piece) is six times longer than this (her piece). So, I thought it takes six to make this. And then, this (the longest piece) is three times longer than this, so, three of these will make this one. So, if I … to make one of these (her piece) to make this (the longest piece), I have to … I just times it. Eighteen.

The salient distinction between Carl’s and Maggie’s ways of reasoning is that Maggie independently and consistently used the mid-sized piece (my piece) to establish the multiplicative relationship among the three pieces, while Carl was limited to making a visual comparison between the shortest and the longest pieces. Her first three comments, “what is yours to hers", “mine to yours is six", and “I can use yours” along with her explanation, “it’s three times longer, I am trying to figure out how many this [her piece] is for that [the longest piece].

So, I can figure … six and three” are solid indications of her effort to find the relationships among the three.

She also wanted to use her pieces and move them around when solving the task. At that time, I asked her to think about it first and manipulate pieces later to explore whether she could find the relationship using her mental images, instead of making direct comparisons. In retrospect, I consider her request as her ways of engaging in quantitative reasoning with her measuring scheme, rather than as an intention to make a direct comparison between the given pieces. In later teaching episodes, she measured given unknown quantities (such as directed segments without numeric values) using her imagery of a unit and estimating another unknown quantity in relation to her unit measure and the given unknown quantity.

In addition, when she was allowed to move the pieces around, she did not actually iterate her pieces on other pieces. After she was told that the pieces were cut to scale, she intently looked at all the pieces and slid her piece a little bit alongside of the longest piece in order to explain why she thought the longest one was eighteen times longer. Hence, I infer that she

107 abstracted the length of her piece, mentally iterating it six times to make up my pieces and recursively iterating my piece three times while monitoring that my piece was composed of six of her pieces. Her consistent uses of multiplication instead of repeatedly adding six three times suggested that she coordinated three levels of units by iterating a composite unit, six, and was aware of the multiplicative relationship among them.

Some might argue that she might have simply multiplied two given numbers because she could rule out other operations; subtraction and divisions can be easily ruled out, and she chose not to add the two given numbers, because the longest strip might have looked much longer than nine times her piece. However, her explanation, “it takes six to make this. And then, this [the longest piece] is three times longer than this, so three of these will make this one. So, if I … to make one of these [her piece] to make this [the longest piece], I have to … I just times it.

Eighteen” along with her pointing actions of indicating each piece, corroborate my hypothesis that she was coordinating three levels of units and was aware of the multiplicative relationships among the three pieces.

The Splitting Task

I started the second initial interview on September 22 with the splitting task. My goal in the second initial interview was to verify or reject my hypothesis by identifying corroborations or contraindications of my hypothesis. Based on the first initial interview, I found the hypothesis that Maggie could coordinate three levels of units while being explicitly aware of the multiplicative relationship between units and composite units. If she had the three levels of units- coordinating scheme, she could successfully complete the splitting task. The splitting task requires simultaneous coordination of disembedding operations that were essential to the construction of a three levels of units coordination scheme.

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For the splitting task, one string was presented to her as my licorice. She was asked to cut off her licorice from a new string, given that my licorice was five times longer than hers.

Protocol 5.11. Maggie’s splitting

Mag: So, yours is five times longer than mine? I: Yes, mine is five times longer than yours. Mag: (Choosing the longest string) So, what I am thinking is I have to take ... make mine to where, it will take five of mine, equals to yours. I: Oh, that’s beautiful. Let’s do it. Mag: So, do you want me to take my string? (I nodded.) Okay, (putting the longest string underneath my string and putting her left index finger on her string at the point that my string ends) so … I’m guessing that this (pointing at about 3 cm away from the right end of her string using her right index finger). I: So, can you cut off your licorice? Now, how can you show me that mine is five times longer than yours? Mag: Make five of these. Five of my pieces. So ... (she put her cut-off piece on top of her lone string and cut off four more pieces by copying the length of her string. Next, she aligns five equal pieces underneath my string.) I: Um … what do you think (looking to see if the length of my string matches the length of the alignment of five of her string)? Mag: I got it right.

Maggie’s uncanny estimation must not be interpreted as a fortuitous event. Her first question, “yours is five times longer than mine?” and the following statements, “make mine to where, it will take five of mine, equals to yours” suggest that she understood the task as I intended, and her goal was producing a hypothetical string so that five of its lengths equaled the length of mine. Next, she put her long string underneath my string and copied the length of mine, which was the only observable action before she marked off her string using her right index finger. I infer that she re-presented my string in her mind, mentally produced a hypothetical string and iterated it five times in order to adjust the length of her hypothetical string to fit in mine.

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According to Steffe and Olive (2010), splitting operation requires “a composition of partitioning and iterating” (p. 99), because the ability to “not only posit a hypothetical stick, but also posit the hypothetical stick as one of the five equal parts (of the teacher’s stick) that had already been iterated five times and see the results of iterating as constituting the teacher’s stick”

(p. 99) is necessary when completing the splitting task. To compose the partitioning and the iterating operations, it is essential to disembed a partitioned whole. Thus, the splitting task requires a simultaneous coordination of disembedding and iterating operations. I infer her actions, cutting off five equal pieces and lining them up side by side right underneath my string in order to compare the length of my string as her enactment of the mental operations that she carried out. Thus, I hypothesize that she could simultaneously coordinate disembedding and iterating operations.

The Recursive Partitioning Task

Based on Maggie’s actions and language that I observed in the splitting task and the three levels of units-coordinating task, I hypothesize that she could coordinate three levels of units and simultaneously coordinate partitioning and iterating operations. In the recursive partitioning task,

I investigated whether she could take three levels of units as a unit prior to activity. My hypothesis was that she could find her share as one-fifteenth of the whole by enacting her three levels of units coordination and reconstitute the whole in relations to her partitioning actions.

The long strip of paper was presented and we pretended that it was a vanilla cake. I covered the strip of paper in order to explore how she mentally solves the given task.

Protocol 5.12 Maggie’s attempt to recursively partition the whole

I: Partition the cake into three equal pieces. Take one piece out. You are sharing that piece equally among five people. Take your share out. What fraction is that of the whole candy?

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Mag: If I just take away that … one piece? I: Uh huh. Mag: One-third? I: Do you remember what I said? Mag: I split it with five people. Then, I will have one-fifth over what I split it. I: My question is that what fraction is that of the whole cake? Mag: I am not sure. I: You are not sure? Can you tell me what you are thinking? Mag: If I already took it away, then it is one-third of the whole cake. But, I split it into five. I think it’s still one-third. Even if I split it into pieces, if I put it back together, it is … three …

From her explanation, “I split it with five people. Then I will have one-fifth over what I split it", I can infer that she understood the partitioning operations that were involved in the task.

However, it was possible that Maggie thought her share was one of the three pieces, and she was sharing her portion with five people. Her comment, “I think it’s still one-third. Even if I split it into pieces, if I put it back together", suggests she was considering her share as one-third regardless of its repartition.

Next, I removed the cover and asked Maggie to cut off her share, hoping my suggestion might clarify the goal of the task. When she cut the strip of paper into three equal pieces, she first folded it into three equal pieces and then asked me if she needed to cut off her share, pointing at one of the pieces measuring a third. Hence, the indication was solid that she understood “take one piece out” as taking one-third piece out and consider it as her share. After she cut off one- third of the strip of paper, she cut off one-fifth of it.

Protocol 5.13 (Cont.) I: Can you tell me what fraction is it of the whole cake? Mag: Two and one-fifth. I: You said two. What does it mean? Mag: Like it was into three. So, it was two. I: Okay. What fraction is this (one-third piece) of the whole? Mag: One-third. I: (Pointing at one-fifteenth piece) and this is …

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Mag: One-fifth of one-third. I: Then, what fraction is this (one-fifteenth piece) of the whole candy (putting all the pieces together to recover the given whole)? Mag: Point twenty? I: Why do you say that? Mag: Because if I put it into a decimal, the fraction … Is this (pointing to the strip of paper) the whole? I: Yes. Mag: Then, it’s three wholes, but it’s not three wholes. So, it will be point zero twenty? I: Why do you say that? Mag: Cause this is one-fifth (pointing at the one-fifteenth piece), and one-fifth is point twenty. So, I think it is point twenty percent.

Her comment, “one-fifth of one-third" suggests that she knew that her share was one-fifth of one-third. Based on her explanation, “one-fifth is point twenty", I infer that she knew that one fifth could be expressed as 0.20. Although I encouraged her to compare the one-fifteenth piece to the whole by putting all the pieces together, it seems as if she only focused on the relation between the one-third piece and one-fifth piece of the one-third piece. I could not find any indications that she could reconstitute the situation in relation to three levels of units while taking the three levels of units as given. From what I observed, I can only infer that she made local comparisons between the whole and three equal pieces and between one-third piece and one-fifth of the one-third piece. It was very challenging as an interviewer to try to “make” her pay attention to the relationship between the whole and the one-fifth of one-third piece while her focus of attention was on the one-third piece. However, rejecting my hypothesis was not appropriate at this point, because I did not observe any indications that she considered the relationship between the whole and the one-fifth of one-third piece. What I can infer from this protocol is that there is an implication that she was able to coordinate three levels of units, because she could monitor the existence of the one-third piece (c.f., Carl did not independently consider one-third piece when estimating how many of his shares would fit into the given

112 whole.). To further investigate her units-coordinating scheme, I decided to proceed to another three levels of units-coordinating task.

The Three Levels of Units-Coordinating Task

The three levels of units-coordinating task was identical to the three levels of units- coordinating task in the previous interview except with some variations in quantities. In the recursive partitioning task, I could not identify indications of her ability to reconstitute the given whole as a unit containing three levels of units. The goal of the task was to help Maggie enact her three levels of units-coordinating scheme and to examine whether her three levels of units- coordinating scheme was permanent. She was asked to express the amount of her cake in terms of the cameraperson’s cake, given that my cake was four times longer than her cake, and the cameraperson’s cake was seven times longer than mine. Three paper strips that were cut to scale were presented for her to use.

Protocol 5.14. Corroboration of Maggie’s three levels of units coordination

Mag: So, can I talk about it with inches? I: Yes. Mag: Okay, so, say I have one inch, and you have four inches. Hers is … seven times … so, twenty-eight inches? I: Why do you say that? Mag: Because I multiplied seven and the other. And, then, it will take twenty-eight of this (her cake) to make this (the cameraperson’s cake).

It is interesting that her first question was to ask my permission if she can consider the length of each cake using inches. Her consistent use of inches as units suggests that she might have re-presented her cake as a measuring unit, mentally measuring my cake using her cake, the cameraperson’s cake using mine, and eventually, the cameraperson’s cake using her cake. Her quick answers along with the indications that were found in the previous interview corroborate

113 my hypothesis that she could coordinate three levels of units, reconstitute the given whole, and review it in the three levels of units.

Although I found several indications that Maggie could take three levels of units as given, it was necessary to confirm my hypothesis by finding solid implications in the recursive partitioning task. In addition, it was also necessary to find more indications regarding Carl’s units-coordinating scheme. So, I revisited the recursive partitioning task with Carl and Maggie for the follow-up of the initial interviews.

The Follow-Up to the Initial Interview

On October 3, 2014, I conducted the follow-up initial interview to find corroborations of my hypothesis regarding Carl’s and Maggie’s units-coordinating schemes. A computer was given to each student, and they were asked to use JavaBars to explain their ways of thinking. It was also the first time that they had used JavaBars, so I guided them in learning how to use the program before I presented the recursive partitioning task. Although they experienced some technical difficulties when using JavaBars, they quickly learned and were able to use the functions that were necessary to solve the task. They first were asked to work separately and then share their answers later.

Maggie’s Recursive Partitioning Scheme and Car’s Repartitioning Action

The recursive partitioning task was as follows: “Make a bar. This bar represents a licorice. Cut the bar into four equal pieces. Pull one piece out. You want to share this mini-piece equally among five people. One of the five pieces will be your share. What fraction is your share of the whole candy?” In the previous interview, neither Carl nor Maggie could independently solve the task. It was possible that cutting strips of paper entailed some practical errors, and those

114 errors might have served to counteract triggering their schemes. Using JavaBars avoids those practical difficulties.

Maggie and Carl made a bar representing a licorice, divided it into four equal pieces, pulled one piece out, and divided the piece they pulled out (one-fourth of the whole) into five equal pieces. They again pulled one piece out (see Figure 5.2) at my request.

a. b.

Figure 5. 2 Maggie (a) and Carl (b) pulled out their shares.

Protocol 5.15 Maggie’s and Carl’s attempt to measure the given whole

I: So, what was the question again? Mag: Um, what is this (pointing to the smallest piece) to all of that (pointing to the largest bar)? I: Yeah. What do you think? Just keep thinking about it and tell me. Mag: Those (the 5 parts of the mid-sized piece) to this (the smallest piece) is one-fifth, and that is (pointing to the mid-sized one and the biggest bar) one-fourth, so… (biting her nail) I would say… (moving a cursor here and there for 21 seconds)...yeah… I: Can you tell me what you are thinking? Mag: I tried to put this (the smallest piece) up there and measure all of them (pointing to the biggest bar), but it’s hard (referring to a technical difficulty using the program). I: Okay. Carl, can you tell me what you are doing? Carl: I am gonna see how these are fitting in here (copying the smallest piece and

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putting each mini-piece on the biggest bar. See Figure 5.3). Mag: Yeah, that’s what I am doing (accidentally deleting the marks on the mid-sized one. See Figure 5.3).

a. b.

Figure 5. 3 Maggie’s and Carl’s distinct ways of measuring the whole.

Note. The screen capture (a) shows that Maggie accidently deleted the marks on the one- fourth piece, and the screen capture (b) shows that Carl put seven copies of his shares on the given whole.

I: So, Okay. Before you actually do that— Can you stop there? Before you put everything in there (looking at Carl’s screen that he was putting the copies of his shares on top of the given bar. See Figure 5.3), can you anticipate...? Mag: (abruptly) Oh, wait. If five of these (pointing to the smallest piece) go into each one, then fives go into the 20. So, it’s one-twentieth. I: One-twentieth? Mag: I guess it’s one over twenty, a fraction. I: (looks at Carl) Does it make sense to you? Carl: Uh huh (nods). I: Why? Carl: Cause if you divide each one into five, it’s five pieces. It’s 20. So one-twentieth. I: So, if you put these pieces on top of this (pointing to the biggest bar), then it’s gonna be what? Mag: It will be twenty of those. ‘Cause this (the smallest piece) is five (pointing to the mid-sized one) … we have five… (Carl interrupts asking technical questions. Then, he accidentally divides the smallest piece into 5 equal pieces. See Figure 5.5. Meanwhile, Maggie partitions the mid-sized one into five as she did before. See Figure 5.3). I: Let’s see if it really works. You (Maggie) said it is one-twentieth, and you (Carl)

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said it makes sense, right? So, can you show me if it really works? Mag: Yeah (tries to overlap the mid-sized piece on top of the biggest bar). Okay, there is five right there (pointing to the mid-sized one. See Figure 5.4), and copy that (trying to copy the mid-sized one, but she cannot. She tries different commands in order to copy it but without success. She puts the mid-sized one on top of the biggest bar). I: (Looking at her) I really like that idea. Mag: Okay, it goes like this (putting the mid-sized one on the first section of the biggest bar and moving it to the second and the third section. See Figure 5.5). It will be all lined up, and it will be— One, two, three, four, five (counting five mini-pieces in the mid-sized piece, then pointing to the four sections of the biggest bar)—twenty. Twenty little ones. I: Ah… Okay.

Figure 5. 4 Maggie’s inserting the mid-sized piece into one-fourth of the whole.

Note. The screen capture shows Maggie’s partitioning the mid-sized piece into five equal

pieces and iterating the mid-sized piece (i.e., five of one-twentieths) on top of the given whole.

Although I first directed Maggie and Carl to pull out one-fourth of the given whole and

repartition it into five equal parts, it was Maggie who independently initiated partitioning the

candy into four parts and the one-fourth part into five equal parts when explaining why the

smallest piece was one-twentieth of the given whole. In addition, Maggie accidently deleted the

117 marks on the one-fourth piece, and there was no visual input that she could instantly apply to the other one-fourth pieces. She independently partitioned the one-fourth piece and iterated the partitioned one-fourth part four times to explain why she said that her share was one-twentieth.

Nonetheless, whether she could envision the whole as a unit containing three levels of units prior to the activity is ambiguous up to this point based on her explanation that she “tried to put this

(the smallest piece) up there and measure all of them". I infer that she had the three levels of units-coordinating scheme available, and she could enact the scheme when explaining how she found one-fifth of one-fourth of the given whole.

Although they both said that they wanted to measure the given whole with their share, it was Carl who copied his piece repeatedly and put them on top of the given whole, while Maggie was experiencing some technical issues. He copied the unknown unit fraction as if he was aware that some number of the unit fractions comprised the whole. Had he constructed the unknown unit fraction as an iterable unit fraction, his actions and explanation might have served as an indication that he had constructed the reversible partitive fraction scheme. Unfortunately, this task alone was not sufficient to determine whether he had constructed the iterable unit fraction.

On the splitting task in the initial interview, I found indications that he was lacking the ability to anticipate the results of iterating actions prior to activity. Hence, I infer that his goal for the given task was to count how many pieces “fit” in the given whole while adding more pieces on top of the given whole. In retrospect, I found that his actions were not different from his counting actions: His intention was to make indefinite numerosity definite, and he counted till he exhausted the whole. Although he later agreed to Maggie’s solutions and restated her explanation, in that it was Maggie rather than he who initiated the actions, his comments do not

118 serve to indicate that he could independently enact the units-coordinating schemes by recursively partitioning the given whole.

Later, Carl accidentally partitioned one-twentieth piece into five equal pieces (See Figure

5.5). This provided a good opportunity to further investigate whether they could take three levels of units as given and recursively use them. To find what fraction one-fifth of one-twentieth is of the given whole, one must recursively insert five units into each one-twentieth piece while being explicitly aware that the whole is a unit containing composite units of composite units. The following protocol illustrates how Carl and Maggie were engaged in partitioning the partitioned whole.

Protocol 5.16 Maggie’s recursive partitioning scheme

I: (Watching as Carl makes one-fifth of one-twentieth accidentally) Carl did something interesting. Carl, can you tell Maggie what you did (See Figure 5.5)? Mag: (She looks at Carl’s monitor and makes one-fifth of one-twentieth. See Figure 5.5.) I: Yes! Tell me what you are seeing right now. Mag: Well, there are twenty of them. I: Yeah. There are teensy little pieces (indicating one-fifth of one-fifth of one-fourth of the bar) there. Can you pull one out? Yeah, what fraction is that of the whole cake? Carl: Um… Mag: Alright (points to the screen and waving her index finger several times), there are twenty… A hundredth? One-hundredth? I: One-hundredth? Okay, you are thinking it is one-hundredth. How about you Carl? Carl: A hundred tenth? I: A hundred tenth? Okay, we have different [answers]. So, what fraction is that (points to the one-hundredth of the bar) of this (points to one-twentieth of the bar)? So, you (means Carl) are saying that this is… Carl: That’s one fifth of that one. But the whole this is one hundred and five. I: One hundred and five? One-hundred fifth? Mag: ‘Cause there’s—how many of those are there...Twenty? Then, you divided it into five right there (points to the middle section that is divided into tiny pieces), so twenty times five. If you make this like this… (divides other sections into five in order to show the whole bar is composed of 100 tiny pieces).

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I: Oh.... Carl, can you see what she is doing? Carl: Guess I counted wrong. I: Uh huh. Um…(looks at Maggie’s dividing each one-fifth into five of one- hundredths) that’s wonderful. Mag: Okay, that one (moves a one-hundredth piece right next to the whole that were divided into a hundred pieces) right there. I: Okay, can you tell him what you did? Mag: You see those lines right there (points to the lines that divided the whole into four equal pieces). So, that’s four. And there is five in each one, so they get down to five (referring to one-twentieth), um, and five little pieces (referring to one- hundredth) in this little one (referring to one-twentieth), and I times these.

(a) (b) Figure 5. 5 Carl’s and Maggie’s repartitions of their share.

Note. The screen capture (a) shows Carl’s screen and (b) shows Maggie’s screen.

Figure 5.5 shows the distinct approaches of the two students. Carl partitioned each one- twentieth piece into five equal pieces. Based on what he previously did and his approximation, one hundred ten, I infer that he tried to count the number of pieces. Maggie partitioned one of the one-twentieth pieces, and her explanation was “there’s four right here [referring to the four sections of the biggest piece], and they get down to five [referring to five pieces with the length of one-twentieth], um, and five little pieces [referring to five pieces with the length of one- hundredth] in this little one [referring to one-twentieth], and I times these". This suggests that she mentally inserted five to each one-piece and found one-hundredth by recursively using her

120 recursive partitioning scheme. She could insert five tiny pieces (each sized one-one hundredth of the whole) into one-twentieth of the whole, iterate this twenty times, and find that one tiny piece was one-one hundredth of the whole. According to Steffe and Olive (2010), reinteriorizing composite units is essential when recursively coordinating composite units. Once it is reinteriorized, each unit of a composite unit becomes a place holder that holds a single unit or composite units (Olive, 2000). Her later explanation while recursively partitioning each one- twentieth piece corroborates my hypothesis that she could take three levels of units as a unit prior to activity. However, I could not find any indications that the same operations were available to

Carl, who was depending on visual estimation.

Summary

Carl’s Two Levels of Units-Coordinating Scheme

I analyze that Carl coordinated two levels of units and was able to produce three levels of units in activity, but he could not review it as a unit containing three levels of units. In the equipartitioning task, he could adjust the length of his share to make five of his shares fit in the whole while obtaining a closer estimation of his share. His estimations, which were approaching one-fifth of the whole on each try along with his independent adjustments, imply that he could envision the given stick as comprising five iterations of his share and disembed his share in order to determine whether five of his adjusted shares fit into the given whole. Thus, I infer that Carl used his number concept five as a template and projected it onto the given whole by coordinating two levels of units.

In the two levels of units-coordinating task, he could count by sevens by progressively uniting sevens but could not reconstitute the given string that was 24 cm long as three of seven centimeters and three centimeters more. His struggle when monitoring the length of the iterating

121 string serves as an indication that Carl was with TNS. In the splitting task, he showed that he lacked the ability to posit his hypothetical stick, iterate it and anticipate that the five iterations of his hypothetical stick comprise a given whole. I analyze that he could iterate his hypothetical stick in activity, but I could not find any indication that his hypothetical stick was iterable in such a way that it was one of the five identical units that comprised the given whole. In addition, I hypothesized that his lack of simultaneous coordination of disembedding and iterating operations along with his lack of anticipation might have served to counteract coordinating three levels of units. In the recursive partitioning task and the follow-up interview, I found the corroborations that he was limited to coordinating two levels of units. In the three different recursive partitioning tasks, Carl could independently insert a composite unit into a unit of a composite unit envisioning the given whole as a unit containing three levels of units. Coupled with my analysis of his actions and explanations in the splitting task, I found that the composite units were not iterable to him, which was one of the constraints in the construction of coordinating three levels of units.

Maggie’s Three Levels of Units-Coordinating Scheme

I analyze that Maggie could coordinate three levels of units and take a unit containing three levels of units as given for further operations. In the equipartitioning task, I analyzed that she could coordinate two levels of units based on the way she adjusted the size of her piece to make a fair share. I found a corroboration that she could coordinate two levels of units in the missing addend task. She found that she needed 15 cm more to make the 14 cm long candy to 29 cm by adding one more 14 cm long candy and one centimeter more. This is an indication that she could count-up-to in order to find the missing addend, which can serve as an indication that she coordinated two levels of units (Steffe & Olive, 2010).

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In the three levels of units-coordinating task, I found indications that the three levels of units-coordinating scheme was available to her. She could iterate three six times and find the relation between the smallest piece of cake was one-eighteenth of the given whole. I analyze that she was able to iterate composite units and assimilate the results of her iterating operations to the product of two given composite units, three and six. In the splitting task, her uncanny estimation along with her explanation suggest that she could posit a hypothetical stick and mentally iterate and adjust it to make five times of her hypothetical stick equal to the given whole. Based on how she assimilated her iterating actions to multiplication in the three levels of units-coordinating task and her anticipation of the length of her piece while mentally iterating the hypothetical piece in the splitting task, I hypothesized that she had the three levels of units-coordinating scheme available. Although I could hardly find any indications that she could coordinate three levels of units prior to activity in the recursive partitioning task in the initial interview, I could find corroboration of my hypothesis in the follow-up initial interview. She showed that she could coordinate three levels of units by iterating five of one-fifth of one-fourth four times to find her share was one-twentieth of the whole. In addition, she could take one-twentieth as an input and recursively use it to find one-fifth of one-twentieth. At this point, I concluded that she could coordinate three levels of units prior to activity and review the whole as a unit containing three levels of units.

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CHAPTER 6

COORDINATING TWO ORIENTED QUANTITIES

To investigate the hypotheses regarding my research questions, I investigated Maggie’s and Carl’s schemes and the operations that were involved when finding (1) sums of two positive quantities, (2) sums of two negative quantities, (3) sums of two oppositely oriented quantities,

(4) missing addends, and (5) differences between two oriented quantities. I presented various tasks that can be modeled by using two oriented quantities, including measures of temperature, account activities, elevation, and distance of travels, for them to represent additive situations. I first introduced hands-on activities to enact their units coordinating schemes. Later, I provided computer programs, such as JavaBars and Geometer’s Sketchpad (GSP) so that they could more efficiently engage in representing the given situation (see Appendix for the complete list of the selected tasks).

Adding Two Positively Oriented Quantities

The goal of the teaching episode of October 27 was to explore Carl’s and Maggie’s strategies when finding a sum of two oriented quantities. My hypotheses were that the two levels of units-coordinating scheme is sufficient to find a sum of two positively oriented quantities (or two negatively oriented quantities) but it is not sufficient to find a sum of two oppositely oriented quantities. I prepared three tasks: (1) a situation that could be modeled as a sum of two positive quantities, and (2) a situation that could be modeled as a sum of two negative quantities. The first task was as follows:

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Today, the measure of the temperature at 5 a.m. in Winder-Barrow County was 45 °F. The change in the temperature over a twelve-hour period was 17 °F. How can you combine the two measures of the temperature in order to find the measure of the temperature at 5 p.m.?

I provided strips of transparencies on each of which a picture of a thermometer was printed (hereinafter, thermometer strips) and various colors of dry erase markers so that the students could represent a measure of the temperature as a directed segment on a thermometer strip. Before starting to work on the tasks, they both agreed to use a directed segment starting from 0 for representing a measure of temperature (see Figure 6.4). I gave them three thermometer strips for each task: two for representing the two given measures of temperature and one for their solutions.

I first guided them in constructing an upward-pointing directed segment for the initial measure of the temperature, 45 °F (see Figure 6.1-(a)). Then, on a separate thermometer strip, I asked them to draw a directed segment for representing the measure of the change in temperature over a twelve-hour period. Maggie constructed an upward-pointing directed segment approximately from 45 °F to 62 °F (see Figure 6.1-(b)). Carl first drew an upward-pointing directed segment from zero to 45 °F, and then made another upward-pointing directed segment approximately from 45 °F to 62 °F (see Figure 6.1-(c)). When asked to explain what they did,

Maggie answered, “I just want [to make] 17 from 45", and Carl said, “I showed what it was before and the 17".

It seems as if they both assimilated the given situation as adding two upward pointing directed segments. Although they both found the same solution, it was possible that the way they treated “the 17” was not the same. It was probable that Maggie mentally disembedded the change in temperature and used the directed segment from 45 to 62 as a representative of it, considering

125 her actions and explanation that she showed in the splitting task in the initial interview.

Considering that he was with TNS, it was possible that “the 17” was embedded in his counting- on acts. To test my hypotheses, I asked them to translate “the 17” to 0 in a separate thermometer strip (see Figure 6.1-(d)). Next, I asked them how they would combine the two thermometer strips (the measure of the temperature at 5 a.m. (see Figure 6.1-(a)) and the change in temperature, “the 17” (see Figure 6.1.-(d)) to find the measure of the temperature at 5 p.m.

Maggie placed the thermometer strip of “the 17” on top of the thermometer strip of Figure 6.1-

(a) so that the two directed segments were lined up as shown in Figure 6.1-(c). Carl, on the other hand, erased the directed segment from 0 to 17, redrew it from 45 to 62, and then placed it on top of the thermometer strip of Figure 6.1-(a). When asked why he redrew it, he said, “because it is down here [referring to the change in temperature starts from zero], and it has to be up here

[referring to the change in temperature has to start at 45] ".

Maggie’s translation of “the 17” suggests that she could disembed 17 and move it around to combine it with the directed segment of 45 as if she envisioned the 17 and 45 as explicitly nested quantities comprising the measure of the temperature at 5 p.m. On the other hand, Carl’s decision of deleting the directed segment from zero to 17 and redrawing it from 45 suggests that

“the 17” and the directed segment from 45 to 62 might have not been identical to him. My hypothesis is that he might have assimilated 17 as a numerical composite, not as a composite unit that was reflectively abstracted so that it could be used as if it were a single unit. His comment that the change in temperature should be “up there”, along with his lack of the disembedding operation that he showed in the splitting task supports my hypothesis. If he did/could not reflectively abstract the change in temperature as a composite unit, it was probable that he could not envision the measure of the temperature at 5 p.m. as a unit containing two composite units.

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However, I need more corroboration in order to analyze their actions. So, I decided to present the next task.

(a) (b) (c) (d) Figure 6. 1 The measures of temperature that were drawn by the students on the thermometer strips (reconstructed by the author).

Adding Two Negatively Oriented Quantities

The second task was designed to investigate how they coordinate two negatively oriented quantities. The task was as follows:

The temperature at 5 p.m. was -12 °C in Jackson county in Wyoming. The change in temperature over a twelve-hour period was -32 °C. Find the temperature at 5 a.m..

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Carl’s assimilation to Counting-On

Carl first asked me whether the temperature dropped to -32 °C or the change in temperature was -32. I told him that the change in temperature was -32, and he was supposed to find the temperature at 5 a.m.. Carl drew a directed segment from 0 to -12 for the measure of the temperature at 5 p.m. Next, he drew another downward segment from -12 to about -20, stopping briefly without withdrawing his pen from the thermometer strip to see Maggie’s answer. After he saw that Maggie drew a directed segment from -12 to -48, he continued to draw the downward segment until it reached -32. He then intently looked at the directed segment, tapping on the thermometer stip three times and extending the length of the directed segment so that it ended at

-42 (see Figure 6.2-(b)). When asked to explain what he did, he said, “I started at -12 and went down 32", and he wrote -32 right beside the extended directed segment. His first attempt to draw a directed segment for the change in temperature from -12 to -32 serves as a corroboration that he did not enact the disembedding operation. In addition, I consider his first attempt to be a necessary error (Steffe & Olive, 2010) that stemmed from his lack of the ability to reflectively abstract -32 as a composite unit. Thus, I consider the extension that he later made as an assimilation to his counting-on action and the 32 units as a numerical composite rather than a composite unit.

To further investigate how Carl coordinated the two given measures of the temperature, I asked him to draw a directed segment separately for the change in temperature. He made the directed segment from 0 to -32. I then asked him to combine the two thermometer strips (one for the measure of the temperature at 5 a.m. and the other for the change in temperature) to show how he found the answer; he lined them up at zero (see Figure 6.2-(d)). At my request to explain, he slid down the directed segment of the change in temperature so that it started at -12 (see

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Figure 6.2-(e)) saying, “it starts at -12 and make it goes down 32". I then asked him how the combination of the two thermometers was related to the measure of the temperature at 5 a.m. He rearranged the three thermometers in such a way that they all lined up at zero (see Figure 6.2-

(f)). When asked to explain how he found the temperature at 5 a.m., he said, “so, showing how it adds up? … I matched them up all down at zero so that I can show it went down 32". Then, he wrote 12+32=44 vertically and said, “[I] need to add the minus sign", and corrected it to 12+32=-

44.

Coupled with the indications that I found in the previous task, I infer that Carl assimilated the sum of the two negatively oriented quantities to his counting-on actions. However, it was not clear whether he treated the given quantities as composite units and combined them, producing a unit containing the two composite units before I saw his final arrangement. His final arrangement of the three thermometers, along with his vertical computation, indicates that he used the directed segments as pointers for the numeric values and used the numbers for his vertical computation.

Thus, it is more likely that he slid down the thermometer strip of -32 in order to show how his vertical computation worked on a thermometer strip.

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(a) (b) (c) (d) (e) (f)

Figure 6. 2 Carl’s directed segments when solving -12+-32, reconstructed by the author.

Maggie’s Translation of the Change in Temperature

Maggie first drew a directed segment from -12 to around -48 and then added another directed segment, which starts at zero and ends at -12, at my request to show the measure of the temperature at 5 p.m. (see Figure 6.3-(a)). I then asked her to draw the directed segments for the two given quantities on separate thermometer strips (see Figure 6.3-(b) and Figure 6.3-(c)).

When asked how she could use the two thermometers to find the temperature at 5 a.m., she put the directed segment of -12 right below the directed segment of -32. Then, she located the directed segment of -48 right beside the two. She said, “I just added. I combined them together".

Then, she pointed to -48 and said that the temperature at 5 a.m. was -48.

Her first representation (see Figure 6.3-(a)) as well as the way she combined the three directed segments by translating the directed segment of -12 to -32 suggest that she treated the

130 change in temperature as a composite unit and used it as if it were a single unit. It was possible for her to review her answer as a combination of two composite unit, considering that she could assimilate the situation with ENS. In addition, her answer, -48, shows that she did not carry out counting actions or computations. Considering that she could coordinate three levels of units, she could have easily found -44 by counting 12 down from 32 if she had enacted the scheme. In retrospect, I found that she used her measuring scheme and used the number lines as if they were rulers. Therefore, I infer that she visualized her answer, -48, as a combination of the two directed segments, -32 and -12, as if she constructed a unit containing the two composite units, -32 and -

12. Her comment, “I just added. I combined them together" is a strong indication that combining the directed segments meant adding the two quantities to her.

(a) (b) (c) (d)

Figure 6. 3 Maggie’s directed segments when solving -12+(-32), reconstructed by the author.

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Maggie’s Composition of Two Composite Units & Carl’s Counting-On

From this teaching episode, I found that they both were capable of finding the sum of two negative quantities and the sum of two positive quantities. However, I found multiple indications that they might have been engaged in different mental operations. I analyze that Maggie reconstituted the given situation by combining two composite units. Maggie’s translation of the directed segments can serve as an indication that Maggie could reflectively abstract the measures of temperature and combine them as if she were combining two composite units. In the case of

Carl, I analyze that he made an association between his counting actions and vertical computations. His way of lining up the directed segments at 0 serves as an indication that he was not coordinating two composite units to find the sum. In addition, when adding two negative quantities, Carl added them vertically and put the “-” sign in front of the sum. It seems like it was not necessary for him to consider the orientation of the given quantities when both quantities have the same orientation.

Adding two Oppositely Oriented Quantities

In the previous teaching episode, none of my students had any problems in finding the sum of two positively (or two negatively) oriented quantities. Although I found some indications of their different ways of finding the sum, I supposed that I could observe much clearer distinctions in their ways of finding the sums when they coordinated two oppositely oriented quantities. I also wanted to further investigate Carl’s conception of the “-”sign. Therefore, I prepared the two situations that could be modeled as a sum of two oppositely oriented quantities for the teaching episode that was held on October 31. The first task was as follows:

In Jackson Hall County, it was -34° F at 5 a.m. The temperature rises 79° F over a twelve-hour period. What is the temperature at 5 p.m.?

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To investigate how they coordinate each given temperature, I first gave them a thermometer strip to make a directed segment modeling the temperature at 5 a.m.. When they finished drawing a downward-pointing directed segment for -34° F, I gave them another thermometer strip to make an upward-pointing directed segment to demonstrate the change in temperature, 79° F. Lastly, I gave them empty thermometer strip to represent the temperature at 5 p.m.

Protocol 6.1 Finding the sum of -34 and 79

I: I want you to find out what the temperature is gonna be at 5 p.m.. Carl: (Puts two thermometers side by side) um… You look at them, and since it is 79 and this is 34, you can subtract it. I: Subtract it? Um… Can you use these two thermometers, like some how…? Mag: We can … um… Start right here (puts the tail of the directed segment of 79 at the head of the directed segment of -34), and it will be right here (marks a hash mark around 45 on the thermometer of -34. See Figure 6.4-(a). She then constantly moves the thermometer strip in order to measure where the mark was.) Carl: Like 40. I understood. I: What do you think (looks at Carl)? Did you see what she is doing? Carl: Yeah, like that. You line them up at zeros (lines two directed segment at zero. See Figure 6.4-(d)) I: Lining up at zero? Carl: Yeah… Hum …You mark it or… (looks at Maggie’s thermometer strips then nods) yeah. What she did. I: So, can you tell me what she did? Carl: Well, she lined them up at zeros. And, then, she … uh… Lined them up at zeros, and then, she… uh… she went up… up here (points at the upper part of the thermometer). I: Okay. So, can you estimate like what the temperature is gonna be at 5 p.m. looking at that? Carl: (Starts at zero and points at the several marks on the thermometer consecutively) um… what… 40, 45 or something? I: 40 or 45? Okay. So, I am gonna give you a new thermometer. I want you to draw the temperature at 5 p.m. here. Carl: 5 p.m.? I: Yes.

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Carl: Alright. I: (Looks at Maggie’s drawing) Maggie, can you explain to me how you got that? Mag: Because I put…um… Because in the morning … wait, I forgot what I was gonna say… I: -34 degrees? Mag: Uh, yeah, it was where it started. I started the zero [of 79] right there, and then, here is forty-seven (indicates the hash mark that she previously made). And, then, I just marked forty-seven right here (points at her new thermometer, see Figure 6.4- (b).) I: Okay. Carl, can you tell us what your method is? Carl: Uh, I matched the two thermometers, and this one is at -34, and this one is… It says it went up 79 degrees. So, I looked at them, and then, I said, “what’s 79 minus 34, and it gave me 45” (See Figure 6.4-(e)). I: So, you did… I think ya’ll reached the same answer with different methods. And I am very interested in your methods, separately. So, what I want you to do before we go any further, I want you to write the equation which represents what you did. Carl: Alright. Mag: Representing the equation? Representing the equation? I just used the pictures. I: Yeah, but can you write an equation which represents what you just did? Mag: Uh huh (writes -34 + 79 = 47 while intently looking at the thermometers, see Figure 6.4-(c).). Carl: (Immediately writes 79 - 34 = 45.) I: Carl, you did 79-34. Can you tell me what you did? Can you make a correlation between this equation and how you combined these thermometers? Carl: How I lined up like that (aligns the thermometers at zero, see Figure 6.4-(d).)? Well, start right here (points at -34), and this goes up 79. You gotta count, five (points at -30), ten (points at -20), fifteen (points at -10), twenty (points at 0), twenty-five (points at 10), thirty (points at 20), thirty-five (points at 30), forty (points at 40)… like that. Mag: (Emphasizes the “-“ part in the “+” sign) I: So, Maggie, can you tell me what you did? Mag: Um, it started at 34, -34, and then… I put added because it went up. But...I think it will be added.

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Figure 6. 4 Maggie’s and Carl’s coordination of the sum of the measure of the temperature at 5 a.m. and the change in temperature for 12 hours.

Note. The screenshot (a) illustrates how Maggie aligned two thermometers; she drew the directed segment of -34 on the top thermometer and 79 on the bottom one. On the top thermometer, the hash mark that Maggie drew is located at around 45. The screenshot (b) shows the temperature at 5 p.m. as drawn by Maggie. (c) is Maggie’s equation. The screenshot (d) shows how Carl aligned two thermometers at zero. The screenshot (e) shows the temperature at 5 p.m. as drawn by Carl. His vertical computation is observed on the top of the thermometer.

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Maggie’s Measuring Scheme

Maggie first lined up the two directed segments of -34 and 79 so that the tail of the directed segment of 79 started at -34. The way she lined up the two directed segments, along with her explanation,

I started the zero [of 79] right there, and then, here is forty-seven [indicating the hash mark that she previously made]. And then I just marked forty-seven right here [points at her new thermometer].

and her equation, “-34 + 79 = 47”, suggest two possible hypotheses;

She interpreted the upward-pointing directed segment, 79, as actions, such as counting-up

or upward movement, that yields an increase in the sum, marking the location at which

the actions stopped, and connecting the mark and 0 to make a directed segment as

requested;

She treated the two given directed segments as composite units, combined them with the

awareness that the combination of the two composite units produces the measure of the

temperature at 5 p.m.

I consider the latter is more probable because (1) the three levels of units-coordinating scheme was available for her, (2) she had found the sum as if she were combining two composite units in the previous teaching episode, and (3) she would have found 45 as a solution instead of

47 if she activated her counting-up scheme.

When asked to write an equation modeling the given situation, she stated that she “used the pictures”. Her use of the directed segments as if she were measuring the sum using two other directed segments suggests that she constructed a conceptual ruler and mentally re-presented each quantities as continuous but segmented quantities. Her vertical translation of the directed segment of 79 can serve as an indication that she could treat 79 as one thing as if it were a

136 composite unit. I think she was surprised, because she might have thought that I was asking her to perform numerical computation (Herscovics & Kieran, 1980), and she did not want to start all over again using a different method. After she was reassured that I wanted her to express what she did in an equation, she wrote “-34 + 79 = 47”. I do not consider her answer, 47, to be a computational error. Rather, I analyze Maggie’s ‘incorrect’ answer 47 as an estimated measurement and her equation as her representation of her coordination of “the pictures” in order to use her measuring scheme.

Carl’s Assimilation to a Take-Away Situation

Carl, on the other hand, did not align the two directed segments as Maggie did. Instead, he lined up all of the directed segment at 0. It was surprising that he did not change the locations of the directed segments while looking at Maggie’s thermometer strips when asked to explain how Maggie arranged the directed segments. His vertical computation 79 – 34 = 45, along with his determination of his original arrangement of the directed segments, suggests that arranging the directed segments in the way that Maggie did was not meaningful to him. Later, to my question as to why he wrote the vertical subtraction, he answered that he just put the bigger number on top and the smaller number on the bottom. Therefore, I infer that his directed segments, -34 and 79, implied numeric values for his numerical computation. In addition, his later counting-up actions serve as a solid indication that he did not combine the two directed segments as Maggie did.

I analyze that he treated the directed segments as pointers to the numeric values for the vertical computation. His later use of counting actions suggest that (1) he regarded adding two oppositely oriented quantities as a take-away situation, and (2) he solved the take-away situation by subtracting two numeric values. His counting spanned the length of 79 units on the

137 thermometer strip, but he counted aloud as though his goal were to count 45 units. His counting did not correspond to the placement of his finger; after counting up the first five, he jumped up by tens with his finger while counting aloud by fives (only tens were numbered on the thermometer, while fives and tens were both marked with tick marks). He stopped counting as soon as he uttered 40 at which he tapped his finger. This disagreement between what he wanted to count and what he actually counted supports my hypotheses; once he assimilated the given situation as a take-away situation, he found the answer by the numerical computation. When he tried to explain his solution by counting, his new goal became finding 45 by counting the units between the subtrahend and minuend. My analysis of the reason why he could not find the apparent conflicts is because he was lacking reconstituting 45 as a combination of two composite units. In other words, the two levels of units coordinating scheme was not sufficient for him to assimilate the situation as the sum of two oppositely oriented quantities. Thus, I infer his answer,

45, was a result of his procedural scheme and he used his counting scheme in an attempt to justify his answer.

Adding Two Oppositely Oriented but Unknown Quantities

Although I found some indications that Maggie could review her answer by reconstituting it as a unit containing two composite units, and Carl assimilated the given situation as a take-away situation, it was challenging to make some inferences about how they coordinated quantities when the problems could be easily solved by mentally computing two numbers. In order to observe how they coordinate quantities without numerical computations, I provided the task excluding numerical values in order to find corroborations of my hypotheses that I made in the previous task. I gave them a transparency with a downward-pointing directed segment of the unknown temperature, a representing the temperature in Athens at 7 a.m. Then, I gave them a

138 separate transparency showing another unknown temperature, b, as an upward-pointing directed segment, representing the increase in temperature over a 12 hour-period. Maggie and Carl were then asked to find the temperature in Athens at 7 p.m. (see Figure 6.5). Empty transparencies on which they could mark their answers were provided to both of the students.

They both arranged the given directed segments as shown in Figure 6.5. Maggie drew a blue downward-pointing directed segment to represent the temperature at 7 p.m. In contrast, Carl made a mark (he always made marks for the measures of the temperature. Only at my request did he drew the directed segments), and then drew an upward-pointing directed segment at my request to represent the measure of the temperature using a directed segment (see Figure 6.5-(c)).

The following protocol begins with their explanation of how they find the temperature at 7 p.m.

Protocol 6.2 Finding the sum of a and b when a<0, b>0, and |a|>b.

Mag: So, this is, I wanna say it’s 45 (points at a), this big thing. So, this (points at b), it only rises 20. So, it will be … (draws the blue arrow in Figure 6.5-(b)). I: Okay. Carl, can you tell me what you did? Carl: I just lined the arrows up and marked the new temperature. Right there (points to the horizontal mark that he made on the blank transparency (see Figure 6.5-(c)), if you want to know. I: Can you line them up for me one more time? Carl: (Carefully lines up his mark at the tip of b.) I: Ok, this is the mark. I want you to express the temperature using arrows. Carl: Using arrows? I: Yes. Carl: (Draws an arrow that was approximately as long as b, and lines it up to show that it is equal to b without putting the marked film on top of anything. He looks satisfied with his answer.) I: Okay, that much? So, if you put them together (referring to arranging all the directed segments) … does it make sense? Carl: (Writes c underneath the arrow that he drew.) I: (To both students) Using the a and b, can you express the temperature at 7 p.m. using a and b? Mag: (Writes a+b=-20)

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I: (Looks at Carl) You are saying it’s c. Can you tell me what it is? Carl: Cause… what? Saying what c is? I: Uh huh. Carl: Oh… (touches three transparencies and puts them on the desk separately), I didn’t know I needed to do that, but… I put a on that one (points to the given a) and b for that one (points to b) and that’s (points to the arrow that he created) c. I: Ok. But, can you express c in terms of a and b? Using a and b? Carl: (Writes a=b=c) I: (To Maggie) …and you are saying …. Mag: I think I know what c is. I can round it. This (writes c on top of -20 in her equation, a+b=-20) is c. I: So, a+b=c? Can you tell me why c is a plus b? Mag: Because, there is a, and then, it increases b, and then (pointing to the blue arrow), this right here is this (pointing to c). I: Uh … (to Carl) can you tell me your method? How did you think about it? Carl: Since the first two… since the first one is a and the second one was b, you do a+b=c. I: How do you do it by composing the two thermometers? I saw you did that (arranges the two thermometers as Carl previously did). This is a, and this is b. How are those two related to this (pointing to c)? Carl: Um… (writing an opposite sign in front of c) I: Negative c? Carl: Because it [a] is negative, and it [the temperature at 7 p.m.] is still negative.

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(a) (b) (c) Figure 6. 5 Maggie and Carl’s distinct ways of arranging the given quantities, reconstructed by the author.

Note. Figure (a) shows the given directed segments modeling the measures of the temperature at 7 a.m. and the change in temperature over 12 hours. Figure (b) shows how

Maggie arranged the two given directed segments to find the measure of the temperature at 7 p.m. The temperature at 7 p.m. was modeled by the blue directed segment. Figure (c) shows how

Carl arranged the directed segments.

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Maggie’s Reconstituting the Sum as a Unit Containing Three Levels of Units

The noticeable difference in the two students seemed to reside in Maggie’s reconstitution of the given situation by coordinating two composite units and her ability to review the given situation in relation to the two composite units and their coordination, which involves three levels of units coordination. Maggie’s assumption of a as being -45 and b as 20 suggests that (1) she mentally measured the given directed segments in her mind (possibly used her conceptual ruler that she used in the previous task), and (2) she regarded a and b as composite units that could be represented with specific numeric values. Her downward-pointing blue directed segment indicates that her answer was a composite unit of a negative quantity. Her equation, a+b=-20, along with her explanation, “there is a, and then, it increases b, and then [referring to the blue arrow], this right here is this [referring to c]", corroborates my hypothesis that she could reconstitute the given situation by (1) assimilating the givens as two composite units, (2) coordinating the composite units to find the measure of the temperature at 7 p.m., and (3) reviewing the result as a combination of the given two composite units. When coupled with how she coordinated two composite units of specific numeric values in the previous task as if she were measuring, the indication is solid that she used her conceptual ruler for estimating the directed segments. Her estimation, -20 instead of -25, corroborates my hypothesis that she used her measuring scheme and found the estimation of the sum. Thus, I infer that she could reconstitute the given situation by coordinating two composite units, producing three levels of units.

Carl’s Finding c as a Place For Stopping His Action

Carl’s way of arranging unknown directed segments is instructive because it provides some indications for how he might have coordinated the given measures of temperature in the

142 previous task. Although Carl arranged the two directed segments as Maggie did, Carl’s solution, c, appeared to have a different meaning than Maggie’s. His initial solution was marking the end of b. It did not seem that he thought it was necessary to construct the directed segment to represent c. He drew an arrow that was identical to b to represent c only at my request. These actions can serve as an indication that he assimilated the end of a as a place where he started counting up (or sweeping up) and the end of b as a place where he stopped counting.

Furthermore, the directed segment of c that was identical to b suggests that Carl was considering c as neither a composite unit comprising two composite units of a and b nor a negative quantity that was accumulated from the reference point. When asked to explain what c was, he explained which directed segments were a, b, and c. His initial equation, “a=b=c”, suggests that he might have indicated a, b, and c as the same because they were all related to his counting-on action: a for where he started counting, b for how much he counted, and c for where he stopped counting.

Or, he might have used the “=” sign to represent the the flow of his action, and wrote “a=b=c” to express that he started counting from a, then counted about b, and reached c. Carl did say, “Since the first two… since the first one is a and the second one was b, you do a+b=c", in referring to

Maggie’s explanation of her equation. Based on his previous vertical computation, it is probable that his equation implied a vertical computation where “the first one” as a number on top, and

“the second one b” as a number on the bottom. Although he later mentioned that c must be negative, I could not find any indications that he reviewed c as an accumulated quantity.

The salient distinction between Maggie’s and Carl’s ways of finding the temperature lies in their schemes, actions and operation they used when finding c, the measure of the temperature at 7 p.m. Maggie modeled the situation using two directed segments and explained c as a result of coordinating the two quantities, a and b. On the other hand, Carl assimilated his counting

143 actions to model the given situation. His c indicated the location where he stopped counting, rather than a quantity representing a result of coordinating two composite units, a and b. I hypothesized that Maggie’s three levels of units coordinating scheme allowed her to model the given situation by coordinating two composite units and to review c as a unit containing the two given quantities. In contrast, I do not find any indication that Carl’s c represents a quantity as a coordination of two composite units. Rather, c seems like the location at which he stopped counting-up. Coupled with what I observed in the previous teaching episode, I analyze that his schemes and operations were not sufficient to reconstitute the given situation as a quantity produced by coordinating two other quantities. My analysis is that Carl’s two levels of units- coordinating scheme served as constraints when reconstituting the given situation as a representation of a quantity that was composed of two other quantities.

The students continued to work on modeling additive situations in the context of a monetary system, including bank account activities, for the following two teaching episodes that were held on November 3 and 7 (see Appendix). Maggie and Carl’s ways of modeling the given situation were consistent with the ways they showed in the previous teaching episodes. I found multiple corroborations of my hypotheses that Maggie used directed segments to coordinate two oriented quantities, which resulted in constructing the sum as a unit containing three levels of units. On the other hand, Carl preferred to use his numerical computation first whenever specific numeric values were provided. He explained his numerical computations in such a way that he put a bigger number on top and the smaller number on the bottom. Then, he later explained his methods using his counting scheme at my request. I did not find any indications that he could review the sum of the two oppositely oriented quantities as a unit containing three levels of units.

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Finding Missing Addends I

For the teaching episode of the 10th of November, I prepared a missing addend task. One of the goals of this teaching episode was to test whether or not Maggie could take three levels of units as an input to review the other quantities in relation to the other quantities. In the case of

Carl, the experiment was to explore whether he could reflectively abstract his counting-up actions and use them for reconstituting the given situations.

Before they started, they looked at an actual bank statement and discussed a net amount as a reference, using a debit card for purchases as adding negative quantities and depositing a checks as adding positive quantities to the net amount. The task was as follows:

You used your debit card and spent $65 at a grocery store yesterday. Some account activity occurred in your bank account overnight. In the morning, you have found that you have $50 more than you had (the net amount before spending $65) in your account. Model the given situation using directed segments, and find the account activity that occurred overnight.

I encouraged Carl and Maggie to model each account activity using a directed segment on a number line and use them to find the unknown account activity that occurred overnight.

Carl’s Nonreversible Counting-Up-To Scheme

Carl and Maggie both counted up either by fives or tens on vertical number lines and said that the account activity that occurred overnight indicated a $115 gain in their account. Carl first counted up from -65 to 50 by fives and then used his numerical computation to explicate and to model his counting actions. Considering that he first engaged in the vertical computation and later he used his counting schemes to check his solution in the previous teaching episode, it seems as if he made a strong association between his procedural scheme and counting schemes.

After the counting, Carl wrote 65, 50 and 115 vertically as if he were adding 65 and 50 to get

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115. He then looked at the number line and saw that it was -65 instead of 65. He placed the “-” sign in front of 65, added “$” in front of 115 (see Figure 6.6-(a)).

(a) (b)

Figure 6. 6 Carl’s vertical computation (a) and Maggie’s vertical computation (b) when finding the missing addend.

His counting actions suggest that he could progressively unite fives to count up to 50 from -65. However, it is unclear whether Carl could reflectively abstract the result of his counting acts while keeping track of the two given quantities. In other word, it is ambiguous whether he could review 115 as a unit containing 50 and the opposite of -65. As he proceeded, I intervened and told him that -65+50 would result in -15. He agreed with me, but he insisted that his equation was also right. His determination to the answer while admitting that the expression of -65+50 would result in -15 supports my hypothesis that his equation was the representation of his counting activity, and 115 was not a unit containing the other two quantities. Also, it serves as a strong indication that he did not construct the concept of negative quantities.

In the previous teaching episode, he justified his vertical computation, 79 - 34 = 45, by counting the span of 79 units as though he was counting 45 units. In this missing addend task, he counted by fives, and then tried to justify his counting by writing the vertical computation. In

146 both cases, he started counting from the first given numerical value, used the orientation of the other quantity as the direction to count. I infer that he placed his solution below the bar of the vertical computations and used his counting scheme (or vice versa) to justify his answer instead of checking whether his answer was compatible with different methods; He counted from -34 to

45 as 45 units for his vertical computation of 79 - 34 = 45 in the previous teaching episode, and, in this teaching episode, he wrote -65 + 50 = 115 for his counting from -65 up to 50. These instances suggest that he was lacking reviewing the given situations using different ways of reasoning. It seems that the association between his counting and his numerical computation is so strong that he treated them as if they were identical operations that inevitably resulted in the same answer.

Maggie’s Use of Inclusion Relations among the Quantities

Although Maggie also counted up to 50 from -65 by tens, her equation and the way she explained the given situation was different from Carl’s. After writing her equation, 115-65=50

(see Figure 6.6-(b)), she explained, “we thought it was this [draws a box around 115], and we started right here [underlining -65]. So, this is… we are like checking the answers, ‘cause we got

50, and [115 is] as much as it went up". I asked her whether “-” in front of 65 meant “minus” or

“negative", she said that it was negative. According to her explanation, her vertical computation should be considered as 115+(-65)=50 instead of 115-65=50, because she considered the “-” sign as an orientation. Coupled with her explanation, “we are like checking the answers", I analyze her vertical computation as her representative of her reversible reasoning; By reflectively abstracting her counting-up actions, she took 115 as an input and reconstituted it using the given conditions. Once reflectively abstracted, these composite units of 115, 50, and the opposite of -

65, became explicitly nested, forming 115 as a unit containing three levels of units, and she

147 could use the inclusion relation when reviewing the given situation by freely disembedding and embedding each composite unit. Thus, I infer that she first engaged in counting up-to actions and reviewed the result of her counting actions in the light of coordinating two composite units.

(More corroborations are found in later teaching episodes that were held on November 14 and

December 5.) I also argue that she was engaged in reversible reasoning when taking the result of her counting action, 115, as an input and reconstituting the given situation. However, whether she could take the result of her action as a unit containing three levels of units prior to activity is ambiguous up to this point.

Finding Missing Addends II

On the 14th of November, I introduced Geometer’s Sketchpad (GSP) to the students. Carl expressed his desire to learn and use “some computer program to do math", several times. Since they have been engaged in modeling the given situation on transparencies using directed segments, I guided them to use GSP for their convenience and to accomplish the additional goals of later teaching experiments. Almost half of the teaching episode on November 14 was devoted to guiding them to how to use the program. They used vertical lines when modeling the measures of temperature and horizontal lines for account activities. Then, the following task was given to the students:

Each of you have a bank account, and the sum of the available amounts in both accounts is always $8. Make problems for each other. Use GSP to model your questions and answers.

The goal of this task was twofold: (1) to encourage Carl to use directed segments instead of numeric values so that he could coordinate composite units rather than using his numerical computation, and (2) to investigate whether Maggie can review her answers by reconstituting the

148 given situation by establishing inclusive relations between the given quantities. In the previous teaching episode, I found the indications that she could reconstitute the given situations from her equation that she wrote to “check” the answer. My intention was to find corroboration of my hypothesis. The directed segment of 8 for the sum was given to the students to encourage Carl to consider $8 as a quantity, rather than a numeric value.

First, Carl made a directed segment of -17 on the number line representing his account.

Maggie constructed a directed segment of 25. She then moved the directed segment of -17 above the directed segment of 25 (see Figure 6.7) in order to show how the combination of the two directed segments produced the given directed segment of 8.

Figure 6. 7 Maggie's solution for Carl's problem

In Maggie’s turn, she made the directed segment of -12 on the number line representing her account. Then, Carl made a directed segment of 8 on the number line that represented his account (see Figure 6.8-(a)). When asked to explain what he was thinking, he said, “I just put eight dollars, ‘cause it is eight dollars every time". I did not think his actions stemmed from his misunderstanding of the given situation. Before this task, they also made two directed segments of four and explained the sum of the available amount of both accounts added up eight dollars. In addition, Carl approved Maggie’s solution in his previous problem for her. Therefore, I analyze

149 that his directed segment of 8 stemmed from his lack of the ability to reconstitute the given situation. In previous teaching episodes, he consistently used his procedural scheme or counting scheme in which he could find the solutions at the end of the execution of his schemes. In addition, he used the counting actions to justify the solution to the numerical computations (see

Protocol 6.1). Similarly, when specific numeric values were missing, he found the solution at the end of the given quantity as if he found solutions where he completes his counting actions (see

Protocol 6.2). His solutions to the given tasks consistently appeared as the last number in his vertical computation as well (see Figure6.4-(e) and Figure 6.6-(a)). Assuming that he had been consistent hitherto in the teaching episodes, I infer that he considered eight dollars as the solution to the given task, and he made the directed segment of 8 for his solution, a numeric value that could be located below the bar of a vertical computation. This also shows his lack of ability to review his answer by reconstituting the given situations.

(a)

(b) Figure 6. 8 Maggie’s problem for Carl.

Note. Screen capture (a) shows Carl’s solution. Screen capture (b) shows Maggie’s

150 solution to the same problem.

After Carl made the directed segment of 8, Maggie took the computer, and she counted from -12 to 0. Then, she started counting from 8 and stopped at 20. Next, she made a directed segment of 12 from 8 to 20 (see Figure 6.8-(b)) explaining, “it should be 20 to be eight".

Coupled with her actions and explanation in the previous teaching episode, her adding 12 more to 8 can serve as a corroboration of my hypothesis; She constructed the missing addend as a unit containing three levels of units. One might argue that she might have used her procedural scheme to find 20 and simply added 12 more to 8 to make 20. However, she never used the procedural scheme before she constructed the directed segment in any previous teaching episodes. Instead, she used her conceptual ruler for measuring quantities. In addition, counting 12 from -12 to 0 and adding as many as she counted on 8 is clear evidence that she was explicitly aware of the inverse relation between positive and negative quantities. Based on how she found the missing addend in the previous teaching episode, I infer that she might have counted up from -12 to 8, then she intentionally counted -12 to 0, taking it as a composite unit, while explicitly being aware that counting backwards would counterbalance -12. She then projected what she had counted right next to 8, constructing 20 as a composite unit containing two composite units, i.e., 8 and the opposite of 12. This is a solid indication that she had the concept of negative quantity as an inverse to positive quantity.

Finding Differences between Two Oppositely Oriented Quantities

On the 5th of December, 2014, I presented tasks that could be modeled with subtraction.

The purpose of the teaching episode was as follows: (1) to explore how Carl and Maggie would use their units-coordinating schemes to model the given situation, (2) to investigate whether their units-coordinating schemes were sufficient when constructing inclusion relations between the

151 given quantities and the difference between them, and (3) to examine whether the construction of inclusion relations are essential when modeling differences between two oriented quantities.

It is important for the teacher-researcher not to consider that the students who can find the sum of two oriented quantities can automatically model the difference between two oriented quantities by considering subtraction as an inverse to addition. According to Steffe and Cobb

(1988), children at the stage of sequential integration, who are limited to producing numerical composites rather than composite units, develop their subtraction strategy independently of their addition strategy. For example, to indicate that 13-6=7 implies 7+6=13, a child needs to treat six and seven as two entities composing the whole, 13. They continued that the construction of a part-whole relation that entails an inclusion relation of composite units is essential when constructing subtraction as an inverse of addition. Following Steffe and Cobb, I hypothesize that the construction of inclusion relations are essential when modeling subtraction as the inverse of addition and reconstituting the given situation with oriented quantities by constructing a unit containing three levels of units. The first task was as follows: Carl has a debt of $35, and Maggie has $20. How much more money does Maggie have?

One computer was provided for them to work together on GSP. Paper and pens were also provided. Maggie made the directed segment of 20, and Carl made the directed segment of -35 on a separate number line.

Protocol 6.4 Maggie’s equation of “-35+n=20” and Carl’s equation of “35+20=55”

I: So, can you all talk about what you are thinking and what you are gonna do? Mag: I marked the line (points to the directed segment of 20) and marked this (the directed segment of -35). So … (makes the directed segment from -35 to 20 on the separate number line, see Figure 6.9-(a).) I: Carl, what do you think? Carl: Uh, that you put them on top of each other, or something? To see? I: Uh huh. So…? Ya’ll gonna tell me more about the situation. So, how much more

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money does she have? Carl: 55 dollars? I: 55 dollars? How did you find that? Carl: ‘Cause you add 35, so, have -35 and plus 20. 55. I: Hm. So, can you write an equation? Carl: (Writes -35+20, then crossing it out, see Figure 6.9-(b)) That won’t work (writes - 35-35= +20, see Figure 6.9-(c)). Oh, you want to know how much more money she has? I: Uh huh. Mag: (Writes -35+n=20, See Figure 6.9-(d)) Carl: (Writes -35+20=55, rewriting 55 and circles around it, see Figure 6.9-(e)) I: Hum… Okay, that’s very interesting. So, can ya’ll talk about what ya’ll are thinking? Mag: Well, he has 35 in debt, and we try to figure out what gets more to get 20. I: Hum… What about this one (points to his last equation, -35+20=55), Carl? Carl: Uh, -35 plus 20, because I am at 35, and she is at positive 20. So, I added them together. I: So… Wait. If you add -35 and 20, isn’t it -15? Carl: (Scoffing) I don’t know. I: (Looking at Maggie) What do you think? Mag: Um… I mean you are adding, ‘cause you want to get 25 to add each one… Carl: (sweeps a mouse from -35 to 20) I: So, what I want you to do is… I see your equation and I want you to explain these equations using all these stuff… Carl: Cause you do like this (sweeps his mouse starting from -35). Five, ten, fifteen, twenty, twenty-five, thirty, thirty-five, forty, forty-five, fifty, and fifty-five. I: Ah… Mag: Then, I wanna measure it (moves the number line with the directed segment starting from -35 to 20 on top of the empty number line and lined them up at zero showing the directed segments ends at 55) I: Maggie’ can you tell me what you are doing? Mag: I tried to make a diagram. So, he said 50. So, 50 is measured right here. I: Ah…it’s 50… it looks like 53. But looks like it’s pretty close. Mag: (adjust the directed segment of 55) yeah. I: It’s close enough. I: So, I am very interested in the way you (referring to Carl) did this. How it is related to the equation? And, I am also wondering how this (Maggie’s equation, - 35+n=55) is related to that (the directed segment from -35 to 20). Mag: This? I wrote how much he had it right there. And, then, I marked it. I: And then, what is that (referring to n)?

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Mag: It’s the answer, right here (points to the directed segment from -35 to 20). I: And, this is (points to 20)? Mag: How much I have. I: Okay. Can you explain your method (looks at Carl)? Carl: Um, I took mine, this amount, and I added hers. And, I got 55. It’s how much she got more. I: But, yours is 35 debt. Carl: Yeah. I: So, it’s negative. So, if you add your negative amount and her positive amount… Carl: You get 55. It’s how much more she has. I: But, I get -15! Cause it is ne… what do you say (points to Maggie)? Mag: I believe they are the same side, and can you the sides subtract? I: So… Carl. I am not saying that you are wrong. I am trying to make sense of it. Like, sometimes it make sense but sometimes it doesn’t because this one literally looks like negative 35 plus 20. Carl: Yeah. I: And what is it? Carl: 55. I: To me, it’s -15! Carl: Well, I mean, yeah, but it’s 55. It’s the right answer.

(a)

(b) (c) (d) (e)

Figure 6. 9 GSP screen captures of the directed segments and Carl’s and Maggie’s equations.

Note. Figure (a) is the GSP screen capture, Figures (b), (c) and (e) are Carl’s equation, and Figure (b) is Maggie’s equation.

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Although Carl was the one who expressed greater interest in using GSP (he seldom used pen and paper in later teaching episodes whenever he had access to the computer), he chose to use a pen to write the vertical computations. On the other hand, Maggie solved all the tasks using graphic representatives on GSP. One might argue that Carl could not access the computer because Maggie was using the computer most of the time. However, the computer was located between the two students so that both students had equal access to the computer. Moreover, Carl did “write” his vertical computations using a mouse on the computer screen in later teaching episodes. Hence, I infer that each student chose the most preferable way for them to solve the given task: Carl chose to use his vertical computations, along with his counting scheme, while

Maggie chose to use directed segments to represent the given task.

Carl’s Conflict Between his Actions and Representation

I infer that Carl found 55 by adding 35 and 20 based on his explanation and his actions.

His explanation, “there are 35 and 20, because I am at 35, and she is at positive 20. So, I added them together", along with his counting up to 20 starting from -35, suggests that he might have indicated that he needed to add the amount (or the magnitude) of his debt and 20 to find how much more money Maggie had. (c.f., In the previous teaching episodes that was held on October

27, he wrote 12+32=-44 vertically to model the sum of -12 and -32.) What is interesting is that his first attempt to write the vertical computation to notate the given situation was -35+20 (see

Figure 6.9-(b)). He then quickly crossed out the expression saying, “that won’t work". Based on the equation that he wrote to represent his counting actions (see Figure 6.9-(b), and Figure 6.6-

(a)) coupled with his later counting-up actions, I infer that he first attempted to express his counting-up actions. However, using his numerical computation, he immediately found that the expression would not result in 55, so he crossed it out. In the case of the next expression, -35-

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35= +20, it is challenging to make any inferences of his thinking. According to his previous expression and his question about whether he was supposed to find how much more money

Maggie had, I can only infer that he might have tried to write a different expression than his first attempt, or he was thinking of finding how much more debt he had. Had he written -35= +20 instead of -35-35= +20, it would have been possible to hypothesize that he was considering how to find how much more debt he had compared to Maggie. In retrospect, I found multiple indications of Carl’s concept of “-” sign3; he treated the “-” sign as a sign standing for subtraction. He then treated all the quantities as if they were positive when finding sums and differences. So, it is possible that he wrote -35 while treating it as 35, and tried to subtract both side by 35.

After he double checked what he needed to find, he wrote the equation -35+20=$55(see

Figure 6.9-e). The way he rewrote 55 and circled it showed his certainty about his solution. Even after I intervened that -35+20 would produce -15, he did not change his answer while admitting that I was right; instead, he said, “yeah, but it’s 55. It’s the right answer". I consider his determination to defend his answer while partially admitting it was flawed stemmed from his (1) lack of ability to explain or review the given situation using a different method other than vertical computations, and (2) lack of the concept of negative quantity. He could find 55 by counting-up-to or using the numerical computation, but he could not reconstitute the result of his actions and operations with the given quantities in order to explain why he got the “right answer”. So, his schemes were one-way. Along with his actions when finding missing addends, the indication is solid that he was not capable of checking his numerical computation with his

3 See Chapter 7 for more details.

156 counting actions or vice versa, due to the lack of the ability to reflectively abstract his counting actions and to disembed them while monitoring the meaning of the orientation of the quantities.

Thus, I analyze that the two levels of units coordinating scheme is not sufficient when reconstituting an additive situation with oriented quantities, because constructing one quantity as a unit containing the other two units requires coordinating three levels of units. My analysis is also consistent with Ulrich’s (2012) analysis that a three levels of units coordinating scheme is necessary when modeling additive situations with integers.

Maggie’s Reconstitution of the Situation Using the Result of Her Operation

Maggie’s equation, -35+n=20 (see Figure 6.9-(d)), along with the way she constructed the directed segments (see the Figure 6.9-(a)) to illustrate the way she found her solution, suggests that she could reconstitute the result of her operation, 55, as a unit containing the opposite of -35 and 20. Her explanation, “I mean you are adding, ‘cause you want to know how much more", coupled with what she showed in previous teaching episodes, I analyze that Maggie constructed n as a unit containing the opposite of 35 and 20 using her three levels of units coordinating scheme, which entails an inclusive relation among three composite units. While

Carl’s solution consistently appeared below the bar of the traditional vertical computations,

Maggie could use her solution, n, to explicate the given situation. Her ability to use the result of her previous operation in order to review the given situation corroborates my hypothesis that she could construct a unit containing the other two composite units and take the unit as an input and use it in order to reconstitute the given situation.

Finding a Difference between Two Negatively Oriented Quantities

To further investigate the ways of Carl and Maggie’s actions and operations when modeling the difference between two given oriented quantities, I provided the following task:

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Carl is in debt of $15 and Maggie was in debt of $52. Find how much less money Maggie has than Carl. Although they both found the same solution, the distinct ways of modeling the situation are instructive.

Carl’s Way of Treating Negative Quantities as if They Were Positive

Carl immediately wrote the vertical computation, 52-15=37. His vertical computation corroborates my previous hypothesis that he did not consider the orientation of the given quantities. Instead, he treated them as the amount or the magnitude of the debt. In other words, it seemed like he was engaging in finding a difference between 52 and 15 rather than -52 and -15.

Moreover, the traditional method of borrowing one from the tenth place to carry out the vertical subtraction can be observed (see Figure 6.10-(a)) in his computation. He found that Maggie had a debt that was $37 greater than that of Carl. This indicates that Carl’s two levels of units coordinating scheme was not sufficient when reconstituting the given situation that can be modeled as a sum of two oriented quantities.

Maggie’s Two Oppositely Oriented Quantities as Inverse to Each Other

Maggie made the directed segments on GSP (see Figure 6.10-(b)) and explained that she had $37 less than Carl (see Figure 6.10-(b)). She also explained that Carl had $37 more than

Maggie and expressed it by constructing an oppositely directed segment of the previous one, the directed segment from 0 to -37. Maggie’s directed segments suggest that she found the difference between -52 and -15 not between 52 and 15. In addition, she explained that not only she had a debt that was $37 greater than that of Carl’s but also Carl had $37 more than her while constructing the directed segment from -52 to -15. This can serve as an indication that she was explicitly aware of the orientation of each quantity and its inverse relation to each other. Her arrangement of the directed segments and her explanation is a solid indication that she could

158 reconstitute the given situation by coordinating two oriented quantity while keeping track of the orientation of each quantity.

(a) (b) Figure 6. 10 Carl's equation and Maggie's directed segments

Finding Differences Between Two Positively Oriented Quantities

For the last task, I prepared the task modeling the difference of two positively oriented quantities to find more corroboration of my hypotheses. The task was as follows: Maggie has

$80 and Carl has $20. How much more does Maggie have than Carl?

Protocol 6.5 Maggie’s ability to review the situation and Carl’s nonreversible numerical computation.

I: Okay, who got more money? Mag: Me. Carl: Her. I: Okay, can you figure it out? Carl: Yeah, (writing 80-20=50, see Figure 6.11-(a)) 80 minus 20 is 50. 50 more. Mag: This (Makes the directed segment from 20 to 80) will fill it to make it 80. I: Nice. So, what is the equation? Mag: So, I will figure it out by moving this (locating the directed segment on top of the empty number line so that the end of the directed segment is located at 60). Carl: (Looks at me when I am looking at Maggie) It’s 50. I: She got a different answer. What do you think? Carl: (Looks at the computer screen while Maggie is moving the directed segments accordingly, see Figure 6.11-(b)) it’s 50 more. I: [Maggie,] what’s your answer? Mag: 60. Cause 80-20 is 60. I: What do you think?

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Carl: (Smiles and then laughs) What was that? (While Maggie and I were turning our heads to look at the place at which he pointed, he quickly changed 50 to 60, see Figure 6.11-(a)).

(a) (b)

Figure 6. 11 Carl’s equation (a) and Maggie’s directed segments (b) on GSP. In Screencapture (a), Carl’s correction of his answer can be observed.

Carl’s Computation as a One-Way Algorithm

Carl’s determination of his solution was solid; while I was looking at the computer screen to observe how Maggie was constructing the directed segments, he repeatedly demanded me to pay attention to his answer. When Maggie said that she had $60, it seemed that Carl did not agree with her until she said, “’cause 80-20 is 60". With two levels of units coordinating scheme, along with what Carl had shown earlier, I do not doubt his ability to find the answer 60.

However, he made a mistake while carrying out his numerical computation, and his way of correcting his mistake was revisiting his computation procedure once more. This suggests that his operations that were involved in finding the difference were one-way.

Maggie’s “Filling” Strategy

Maggie, on the other hand, constructed the directed segments to represent the relations

160 between the quantities. Her way of checking her solution was putting the directed segment from

20 to 80 on top of the empty number line and reconstructing the identical directed segment that started from 0. Her comment, “this [makes the directed segment from 20 to 80] will fill it to make it 80” can serve as a corroboration of my hypothesis that she has been using her conceptual ruler in order to measure her solutions. Her explanation, along with the way she constructed the two directed segments—one from 20 to 80 and one from 0 to 60— suggests that she could review her solution (the directed segment that was from 20 to 80) in relation to other quantities, and she perceived it as a quantity that can be represented from 0 to 60 as well.

Throughout this whole teaching episode that was held on December 5, Maggie did not write any equations unless I asked her to; the only equation that she wrote was -35+n=20, where n was the solution that she found in the given task. She checked her answer by measuring the directed segment using an empty number line or by explaining the given situations as a relation between the given quantities and her solution. My analysis is that she could reflectively abstract her solution as a unit containing three levels of units and disembed each composite unit with explicit awareness of the inclusion relations among the quantities. In addition, she could “check” her solution by taking the solution as an input and reviewing it in relation to other quantities. On the other hand, Carl wrote his vertical addition or subtraction algorithm to find the differences.

He revisited his computation procedures or counting schemes to check his solutions. Hence, I conclude that his operations including his procedural counting schemes were one-way.

Finding the Missing Addend of Oriented but Unknown Quantities

On the 23rd of January, I prepared a task with unknown quantities and asked Carl and

Maggie to find the missing addend and express the situation using an equation. They had a similar task on October 31, 2014, which was adding two two oppositely oriented but unknown

161 quantities. I could infer that, without the specific numeric values, Maggie used her conceptual ruler to estimate the unknowns and justify her strategy, while Carl associated his counting action with one of the addends (See Figure 6.5; Protocol 6.2). In the teaching episode that was held on

December 5, 2014, the task was given with quantities of specific numeric values. Carl wrote vertical computations to illustrate his counting actions, and Maggie wrote the equation taking the result of her operations as an input (see Figure 6.9; Protocol 6.4). Based on theses teaching episodes, I designed a task to find corroborations of my hypotheses that (1) Maggie could take the result of her operations as an input and reconstitute the given situation, and (2) Carl was yet to reflectively abstract the addends as a composite unit. The task was as follows:

The green directed segment represents the sum of the measures of the temperature of Winder-Barrow and New York city. The red directed segment represents the measure of the temperature at Winder-Barrow. Represent the measure of the temperature in New York city on the vertical line in the middle (see Figure 6.12). Then, write an equation to represent the situation.

Figure 6. 12 Carl's modeling the given situation

Note. Maggie also constructed the identical directed segments to those of Carl’s. The red and green directed segments (a and b) were given as the measures of the temperature in Winder

Barrow and New York city. Maggie and Carl constructed the blue directed segment (c) as the sum of the measures of the two temperatures.

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Maggie’s Equation, a-(-b)=c

Both Maggie and Carl lined up the given directed segment (say the red one a and the green one b) at 0 and constructed the blue directed segment, say c (see Figure 6.12). Then, they were asked to write an equation to represent the situation. Maggie’s first attempt was “a-b=c”.

While explaining how the equation represented the given situation, Maggie changed her equation to “a-(-b)=c saying, “I do not think that’s right...because this is not how you solve it. ’Cause it

[referring to c in the equation of a-b=c] will be positive, but it [the directed segment of c] is not positive".

Maggie’s first attempt, “a-b=c", should be interpreted based on their previous actions and explanation in the previous teaching episodes. In those episodes, Maggie coordinated two directed segments to model the additive situation with quantities with numeric values. Thus, I infer that “a-b” symbolizes the combination of the two directed quantities, and the subtraction operation implies the decrease in the measure of the temperature. While explaining her equation, she found a conflation between her equation and the directed segment, c. She then placed the “-” in front of b to alter the directionality of c. Although it is unclear whether her units coordinating scheme was involved when she added the opposite sign, her revision of the initial equation indicates that her way of modeling the situation was self-regulatory. Based on previous teaching episodes, she might have associated some unknown quantities with a, b, and c, and found c must have negative quantity. She, then, might have changed the orientation of b in order to change the orientation of her solution. However, she did not change the orientation of c. Instead, she chose to change either the operation of the equation or the orientation of b. Thus, I infer that she understood c as the result of her operations with given quantities (i.e., in a relation with other

163 quantities) and knew its orientation would be changed in accordance with the changes that she made in her operations or the orientation of b.

Carl’s Equation, A-B=B-C=C

On the other hand, Carl changed his equation to “A-B=B-C=C". He explained that subtracting B from A produced B, and B-C gave him C. Then he added, “A-B makes A goes to

0". Although it was surprising that Carl constructed the directed segment of c for the first time, it should not be interpreted in a way that Carl made some accommodations in his way of operating.

Rather, I infer that he lined up the two givens at 0, starting to count from the end of the first addend to the given sum. Then, he assimilated the decreasing action as if he were counting- down-to. In case of equation, he first wrote A-B-C=C, and then quickly changed it to A-B=B-

C=C. It is ambiguous what his first equation symbolizes. It is possible that the “-” sign in front of

B refers to the decrease and he put the”-” sign in front of C because it was below 0. He then put

C on the right side of the equation because he knew that the result of his scheme would be C.

After he changed the equation to A-B=B-C=C, he explained that subtracting B from A produced

B and that B-C produced C. Thus, I infer that he used the equal sign to express the flow of his actions, and he assimilated A-B as a counting down (or sweeping down) action regarding A as where he started counting and B as his counting-down (or sweeping-down) actions. So, from his perspective, it was reasonable to consider A-B gives B because his counting actions would stop at B. Similarly, I infer that he wrote B-C=C to express that he activated his scheme or action at

B. He knew that the result would be C, so he wrote C on the right side of the equation. I argue that his solution indicates the location at which his actions were exhausted instead of taking C as an unknown quantity that is accumulated from 0. In addition, it is possible that he wrote “-C” to express that C was below 0 considering his inconsistent use of the “-” sign. These instances serve

164 as a corroboration that he was yet to reflectively abstract his counting actions and to reconstitute the given situation as three levels of units.

Summary

Carl’s Counting Schemes and Nonreversible Numerical Computations

I found that Carl’s two levels of units coordinating scheme is sufficient to find the solution of sums, missing addends, and differences of two positively oriented quantities, but it is not sufficient to coordinate two oriented quantities to model or the sums or differences. I infer that the lack of the ability to reflectively abstract his actions and the disembedding operations served as constraints when reconstituting the given situations or when “checking” his solutions.

In addition, I found multiple indications that he made a strong association between his numerical computations and his counting actions when finding sums, missing addends, and differences of two oriented quantities. He assimilated the given situations as adding or take-away situations, computed numbers to find the solution, and use his counting actions to support his solutions.

Furthermore, I found that he did not have the concept of negative quantity as an inverse to positive quantity.

When asked to model the situation that can be modeled as a sum of two oriented quantities, Carl almost always used his procedural scheme first to find the solution and later explained it by either counting fives or tens. When adding or subtracting two negatively oriented quantities, he wrote equations such as 12+32=-44 (for the situation that can be modeled as -12+(-

32)=-44 in the teaching episode of October 27, 2014) and 52-15=37 (for the situation that can be modeled as -52-(-15)=-37 in the teaching episode of December 5, 2014) as if he were ignoring the orientation of the given quantities. Similar patterns were observed when he found the sum or difference between oppositely oriented quantities. He said that he would put a bigger number

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(meaning a number that had greater magnitude) on top and the smaller number on the bottom. He then decided the orientation of the solution accordingly. Thus, I analyze that he found the solution of a given situation after assimilating it as an adding or take-away situation and then used either numerical computations or his counting scheme while treating them as positive quantities, and only then did he decide the orientation of the solution.

The task with unknown quantities provided corroborations of my hypothesis about how he would coordinate two quantities when he did not use his numerical computations as his method to find the solution. In the teaching episodes that were held on October 31, 2014 and

January 23, 2015, I found multiple indications that (1) he treated the first given unknown quantity as a location from which he started his counting actions or sweeping up/down actions,

(2) his actions represented increases or decreases, and (3) he found the solution as the location where his actions were completed. The equations, a=b=c, and A-B=B-C=C modeling a+b=c corroborate my hypothesis.

Although he often found the “correct” solution to the given tasks, his schemes, actions and operations were not sufficient for him to reconstitute the given situation by coordinating two composite units. In addition, His counting actions were not reflectively abstracted in such a way that he could use them as an input to review his solution in relation to the other quantities. In other words, his schemes were one-way. His solutions were located below the bar of the vertical computations, and he reenacted the same scheme whenever he was asked to explain or check his answers. His conflation when he wrote the vertical computation of -35+20=55 while admitting that -35+20=-15 (on the teaching episode on December, 5, 2014) suggests his lack of the ability to review the results of his schemes.

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Maggie’s Reconstitution of the Given Situations by Coordinating Two Quantities

I found that Maggie’s three levels of units coordinating scheme, her disembedding operation, as well as the ability to reflectively abstract the given quantities and her actions allowed her to find the sum, missing addends, and difference of two oriented quantities as well as to review her findings by reconstituting the given situations as a unit containing three levels of units. When asked to find a sum of two oriented quantities, Maggie first constructed directed segments to express the given quantities and found the solutions by combining the two directed segments. I hypothesized that she coordinated two given quantities rather than she was merely engaged in numerical computations, and I found multiple corroborations of my hypothesi; she used number lines as if they were rulers and translated the directed segments to explain her solutions and used her “filling” strategy to find and to represent the situations. These instances indicate that she constructed her conceptual ruler and used her measuring scheme to reconstitute the given situations. I consider that her measuring scheme (the “filling” strategy) was fundamental when reconstituting the situation as a unit containing three levels of units (c.f.,

Carl’s vertical computation and his counting schemes)

In addition, she coordinated two unknown quantities in the similar way she did with known quantities. She was able to express the coordination of the unknown quantities as the equation of a+b=c in the teaching episodes of October 31, 2014 and January 23, 2015. She also used her counting schemes when finding missing addends in the teaching episodes of November

10, 2014. She started counting-up by tens from -65 to 50, then she reflectively abstracted her counting actions and used them in a review of the given situation. Her equation, 115-65=50, and her explanation that she wrote of the equation to check the answer supports my analysis that she could reflectively abstract her counting actions, 115, and reconstitute the situation by

167 coordinating the result of her scheme and the given -65. I infer that her schemes and operations that were involved in modeling the situation, 115-65=50, was reversible because she used the result of her scheme as an input and checked the given, 50. Corroboration of my hypothesis can be found in the equation of -35+n=20, in the teaching episode that was held on December 5th.

My analysis is that she reconstituted the missing addend, n, as a unit containing the composite unit of 20 and the inverse of -35, disembedded n and rearranged the other composite units using her three levels of units-coordinating scheme. Taking a unit containing three levels of units as given, I found that she could disembed one quantity without destroying the three levels of unit structure. On November 14, she counted from 0 to -12 and added 12 more to 8 in order to show why Carl needed to add 20 to -12 in order to find the sum, 8. The way she constructed the directed segment of 20 as a combination of the directed segment of 8 and 12 suggests that she could construct 20 as a unit containing two composite units, 8 and the opposite of -12.

Although Maggie could coordinate three levels of units prior to activity based on actions and explanations that she had shown in the initial interview while recursively partitioning the given whole, I did not find indications that she could anticipate the result of the given task as a unit containing three levels of units prior to activity. However, I could find many indications that she took the result of her operations for an input to reconstitute the given situations, which was not available to Carl, who was limited to coordinating two levels of units.

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CHAPTER 7

SCHEMES AND OPERATIONS WHEN CONSTRUCTING x+y=a

In the previous chapter, I made my explanatory model of how Maggie and Carl used their schemes and operations when finding sums, missing addends, and the difference of two oriented quantities. The two levels of units coordinating scheme was sufficient to find the solutions to the given additive situations. However, when modeling the given situations, the ability to reflectively abstracting the quantities or their counting actions and the disembedding operation were essential. Thus, I concluded that the three levels of units coordinating scheme is essential when reconstituting the given situation as a unit containing two composite units.

In the latter part of this study, I focused on how they reorganized their existing schemes and operations when constructing x+y=a. My hypothesis was that the students would have different constructions of x+y=a in accordance with their distinct units-coordinating schemes.

Distinct Meanings of the “-” Sign

In the teaching episode that was held on December 8, 2014, I posed the task as follows:

The sum of the measure of the temperature of Carl’s city and Maggie’s city is always 10 °F. Create problems for your partner to solve.

The goal of this task was to investigate how they coordinate measures of changes in temperature of both cities while helping Carl reconstitute the given situation by coordinating two quantities instead of using numerical computations. There were four horizontal number lines on GSP, three of which were blank. On the very bottom of the number line, I preconstructed the direct segment

169 of 10 to represent the sum. I labeled the first number line “Carl", and the second “Maggie” for each student to construct the directed segment for the measure of the temperature of each city.

Maggie first made a directed segment of 20 on her number line for Carl. Carl made the directed segment of -10 on his the number line (see Figure 7.1-(a)). Then he deleted the directed segment that he just made at my request to explain.

Protocol 7.1 Carl’s assimilation to a take-away situation.

I: Carl, can you tell me why the sum of the temperatures is 10? Carl: Um… I don’t know (with an inaudible whisper, he deletes the directed segment of -10). I: You can explain or you can’t … What you are thinking? Carl: Um… So, it has to equal 10 (making a directed segment from 0 to 10, see Figure 7.1-(b))? So, like that (pointing to the directed segment of 10 that he just made). I: That? What does that mean? Carl: It’s 10 °. I: So, you have to get to 10 ° and Maggie said 20 °, right? And what is the sum of the measures of the temperature of the two cities? Carl: Oh, it has to be a sum? I: Yes. Carl: (Deleting the directed segment of 10 and making the directed segment of -10 instead) it will be negative ten, right? I: Yeah, can you tell me why? Carl: ‘Cause 20 minus 10 is 10. I: Can you move these around (means translating the directed segments) to show me that the sum is 10? Mag: (Looking at Carl who was having some technical issues on GSP, she moves Carl’s number line where his directed segment of -10 was located and placed it on top of her number line so that the directed segment starts from 10 to 0, see Figure 7.1- (c).) I: (To Carl) Is this what you wanted to do? Carl: Yeah. I: Uh, can you explain why? Carl: Cause it shows 10, I guess? Cause this is 20, and that’s 10 and 10. I: So, 10 in which way? Can you use your mouse to show me if it is this way (points to the right) or this way (points to the left)? Carl: What do you mean? I: So, you said 10. What does 10 mean? Is that the sum of the temperature of your

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city or… Carl: The sum (circles 10 on the number line). I: Maggie, did you want to add more explanation? Mag: Uh uh (Means “no". She then moves Carl’s number line right underneath her number line). It’s like a half (then moves Carl’s number line so that the zero of the directed segment of -10 is placed right underneath 20 on her number line, see Figure 7.1-(d)). Carl: Yeah, (counts down from 20) five, ten. I: How are these related to the sum? Mag: (Tries to move the number lines.) Carl: (Intervening) ‘Cause this is five, ten (counting down from 20 to 10 by fives), so it’s ten. 20-10 is 10. Mag: (Makes the directed segment of 10 underneath the two number lines.) This is the same with that (comparing it with the premade directed segment of 10) and it fits right here.

(a) (b) (c) (d)

Figure 7. 1 The screen captures of Carl’s and Maggie’s directed segments

Note. The screen capture of (a) and (b) show the first directed segment that Carl constructed, the second directed segments that Maggie made for Carl, and the last given one. The screen captures (c) and (d) show how Maggie arranged the directed segments when helping Carl.

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Carl’s Meaning of the “-” Sign

Carl first made a directed segment from 0 to -10, deleted it, and then made a directed segment from 0 to 10 after he failed to explain why he made a directed segment of -10. I do not consider that he simply misunderstood the problem. In previous teaching episodes, he consistently used the take-away strategy when adding two oppositely oriented quantities. Thus, it is possible that he took 10 away from 20 and made a directed segment -10 to express the reduction. He then got conflicted when he saw the directed segment that he just made did not match the sum, 10. So, he deleted the directed segment of -10 and made the directed segment of

10. When explaining, Carl pointed to 20 while saying, “this is 20, and that’s 10 and 10". It seems as if the orientation of the directed segment was not important to him. Rather, it seems like he was treating 20 as two tens and subtracted 10. He could review the sum of -10 and 20 as take- away 10 form 20, but he was unable to review the take-away situation as a sum of oppositely oriented quantities. I infer that his confusion stemmed from the lack of the concept of negatively oriented quantity as an inverse to positive quantity.

Maggie’s Negative Quantity as an Inverse to Positive Quantity

It is interesting how Maggie arranged the directed segments when helping Carl. In previous teaching episodes, she consistently reconstituted the given situation by coordinating two quantities as if they were continuous but segmented units by using her “filling” strategy. In this teaching episode, she translated the directed segment of -10 ten units to the right so that it started from 10 and ended at 0. Considering that Maggie had seldom interrupted Carl’s explanation and had been respectful whenever he explained his reasoning in his turn, I infer that she arranged them to model Carl’s take-away method. Her later rearrangement shows that she was also capable of reconstituting the given situation as three levels of units. These instances suggest that

172 she could not only arrange the directed segment to illustrate a take-away situation but also reconstitute the take-away situation as the sum of two oppositely oriented quantities. Maggie’s last comment, “it fits in there", along with her way of moving the directed segments corroborates the hypothesis that she used her “filling” strategy and reconstituted the take-away situation as the sum of two oppositely oriented quantities.

For the next task, Carl made a problem for Maggie. He made a directed segment of -33 on the number line of Carl’s city.

Protocol 7.2 Maggie’s modeling -33+n=10.

I: Alright. What is that? Carl: Uh, it is 33. I: 33? Carl: Negative thirty-three. Mag: (Making a directed segment of 10) so, it’s 10? I: Yeah, the sum of the temperature is 10. Mag: (She moves the directed segment of 10 right underneath the directed segment of - 33. She then makes another directed segment on the blank number line by tracing over the directed segment of -33 and connecting it to 10, see Figure 7.2.) I: That’s very interesting. Maggie, can you tell us what you are thinking? Mag: He said -33, and I need to get 10. So, I took it from -33 to get all the way to 10. I: Can you tell me the value? Mag: (Makes another directed segment of 40 and places it right underneath the directed segment that starts from -33 and ends at 10.) I can measure it. It’s about 40 … (rearranges it) it’s 43. I: Carl, what do you think? Did she get it right? Carl: Ah, yeah. I think. I: Is she right or wrong? Carl: What was it again? 33? I: -33. Carl: Yeah, it should be like 43. Yeah, 33, that would be zero, and plus ten, It will be 43.

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Figure 7. 2 Maggie's solution to Carl's problem

Maggie’s first action was moving the directed segment of 10 right underneath the directed segment of -33 after double-checking that the sum was 10. Her actions suggest that she monitored the orientations of the given quantities, treated them as composite units and tried to find the solution by coordinating them. It is possible that she anticipated the solution as a combination of the opposite of -33 and 10, but whether she actually made an anticipation is ambiguous at this point. When asked to find the numeric value for her solution, she independently took a number line and measured it. This measuring action indicates that she did not enact her counting scheme, but visually coordinated the two given quantities to find the solution by enacting her measuring scheme. In addition, this serves as a clear indication that she was operating with at least two number lines: one for generating the solution by coordinating the givens, and the other for measuring the unit containing the other two composite units. Her comments, “He said -33, and I need to get 10. So, I took it from -33 to get all the way to 10", and

“I can measure it. It’s about 40 … [rearranges it] it’s 43", as well as her translation of the directed segment of 10 suggest a possibility that she anticipated that the given situation could be modeled as a unit containing three levels of units. Couple with her “filling” strategy that she had shown in the teaching episode that was held on December 5 (Protocol 6.5), I infer that she could mentally re-present the given situation in such a way that two quantities were filled in the other

174 quantity while keeping track of their orientations.

Carl’s confirmation of Maggie’s answer, “yeah, it should be like 43. Yeah, 33, that would be zero, and plus ten, It will be 43” suggests a possibility that he could review 43 as a composition of 10 and the inverse of -33. However, in that it was Maggie who constructed the representations and lined them up, I infer that Carl was able to review the missing addend as a unit containing three levels of units with a visual input.

Toward the end of the teaching episode, I asked them whether the measure of the temperature of Maggie’s city falls or rises when the measure of the temperature of Carl’s city rises 30 °F, given that the sum of the measures of the temperatures of the two cities is always 8

°F. Maggie immediately said, “It’s gonna fall", while Carl said, “It will be 22, because 30-22 is

8". I intervened and told Carl that the sum of 30 and 22 would be 52 not 8. He insisted that 22 needed to be subtracted, and eventually said, “it’s -22". It seems like Maggie was aware of the counterbalancing relation between the changes in temperature of two cities while Carl was reasoning with specific quantities and considered “rises 30 degrees” as a 30 ° temperature. He then focused on the difference between two quantities and assimilated it as a take-away situation instead of considering covarying changes in each quantity. In addition, it was possible for

Maggie to envision the counterbalancing relation while covarying the two quantities, if the sum was re-interiorized in such a way that she could take it as an input. Unfortunately, I could not investigate any further due to lack of time. However, I found that asking questions about the counterbalancing relation might be too challenging to Carl, who was lacking the ability to reflectively abstracting his counting actions as a composite unit. Thus, I prepared more tasks to help Carl abstract the changes in quantities in the context of elevation in the teaching episode of

February 2 (see Appendix) and coordinating unknown quantities on January 23 (see Figure

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6.12). I saved the questions regarding the counterbalancing relation for the teaching episode that was held on March 2.

Anticipating the Relationship between Two Covarying Quantities

From the previous teaching episode, I learned that coordinating changes in temperature would be challenging to the students, especially to Carl, who did/could not reconstitute the given situation as a unit containing three levels of units. Thus, I designed the task so that they could

“see” the covarying directed segments instead of mentally envisioning them, and investigated whether or not they could find the relationship between two covarying quantities.

In the teaching episode that was held on February 20, 2015, I showed Maggie and Carl two mobile directed segments on GSP that moved in tandem to the opposite directions, which could be modeled as a+b=0, where two directed segments a and b represented the measures of the temperatures of Maggie’s and Carl’s city. For the second task, I prepared the two directed segments of a and b that covaried in relation to the equation of a+b=8. There were two buttons:

One for stopping and resuming the movement, and the other for adjusting the speed of the animation so that the students can stop or resume the animation and change their speed whenever necessary. The students shared one computer with two separate mouses so that both of them had an equal chance of accessing the computer.

After playing with the mobile directed segment for a while, they both said that each directed segment moved to the opposite direction from each other. Next, they labeled each directed segment a and b. When they completed labeling, I asked to express the relation between the two directed segments in an equation. Each end of the directed segment was placed approximately at 3.5 and -3.5 at that time (see Figure 7.3-(a)).

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Protocol 7.3 The different meanings of letters in Carl’s and Maggie’s equations.

I: Can you write a relation between the two in an equation? Carl: An equation? I: Yeah. Carl: (He writes a=B, A=3.5 and B=3.5 vertically (see Figure 7.3-(b) ) Mag: (She writes a-b=b+a and a+b=a-b, see Figure 7.3-(c).) I: Can y’all talk about what you are thinking? Mag: Like, cause they are opposite, so I’m trying to figure out if I put a-b, it will be the same… it will be the same as this (pointing to a+b). Or, I was gonna put, like, (writes a+b=a-b). I: So, a+b=a-b? Mag: Yeah, or (writes a-a=b-b and a+a=b+b, see Figure 7.3-(c)). I: Hm. Okay, what you do think, Carl? Carl: I put a=B, then I put an absolute sign. I: Why? Carl: ‘Cause they equal to same. So, I just put that. I: Okay, so, you said a=3.5 and b=3.5 (looking at his memo, A=3.5 and B=3.5)? I thought a=3.5 when b=-3.5. Carl: It’s negative B (adding “-” in front of 3.5, see Figure 7.3-(b)). I: What if a is -5 (dragging the directed segment of a to -5)? Mag: b is 5, ‘cause they are opposite. I: Can you somehow show it? Mag: Okay, so, if you check … subtract …(takes the directed segment of b and places it on top of a so that the b ends at zero. She then separated the directed segments and placed them parallel to each other.) This is a (grabs b), because that’s gonna have, um, 2b that [is] added onto it (drags the directed segment of b to approximately to 15, drags it back to zero, pauses for 2 seconds, and put it back to 5). If you subtract, they are the same amount, so it’s gonna be zero.

(a) (b) (c) Figure 7. 3 The model of two quantities of a+b=0

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Carl’s Use of Letters for Specific Numerals

I analyze that Carl used the letters as if they were specific numeric values instead of treating them as representatives for varying or unknown measures of the temperature. After he stopped the animation of the two directed segments, he saw that each end of the directed segments were at 3.5 and -3.5. He first wrote his equation a=B almost instantly with A=3.5 and

B=3.5 and made them as a vertical computation. He then wrote 0 below the bar of the computation. Considering that he wrote all the expressions and equations while looking at the two directed segments that were fixed at 3.5 and -3.5, I infer that “a” and “B” were mere substitutions for 3.5 and 3.5. In other words, the letters represented specific numerals instead of unknowns that entails a potential quantity where the specific numerical values are yet to be known. In multiple teaching episodes including the latest one, Carl never said “negative” whenever he referred to the negatively oriented quantities unless the directionality was questioned by me (for example, he said 33 when he made a directed segment of -33, on

December 8, 2014). Thus, I infer that he wrote his first equation, “a=B”, as if he had said 3.5 was equal to 3.5 instead of -3.5. He later placed the “-” sign in front of 3.5 and carried out the vertical computation. I do not think he found 0 by adding -3.5 to 3.5. Considering that he consistently used the take-away method, I infer that he subtracted the magnitude of -3.5 from

3.5. His later equation, a=|B|, along with his explanation, “I put an absolute sign. ‘Cause they equal to same. So, I just put that", corroborated my hypothesis. Later, he wrote a new equation, a=-B, after he placed the “-” sign in front of 3.5 in B=3.5. This action can serve as another indication that both a and B were all 3.5 to him, and he placed the “-” sign in front of B just as he placed the “-” sign in front of 3.5. I also observed that he found his solution quickly, and he looked as though he was pretty bored when he was waiting for Maggie to solve and explain it.

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Along with his confident attitude when explaining and the short amount of time that he spent solving this task, it is highly probable that he was limited to focusing on one case where a=3.5 and B=-3.5, and his equation, a=-B meant 3.5=-3.5. Although he watched smoothly covarying two directed segments, he might have not found any necessity to coordinate the changes because he focused on the numerals that were located on the other side as well as equidistant from zero, which are, according to him, “equal to same".

Carl’s Concept of Negative Quantity

His later equation, a=|B|, along with his explanation, “I put an absolute sign. ‘Cause they equal to same” suggests that he considered the negative quantity as a number that became “equal to the same” with the absolute signs. Considering Carl’s use of letters for specific numerals and the way he treated the directed segments as pointers for numbers, it is unlikely that he constructed the concept of negative quantity as a scheme involving the inverse relation. In retrospect, I found Carl’s way of treating negative quantities was compatible with the guidelines of the Mathematics Georgia Standards of Excellence. The standards are as follows:

Thus, his equation, a=|B| represented that -3.5 as a number of the same absolute value that was located on opposite side of 0. Based on theses instances, I infer Carl’s meaning of negative quantities as points on “the other side” and treated them as positive quantities.

Maggie’s Attempt to Use Unknown Quantities to Model the Situation

Without mentioning any particular numeric values, Maggie explained the relationship by moving the directed segments. Coupled with how she combined the directed segments to reconstitute the given situations in previous teaching episodes, I infer that she was coordinating

179 two directed unknown quantities rather than being confined to an example with specific numerical values. It is possible that she mentally re-presented that b compensated a, so she translated b a units to the right then placed it on top of a to illustrate her mental re-presentation.

Her dragging b while saying, “this is a [grabbing b], because that’s gonna have, um, 2b that added onto it” suggests that she envisioned 2b as a unit containing the opposite of a and b.

However, whether she coordinated the changes in each quantity and was able to generalize it are unclear at this point. What I can infer is that her dragging motion was involved, and her explanation was not confined to a specific example.

It is unclear what her equations, a-b=b+a and a+b=a-b implied. It is possible that she wrote four equations in attempt to show that they all became 0. At that time, I was very confused and Carl looked so bored when Maggie was explaining her equations. So, I did not ask further questions. Instead, I proceeded to the next task hoping to find more implications of their ways of thinking in retrospect.

The latter part of the teaching episode was devoted to exploring how they describe and represent the situation that can be modeled as a+b=8. The setting was identical to the first task.

Two mobile directed segments that smoothly covaried formed a constant sum of 8. The students were asked to freely animate or manipulate the directed segments to find the relationship between the two varying directed segments. They both took time to see the animation, and

Maggie stopped the animation after some time and manipulated the directed segments. She then dragged b to 0 (see Figure 7.4).

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Figure 7. 4 Maggie's dragging b to 0

Protocol 7.4 Maggie’s coordination of two covarying quantities and Carl’s gross comparisons.

Mag: (Slows down the speed of the movement of the directed segments. She then stops the animation and slowly drags the directed segment of b that was placed around - 2 to zero.) So, it’s 8. Always 8 more, because when this reaches 8 (moves b from zero to 8, see Figure 7.4), which is right here. It’s always 8. And this is opposite (moves b around -8, and then moves it back to 8), I mean the opposite is right here. I: So, which one is 8 more? Mag: I mean, they are just, they are both 8 more than each other, so I just added it. a is 8 more (places a at 8). I: They are both 8 more than each other? Mag: Because if this is 8 (moves b), positive 8, then it goes to zero (looks at a). I: Then, what about if you put it in the middle way (stops the directed segments so that both a and b are placed at 4). Are they still 8 more each other? Mag: Oh. Um… Carl: (Clicks the animation button, and asks some questions relating to some technical issues. Carl and I talk about how to extend the screen while Maggie was quietly thinking.) I: It’s kinda tough, right? Why don’t you think about the amount of change? Let’s say that when a changes one, what happens to b? Mag: Uh… I mean they are still opposite (clicks the animation button and stops it), so… There’s growing one way and (inaudible…) I: So, when a increases by one, what happens to b? Carl: Decreases by… 8? Mag: Decreases by… 8… (moves the segments, stopping at various places including where both quantities were positive) I think they are still 8 away from each other. I: Still 8 away from each other? Why do you say that?

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Mag: Because that’s two (looks at b that was placed at -2), then you subtract [from 10]. That’s 8. And I can vary where a is at. Here it’s five and three. You add that, since you have five and three (continues dragging the directed segment of a here and there). Looks like they are 8 away each other. I: 8 away from each other? Mag: This is 15 and that’s 7 (looks at -7). I: What do you think, Carl? Carl: Um, when blue (referring to b) is negative, it’s always more. But when red is positive, it’s always more. I: Red is always more? Carl: No. When blue is positive... Mag: Red is more away… (continues dragging b) Carl: Red is… Um… I don’t know. Mag: When blue is positive, it’s more… I: That’s a good observation. And I really like the connection that you made that it has to do something with 8. You can also move these around (meaning the number lines), and put them together and see… Mag: Okay, so, put that 9 and it’s at 1. I subtract 1. It’s 8. I started right there (place the blue directed segment at 0). It will be 8 and not 9 I: Can you show me the other… Mag: Another? Um… Put it at 14, and this is…subtract. 8. It’s like always 8 (see Figure 7.5). I: What do you think, Carl? She says it’s like always 8. Carl: Um...Yeah. Wait, it’s not. Cause remember the one was at five and one was at three? So, I guess it’s not always, but... I: Which one? Can you show me? Mag: (Moves the directed segments so that a is at 3 and b is at 5.) Carl: Yes, see? Mag: You add it. They are both positive. Carl: Yeah, okay, if you do add it, I see they are always 8.

Figure 7. 5 The screen capture that was taken when Maggie dragged the lower directed segment, b, to -6 and moved its origin to 15.

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Maggie’s Anticipation

My conjecture is that Maggie could anticipate her solution while she was watching the animation of two mobile directed segments, so she placed the directed segment b at zero in order to verify her anticipation. In the past teaching episode, I found an indication that she was aware of the counterbalancing relations between changes in temperature of two cities. It is probable that she used her “filling” strategy with her conceptual ruler to find that two mobile directed segments always produced 8. After stopping the animation, she dragged the directed segment of b to zero without stopping at any other points. Coupled with her explanation, “it’s always 8", I hypothesize that she anticipated the relationship, mentally generalized it, and used a particular case that most clearly represented the relationship to show or verify her anticipation. She then used her several examples to justify her hypothesis.

I also found solid indications that Maggie could distinguish the orientation of the change in each directed segment and that of the directed segments. When I placed the two directed segments at 4 where both were positively oriented and moving in opposite directions, she said, “I mean they are still opposite… They are growing one way...". Furthermore, when Carl made an objection to her conjecture that “it’s always 8", when a is 5 and b is 3, Maggie said, “you add it.

They are both positive". Considering that she could take three levels of units as given, I analyze that she reconstituted the given situation as a unit containing three levels of units, took it as an input, and disembedded changes in each quantity.

Although she said, “it’s always 8", she first said, “always 8 more". Thus, it is possible that she took b as an input and attempted to find how b affected a instead of looking for the mutual relationship between a and b. So, I asked her which one was 8 more, she said, “they are both 8 more than each other, so I just added it". This is an indication that she tried to express

183 their mutual relationship. I noticed that Maggie used the word “add” to describe the coordination of the two directed segments as a combination rather than implying a counting-on action or vertical addition of two positive numbers. In addition, she constructed the concept of negative quantity as an inverse to positive quantity. Thus, I infer that she was explicitly aware that she needed to subtract when “adding” two oppositely oriented quantities because one quantity removed a part of the other quantity.

I also considered that Maggie constructed her anticipation as a self-regulated system. In the teaching episode of January 23, she showed that she had an ability to make a hypothesis, verify it using specific examples and adjust it by feed backing a counterexample into her anticipation so that she could make more compatible hypothesis. Likewise, I infer that she made a hypothesis that the sum of the two directed segments was 8, and dragged the directed segments to test her hypothesis. I argue that reversible reasoning was involved when she constructed her hypothesis as a self-regulating system. In Chapter 3, I hypothesized that a child is engaged in reversible reasoning when she uses either the result of a scheme or the anticipation of the scheme as an input and recursively uses the scheme to return to its starting point by reconstituting the given situation. I infer that she used her measuring scheme (or her “filling” strategy) to anticipate the relationship, took her anticipation as an input and recursively used her measuring scheme in order to test and confirm her hypothesis.

Carl’s Contradicting Prediction

The salient difference in Carl’s and Maggie’s ways of finding the relationship can be found in their actions and language. Maggie manipulated the speed, dragged b here and there while Carl did not move anything while they both had access (one mouse for each student) to the computer. When asked to find the change in b when a increases by 1, he answered that it would

184 decrease by 8. This is an indication that he was not considering the changes or the movement of the directed segments. In addition, he said, “When blue [referring to b] is negative, it’s [referring to a] always more. But when red is positive, it’s [referring to a] always more”. Considering that he conflated later when he saw the case of a and b were five and three, I infer that he made a local generalization of the cases when two quantities are oppositely oriented and assimilated it to a take-away situation. When Maggie explained that it was always 8 with two occasions where the two quantities were oppositely oriented, he suggested a counter example where a was 3 and b was 5. When Maggie disputed this by saying that they needed to be added when both were positive, Carl responded that “if you add it, they are always 8". His comment can serve as a corroboration that he used his take-away strategy when finding the relationship.

In previous teaching episodes, I found multiple indications that he was limited to coordinating two levels of units due to the lack of the ability of reflectively abstracting his actions and disembedding them. In addition, he used a number line as a set of discrete number sequences rather than treating directed segments as an accumulated quantity from zero.

Similarly, I found that his lack of the ability to reflectively abstract changes or movements of the directed segment constrained him when coordinating changes. He used his take-away strategy, but I could not find any indications that he compared them by coordinating the changes in them.

Although he answered that b would decrease when a increases, it is hard to say he found it by coordinating the changes because he was watching the directed segments that were moving in opposite directions at that time. In addition, he had two different strategies for two different occasions: adding the numeric values of the directed segments when both were positive and subtracting one from the other when they were opposite to each other. These separate strategies serve as an indication that he could not construct his solution as a unit containing the other two

185 composite units. Rather, he consistently assimilated the given situation as a take-away situation whenever necessary.

Carl’s Consistent Use of Letters as Specific Numerals

On February 23, Maggie was absent. So, I decided to investigate how Carl coordinates two covarying quantities in depth. I prepared three situations that can be modeled as a=b, a+b=0, and a+b=8. On GSP, two mobile directed segments, a and b, which moved in tandem in relation to each equation, were presented. He first watched the two directed segments that were moving from zero to the same direction and then stopped the animation.

Protocol 7.5 Carl’s a=b as 10=10.

I: What can you say about the measures of the temperature of the two cities? Carl: They are exact. I: Yeah? Carl: Yeah. When the temperature of a is at 10, b is at 10, and so on. I: So, can you express it in an equation? Carl: Um. what do you mean by an equation? Like an actual equation? I: Yes. You use [an] equal sign to represent the relation between the quantities. Carl: Alright, so maybe something like this, 10 equals 10 (writing 10=10). I: Yeah, 10=10. But what if I say… ‘Cause we put this is a and… so like… Carl: 10a=10b or something? I: What if I say that the temperature of city M is a, then what is the temperature of city C? Carl: a? Or b? Like a, b, or something? I: Yes, so you can say that a equals to b, right? Carl: Right (writing a=B). I: Does it makes sense? Carl: Yes, ma’am. I: So, if you see that (pointing at the equation), how can you interpret that in terms of the measures of the temperature? Carl: Make it like a key that also says that a=10 and B=10 (writes a=10, B=10, see Figure 7.6). I: Yes, so what can you say about it? If you see this equation (pointing to a=B), what can you say about the temperature? Carl: That they are equal? I: Yeah, they are equal, right?

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Carl: Oh, you could, um… would you like the… um… like this (writing an inequality sign of ≤, and then adding 10≤12)? I: Yeah, but this is different. It means this (referring to 12) is bigger than… Carl: Yeah, but when they are equal, you put it like that (points to a=B).

Figure 7. 6 Carl’s screen when finding a relationship between a and b that can be modeled as a=b

The purpose of this task was to explore his focus of attention when watching two identical directed segments moving in tandem. When asked what he could say about the measures of the temperature, he said, “they are exact". Later, he mentioned the case where both were “at 10” and implied that there were multiple cases by saying, “so on". I infer that he focused on the numeric value at which the head of the directed segments was placed rather than focusing on their movements. Coupled with the indications that were found in the previous teaching episode and his comment, “make it like a key that also says that a=10 and B=10", the indication is solid that he was focusing on the numerical values and used a and B for 10.

I noticed that Carl’s meaning of equation was a representation of a true statement. He mentioned “actual equation” and wrote 10=10. He then later provided an expression using an inequality sign, 10≤12, to explain what the equation meant by saying, “When they are equal, you put it like that [points to a=B]". Considering the way he treated letters in an equation as a substitution for numeric values, I reckoned that “actual equation” implies a statement that turns out to be true when he substitutes numbers with letters. Hence, an equation that is composed of

187 letters without numbers might have not been an “actual equation” to him.

The next task involved revisiting the task that he did in the previous teaching episode.

The goal of posing the same task was to find corroborations to the analysis that I have previously made and to observe if he could take account of variance of the given quantities. When asked to talk about the relationship between two mobile directed segments which moved in tandem in relation to the equation m+c=0, Carl wrote m=-c, m=10, and c=-10 on the computer screen using the mouse (see Figure 7.7).

Protocol 7.6 Carl’s m=-c as 10=-10 where m=10, c=-10.

I: So can you tell me about the relationship [between m and c] in terms of the temperature? Carl: (Writes m=-c, m=10, c=-10) I: Yeah, why did you say that m [is] equal to the opposite of c? Carl: Because um… this is at -10 and this is at 10. So, if you keep going, they are always gonna be the opposite of each other. Unless you get to zero, but (clicks on an animation button and watches the two moving directed segments)... They always gonna be… see that thing hits 15? And if you stop it at zero, that’s the only time that they are the same. I: Yes, exactly. Does this equation (pointing to m=-c) work when these are all zeros? Carl: Um… yes… I: That’s right. Why do you say that? Carl: Because they are… um… same, I guess?

Figure 7. 7 Carl's screen when finding m=-c

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His equations, m=-c, m=10, and c=-10 serve as a corroboration to my hypothesis of his meaning of letters and equations. This time, a clear indication that he considered many different cases other than m was 10 and c was 10 can be found in his comments, “if you keep going, they are always gonna be opposite of each other". However, there is no basis to infer that he coordinated changes in quantities. Considering that he had been placing the “-” sign in front of letters (on January 23), whenever he wanted to specify a negatively oriented quantity, I infer that he wrote the equations, m=-c, m=10 and c=-10 to express the situation such as a positive value on one side and a negative value in the other. His comment that zero would be the only exception supports my hypothesis. In retrospect, I found another occasion that can serve as a corroboration to my hypothesis. In the teaching episode that was held on February 20, he wrote a=B, A=3.5, and B=3.5 to represent the situation that can be modeled as a=-b. When it was pointed out that b was -3.5, he added the “-” sing in front of 3.5 and wrote A=-B, (see Figure 7.3-(b)) to show that the right side represents a negative number. The fact that he conflated “c” as a negative quantity only when writing it as “-c” serves as an indication that he could review “c” neither as an unknown quantity nor as a variable that can possibly be a negative quantity. Rather, it was a symbol of a fixed number 10.

Carl’s Contradicting Two Strategies

For the last task, he clicked the animation button and watched the two directed segments moving in a relation of a+b=8. He then stopped the animation when b hit 0 and a hit 8. He changed his strategies three times: Firstly, he compared the length of the two directed segments

(Protocol 7.7), secondly, he attempted to compare the changes or the movement of the directed segments (Protocol 7.8), and lastly, he used several ordered pairs to prove his hypothesis

(protocol 7.9). I separated the whole interactions based on his three different strategies.

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Protocol 7.7 Carl’s comparison of the length of the two directed segments.

I: Just think about them in terms of temperature and tell me what you are thinking about. Carl: Um, that the city C went zero. m is um… 8 greater (looks at the directed segments that were located at 0 and 8)? I: 8 greater? Yes, I see that. And? Carl: I am thinking like, when it does like this (clicks the animation button to see the movement). I: What happens? Carl: It gets larger (stops the animation so that b was placed at -7 and a was placed at 15). That’s (points to b) five, six, seven. That’s seven. And it’s 15. So, 15 plus 7 is… you know whatever that is… I: What is it? Carl: 22, I think? I: 15 and 7? Yes, if you add 15 and 7, it’s gonna be 22. But look at this (pointing to b). Look at the temperature. Carl: Negative? I: See the direction. Yeah, this is negative. Carl: Um… (clicks the animation button and stops it so that a is approximately at 13 and b is approximately at -5). Oh! Whenever, um, this is, um, even though this is negative, if you put it on this side (points to the opposite side where a is placed)... Five was right here (marks five on the number line in which b lies), it’s 8 more anyways. I: What do you mean by 8 more? Carl: So, (confidently counting from 5) one, two, three, four, five, six, seven, eight! I: One, two, three, four, five, six, seven, eight more? Carl: Yes! Even though it’s on this side (meaning left side of zero), it’s always 8 more than the other side. I: Can you move them and put in on other places? Can you show me if that works in all the other places, too?’ Carl: Like here (stopping where b is a -12 and a is at 20). This is 10, 11, 12, so this is 12. So, (starting counting from 12) one, two, three, four, five, six, seven, eight! I: I like that. Think about it in terms of the temperature. So, when you want to explain this in terms of the temperature, what could you say about it? Carl: m is always…Wait (subvocally starts counting from -12), one, two, … I: So, what did you count? Carl: 32 or 33. I don’t know. I: Oh, from here (points to -12) to all the way here (points to 20), is that what you counted?

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Carl: Yeah. So, city M is always 32 more than city C? I: Is that true? Can you show me why that is true? Carl: Um, because it’s (dragging b to back and forth quickly)... I guess it’s not true (slows down his dragging motions and slowly drags b back and forth and intently looking at the screen then placing it at 4). I: Hum! Tell me about this. What can you see about the relationship between the two temperatures of the two cities? Carl: They are the same. I: Sometimes they become the same? Carl: Yeah…Guess it varies. I: It? What do you mean by it varies? Carl: The temperatures aren’t always the same length away (marks the length of the directed segment of 4 by drawing an arc from 0 to 4, see Figure 7.8).

Figure 7. 8 Carl’s screen when finding a relationship between a and b that can be modeled as a=b

Note. The screen capture shows his attempt to mark the length of the directed segment.

The shifts of Carl’s attention to the orientation of the given quantities are quite dramatic.

First, he stopped the animation and saw that m was 8 and c was 0. He then said m was 8 more.

Next, he continued the animation and stopped it where c was at -7 and m was at 15. He counted from -7 to 15 and found 22 by adding 15 and 7. It is possible that he was able to view 22 as a combination of 15 and the inverse of -7. However, I could not find any indication that he could review 22 as a unit containing three levels of units while taking 7 as an inverse of -7. Rather, it seems like he counted on. He got confused, because what he just counted contradicted his previous hypothesis. He moved the directed segments again, and confirmed his hypothesis using the example of -5 and 13. He placed -5 “on the other side” of the number line and compared the

191 magnitudes of each quantity by counting from 5 to 13. Had he kept track of his action of placing negative quantities to “the other side” and the orientation of a negative quantity in other occasions, I could hypothesize that he could review the given sum as a unit containing three levels of units. However, he lost track of his action when he relocated the directed segments at

-12 and 20. He found 8 by counting from 12 to 20, but he was confused when asked to explain it in terms of temperature. He counted from -12 to 20 and found 32, which was not consistent with his previous method. In addition, he rejected his previous conjecture and said, “it varies” when he saw the two directed segments of 4.

Whenever he was reminded of the orientation of the quantities and asked to explain his method in terms of temperature, he used his counting schemes. In the occasions where he tried to justify his reasoning using other examples, he compared the lengths of the directed segments using his take-away method. These two contradicting strategies suggest that he did not keep track of the orientation of a negative quantity. In addition, no indications were found that he was coordinating changes or movements at this point.

Carl’s Confusion when the Orientation of the Change and that of Quantity Disagree

I continued to explore Carl’s ways of finding the relationship of two mobile directed segments. Although he said, “it varies", I decided to encourage him to investigate further by emphasizing that there exists a certain relationship between the two co-varying quantities.

Protocol 7.8 (Cont) Carl’s attempt to abstract the changes in each quantity.

I: Yeah, it varies. The temperature of city C varies and the temperature of city M varies, but there’s a certain relationship. That’s why they vary in tandem. Carl: (Clicks the animation button and watches them moving for about 3 minutes.) Carl: They are always opposite (keeps looking at the animation)? I: They are always opposite? Carl: Uh huh. I: Can you show me if that is true? Oh, what about that part (points to the part where

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0 < a< 8 is on the number line)? Carl: They are still opposite, though. I: What do you mean? Carl: When they are here (stops the animation and drags b at -15) , they are opposite, like on each side. Even right here (placing b at 4), they are still opposite. I: What do you mean? Carl: Cause this (a) is positive and this (points to b) is positive... Ah… I don’t know. Never mind. I: Are they opposite? Carl: (Shakes his head and drags b to 4 and paused. Then he moves it to 0).

At the end of Protocol 6.7, he used the word “varies” for the first time. So, I decided to take advantage of this opportunity for him to think about changes or movements of the directed segments. He intently watched the animation for 3 minutes and found that the directed segments moved to the opposite directions. However, once the animation was stopped, he got confused between the orientation of the directed segments and the orientation of the changes in the directed segments. When he saw that the two directed segments were fixed at 4, he failed to envision their movement. This is a strong indication that he could not reflectively abstract changes in each quantity. This finding is aligned with my previous analysis of Carl’s lack of ability to reflectively abstract his counting actions.

Carl’s Meaning of “-” Sign

Protocol 7.9 (Cont.) Carl’s generalization based on fixed number pairs.

I: You can move them (means translating number lines) if that makes the comparison easier. Carl: (He grabs the number line on which b was placed and places it on top of the other number line so that they all lined up at zero. He then animated the two directed segments, intently looked at them moving for about 3 minutes and stopped them where b was at -5 and a was at 13. Then he wrote -5 and 13 vertically and “=-8” right next to the two numbers (see Figure 7.9-(a)) I: Uh…why did you get -8? Carl: I just have 8 difference. So I just put ‘-’ here. I: Oh, to indicate the difference!

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Carl: That’s right. So… Um… I: Can you tell me more about what ‘8 difference’ means? Carl: If you subtract this, 5, 13, that’s what you get, 8. I: I thought if you subtract 13 from -5, that’s gonna be -18. Carl: ‘Cause the different signs, you subtract. I: But they are already in different signs, though. Carl: So, we should subtract them. I: So, can you write an equation? You are subtracting what from what? Carl: Thirteen minus five (writes 13-5=8 vertically). And that (points to the “-” sign”) represents the negative right here (He then added 1 in front of 8 so that it looks like 18, see Figure 7.9-(b)). I: (Points to 18) what is that? Carl: Oh, you subtract it but…You subtract. I: Okay, so… Carl: It’s 8. I: Is it 8? Carl: Yeah (crosses out 18 and rewrites 8). I: So, what did you do? Did you add -5 on… Carl: Subtracted. I: What did you subtract from what? Carl: Subtracted 5 from 13! I: Okay, now I got it. That is true. Carl: Right. Then (clicks the animation button and stops it when b is at -10 and a is at 18. He then writes -10, 18, 8 vertically, see Figure 7.9-(c)) … that’s 8. I: Can you tell me what you did and why you did that? Carl: ‘Cause 10 is negative, so you subtract. It gives you 8. I: Okay. Carl: And… (clicks the animation button and stops is when b is at -15 and a is at 23) 23 minus 15 (performs vertical computation using the traditional methods), that gives you 8. I: Can you move it to… right there (looks at b which is at 3 and a which is at 5, see Figure 7.9-(d))? Yeah. Carl: Like this? Alright (writes a vertical computation of 5+3=8, see Figure 7.9-(c)), 5+3, which equals to 8. It always gets like a sum of 8. I: Oh, so can you talk about it in terms of the temperature? Carl: Um, the sum of the two… the sum of the two temperatures always equals to 8. I: Yeah, I like that. Does it make sense? Carl: Yeah, it makes sense now. I: Can you write in an… Carl: An equation, you mean?

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I: Uh huh. Carl: Alright. So, equation for them? I: Uh huh. Say that m represents the temperature of city M and c represent[s] the temperature of city C. Carl: m could equal to 0 and c could equal to 8 (see Figure 7.9-(e)). So, that equals to 8. I: I like that. But this only explains particularly when m is 0 and c is 8. I want an equation that represents all the cases. An equation which works all the time. Carl: Uh… m=m and c=c. I: Okay, that works, too!

(a)

(b) (c) (d) (e)

Figure 7. 9 Carl’s memos for the relation between a and b that can be modeled as a+b=8

Note. The screen capture (a) shows how he lined up the number line and his equation.

Pictures (b) and (c) show Carl’s numerical computations for 13-5=8, -10+18=8, 23-15=8 and

5+3=8. The screen capture (d) shows that a is at 5 and b is at 3. Picture (e) shows Carl’s equations for the relationship between a and b that can be modeled as a+b=8.

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I suggested Carl to move the directed segments around so that he could line them up and construct three levels of units structure. However, he consistently lined up the directed segments at zero, and he used the numerals in his vertical computation. He then assimilated the addition of two oppositely directed quantities as take-away situations, i.e, comparing the magnitudes of each quantity, taking the shorter length away from the longer one. His comment, “‘cause the different signs, you subtract” supports my analysis. In addition, he placed the “-” sign in front of 8 while saying, “8 difference. So I just put ‘-’ here". When asked whether he added -5, he firmly said he subtracted 5 from 13. His answer suggest that he did not consider adding -5 and subtracting 5 were identical in the given situation. Rather, “-” meant “minus” or difference to him. According to Ulrich (2012), the construction of the concept of signed quantity requires the concept of an

Explicitly Nested number Sequences (Steffe & Olive, 2010). Considering Carl was with TNS (he lacked the ability of reflectively abstracting and disembedding his counting actions), I infer that he was yet to construct the concept of a negative quantity as an inverse to a positive quantity.

I found an indication that he was momentarily confused when he tried both adding and subtracting given numerals, 13 and -5. He wrote 18 first then crossed out 18, and wrote 8. I infer that he tried both addition and subtraction and chose to subtract 5 from 13, because it led him to

“a right answer”. Thus, I found that his scheme was not anticipatory. In case when b=3 and a=5, he added them. It seems like he was not aware that his operation that he just carried out was not consistent with his previous operation. Rather, it seems like he was satisfied with his strategy which provided him the right answer, 8: subtracting magnitude of the two oppositely oriented quantities and adding two positively oriented quantities. Without the construction of the concept of negative quantities, I infer that he associated the “-” sign to a take-away situation. His equation, m=m and c=c suggests that he thought about placing a numeric value on the right side

196 to make “actual equations” that looked like answer keys as he previously did instead of reviewing his previous examples as a unit containing two quantities. Moreover, I did not find any indications that he coordinated changes.

I found that Carl, who was limited to coordinating two levels of units, was yet to construct the concept of negative quantities. His confusion when shifting his focus from the changes in each quantity to the orientation of the quantity itself supports my analysis. In addition, I found multiple indications that he used the “-” sign as either a pointer indicating a numeric value that located “on the other side” or assimilating to take-away situations. Without the concept of negatively oriented quantity, I found that he could not consider addition and subtraction as inverse to each other. Thus, he developed separate schemes for adding oppositely oriented quantities and for positively oriented quantities.

Graphing x+y=a

The purposes of the teaching episode that was held on March 2 was to investigate how

Maggie and Carl graphed the given situation of a+b=8. The first part of the teaching episode focused on exploring their responses to the request of “graphing the given situation” without any guidance of how to coordinate two quantities in order to explore the inferences and indications of their understanding of graphing itself. The latter part of the teaching episode was designed to explore schemes and operations involving the construction of the relationship between the changes in the two quantities, which is the counterbalancing relation in this case.

Carl’s and Maggie’s Graphs: a Line Connecting Two Points

To investigate their notion of graph, I asked them to “graph” the situation that can be modeled as m+c=8. I intentionally chose to say “graph it” instead of “represent the relationship” to explore their notion of graphs without interaction with me.

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Protocol 7.10 Maggie and Carl’s attempts to graph m+c=8.

I: The sum of the measures of the temperature of two cities is always 8. What I want you to do is that, I want you to graph it. Mag: I was gonna do what we were working on. What are you doing (looks at Carl)? I: Carl, what are you doing? Carl: I don’t know what the heck I’m doing. I: Okay, keep working on… Carl: I will just count up and then I will put…kinda like a bar graph (see Figure 7.10- (a)). I: Okay. Mag: It was m and a? I: m and c. Carl: (Draws a vertical bar of height 8 for m) so, this is a bar graph (draws another vertical bar of height 3 for c, see Figure 7.10-(a)). Then you want me to make an actual graph now? I: Yes. Carl: What kind of graph? I: You know, some kinds of graph that you learned at school… like… Carl: x and y? I: Yeah. Carl: Okay (draws two perpendicular lines, see Figure 7.10-(b)) Mag: Um, (draws double number lines) I don’t know what was it… What was it… I: The sum of the measures of temperature is always 8? Carl: (He writes an equation of y=8x at the end of the vertical line he made.) Mag: (Marks m and c for each number line) I can make it uh… I can make it whatever I want to either way? I: Just do what you wanted to do first, and then we can work on the graph that you learned at school. Mag: (First she makes a horizontal bar from 0 to 5 on the number line m and a bar from 0 to 3 on the other number line, c. She then shortens the first bar by one unit and extends the other bar by one unit so that both bars end at 4.) Then I will do like that (looks at Carl’s coordinate plane). Now, do you want us to do a bar graph? Carl: (Completes his second graph, see Figure 7.10-(b)) Guess it looked better right now. I: So, can you explain to me why you drew this line (points to the vertical line from 5 to 18)? Carl: It’s kinda messy, but, zero is supposed to be right here (circling (0,5)). You know where zero is? And this one (draws a vertical through (1,0) to (1.16)) is supposed to be right at 1. It supposed to be like one sixteen…

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I: One sixteen? Carl: Yes. I: What do you mean by one sixteen? The point is gonna be at one sixteen? Like one comma sixteen? Carl: Uh, (looks at Maggie’s horizontal bars on her double number lines) I remember it should be at 4… I don’t know. I: Where did you get that equation (points to the equation of y=8x)? Carl: Since it equals to 8, so, it’s 8. I: Alright (turns her head to Mag). Mag: I don’t know. I just… the y will be (writing y= )… I: So, what should be on the x axis and what should be on the y axis? Mag: Uh, at least up to 8, I guess? I: Huh? Mag: At least up to 8? I: You are thinking about the relationship between the temperature of city C and city M. Carl: Alright. Mag: I can put different colors. I can get [a]different color for that. Carl: Ah! (Starts to draw two new perpendicular lines.) I: A different color? Can you tell me more about it? Mag: Like, if I need a mark, see m? I can mark it. If I need a mark, see the c is at… and mark it in different color. I: Oh, can you do that? I’ll get you a color pen. Mag: (Adds a negative sign to each of the numbers on the number line of c) Do you want me to mark what I did right here (points to the double number lines, see Figure 7.10-(a)) in here? I: Yes! You can do whatever you want to do. (Looks at Carl.) Carl, can you tell me what you are thinking? Carl: Um, I put my graph that will make one (referring to either m or c) equal to 5 and one equal to 3. I guess I put like y=5 and x+1. Graph that, I guess? Guess that’ll work … (writes y=5x+1 and y=3x+1). I: Okay. (Looks at Maggie.) Mag: I just changed all these negatives (points to the number line c),.... and seven down …temperature. I: Ah, so this (points to (0,4)) represents…what does each one represent?… Mag: (Labels (0,4) as m, (0,-4) as c.) I: Oh, but c is also 4. Mag: I changed it. I: You changed it! But the sum of the measures of the temperature is 8. If you add them up, what do you have?

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Mag: Dang. Carl: (Plots (0,3) and (0,5) on the coordinate plane.) I: (Looks at Carl’s two points.) What if I ask you to put city M on a x-axis, and city M on a y-axis? Will that be different? Carl: Wait, so, city M for both? I: No, city M on the y-axis … Mag: Like this (writing city M for the vertical axis and city C for the horizontal axis, see Figure 7.10-(g)). I: Yeah, like that. Is that familiar to you? Carl: Yes. I: And what is the relationship between city C and city M again? Mag: They are gonna be always 8. I: Yeah. Mag: Alright. Carl: (Plots two points of (3,0) and (0,5) on each axis and connected them with a line, see Figure 7.10-(d).) I: Carl, can you tell me what you did? Carl: Um, started at 5 (points to (0,5)), then I put a point at three (points to (3,0)). Then I lined up a little. I: Do you think that represents all the cases? Carl: Yeah, it could, yeah. This is for five and three. I: Can you check other cases to see if this graph represents the whole situations? Mag: So, what did you say? I: Just think about ways to represent the measures of the temperature of two cities in all cases. Mag: So, that will be what? If it equals to 8 (writing “=8” on a coordinate plane), then… oh, never mind. I: (Looks at Carl) So, can you tell me what you are thinking about the second line (pointing to a line with negative slope, see Figure 7.10-(c))? Carl: Um, I started at -1, then I went up to 7. 7 is up here, but you just can’t see it. I: So… what is -1? Temperature? Carl: Yes. I: City C’s or City M’s? Carl: City C’s. I should have wrote farther. (pointing to the tip of the vertical axis). I: So, 7 means city M’s? Carl: Yeah, but it should have been 9. I: If you need, we have more paper (handing another piece of graph paper to Carl). Carl: Alright. I: (Looks at Maggie drawing a green line that passes through (0,1) and (7,0), see Figure 7.10-(g)) Can you tell me what you are thinking?

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Mag: Um, m is 1, and then c is 7. So, I marked it and then (connecting the two points)... I: Right. But, um… Mag: And it equals to 8. I: Yeah, but does it also work in this case where all are 4 and 4? Mag: Uh huh. I: Can you show me how it works… Mag: If that wasn’t negative (points to -4), like one (marks (0,4) and (4,0) and connecting them forms a line... I: Then what does this point (plots (2,2) on the line that Maggie just created) represent? The point in the middle? Mag: Um… A half of them? I: A half of them? Mag: I don’t know. I: So, what is a relation between m and c? Can you write an equation? Mag: (Writes m+(-c)=8, then tried to erase “-” in front of c) I: Does it also work or this (points to her writing, m=1 and c=7, see Figure 7.10- (g))? Does it also work when m is negative? Mag: That goes to right here (points to c). I: Okay. Carl: (Finishes up his new graph, see Figure 7.10-(d)) I: Carl, so you came up with two different graphs. Can you tell me what you were doing? Carl: I just did it where they always equal 8. So, I started at -1 and I went up 9. So, this will be 9 minus 1 equals to 8. And then, 5 and 3, that equals to 8. I did the line but I don’t know what the line means. I: What did you say? Carl: I did the line, but I don’t know what lines mean. I: Oh… What do you think, Maggie? What do you think the lines represent? Mag: This or this (traces each line on her coordinate plane, see Figure 7.10-(g))? I: This and this. Mag: It could be like, um… This is morning, or this is just a night (writing “night” above m=7, see Figure 7.10-(g)). In the morning, it was 4 and it went up. I: What do axes represent? Carl: Degrees (labels c and m for the horizontal and vertical axes)? I: Yes, degrees. That’s right.

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(a) (b) (c) (d)

(e) (f) (g)

Figure 7. 10 Carl's and Maggie's graphs

Note. Picture (a) shows Carl’s bar graph, (b) shows Carl’s vertical graph of y=8x on a xy- coordinate plane, (c) shows Carl’s graphs of y=5x+1 and y=3x+1, (d) shows two lines: one goes through (-1,0) and (0,9), and the other goes through (0,3) and (0,3). Picture (e) shows Maggie’s number line graphs, (f) shows Maggie’s graph demonstrating one example of m+c=8, (g) shows

Maggie’s two lines that shows two examples of m+c=8.

The Protocol 7.10 provides many implications about how students learned graphs in school. I began asking Maggie and Carl to graph the given situation without discussing their meanings of graphs in order to explore their understanding of graphs. Maggie drew two number lines (see Figure 7.10-(e)) and Carl made two bar graphs (see Figure 7.10-(a)). It seems as if

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Maggie thought the double number lines that she had been working on was one way to represent the given situation while Carl tried to recall something that he learned in school. Nonetheless, I observed some hesitation in both students’ attitudes: Maggie glanced over Carl’s graphs and Carl said, “I don’t know what the heck I’m doing". One distinction that can be found in their first graphs is that Carl used bar graphs to show one particular example of the given situation while

Maggie first made horizontal bars of 5 and 3 and then she changed it to 4 and 4 by reducing and extending one unit of each horizontal bar (the“-” signs were later added in front of each number of the number line for C). I infer that Maggie was aware that she could demonstrate multiple examples that satisfied the given situation using the double number lines. Although she showed that she could coordinate changes on double number lines, whether she considered changes in temperature is ambiguous at this point.

After completing the bar graphs, Carl asked, “You want me to make an actual graph now?” He then asked me what kind of graphs I wanted him to graph. In retrospect, I should have asked them their meanings of graphs, but instead, I asked him to draw some graphs that he learned in school. His question, “x and y?" along with his construction of a set of perpendicular lines that he drew on the coordinate plane suggest some indications of what he meant by “an actual graph". His later comments after he drew a vertical line on a coordinate plane, “Guess it looked better right now", suggests that he might have regarded lines on a coordinate plane as

“actual graphs". His first “actual” graph (see Figure 7.10-(b)) was composed of one line segment from (0,5) to (1,16), a ray from (1,-∞) to (1,16) and the equation of y=8x. Although it is unclear why he plotted those points, I guess that (0,5) might have been another demonstration of his first bar graph that was vertically erected from 0 to 5. The number 1 in (1,16) might imply that he was thinking the next sequence, and 16 as the value of the second sequence that he got when he

203 plugged 2 into 8x, based on his equation, y=8x. After he saw Maggie plotting (0,4) and (4,0), he said, “it should be at 4". At this point, he was confused and looked incapable of explaining further. So, I asked him about his equation, and he said, “Since it equals to 8, so, it’s 8 the equation’” to explicate y=8x. It seems as if he wrote y=8x, because he learnt it at school that an equation for a graph starts with “y=” along with some number and x on the right side of the equation. Knowing that the sum “equals to 8”, I hypothesize that he added “=8” and then placed x. Although confusing, the incoherence between his graphs and his equation suggests that the absence of coherent meaning between his graphical representation and an equation along with his effort to make an association between the given situation and what he learned in school.

Carl’s last two graphs suggest strong implications of how he learned to draw graphs, and how he related the graphs to the data and equations. He plotted two points, (3,0) and (0,5) for y=5x+1 and y=3x+1, and two points, (-1,0) and (0,7) for another line that he later changed (0,7) to (0,9). I consider his primary choice of (0,7) for two numbers together gets 8 as a necessary error rather than a mere mistake. In previous teaching episodes, multiple indications were found that Carl lacked the concept of negative quantity. Thus, I infer that he thought of a magnitude of

-1 when he was plotting -1. Later he changed it to (-1, 9). He explained the first graph, “I put my graph that will make one [referring to either m or c] equals to 5 and one equals to 3. I guess I put like y=5 and x+1". Thus, I infer that his equation did not represent the relationship between x and y. Rather, the coefficient shows two numbers that together make 8. Although it is unclear at this point what x+1 meant, I guess it might imply that he was thinking the ordered pairs sequentially. In addition, connecting two points was meaningless to him, based on his comments,

“I did the line but I don’t know what the line means". Hence, I infer that he was engaged in sequencing examples and used axes to plot them.

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When Maggie tried to graph the situation in a way that she learned in school, it seems that she made two perpendicular lines in attempt to make another representation of her double number lines. She purposefully constructed a coordinate plane where both x and y expanded from -8 to 8 because it had to be “at least up to 8". She also wrote an equation starting with “y=", but it seems as if she could not make any meaningful connections between her graphs and the equation that was starting with “y=". She attempted to demonstrate what she showed in the double number lines, but it seems as if she felt conflicted, because both of the number lines looked identical (see Figure 7.10-(e)). So, she chose to plot (0, 4) for m and (0,-4) for c (see

Figure 7.10-(f)). It is surprising that she plotted points while she treated the measures of the temperature as composite units in the past teaching episodes. Considering that they were from the same mathematics class, it was probable that they share a similar notion of graphs, and used each other’s graphs as references when drawing graphs that they had learned in school.

Maggie’s third graph (see Figure 7.10-(g)) was similar to Carl’s last graph in a way that she plotted two points for two numbers that make 8 when adding, and connected them to form a line. It was probable that she connected the two points following how Carl did without considering meanings to it. When asked of a point (2,2) on a line, she described the location of the point as “half of them” and said that she did not know. In addition, she also explained that each line represented distinct occasions such as the temperature that was measured in different times. Maggie could not write an equation, and I interpret her hesitation after writing “y=” stemmed from a lack of coherent connection between graphs and her understanding of the situations, considering how she coordinated changes or movement in the previous teaching teaching episode that was held on February 20.

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Considering Maggie’s and Carl’s distinct ways of operating when finding x+y=a, it was surprising that their graphical representations and their explanations were similar. Based on that there was little meaning of the lines and the lack of coherence connection between the equations and the lines along with Carl’s comment, “I did the line, but I don’t know what lines mean”, I infer that their existing concept of graphs, such as plotting two points and connect them to make a line, was not sufficient when they meaningfully coordinate two co-varying quantities.

Exploring the Counterbalancing Relation between Two Quantities

Since graphing was a constraint when thinking about changes in temperature, I asked them to think about changes in each quantity rather than thinking about discrete number pairs.

My goal was to help them consider changes and movement to make a more meaningful representation.

Protocol 7.11 (Cont.) Maggie’s construction of counterbalancing relations and Carl’s counterexample.

I: Let’s think in heads instead of doing with numbers. The sum of the measure of the temperature of the two cities is always 8. Let’s say the temperature of city M increases by 1. What happens to the temperature of city C? Carl: It increases. Mag: It decreases. I: You said increases and you said decreases. One person is right. Can you tell me what you are thinking? Mag: Wait, if M decreases, what is C? I: If M increases, what does C do? Mag: Decrease. I: You (referring to Maggie) said decrease and you (referring to Carl) said increase. Can you tell me why? Carl: Since this increases (traces the line that goes through (-1,0) and (0,9), see Figure 7.11-(d)) this decreases (points to the right side of the line that goes through (3,0) and (0,5), see Figure 7.11-(d)). Since this. If this decreases (traces the line that goes through (-1,0) and (0,9)), this gotta increases (points to the right side of the line that goes through (3,0) and (0,5)). ‘Cause one’s got to.

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I: One’s got to? Can you tell me more about it? Carl: Cause… Mag: Wait, I said decrease. I: Yeah, you said decrease (referring to Maggie) and you (referring to Carl) said increase. Carl: Okay, what is increasing then? I: M increases by 1. What happen to city C? Mag: (Almost at the same time as Carl) It goes down. Carl: Increase. Especially in my case, it will be increase (looks at his last graph). Mag: What (looks at Carl)? I: Okay, why? Carl: (Smiles.) Mag: (Abruptly) I can explain. (Draws a double number line. See Figure 7.11-(b).) Okay, say this is M that we were working on (writes 1,2,3,4,5,6,7,8 and draws a line from left to right, above the numbers). And then (Draws above the number line for M)... So, this is M (traces from 1 to 8 on the number line for M) Say it has… it increases one, right? Okay. (draws a segment from 7 to 8 the number line M and a short segment above the number line for C, see Figure 7.11-(b)). I: Uh huh. Carl: She is right, isn’t she? I’m pretty sure she is. Mag: Okay (traces line C from left to right in Figure 7.11-(b)). And if it goes to 1 (meaning the increase of 1 in M), and this (meaning the sum) is always 8, then… I don’t know. I: (Looks at Carl) Well, what do you think? Carl: I am pretty sure it decreases now. ‘Cause if it increased, it’s like a number line (draws a number line as Maggie did). So… Mag: Oh, yeah, it goes like that. ‘Cause it’s always 8, when it increases this much (draws a short line segment from left to right on the number line for M) it decreases this much (draws a segment of similar length from right to left above C). Carl: (Continues) If this is five and this was three, you add one to that, It’ll make that six and it’ll make that four (writes the equation, see Figure 7.11-(a)). That will be ten. I: Ah… Mag: So like, if M is right here (points to 6) and it increases past eight (moves from 6 to 9 or 10), then it (meaning C) must decrease to make eight (moves from around 10 to 8). I: Ah… Carl: So, decrease. I: decreases how much?

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Mag: What amount of was the increase? I: So, if M increases by 1… Carl: It has to decrease by 1 or 2. It’ll be one. I: 1 or 2? Mag: 1. Carl: 1. Yeah. Mag: Okay, if M increases… [it decreases] 1 (draws another double number line). Say it’s at 8 already (draws a line segment from the very left to 8). So, increases 1. So, now it’s at 9 (marks 9 right next to 8 and draws a line segment from 8 to 9). This decreases 1 (draws the lower number line and makes a segment from 9 to 8 above the line. Carl leaves early.) to be 8.

(a) (b)

Figure 7. 11 Carl’s (a) and Maggie’s (b) notes when finding changes in c

Carl’s Counterexample for the Counterbalancing Relations

One of the differences in the ways Carl and Maggie found the changes in temperature lies in that Carl was looking at the last graph that he made (see Figure 7.10-(d)) and Maggie used her double number lines while completely disregarding the other two graphs that she drew. It seems as if Carl made an association between the slopes of the lines and the changes in temperatures.

His explanation that if one increases the other decreases along when referring to the lines, as well as his later comments such as “one’s got to", and “especially in my case” suggest that he was

208 referring to the changes of the slope of the line graphs. When he changed his answer, he did not use the graphs, but performed numerical computation. He knew that the sum must be 8, and rejected his initial answer by showing that increase of 1 in both m and c would result in 10, not 8.

I infer that he chose to add 1 to each number and compared the result of his computation because he lacked the ability to reflectively abstract changes or movements of the quantities. Although self-corrected, I found another indication that he was not coordinating changes in each quantity.

His answer to the question of how much it would decrease was one or two. Considering that he added one to each quantity, and 10 is two more than 8, his answer is a solid indication that he did not coordinate changes in each quantity. Unfortunately, I could not ask further questions because he needed to leave five minutes early to be on time in his class, for he injured his leg and got his leg in a cast.

Maggie’s Construction of Counterbalancing Relations

To the first question, she instantly answered that it decreased. She then used the double number lines to draw line segments with opposite orientation in order to show the opposite movements. However, it is ambiguous whether she simultaneously coordinated changes in each quantity or sequentially coordinated them. Based on her first explanation, “Cause it’s always 8, when it increases this much [draws a short line segment from left to right on the number line for

M] it decreases this much [draws a segment of similar length from right to left above C]", it seems as if she operated with two double number lines side by side and envisioned the simultaneously compensating changes in each quantity. However, her later explanation, “so like, if M is right here [points to 6] and it increases past eight [moves from 6 to 9 or 10], then it

[referring to C] must decrease to make eight [moves from around 10 to 8]", suggests that she took 8 as a reference point and sequentially coordinated the compensating movements. In either

209 ways, I found a solid indication that she was explicitly aware that 8 was an invariant and coordinated changes while keeping track of the invariant. Whether the invariance, 8, was a unit containing two dynamic changes or a reference point for sequential changes is not clear at this point.

Based on how Maggie coordinated changes in temperature, I hypothesized that she could interiorize the counterbalancing relation of the changes in each quantity. To test my hypothesis on the spot, I asked her a question to investigate whether she could coordinate two consecutive changes by recursively applying the counterbalancing relation that she had constructed.

Protocol 7.12 (Cont.) Maggie’s recursive coordination of changes.

I: What if M increases… no. M decreases by two and then increases by three? Then can you tell me what happens to the temperature of city C? Mag: (Draws a line segment from left to right and made another one from right to left) Ok, so… it always equals eight? I: Yes. Do you remember the situation? Mag: Uh huh… It will be like seven (writes 7=c), then decreases two, then increases three, 8. I don’t know what c would do. I: You don’t know what c would do? Mag: And… uh… right here, 8. It decreases two and then… so… I don’t know what c will be. I don’t know like how much it moves. I: Can you tell me more about it? What do you mean by… Mag: Like, you remember when you touch [the animation button] (refers to animation that she saw in the previous teaching episode)... I don’t know if it goes to here (draws a line from right to left on the number line for m) or here (draw a line from left to right on the number line for c) or here (draws a line from left to right on the number line for c)... I: Ah… you just told me that when m increases by 1, then c decreases by 1. Mag: Oh, yeah. Okay, um... So, 8, it decreases 2 (draws a line segment from 8 to 6, see Figure 7.12-(b)), so, this will go to 9 (draws a line segment from 7 to 9 on the number line for c). So this is 6 (points to 6 on the number line m), and then… one, increases one (referring to increases one from 8) and then this will go to 8 (referring to c moves from 9 to 8). I don’t know. It’s won’t equal 8. I: It won’t equal 8, You are right. You remember the question? When m decreases by 2, and then increases by 3. Mag: Oh, I see it. (some inaudible whispering) So, increases 2 and decreases 3. So,

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here, I guess, 7. Or at 6. I: So, can you tell me the sequence of the changes in the temperature of city C? Mag: Um… it drops. So, it’s four away… subtract? Cause this is at 9 (starts countdown from 9)... I don’t know if I… I: I think you almost got it, but… Mag: I’m not explaining right. I: So, think about… Let’s stop thinking about everything here and just think about the changes, okay? M decreases by 2 (moving my hand down) and increases by 3 (moving my hand up), what happens to the city C? Mag: It decreases. I: It decreases? How much [does] it decrease? Mag: Every time M increases, C decreases. And for at eight (meaning M) if it decreases 2, so eight is nine now. I: Uh huh. So, okay. Where does C start from? Mag: (Draws a double number line in Figure 14) Okay, here’s C, and there’s eight. It’s (meaning M) at eight. If it decreases by 2 (draws a line segment from 8 to 6), it (meaning C) goes up 1, no, 2, ‘cause every time it increases, it decreases. So, 2 (draws a line segment from 8 to 10). And if it increase by three, like one, two, three (draws curves under the lower line segment from 6 to 7, 7 to 8 and 8 to 9), it decreases by three (draws curves under the upper line segment, see Figure 7.12), like one, two three.

Figure 7. 12 Maggie’s way of coordinating two consecutive changes in the measures of the temperature of city C and city M.

Although the question was about the changes in temperature of city C, Maggie found the measure of the temperature 7. Based on her explanation, “It will be like seven [writes 7=c], then decreases two, then increases three, 8", it seems as if she envisioned moving two units to the left and three units to the right, found that it resulted in one unit farther than she started. In other

211 words, she found 7 while treating 8 as a reference point. This suggests that she could start counting from an unknown number, coordinate changes to the unknown, finding the numerosity of it by taking 8 as an input by reversibly coordinating the consecutive changes. It was also possible that she started at 8, reversed the operations and found 7, and carried out the operation again to confirm her answer. Considering that she was able to reconstitute the given additive situation while generating a unit containing three levels of unit and that she could construct the counterbalancing relation by coordinating changes, it is probable that she coordinated the changes as a unit containing three levels of units, placing it at 8 and found 7 by envisioning the coordination of the changes. These are strong indications that she could freely reverse the operation, take the amount of increase and decrease as an input and coordinate them forming three levels of units. However, after being reminded that the sum should be 8, she got confused while manipulating numbers and changes. Later, when prompted to focus on changes with motions, she quickly found that the measure of the temperature of city C would decrease, because “every time M increases, C decreases". Then she showed how she coordinated changes in each quantity on the double number line (see Figure 7.12). Her last explanation is a solid indication that the counterbalancing relation between the changes was interiorized so that she could recursively use it for further coordination. Maggie’s construction of the counterbalancing relation suggests that three levels of units coordinating scheme was essential when coordinating changes in each quantities that were in a counterbalancing relation. In addition, I argue that she engaged in reversible reasoning when she coordinated consecutive increase and decrease to find

7, which was the measure of the temperature that became 8 after applying the two changes in temperature.

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Carl’s Attempt to Sequentially Coordinate Changes

On March 20, Maggie was absent. So, I focused on investigating how Carl might be engaged in reflectively abstracting movement or changes when coordinating two quantities. I noticed that simultaneously coordinating two co-varying quantities was not in his zone of potential construction, based on the last teaching episode. So, I prepared the task for him that can be solved by sequentially coordinating changes. The given situation is as follows:

A train makes two trips to get to station 6 that is 6 miles away from where you started the first trip. Describe the second trip when the first trip is given (shown as an animation).

Three horizontal lines were given; on the top line, the first trip is shown as an animation. On the second line that was empty, Carl was asked to construct the second trip. On the bottom line, the directed segment from 0 to 6 was embedded (see Figure 7.13).

(a) (b) (c) (d)

Figure 7. 13 Carl's way of finding the second trip to station 6 Note. The red directed segments representing the first trip and the green directed segments representing station 6 were given. The thin directed segments between the red and the green one are Carl’s solutions. The screen capture (a) shows his initial estimation, (b) shows how he tried to justify his initial answer, (c) shows his second try, and (d) shows the way he explained the second trip.

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To help him engage in abstracting motions or changes rather than engaging in numerical computations, the blank number lines were provided (the only numerals that as appeared on the number lines was 6 for station 6). I also had made an action button for the given trip so that Carl could see it as a directed segment departing from zero to a certain location, not as a static arrow pointing at a particular a numeric value.

Protocol 7.13 Carl’s disembedding the distance of the second travel.

Carl: (Moves a mouse around the first number line.) I: What do you want to do? Carl: (Clicks the animation button for the first trip) Trying to see how far because I can’t really see it. (Traces from 0 to the end point of the first trip) See that there like 3? 4? I: (Points to the bottom number line and the directed segment of 6) You see this? This is where the station 6 is at. Carl: (Makes a directed segment that was equal to the directed segment of the first trip, see Figure 7.13-(a).) I: Can you tell me why you made that? Carl: I said this is probably about like 3 (points to the given directed segment), so put it kinda like the same. ‘Cause that will be six. I: Can you tell me, no, show me why that is true? Carl: Well, I don’t even know. Putting them together (places the two directed segments together and then comparing them to the directed segment of 6, see Figure 7.13- (b)). Not even close. Maybe like two more of these. I: You want to fix it? Carl: Yeah (makes a directed segment starting from the end point of the given directed segment and ending at 6, see Figure 7.13-(c)). I: Can you tell me what you are trying to do? Carl: I’m trying to make it start right there (points to the end of the given directed segment of the first trip) and get it right here (points to 6). And I’m gonna… see if I can do this (translates the directed segment that he just made to 0, see Figure 7.13-(d)).

Carl’s first estimation for the given travel was “three or four", but he made a directed segment that was much shorter than 3. It seems as if he had his own unit length in his mind and used it to measure the given trip instead of making a relative measure to 6. I infer that he was

214 engaged in neither disembedding the given travel nor mentally iterating it to estimate the relative length compared to the length of the directed segment of 6. Thus, he could not anticipate the second trip. After estimating the length of the first trip as three miles long, he made another directed segment that was equivalent to the given one, probably because two threes make six.

Only after he placed the directed segments together did he find the second trip needs “two more of these [referring to the first trip]”. This corroborates my hypothesis that he did not anticipate the second trip. Rather, I infer that he assumed the magnitude of the first trip would be 3 and found the second trip to be identical to the first one using his numerical computation.

The way he started the second trip from the end of the first trip to the station 6 seems similar to the way he counted up-to when he found missing addends (see Figure 7.13-(c)).

However, it was surprising that he translated the directed segment that starts at the end of the first trip to zero to show the second trip (see Figure 7.13-(d)). This might serve as an indication that he was beginning to disembed his sweeping or dragging action and composing the given directed segment with two explicitly nested quantities. However, considering that he also lined up all the directed segments at 0 when he did vertical computations, more corroborations were necessary to infer that he began to compose two quantities and to construct three levels of units.

Thus, I infer that he could coordinate two positively oriented quantities as three levels units in activity.

To further investigate whether he could also coordinate oppositely oriented quantities as he did with positively oriented quantities, I presented the situation where the first trip was far more than 6 miles to the east (about 15 miles away from the reference point). He was asked to find the second trip to get to the station 6. He clicked the animation button and I asked him to describe the first trip.

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Protocol 7.14 Carl’s numerical computation.

Carl: Uh… it’s almost about… 30? I: To [the] east or west? Carl: East (writes 30e below the directed segment of the first trip). And then, it means you have to do 30-6, which is 24 (moves his fingers on his head), right? I: Yes, 30-6 is 24. Carl: So, go approximately 24 (points to the left side of 0 on the number line)... 24 back… about here (makes a right-side pointing directed segment from where he pointed at to 0). I: Does it go this way (referring to west) or this way (referring to east)? Carl: I guess this way (changes the direction of the directed segment). Negative 24 (writes -24w below the directed segment, see Figure 7.14).

Figure 7. 14 Carl’s solution to 30+ x =6.

I have found a corroboration to my hypothesis that he measured the distance of the given travel using his own measuring unit that was irrelevant to the distance of station 6. In this task, he estimated the distance of the first travel as 30 miles while it was about two and half times longer than 6 miles. In retrospect, I compared his directed segment of 3, the initial estimation of the first travel in the first task, and the directed segment of “30e” in this task (see Figure 7.15). I found that the latter was ten times longer than the former one. So, the indication is solid that he was iterating his own unit measure instead of constructing a unit measure that was relevant to the given directed segments.

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Figure 7. 15 The comparison of Carl’s estimations of the length of the directed segment of 3 and the directed segment of 30.

Note. The very top directed segment represents a 3 miles long trip to the East, and the very bottom one represents a 30 miles long trip to the East. The 11 vertical lines were added by the author.

When negative quantity was included, it seems as if he did calculate the difference between 30 and 6 rather than coordinating the orientation of the quantity with the distance traveled. After he said the first trip was approximately 30, he pointed to the spot that he estimated -24 instead of starting the second trip from 30 and dragged it to 6. Based on what he has been doing in the past teaching episodes, it is probable that he was engaged in numerical computation using 30 and 6; he knew he needed to subtract 24 from 30 to get 6, and placed the “-

” sign in front of 24 in order to indicate that 24 needs to be subtracted, not to indicate the directionality of the quantity of -24. Then he found the location at which 24 should be placed, pointed to it, and connected it to 0, which resulted in the directed segment from -24 to 0. Only after being reminded of the directionality of the second travel did he alter the direction of the segment. Later, without my request, he volunteered to move the directed segments to show that he made a good approximation (see Figure 7.14-(b)). He smiled with satisfaction when he saw

217 his estimation and said, “That’s pretty close". However, it was unclear whether he reviewed 30 as a unit containing 6 and the reverse of the second trip. Had he constructed a directed segment from 30 to 6, it would be reasonable to assume that he reviewed 30 as a unit containing three levels of unit. At most, I infer that he assimilated the given situation as a take-away situation, found 24 using a numerical computation, and checked 30 was composed of 24 and 6, forming inclusion relations in activity.

Carl’s Attempt to Express y Using x and 6

Next, I provided other situations to investigate Carl’s ways of finding the second trip when the first trip was toward the West. Carl successfully solved two more situations using estimated numerical values. When solving the task shown in the Figure 7.16-(a), he said, “this

[points to the red directed segment] is -2 or 3. Maybe 3. -3… Maybe -3, yes. And then this gotta have to go couple more before 6". When solving the task shown in Figure 7.16-(b), he said,

“that’s [referring to the first trip] probably about -6, isn’t it? So, 12". In both cases, it seems as if he used the estimated number to find the solution rather than constructing the second trip by coordinating the reverse of the first trip and the distance of the station. Nonetheless, it seems as if he could find one numeric value whenever the other numeric value is given. Thus, I decided to test my hypothesis on the spot.

(a) (b)

Figure 7. 16 The two other task that Carl solved.

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Protocol 7.15 Carl’s attempts to find one quantity in terms of the other quantity.

I: You have been doing so good. Now, whatever the first trip is given, you probably can find the second trip, right? Carl: Yeah. I: How can you find it? Carl: Station 6? I: Yeah, station 6. Carl: On this one (points to another prepared problem)? I: No, so, just now I gave you some of the examples. What I want you to do is generalizing. Carl: What do you mean? I: So, say that the first trip is x. Can you think of what is the second trip? Carl: Guess you have to determine what x was, right? I: Yes, whenever x is given. Carl: Alright, I guess it would be… uh… whatever you guys call… y? I: Uh huh. Carl: It will be 6. I: y will be 6? Carl: (Looks at the screen in a deep thought.) I: Let’s say that you have the first trip as a hundred miles to [the] East. What’s going to be the second trip? Carl: Um… (writes 100 and 6 (without “-” sign. He adds it later, see Figure 7.17-(a)) I don’t know why I did this… (moving his fingers as if he is counting) 94 (writing 94)? I: To [the] east or west? Carl: West (moving his head to the left and adding a negative sign in front of 94 and “w” behind 94). I: Okay, if I give you a really complicated number, like 2,436… Carl: I can do this! I: You can do it? So, whatever number I give you, how can you get the second trip? Carl: Find it subtracting by 6. I: Subtracting by 6? Okay, if I give you 100 and subtract 6, what do you get? Carl: 94. I: But it’s positive, so it will be to [the] east. Carl: Yeah... But, you make six negative. I: Like this (placing the negative sign in front of 6)? Carl: No, you make the 100 negative. I: Let’s think about it. So, you travel 100 to [the] west… Carl: (Makes a directed segment pointing to the left side.)

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I: To get to station 6, how far do you have to move? Carl: (Making a directed segment pointing to the right side) Obviously you need to add. I: Add? Carl: No, ‘cause this is not how it looks. It goes that way (erases the directed segment to the right side and made a directed segment to the left side. See the first directed segment in the Figure 7.17-(b)). And then you have to do whatever it gets back to 6 (making a directed segment pointing to the left side that is longer than the first one, see the second directed segment in the Figure 7.17-(b)). I: So, what does the first segment mean? Carl: (Writes 100 underneath the first directed segment.) I: What does the second segment mean? Carl: (Writes -94 underneath it.) I: But this (pointing to the second directed segment) looks longer than that (pointing to the first directed segment). Carl: (Scoffs and laughs. He then shortens the second directed segment, see Figure 7.17-(c).) I: So, there is some special relationship. How can you describe it? Whatever the movement I give you, let’s say that it is x… then what is y? Carl: y? I: Whatever the second travel is. What it the first travel? Carl: x. It will be like x-6. I: Here, when x is 100, and you subtract 6. Then, you get 94. But you said this it not it. Because when you do 100 to the east (moves a hand from left to right), you have to come back (moves a hand from right to left)… to the west… Carl: (Yawning) I don’t really know. But it really works. If you put the negative sign on it.

(a) (b)

(c)

Figure 7. 17 Carl’s attempt to find the second trip whenever the first trip is given in order

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When asked to find the second trip given that the first trip was unknown, x, Carl said,

“you have to determine what x was". His comment serves as an indication that he regarded x as a quantity that was yet to be given, but he needed a specific numeric value or quantity to take it as an input for further operations. Coupled with the analysis that I made for his meaning of letters in his equations in the teaching episode of February 23, the indication is solid that he could not take an unknown quantity as an input. Although he later said, “find it subtracting by 6” and “x-6” later, I could not find any indications that he constructed “x-6” as a quantity that was 6 less than the unknown quantity. Rather, it seems like he would start counting 6 down only when a specific numeric value for x was given. His difficulty when reconstituting a missing addend situation in the teaching episode of December 5, 2014 serves as a corroboration.

I also found the corroboration of my previous hypothesis regarding the meaning of the “-

” sign to Carl. My analysis was that he considered the “-”sign as subtraction, and he was yet to construct the concept of negative quantity, due to the lack of reflective abstraction of changes or movements. Instead, he assimilated the given situations as take-away situations when two quantities were oppositely oriented. In this teaching episode, he used “add” or “subtract” when explaining how he would find a second trip. When the first trip was given as 100 miles to the

East, he found 94 and said, “find it subtracting by 6”. After I pointed out that 94 was the opposite of -94, he wanted to “make six negative". Then he changed his answer and wanted to “make the

100 negative” to get to his original answer, -94. His frequent changes of signs of the given quantities suggest that he assimilated the given situation to a take-away situation, subtracted one from the other, and later assigned to the result the sign of the number with the greater magnitude.

Thus, my analysis is that his solution, “x-6”as y, along with his comment, “I don’t really know.

But it really works. If you put the negative sign on it" implies that he first found y where the

221 magnitude of y is 6 units smaller than that of x and placed the the “-” sign accordingly. Based on his strategies, “x-6” works from his perspective. However, I did not find any indications that he considered y as a unit containing x and -6.

There exists a discrepancy in Carl’s construction or the meaning of 6 when he solved the first task of Protocol 6.13 and when he solved the rest of the tasks. When solving the first task, I found an indication that he reconstituted 6 as a unit containing two positively oriented composite units. However, when he solved the other tasks, he either found 6 as a result of his numerical computation, or a difference in length between the two directed segments. In other words, I could not find any indications that he reconstituted 6 as a unit containing the other quantities.

Considering that he operated with two levels of units, it is probable that he could coordinate two positive composite constructing three levels of units in activity. However, his units-coordinating scheme was not sufficient to construct 6 as a unit containing oppositely oriented composite unit generating a unit containing three levels of units.

Representing Covariation on a Coordinate Plane: a Bouncing Ball

Based on previous teaching episodes, I could made the hypothetical models of Carl’s and

Maggie’s schemes and operations when constructing x+y=a. Maggie, with the three levels of units-coordinating scheme, could reconstitute the given situation as a unit containing three levels of units. Taking the three levels of units as given, she could also coordinate changes in each quantity that were in the counterbalancing relation. On the other hand, Carl, who was limited to coordinating two levels of units, assimilated a sum of two oppositely oriented quantities as a take-away situation, and placed the “-” sign accordingly. However, their previous graphs (see

Figure 6.9) were not sufficient to represent their ways of coordinating quantities. So, I decided to promote coordinating two covarying quantities on a coordinate plane using different contexts.

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On April 10, I investigated how Carl and Maggie coordinate the height of a ball to the time spent for the ball bouncing. The purpose of this teaching episode was to explore how they coordinate vertical motions or changes to horizontal ones, which is essential when graphing x+y=a on a coordinate plane. A computer was provided to each of the students, and they were asked to watch a 13 second-long slow motioned ball bouncing video. In the video, the ball bounced once for each 30.3 milliseconds. Students could see the time spent for the ball bouncing on the upper right corner of the screen (see Figure 7.18). They could stop, reverse, or restart the video whenever necessary. I first asked them to explain how the ball moves and how time flows.

They both said that ball moved vertically and time flowed from left to right. Next, I asked them to coordinate two movements and represent it. The entire teaching episode was devoted for them to explore the ball movement and to represent the relationship between the height of the ball and the time spent on a time-height coordinated plane. This teaching episode provides some insight of how each student might have envisioned the coordination of the movements.

Figure 7. 18 A screen capture of the ball bouncing video.

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Carl’s Interpolation

Both students watched the video intently, frequently stopped the video to “see” the relationship. Carl confidently created a pair of perpendicular lines and labeled the horizontal one

“time” and the vertical one “height”. Then he plotted (0,30), stopped the video when it hit the ground and plotted approximately (15,5). Then he plotted the (30,30). He replayed the video, carefully stopped when the ball was halfway from the ground, and plotted two points approximately (5,20) and (25,20). After he plotted five major points (the bigger dots in Figure

7.19-(a)), he stopped with satisfaction. So, I asked him to explain his representation. He said,

“this is where it started, and this is when it went midway, this is when it bounces, this is a half way up…” When asked the meaning of the space between the points, he said, “There are many points in between but they are the easiest ones". I asked him to find a strategy to represent “many points". He watched the video once more and added as many dots as possible between the five points. He did not replay the video while he was plotting the smaller points. Coupled with my analysis about his meaning of number line as a set of discrete sequences, I infer that he was coordinating discrete sequences of time and height.

(a) (b)

Figure 7. 19 Carl’s graph (a) and Maggie’s graph (b).

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Maggie’s Connected Segments

Maggie frequently stopped the video and replayed it when creating her graphs. The difference was that she chose to use a line segment tool instead of a point tool and put height on the horizontal axis and time on the vertical axis. She first drew a vertical axis for the time spent, placing 0 on the bottom, 30 on top, and making total 15 marks. She then drew a horizontal line on top of the vertical axis. She said that the left end of the horizontal line stood for the case when the ball hit the bottom and the right end implied where the ball dropped. Next, she plotted a point at the lower right corner of her coordinate plane. Then, she chose a line segment tool instead of a point tool and continued producing the connected line segments while frequently pausing the video (see Figure 7.19-(b).) When asked to describe her graph, she said, “they are line graphs".

In previous teaching episodes, I observed that Maggie could reflectively abstract her counting actions, disembed it and use it as a composite unit to coordinate it with other composite units. In addition, she could isolate changes in each quantity and construct the counterbalancing relation between the changes in each quantity in the teaching episode that was held on March 2, (Protocol

7. 10). Thus, I infer that her line segment implies the coordination of reflectively abstracted vertical and horizontal motions.

Representing a+b=0 on a Coordinate Plane

In the previous teaching episode, I investigated how Carl and Maggie coordinate covarying quantities when representing the height of a bouncing ball in relation to the time spent.

While their representations were different, I found strong indications that they both could coordinate values on a horizontal axis with values on a vertical axis and plot points to represent the coordination. So, in the teaching episode that was held on April 24, I planned to investigate how they would represent the situation that can be modeled as a+b=0. First, I needed to confirm

225 whether or not they knew the meaning of a point on a coordinate plane because in the teaching episode that was held on March 2 (Protocol 7.10), it seemed like they were not aware of the meaning of points on a coordinate plane and simply plotted points on each axis to form a line.

Next, I planned to ask questions promoting coordinating changes in each quantity and to encourage them to represent them on double number lines. Finally, I planned to ask them to construct a+b=0 on a coordinate plane while suggesting to rotate one of the double number lines of 90 °.

Carl arrived first, and I asked him to plot a point that represented the measures of the temperature of city A and city B when their measures of the temperature were 5 °F and 2 °F. He plotted (5, 2) on the coordinate plane on GSP. Then Maggie arrived to class, and Carl asked

Maggie to plot a point representing 20 °F for city A and -7.5 °F for city B. She plotted (20, -7.5) on the coordinate plane. Then I plotted (-4, 6) and asked them what the point meant. Maggie said, “It represents the temperature of city A and city B", and Carl added, “A is negative 4 and B is 6". Thus, I confirmed that they understood a point on AB-coordinate plane represented the measure of the temperature of each city.

Next, I asked them to make a dynamic model representing a+b=0 on double number lines, where a and b are the measures of the temperature of city A and B. The purpose of this task was to investigate how they abstract changes and coordinate them.

Protocol 7.16 Coordinating changes.

I: The goal is to make a dynamic model of the situation that the sum of the measures of the temperature of two cities is always 0. Say that the temperature of city A changes from 0 to 10. Can you tell me what is the change in temperature of city B? Carl: They are equal to 0? I: Yes. They are always equal to 0. Mag: (Makes two directed segment: one from 0 to 10 and the other from 10 to 0, see

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Figure 7.20-(a)) Carl: (makes two directed segments: one from 0 to 10, and the other from 0 to -10, see Figure 7.20-(b)) I: So, what’s the temperature? Mag: 0. I: Yeah, the sum of the temperature is 0. So, does the temperature of city B increase or decrease? Mag: Decrease. Carl: Of B? Yes. I: How much? Mag: 10. I: 10, right? (Looks at Maggie) Does it start from 0 or 10? Mag: Oh, yeah. A starts from 0 and B starts from 0.

(a) (b)

Figure 7. 20 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first number line) when A changes 0 to 10 (the second number line).

I: What if A changes from here to here (moving my index finger from left to right)? Mag: Then it moves from here to here (moving her index finger from right to left). Carl: This way (tracing the directed segment from 0 to -10). I: What if A changes from 5 to 7? Then what is the change in temperature in city B? Carl: (He deletes the directed segments and makes a new directed segment from 0 to 5. Then he tries to make another one right next to it). Wait, B is 7? I: Yeah, city A changes from 5 to 7. What is the change in temperature in city B? Carl: (Makes a directed segment from 5 to 7 right next to the directed segment from 0 to 5 on the number line for city A. Then he makes another directed segment from 7 to 0 on the number line for city B, see Figure 7.21-(a).) Mag: (Shortens the directed segment that she made for the previous problem that was from 0 to 10 to 0 to 5) Increases to 7? I: Yeah, it starts from 7 (misspoken), and it increases, how much does it increase? Carl: 7?

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I: So, 5 to7. How much? Carl: Oh, 2. I: 2, right? So, when it increases from 5 to 7, what happens to the change in the temperature of city B? Carl: It has to go back. Mag: So, it hits the 5, and this sends me to 5 (tries to shorten the directed segment that she had made for the previous problem that was from 10 to 0. She drags the head of the directed segment and stops at 5), and in goes to 7 (continuing the directed segment to 3, see Figure 7.21 -(d)), so it’s gonna go… I: So, can you tell me, let’s say that we measured the temperature of city A at midnight. At that time, the temperature was 5 °F. For an hour, [it] increases 2 degrees, right? At midnight, what temperature was it in city B? Mag: It decreased as the same amount that A increased? So, it will be 7. I: So, from where to where? Mag: From 10 to… from right here…. It decreased... Oh, I don’t know why I did… Carl: (Deletes the directed segment from 7 to 0 and makes a directed segment from 0 to -7, instead, see Figure 7.21-(b).) Mag: So, it’s 0, right? I: No, this time, it did not start from 0. Mag: So, it’s 5. I: It started from 5. 5 to 7. Mag: So, it’s 5 to 3 (makes a directed segment from 5 to 3, see Figure 7.21-(e). Cause it starts from 5, and it increases 2, and this decreases 2. I: So, it (meaning the measure of the temperature of city A) starts from 5 to 7. It (pointing to Maggie’s directed segment for city B) starts from 5 to 3. If I add 7 and 3, it’s 10. But the sum of the temperature is always 0. Mag: That’s what I thought I started (makes a directed segment from 0 to -7, shortly pausing at -5)... Carl: (Writes -7 for the directed segment for city B and 5+2=7) for the directed segment for city A.) I: (Bringing the notebooks together) Okay, Carl, can you tell us what you were thinking of? Carl: I just started B from 0, and it went to -7 since both equals to 0. It’s a math. I: Okay, so 5+2=7, this much. So, how much [does] it decrease there (points to the directed segment of city B)? Carl: From here (pointing to the number line of city B)? I: Yeah, in city B. Carl: What do you mean? Mag: Seven. Carl: If it starts from 7 and goes to?

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I: So, let’s say that, at midnight, the temperature of city A was 5. Then what is the temperature of city B? Carl: -5. I: Yeah, -5. And then for an hour, in increases 2 degrees. Then what happens to city B? Mag: It decreases 2 degrees. I: It decreases 2 degrees. So, what is the change in temperature of city B? Carl: (Adds a directed segment from 0 to -5, see Figure 7.21-(c)) I: Is it 0 to -7? Mag: No, it’s from -5 to -7 (see Figure 7.21-(f)). Carl: (Writes 5 in front of -7, see Figure 7.21-(c))

(a) (b) (c)

(d) (e) (f)

Figure 7. 21 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first number line) when A changes 0 to 10 (the second number line).

Maggie’s Disembedding Changes in Each Quantity.

When Maggie was looking for changes in B, given that the measure of the temperature of city A changes from 0 to 10, it seems as if she coordinated the changes sequentially because she made a directed segment from 10 to 0 not from 0 to -10. When asked to find the changes in the temperature of city B when the temperature of city A rised from 5 ° to 7 °, she found that the

229 directed segment from 10 to 0 was not appropriate to illustrate the previous situation. She said,

“From 10 to… from right here…. It decreased... Oh, I don’t know why I did…” and then said,

“So, it’s 0, right?” In retrospect, I found that she meant the sum to correct herself. However, at that time, I understood that she was asking about the second situation, so I told her that the measure of the temperature of city A was 5 °. Thus, I infer her directed segment from 5 to 3 was her representation of the decrease in the temperature of city B. When the sum was reminded, she made a directed segment from 0 to -7 while shortly pausing at -5. Coupled with her explanation,

“it’s from -5 to -7", and her comments, “it decreases 2 degrees", the indication is solid that she envisioned that the decrease from -5 to -7. Maggie has shown her abilities to reflectively abstract changes, disembed it, and coordinate it with other quantity or changes while reconstituting the given situation generating a unit containing three levels of units, which supports my hypothesis.

Carl’s Search for Corresponding Values

On the other hand, Carl immediately made a directed segment from 0 to -10 while confirming the sum, 0. Based on his schemes and operations that he had shown in the past teaching episodes, it was possible that he used the directed segment to indicate the numerical value, -10, instead of the change in B. To test the hypothesis, I asked them to find how the measure of the temperature of city B changed when the measure of the temperature of city A changed from 5 to 7. Car’s two connected directed segments, one from 0 to 5 and the other from

5 to 7 show that he was explicitly aware of the situation. It seems as if he represented 7 by uniting two composite units, 5 and 2. His solution, the directed segment from 7 to 0 suggests that he coordinated changes sequentially. Then he corrected himself, making a new directed segment from 0 to -7 and wrote -7 above it. He was satisfied to see that -7 was the opposite of 7=5+2.

This is an indication that he was not focusing on the changes, but looking for the opposite value

230 to make the sum 0. After he said the measure of the temperature of city B was -5, he added an additional directed segment that starts from 0 to -5, not from -5 to -7. This is a solid indication that he was looking for a matching number of 5 that makes 0 when added together. It is unclear why he placed 5 in front of -7 in his notation (see Figure 7.21-(c)). It is possible that he actually meant -5 but wrote 5, considering that he has been saying or writing a negative quantity without the “-” sign. One might argue that he was thinking about -2 as the change in B because 5-7 results in -2. However, whenever he added a negative quantity and a positive quantity, he assimilated it to a take-away situation, found the difference between the magnitudes, and added the “-” accordingly. Thus, I do not consider that he envisioned -2 as the sum of 5 and -7, which was also irrelevant to the context of the given situation.

Based on Carl’s way of finding the measures of the temperature of city A instead of finding the change, I found that this sequential coordination of changes or movement was not sufficient to reflectively abstract and to disembed changes in the temperature. To understand one quantity decreases as much as the other quantity increases regardless of its starting points, I hypothesize that one must reflectively abstract the changes or movements in quantity, disembed them, and placed them side by side to compare them. In doing so, one must be able to take three levels of units as given and engage in reversible reasoning, i.e., reconstitute the given situation as a unit containing three levels of units and compare changes by coordinating the changes and the inverse of the changes. Thus, I found that his two levels of units-coordinating scheme was not sufficient to construct the counterbalancing relation between changes in each quantity.

Maggie’s a=-b and Carl’s -B=A, -A=B

At this point, I was curious about how they would model two oppositely moving directed segments, a and b. I showed them two moving directed segments that can be represented as

231 a+b=0 on double number lines, where a and b were the measures of the temperature of city A and B. The goal of this task was twofold: To investigate how they abstract the given relation that is shown as two co-varying directed segments, and to make sure they experience (at least) watching two co-varying quantities so that they could regenerate their experience when representing it on a coordinate plane. Both said that the directed segments moved to the opposite direction. At my request to write an equation for the relationship between the measures of the temperatures of the two cities, Maggie wrote “a=-b” and Carl wrote “-B=A, -A=B”, saying,

“Cause one [is] negative and one is positive". Carl’s two equations and his explanation, along with his early comments, “it’s math", suggest that he was not coordinating changes. I noted that two equations were necessary for him to represent two distinct situations: the cases of the right side of the equation was positive and negative. I do not find any indications that he considered the two equations identical. Thus, I infer that he represented the way he matched the same numbers with opposite signs by separating the cases into two categories.

In contrast, I infer that Maggie expressed the counterbalancing relation, that whenever a increases, b decreases as much as the increase, using her equation, “a=-b”. In the past teaching episodes, she showed multiple indications that she could reconstitute the given situation as a unit containing three levels of units, and she engaged in reversible reasoning when coordinating changes (Protocol 7.12 and 7.16) constructing the counterbalancing relation. In addition, she wrote the equation was right after she coordinated changes in the previous task, saying, “it decreases 2 degrees” and “it’s from -5 to -7". So, the indication is strong that she was not confined to express the opposite values of a and b.

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Maggie’s Continuous Line Graph

Based on the previous tasks, I learned that both of them could coordinate a value on the horizontal axis and a value on the vertical axis and plot a point to represent them. In addition, I found indications of how they envisioned and coordinated two co-varying quantities on the teaching episode that was held on April 24; Carl looked for corresponding values for each given quantity (e.g., -10 for 10 and -7 for 7) while Maggie could disembed changes in each quantity constructing the counterbalancing relations. So, I decided to investigate how they would use their existing schemes and operations that they used on double number lines when they represent the same situation on a coordinate plane.

I asked them to imagine how the directed segment of city B would change when the number line for city B was rotated 90 °. Maggie said, “It goes down", and Carl agreed with her. I then asked them to rotate the number line of city B and explore whether their anticipation was right. They both played with it for a while and said it appeared to be what they had envisioned.

On the new page, I prepared a premade coordinate plane and two perpendicular mobile directed segments that can be modeled as a+b=0 (see Figure 7.22). Next, I asked them to plot points that represents a measure of the two cities’ temperature. Carl plotted (-10, 10), but he had to leave early that day. Maggie also plotted (11, -11) and continued to plot other points that satisfied a+b=0. She first plotted some points on the second and the fourth quadrants and deleted all the points on the second quadrant except (-4, 4) saying, “Oh, I didn’t realize that a has to be the same amount over here. So, it’s 4 and 4 here". She then moved the directed segment of b from 4 to 2 and plotted (-2, 2). Next, she deleted all the points on the fourth quadrant, moved the directed segment of b to 6, and plotted (-6, 6). She repeated the same processes and additionally plotted (7, -7), (8, -8), (9, -9), and (10, -10).

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(a) (b)

Figure 7. 22 Maggie’s (a) and Carl’s (b) screen captures when finding the changes in B (the first number line) when A changes 0 to 10 (the second number line).

Protocol 7.17 (cont).

I: How about between this point and this point (pointing to (7,-7) and (8,-8))? How many points do you have? Mag: I will right here. It’s like a half (plotting approximately (7.5,-7.5)). I: So, imagine that you plot all the points on this plane. Then, what’s that gonna form? What kind of form is it gonna make? Just think about that you plot all the points for all the cases. Mag: Like a line graph or something? I: Do you think it is a line graph? Then can you sketch a line? Mag: (Makes a line stating approximately from (-9,9) to (15,-15).)

Later, I showed how a locus of the (a, b) formed a line graph of a+b=0 on GSP using the tracing function. I then asked Maggie what the trace meant. She asked me back whether I meant the line of the trace of the points on the line. Unfortunately, I asked her only about the points, not about the line because, from my perspective, a set of points formed a line. In retrospect, I realized that the meaning of the line and the points on the line might have distinct meanings to

234 her, based on her ability to disembed changes in quantities. To my question, she said that they meant the temperature of the two cities. When asked to write an equation of the graph, she wrote

“a=-b” and smiled. After I saw her smile, I showed her how tracing (a,-a) formed a line of a=-b using the trace function on GSP. She identified that her graph was identical to the trace and labeled it a=-b. Although I had strong indications that she previously coordinated changes in each quantities, I could not find a clear indication of whether or not she was envisioning covarying changes in each quantity when she said it was a line graph.

On the teaching episode that was held on April 27, I showed her the locus of (a, -a), the very trace that I showed her in the previous teaching episode (see Figure 7.23) to continue my investigation. At this time, she did not see the movements of the trace, because the locus was preconstructed.

Protocol 7.18 Maggie’s explanation of the line graph.

1: You can see what you did last time. So, can you tell me what you remember… Mag: That, um… We did if B decreases… I mean if A increases B decreases. So it was…um… I remember this (points to the locus) will be like the same. Like this temperature.

Figure 7. 23 The locus of (a, -a)

Note. On April 24, Maggie watched the tracing movement of the locus. On April 25, the locus was premade and she did not physically see the tracing movement.

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To be honest, my intention was not to find whether she envisioned coordinating changes in each quantity at that point of time. I was excited that she said her graph formed a line in the previous episode, so I wanted her to simply recall her line graph. To my relief, in retrospect, I found a solid indication that Maggie regenerated her previous experience of coordinating changes in each quantity forming the counterbalance relation based on her first response, “we did if B decreases… I mean if A increases B decreases". Her second response, “I remember this

[points to the locus] will be like the same. Like this temperature” suggests that she regarded the representation of the counterbalancing relation (the locus) to be identical to the set of points, which implied the measure of the temperature of each city.

Carl’s Pointwise Graph

Carl had to leave early in the previous teaching episode, so I did not have much chance to investigate how his schemes and operations impacted his representation. He plotted a point (-10,

10) right before he left on April 24, so, in the teaching episode that was held on April 27, I decided to start with reminding him of what he did last time in order to continue my investigation. In this teaching episode, Maggie was working on a different task. The following protocol is edited to focus on Carl.

Protocol 7.19 Carl’s interpolation.

I: Do you remember why you plotted the point (pointing to (-10,10))? Carl: Cause it will be -10 and 10, that equals to 0. I: So, I told you that that explains only one situation. I want you to find out all the cases. What if the temperature of city A is 100? Carl: (Plots (-9, 9), (-8, 8), (-7, 7), (-6, 6), (-5, 5), (-4, 4), (-3, 3), (-2, 2), (-1, 1) and stops. Then deletes all the points that he plotted.) No, it will be down here (circling around the fourth quadrant). No, it’s not (replotting all the points.) I: Okay. What about this side (pointing to the fourth quadrant)? Carl: I could not do it because both are positive. I: You cannot do it? Why do you say that? Carl: Cause it’s positive.

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I: What is positive? Carl: This will be negative, too, then (points to the negative part of the vertical axis). For this one, it’s positive, too, right? For this one (points to the vertical axis), this is all city B, right? I: Yes, this is city B, the vertical. And the horizontal one is city A. So, what does that point (points to (1,-1)) mean? Carl: One, negative one, I guess? I: Uh huh, which one is one and which one is negative one? Carl: This is one (points to (1,0)) and this is negative one (pointing to (0,-1)). I: Which one? City A or city B? Carl: Um, city A is positive and city B is negative? I: Okay. Carl: And… (plots (2 ,-2), (3, -3), …, (20, -20)) I: So, good job. So, it looks like… So, can you tell me how many points you can plot? Carl: (Keeps plotting) A lot. Guess infinity? I: Infinity? Okay, I want to know what’s gonna happen between this point and this point (points to (-3, 3) and (-2, 2)). Carl: It’s gonna go down one. This is negative three (points to (-3, 3)) and this is negative two (points to (-2, 2)), one, zero. I: Do you think there are any points between these two points (points to (-3, 3) and (-2, 2)). Carl: Ah, yeah. I: Yeah? How many do you think? Carl: Four? I: Four? Then can you plot all the points between those two points? Carl: (Plots four points between (-3, 3) and (-2, 2) and labels them as 1, 2, 3, 4, See Figure 7.24-(a)) I: Okay, so you are saying… Carl: Uh, there’s like 99 points in between. I: 99 points? Carl: Yeah. Cause like .99? I don’t know. I: Can you tell me the coordinates of point 1 and 2 and 3and 4? Carl: What… I: Coordinates. Like x coordinate and y coordinate. So, what is the temperature of city A and city B? Carl: Like (3.1, -3.1)? I: What about 2? Carl: (-3.2, 3.2) I: What about 3?

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Carl: (-3.3, 3.3) I: What about 4? Carl: (-3.4, 3.4) I: And there’s only four points in between? Carl: I don’t know. Probably it’s 99 or something. I: Then what is the first point and the last point of those 99 points? Carl: 3.1 and -3.1 and whatever it took 99? I: You said 3.1. So, what about 3.09? Is 3.09 is bigger that 3.1 or smaller than 3.1? Carl: It’s uh… smaller than 3.1? I: Smaller? Alright, then, what if the temperature of city A is 3.09? Then, what is the temperature of city B? Carl: -3.09? I: Yeah, so there is a point before 3.1. You said that 3.1 is the first point. Then how many points will fit in between those points? Think about it. Carl: (Labels each point 3, 3.1, 3.50, 3.90, and 4, see Figure 7.24-(b)) I: So, can you explain what you did? Carl: Uh, I tried to get some random points. This is 3, and 3.1. And 3.50, which is a midway, 3.90 and 4 (points to the points accordingly, see Figure 7.24-(b)). I: So, how many points are there in between? Carl: Um… Guess… I don’t really know. Guess infinity? I: Yeah, cause there are infinitely many points. Think about that you are plotting all the infinite points. Think about the set of points of this relationship. What do you think that the set of points looks like? Carl: Just a lot of dots? I: Lot of dots? And that’s gonna compose… Carl: Dots.

(a) (b) (c) Figure 7. 24 Carl's point wise graphs

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Next, I showed how a locus of (a, b) formed a line graph of a+b=0 on GSP using the trace function and told Carl that the points that he plotted formed a line (see Figure 7.24-©). He nodded, so I asked what the line meant.

I: Can you tell me what this line means? Carl: Um… I: What I want you to do is to find the meaning of this line and the equation of this line. Carl: (Writes y=0x+1) I: Tell me why you write this 0 and … Carl: ‘Cause it starts at 0 and this [is] gonna be the y. And then it goes one every time. It perfectly gets to the next point every time.

When asked how many points there were between two points, he said that there were infinitely many points. His sequential labels such as 1, 2, 3, and 4, along with his later labels with decimals such as 3, 3.1, 3.50, 3.90, and 4, suggest that he sequentially plotted points. In addition, he identified the four points between (-2, 2) and (-3, 3) as (-3.1, 3.1), (-3.2, 3.2), (-3.3,

3.3), and (-3.4, 3.4). My analysis is that he was not considering the orientation of the quantity and sequentially labeled the points in an increasing order. Although he knew that there were infinitely many points, it is interesting that he did not envision the set of infinite points as a straight line. From his perspective, they were “just a lot of dots”. To convince him that the set of infinitely many points formed a line, I showed him the animation of tracing (a, -a). He nodded, but it was unclear whether he assimilated his pointwise graph to the line. To investigate further, I asked him to write an equation of the line. He wrote “y=0x+1” and said, “Cause it starts at 0 and this [is] gonna be the y. And then it goes one every time. It perfectly gets to the next point every time”. His equation and the comments serve as a solid indication that he did not consider the given trace of (a,-a) and his pointwise graph to be identical. His equation “y=0x+1” along with his comments suggests that the line implied neither a set of infinitely many points nor a

239 coordination of changes to him, based on the previous teaching episode that was held on March 2

(Protocol 7.10); he used the line to connect the points that he plotted and said he did not know why he did that. My analysis is that Carl’s two levels of units- coordinating scheme was not sufficient to coordinate changes in each quantity, so his focus was on finding a corresponding value of b for each given value of a. Thus, he constructed the pointwise graph.

Summary

Maggie’s Construction of the Counterbalancing Relation and the Line Graph

Maggie could take three levels of units as given and review the sum as a unit containing three levels of units. In the teaching episode that was held on December 8, 2014 (Protocol 7.2), she rearranged the given directed segment of -33 and 10 to find the missing addend. Using her measuring scheme and the “filling” strategy, she consistently rearranged the given directed segments to represent the other situations that can be modeled as a sum or difference of two oriented quantities throughout the entire series of teaching episodes. This is a solid indication that she constructed the concept of negative quantity as an inverse to positive quantity and was able to use it as a composite unit while keeping track of the meaning of its orientation. In the teaching episode that was held on March 2 (Protocol 7.10 & 7.11), she constructed the counterbalancing relation between the changes in temperature. She explicitly stated, “every time it increases, it decreases” when asked to find the change in temperature of city C when the temperature of city M increases, where the sum of the measurements of the temperatures of two cities was always 8. Furthermore, she could take two consecutive changes in temperature of city

M as an input, and could coordinate corresponding changes in temperature of city C. My analysis is that she first reconstituted the given situation as a unit containing three levels of units, took it as an input, and disembed changes in each quantity in order to compare them. I argue that she

240 engaged in reversible reasoning when disembedding changes in each quantity and compared them because she had to embed and disembed the changes in order to compare the effect of changes in temperature. When representing the height of a bouncing ball in relation to the time spent for the ball bouncing on April 10, she constructed a set of connected line segments instead of plotting points. I do not consider her choice to use line segments instead of points was accidental. In retrospect, I infer that she was focusing on movements of the bouncing ball and the time spent rather than focusing on the relative location of the ball and the time that was presented as numerical values on the top right corner of the ball bouncing video. Finally, in the teaching episodes of April 24 and 27 (Protocol 7.16, 7.17, and 7.18), I found solid indications that she constructed the counterbalancing relation between changes in each quantity. After plotting a few points satisfying a+b=0, she said that the points formed a straight line. In addition, she could envision the line graph in two ways: as a representation of the counterbalancing relation and as a set of points satisfying a+b=0.

I found that the three levels of units-coordinating scheme was essential when disembedding changes in each quantity in order to construct the counterbalancing relation so that she could focus on coordinating changes in each quantity. In order to construct the counterbalancing relation, I argue that Maggie engaged in reversible reasoning. In the teaching episode of February 20, I found another occasion that she engaged in reversible reasoning when finding the relationship between the two covarying directed segments. My analysis is that the three levels of units coordinating scheme and reversible reasoning were essential in the construction of the counterbalancing relation, which was necessary for Maggie’s construction of x+y=a as a continuous line.

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Carl’s Vertical Computation and His Pointwise Graph

Limited to coordinating two levels of units, Carl could not reconstitute the sum of two oppositely oriented quantities as a unit containing three levels of units. Consequently, he developed distinct strategies to find the sum, difference, and missing addend. For example, in the teaching episode of February 20, 2015 (Protocol 7.4 & 7.7), he was asked to find the relationship between two directed segments, a and b, that were colored red and blue and covaried in relation to the equation a+b=8. Carl categorized the given situation into two types: the case when a was greater than 8 and the case when b was negative. Then he said, “When blue [referring to b] is negative, it’s [referring to a] always more. But when red is positive, it’s [referring to a] always more ”. Later, he found a conflict in his hypothesis when he saw that a was 3, and b was 5. He eventually failed to find the relationship of a+b=8. In addition, he had yet to construct the concept of negative quantity (Protocol 7.8). He used the “-” sign to indicate the difference or numbers that were located “on the other side”. My analysis is that he did not construct negative quantity as an inverse of positive quantity, which served as a constraint in the construction of the sum of two oppositely oriented quantity as a unit containing three levels of units. Thus, he frequently engaged in numerical computations in which he placed the number of greater magnitude on top and that of smaller magnitude on the bottom, found the difference, and placed the “-” sign accordingly. When asked to consider changes in each quantity that could be modeled as m+c=8, he became confused when he saw that both m and c were positive, while the changes in m and c were opposite to each other (February, 23, Protocol 7.8). My analysis is that his two levels of units-coordinating scheme was not sufficient to reflectively abstract and to disembed changes in each quantity. Thus, I do not consider that it was a coincidence that he chose to plot points when representing the relationship of height of a ball and the time spent for the ball

242 bouncing in the teaching episode of April 10. I hypothesize that he chose to plot points to represent the relative locations of the ball and the numbers that appeared on the top right corner for the time spent, instead of coordinating the changes in height of the ball and the time spent.

Therefore, I found that his pointwise graph (April 27, Protocol 7.19) that was composed of infinitely many points could not be a continuous line to him because he focused on finding corresponding number pairs satisfying a+b=0.

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CHAPTER 8

CONCLUSIONS

In this chapter, I will revisit my research questions and discuss my findings. The research questions were as follows: