The purpose of this qualitative study was to explore how young children conceptualize the relationships among positive and negative numbers and zero within contextual situations. Data were collected through video-taped interviews between the children and the researcher and from written work completed by the children.

Twenty-four four- to eight-year-old children with no formal instruction on negative numbers participated in the study. The children encountered situations of negative number context in modified number line and collection activities. They demonstrated insights and intuitions of negative numbers. They were encouraged to make representations for negative numbers encountered as deficit or directional situations. The findings indicate that young children conceptualize relationships among only positive numbers, positive numbers and zero, and positive and negative numbers in contextual situations, but not between two negative numbers. Few children attempted to create representation for negative numbers, and only one child utilized her notation to solve problems. The children demonstrated more conceptual understanding of a deficit situation, or negative number context, in collection-type activities than in modified number line activities.

ACKNOWLEDGMENTS

I would like to take this opportunity to thank both of my committees for supporting me and encouraging me to complete this dissertation. I am very grateful for the efforts put forth by Dr. Gen Davis and Dr. Trish Koontz during the formative period of this writing. They forced me to meet their expectations and prepare a quality paper. I greatly appreciate Dr. Michael Mikusa and Dr. Anne Reynolds for accepting me as a doctoral candidate in the final stages of my writing after the retirements of Drs. Davis and

Koontz. Dr. Kristen Figg and Dr. David Dees are also appreciated in their roles of outside members of my dissertation committee.

I would especially like to thank the children, and their parents, who welcomed me into their homes and willingly shared their ideas about numbers, including negative numbers. Their responses were refreshing and insightful, and the children were delightful to work with.

Special thanks to other friends and family who supported and encouraged me throughout this lengthy process of writing this dissertation. So many people believed in me and urged me toward completion of this goal. I want to especially thank my husband,

Ted, and my children, Jennifer and Chad, for their understanding of the commitment involved in this undertaking. I also want to thank my colleagues, Cathy Mills, Kathy

Johnson, and Penny Harrold, for their continued encouragement. Jennifer Murphy, Rosa

Davis, and Sherri Shehan are greatly appreciated for their assistance in validating the data and sharing their perspectives.

The journey has been long, at times nearly defeating, but ultimately rewarding.

An inner strength emerged and my love of learning from others was renewed.

TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ...... iv

LIST OF TABLES ...... ix

CHAPTER

I BACKGROUND OF THE STUDY ...... 1 Introduction ...... 1 Contexts of Negative Numbers ...... 3 Statement of the Problem ...... 6 Rationale for the Study ...... 7 Purpose of the Study ...... 9 Research Questions ...... 11 Significance of Research ...... 11 Summary ...... 12 Definition of Terms ...... 12

II REVIEW OF THE LITERATURE ...... 14 Introduction ...... 14 Number Sense and Learning Number ...... 14 Introduction ...... 14 Children and Number ...... 15 Zero ...... 18 Constructivism ...... 19 Meaning-making of Number...... 21 Role of Representation ...... 24 Notation ...... 27 Play and Learning Mathematics ...... 31 Games and Learning Mathematics ...... 33 Summary ...... 37 Negative Number Instruction ...... 38 Introduction ...... 38 Models of Integers ...... 39 Number Line Model ...... 41 Collection or Charged Particle Model ...... 43 Summary ...... 46 History of Negative Numbers ...... 46 Introduction ...... 46 Third to Eleventh Centuries ...... 47 Twelfth to Sixteenth Centuries ...... 49 vi

Seventeenth to Twentieth Centuries...... 51 Summary ...... 56 Chapter Two Summary ...... 57

III METHODOLOGY ...... 59 Introduction ...... 59 Qualitative Research ...... 59 Research Design ...... 63 Participants ...... 63 Data Collection Procedures ...... 63 Analysis of Data ...... 65 Pilot Study ...... 66 Issues of Trustworthiness ...... 67 Current Study ...... 68 Limitations ...... 69 Summary ...... 71

IV RESULTS ...... 72 Introduction ...... 72 Interview Activities...... 74 Data Addressing Conceptualization of Relationships of Number ...... 78 Cardinality of Set ...... 78 Ordinality of Set ...... 80 Zero as a Referent...... 82 Zero as a Quantity ...... 84 Relationships Among Positive and Negative Numbers and Zero ...... 86 Relationships Among Positive Numbers ...... 86 Relationships Among Positive and Negative Numbers ...... 87 Relationships Among Negative Numbers ...... 90 Relationships Among Opposite Numbers ...... 91 Data Addressing Representation of Number ...... 96 Number Line Representation ...... 96 Representation of Deficit ...... 104 Making Sense of Number in the Context of a Story ...... 111 Summary of Findings ...... 115

V ANALYSIS ...... 118 Introduction...... 118 Overview of Results ...... 118 Results of the Study ...... 119 Relationships Among Positive and Negative Numbers and Zero ...... 119 Zero as a Quantifiable Set ...... 121 Zero as a Referent...... 126 vii

Contextual Situations as Interactive Stories ...... 129 Representation of Intangible Numbers ...... 132 Rote Learning ...... 134 Summary of Analysis ...... 138 Limitations of the Study ...... 141 Implications of the Study ...... 143 Recommendations for Future Research ...... 148

APPENDICES ...... 150 Appendix A: Activity One: Streets with Houses Modified Horizontal Number Line ...... 151 Appendix B: Activity Two: Ladder from Cave to Treehouse Modified Vertical Number Line ...... 154 Appendix C: Activity Three: Collecting Bears, Pigs, or Frogs ...... 157 Appendix D: Activity Four: Collecting Pennies ...... 160 Appendix E: Consent Form for Parent of Minor ...... 163 Appendix F: Assent Form for Minor ...... 166 Appendix G: Audio/Videotape/Photograph Consent Form ...... 168 Appendix H: SH7 Record and Representation ...... 170 Appendix I: NB8 Record and Representation ...... 172 Appendix J: CH8 Record and Representation ...... 174

REFERENCES ...... 176

viii

LIST OF TABLES

Table Page 1. Presentation of Children’s Representation of Horizontal Number Line ...... 97

2. Presentation of Children’s Representation of Vertical Number Line ...... 103

CHAPTER I

BACKGROUND OF THE STUDY

Introduction

Operations on integers, especially when negative numbers are involved, are a

major point of contention for students beginning algebra. Not only do the students

encounter a new set of numbers representing quantities that do not fit the cardinality

principle of the concept of number (Goldin & Shteingold, 2001), but they are expected to

learn a new set of rules of operations that contradict the rules of operations for whole

numbers previously learned. Cathcart, Pothier, Vance, and Bezuk (2003) argue that since

in later schooling the students will work with negative numbers in algebra, they should

start by exploring negative numbers in a meaningful and unhurried manner. Traditionally

formal instruction on negative numbers and operations with negative numbers initially

appears in the grades five through seven mathematics curriculum (Davidson, 1992). The

National Council of Teachers of Mathematics (NCTM) states in the Number and

Operations standard that children in grades three to five should be expected to “explore

numbers less than zero by extending the number line and through familiar applications.”

(2000, p. 148) However, children in these grades may have previously encountered negative numbers in the context of temperature, e.g. two below zero, or scoring in a game, without symbolic quantification (Ball, 1993). Hativa and Cohen (1995) argue that even young children with whom they have conducted research have an intuition about or informal knowledge of negative numbers prior to formal instruction, and Ball argues that

“Teaching them [third graders] about negative numbers is an attempt to bridge their

everyday quantitative understandings with formal mathematical ones.” (1993, p. 378)

Wilcox (2008) argued that “young children have logical thoughts and ideas about

mathematical concepts to which they have never been exposed.” (p. 205) Young children

are naturally inquisitive and may be intrigued by the sheer magnitude of numbers – both

positive and negative – or by negative numbers generated on a calculator. Teachers and

researchers have found that children younger than children in grades three to five are naturally inquisitive about negative numbers. Behrend and Mohs (2006) relate a

classroom experience in which a first grade child asked whether numbers ever stop, and

the conversation continued to question whether numbers ever stop when you count

backward. This question initiated a two-year classroom exploration of negative numbers

with young children in first and second grades. Aze also related an experience with his

six-year-old daughter who asked why the calculator didn’t display or said “nothing when

I do three take away four?” (1989, p. 16) Children like these, considerably younger than

third to fifth grade children, appear to intuitively sense numbers other than the counting

numbers that they use in class and daily life. In a longitudinal study of children in grades

two to four, Schliemann, Carraher, Brizuela, Earnest, Goodrow, Lara-Roth, and Peled

found that these children were able to “accept the idea of negative numbers”

(2003, p. 4-128) but were challenged by operations with negative and positive numbers.

Schliemann et al. argued that “given the proper conditions and activities, elementary school children can reason algebraically and meaningfully use the representational tools of algebra.” (2003, p. 4-128) Experiences such as these raise the question of whether

3 young children may be developmentally ready to explore the concept of negative numbers before third to fifth grade.

Conceptualization of negative numbers has been historically challenging to learners of all ages because of the duplicity of definition of negative numbers as both magnitude and direction and because of the inability to quantify in a concrete manner.

Mathematicians from as far back as the third century have recorded struggles with the concept of negative numbers (Beery, Coschell, Dolezal, Sauk, & Shuey, 2004). Early mathematicians were said to have used negative numbers in algebraic computations following the same rules of operation that students use today, but they argued, often vehemently, about the meaning and the existence of negative numbers (Beery et al.,

2004; Katz & Michalowicz, 2005; Prather & Alibali, 2008). Therefore, if early mathematicians found the concept of negative numbers difficult to conceptualize, then how can current educators expect students, both children and adults, who lack the mathematics background of their ancestral mathematicians to master the concept of negative numbers as more than an implementation of memorized rules? (Carraher,

Schlieman, & Ernest, 2006)

Contexts of Negative Numbers

One difficulty in the conceptualization of negative numbers is that negative numbers represent both a cardinal context and an ordinal context. While the cardinal context of negative numbers refers to the concept of deficiency with respect to zero, the ordinal context of negative numbers refers to the concept of direction with respect to zero. The earliest recorded work of mathematicians focused on the cardinal context

(Beery et al., 2004; Katz & Michalowicz, 2005). Diophantus and other early mathematicians who attempted to define negative numbers in a context of magnitude, or cardinality, met strong opposition (Beery et al., 2004), but when Descartes and Girard interpreted negative numbers in the geometric context of directed numbers, they met less resistance (Janvier, 1985; Katz & Michalwicz, 2005). In their quest to make sense of negative numbers early mathematicians created representational notations and models for negative numbers in both contexts.

Educators still search for models of operations on integers to assist students in understanding the concept of negative numbers. These models, which include variations of the number line model to address the ordinal context of negative numbers and variations of the charged particle model to address the cardinal concept of negative numbers, are formally presented in the grade five to seven mathematics curriculum with the introduction of negative numbers as referenced to temperatures below zero, altitude, football gains and losses, and scoring of games (Van de Walle, 2001; Davidson, 1992).

These models of representation for operations with integers are demonstrated in conjunction with, or sometimes following, the introduction of the rules of operations which students are expected to memorize and apply in calculations or problem-solving.

Another difficulty in the conceptualization of negative numbers is the inability for representation with concrete objects. Although Piaget (1964) did not address the development of number sense in regard to negative numbers, he did argue that children, in the concrete operational stage of development, develop a number sense as one-to-one correspondence, ordinality, and then cardinality of set. One-to-one correspondence

implies that objects are tangible and may be manipulated. While one-to-one correspondence is traditionally associated with pointing to an object in a set while simultaneously assigning the sequence of counting numbers to the objects, the correspondence may also be established by comparison of sets. Ordinality assumes serialization of objects by characteristic (e.g. ordering by size, and assigning a number to each object in regard to its position in the set). Cardinality implies that there is a set of tangible objects that has some numerosity, and the cardinality of a set is determined by the logico-mathematical thinking process of developing relationships within the context of number such that the number of the last item counted in a set of objects determines the cardinality of the set or how many in the set (Piaget, 1964).

However, when encountering negative number concepts, children can no longer make these tactile one-to-one correspondences or model a set with concrete manipulatives (Hersh, 1997). In early research Galbraith (1974) argued that working with negative numbers requires the more abstract thinking in Piaget’s formal operational stage to make sense of a numerosity that can not be represented tangibly. Furthermore,

Davidson (1992) argued that the ordinal context of negative numbers demanding an image of ordered units along an established path containing an origin may be more abstract than the cardinal context demanding actions of grouping or separating discrete objects. However, Davidson (1987) argued that children at the age of seven are able to conceptualize negative numbers in an ordinal context when the actions are presented in game-like activities. In more current research Aze (1989), Behrend & Mohs (2006), and

Mukhopadyay, Resnick, & Schrable (1990) argue that younger children, in Piaget’s

concrete operational stage of development, are intuitively inquisitive about negative

numbers and capable of constructing meaning for negative numbers in both cardinal and

ordinal contexts with reference to a number line. Therefore the purpose of this study is to determine whether children younger than third to fifth grade as proposed by NCTM

(2000) are able to make sense of the relationships among positive numbers, negative numbers, and zero in a cardinal and/or an ordinal context and to what extent they carry out the sense-making process.

Statement of the Problem

Negative numbers are not formally introduced until ages nine to thirteen setting

up conflict with the child’s imbedded intuition of operations on whole numbers

(Davidson, 1992). However, recent research findings (Behrend & Mohs, 2006;

Davidson, 1987; Goldin & Shteingold, 2001; Human & Murray, 1987; Peled,

Mukhopadyay, & Resnick, 1989; Schliemann et al., 2003) suggest that young children in

early primary grades may be capable of exploring the concept of negative numbers

imbedded in a real-life situation context. Children at this level of development may

intuitively make sense of the concept of negative numbers without reference to the rules

of operations. (Behrend & Mohs, 2006). Mukhopadyay, Resnick, & Schrable (1990)

found that second through fifth grade children were more able to make sense of negative

number situations when presented in a familiar social situation such as a debt problem

than in an equation. An early exploration of the relationships among positive and

negative numbers and zero by young children may enable the children to develop a

conceptual understanding of negative numbers that will not hinder their progress in

7 computations later (Davidson, 1992; Davydov, 1975). Davidson (1987) suggests that teaching the concept of negative numbers to young children by using manipulatives is possible and may not lead to the future misconceptions brought forth by the traditional curriculum. Furthermore, Davydov argues that learning operations with only whole numbers may lead to difficulties when children begin to work with rational or real numbers. Cathcart et al. (2003) also argue that learning about integers and operations with integers is less difficult for children than learning about positive rational numbers.

These arguments may indicate the need for more research on rational and negative number instruction within the mathematics curriculum.

Rationale for the Study

The rationale for this study is to explore the thinking of young children as they encounter circumstances of deficit in game-like activities that necessitate the consideration of a negative number context. In the traditional mathematics education curriculum, young children are not formally instructed on negative numbers, and little is known about their insight into the concept. Research on young children and their conceptualization of negative numbers is sparse. A search of ERIC documents for research pertaining to young children and their conceptualization of negative numbers yielded only a few results. More studies were available from the proceedings of the

International Group for Psychology of Mathematics Education, but almost all of these studies focused on older children (ages twelve to thirteen predominantly) who did not live in the United States. A search of articles pertaining to negative numbers in The

Journal for Research in Mathematics Education yielded only a couple results, but none

8 were studies with young children. Therefore, the focus of this study is on the insight, intuition, connections, and strategies used by young children as they attempt to conceive the meanings of negative numbers and zero and formulate the relationships among positive and negative numbers and zero through application.

The participants for this study will be ages four through eight, younger than the children predominantly studied in regard to negative number concepts. Studies in the

1970s and 1980s used middle-school-aged children (twelve to fifteen years old) as their participant populations and focused on the effectiveness of particular teaching methods or models for integer operations (Peled, Mukhopadhyay, & Resnick, 1989; Gallardo &

Rojano, 1987; Murray, 1985). This study, like more recent studies of children and negative numbers (Aze, 1989; Behrend & Mohs, 2006; Peled, Mukhopadhyay &

Resnick, 1989) will focus on younger children who may not have had prior experience with negative numbers.

The research focus of this study will be based on observation of the children’s physical and verbal actions, interactions, and reactions to purposefully selected tasks during which conceptualizations of relationships among positive and negative numbers and zero may develop. Prior studies focused on measurement of the performance of children with operations on positive and negative numbers and zero as the result of a certain method of instruction or method of implementation. Unlike studies in which researchers such as Peled (1991), Hativa & Cohen, (1995) and Kuchemann, (1980) focused on the effectiveness of a standard number line model for teaching the operations on integers, this study will use a modified number line model more appropriate for young

children to conceptualize the ordinal context of negative numbers. Collection and loss

activities rather than models of operations based on the charged particle model will be

used with the children. The children will be observed as they engage in the activities that

support both cardinality and ordinality concepts of negative numbers.

More recent studies by Behrand & Mohs (2006), Borba & Nunes (1989),

Ginsburg (2002), Streefland (1996), and Towers & Anderson (1998) support the

argument that children aged four through eight may be able to develop some conceptual

understanding of negative numbers prior to formal instruction. Children as young as

seven or eight inquired about the finiteness of numbers and demonstrated an intuitive

understanding of negative numbers on the number line without exposure to the rules of

operations on integers (Behrand & Mohs, 2006; Davidson, 1987). Therefore, this study

will explore the readiness of young children for making sense of the relationship among

positive and negative numbers and zero using conceptual models.

Purpose of the Study

The purpose of this study is to explore how children make sense of the relationships among positive and negative numbers and zero and the extent to which the sense-making extends. Since the children in this study are younger than the children who traditionally encounter these relationships of number in the current mathematics curriculum, the information gathered from this study may provide insight for teachers, administrators and local, state, and national education policy makers into the mathematics curriculum to facilitate students’ learning of algebraic principles involving operations with negative numbers.

Unlike many of the previous studies (Battista, 1983; Bruno & Martinon, 1999;

Galbraith, 1975; Janvier, 1985; Kohn, 1978; Schwartz, Kohn, & Resnick, 1993/1994)

that focused on the effects of using particular number line and charged particle models of

integer operations to assist the students in mastering integer addition, subtraction,

multiplication, and division, this study will explore the conceptual aspect of making sense

of the relationships of positive and negative numbers and zero through game-like

activities that address the cardinal and ordinal aspects of negative numbers. Researchers

and mathematics educators argue that models using colored or +/- marked chips, number

lines marked with positive and negative numbers and zero, charged particles icons, and

other representations of positive and negative integers are essential in the understanding

of the operations (Janvier, 1985; Ost, 1987). However, these models are frequently

presented in the classroom after the rules of operations have been introduced without

initial conceptual development of negative numbers (Wilcox, 2008). From this

perspective, modeling of integer operations has the potential of becoming little more than

rote memorization of the modeling rather than a means of conceptual development.

Since many of the previous studies focused on children in the middle school or junior

high grades (ages 12-16), the researchers risked including children who had previous

experience with negative numbers in a formal or informal situation. Children of this age

have usually encountered negative numbers in some context of real-life situations including temperatures below zero or scoring of games, and some of the children had previously been introduced to the rules of operations for signed numbers. Like mathematicians of centuries before, these children may have been able to manipulate

positive and negative numbers in solving equations but lacked conceptual meaning for

negative numbers. And like mathematicians of centuries before, older children may be

content with being able to apply a set of rules to solve algebra problems involving

negative numbers and produce negative solutions even though the solutions do not make sense in application. Therefore, younger children who may not have encountered negative number context or representation will be selected for this study.

Research Questions

Young children will be studied for making sense of relationships among positive

and negative numbers and zero as they experience the concept of negative numbers

through models of ordinality and cardinality of number. With these points in mind, this

study will address the following research questions:

1. Do young children make sense of the relationships of positive and negative

numbers and zero encountered in contextual activities? If so, how?

2. What representation, expressed verbally and/or expressed as pictures, tally

marks, numerals, etc., do young children use to designate numbers, negative

integers in particular, in recording results of task-related activities?

Significance of Research

This study is significant because of the impact the findings may have on typical

mathematics curriculum in which initial formal instruction of negative numbers is

presented to children in grades six to eight (NCTM, 2000). Findings from this study may

suggest that younger children can intuitively make sense of the relationships among

positive and negative numbers and zero, enabling them to initialize the development of

conceptual understanding of negative numbers prior to middle school and potentially

encounter fewer difficulties in later mathematics. If further research substantiates these

findings and uncovers evidence that an earlier initial encounter with the negative number

concept facilitates learning operations with negative numbers, then mathematics

educators, administrators, and local, state, and federal education departments may need to

adjust the current mathematics curriculum to benefit the learners. Additionally, development of mathematics concepts in textbooks may need to be realigned to allow for exploration of the relationship of positive and negative numbers and zero by young children.

Summary

Traditional mathematics curriculum places instruction on integers in grades six through eight. However, the findings from recent studies suggest that children younger than middle school age are intuitively able to conceptualize negative numbers in a meaningful context and develop a conceptualization of the relationships among positive and negative numbers and zero. Children need to explore the concept of negative numbers in both cardinal and ordinal contexts. Further study of this inference is warranted to explore the developmentally appropriate placement of this concept in the mathematics education curriculum.

Definition of Terms

Cardinality – Objects in a set are counted with one-to-one correspondence; the cardinality

of the set is determined by the number of the last object counted (Piaget, 1952).

Integers – Integers are defined in two contexts: magnitude or cardinality and direction or

ordinality. The set of whole numbers (1, 2, 3, …), their opposites (… -3, -2, -1),

and zero represents magnitude. The relationship of the number in respect to

zero represents direction e.g. – 2 means two units below or left of zero.

Ordinality – Objects in a set are seriated by a defining characteristic; the ordinality of an

object is identifying placement of an object in regard to other objects in the set

(Piaget, 1952).

Seriation – Objects in a set are placed in order by a qualitative characteristic such as

length or size.

CHAPTER II

REVIEW OF THE LITERATURE

Introduction

Since children learn positive number concepts prior to learning negative number concepts, this chapter will first provide a review of literature on the development of children’s number sense and their learning of number. Because the study will be conducted with children engaged in game-like activities, a review of play and games in conjunction with learning mathematics will also be presented. This review of literature also provides an historical overview of the difficulties that mathematicians had conceptualizing the negative number concept as they used negative numbers in algebraic calculations. These historical overtones repeatedly present themselves in the review of studies on the teaching and learning of the relationships among positive and negative numbers and zero. Since much negative number instruction utilizes several different models of operations, a discussion of these models of operations is included.

Number Sense and Learning Number

Introduction

Young children learn number as a social phenomenon. They learn to recite the counting number sequence before they begin to conceptualize number as a quantification of set or a directional referent. The learning theories of Piaget are predominant in the current mathematics education reform movement, and in this constructivist learning

approach, children are encouraged to construct their own meaning as they encounter

situations of number.

Children and Number

Although Piaget (1952/1964) studied young children and their conception of

number, he did not expand his studies to the young child’s conceptualization of negative

numbers. However, in the study of young children and negative number

conceptualization, the inclusion of Piaget’s studies of young children and positive

numbers is necessary.

Piaget (1952/1964) categorizes three stages in which young children understand

number. Children in stage one use global comparisons of objects but lack exact

quantification, determining whether one group of objects has more or less than another

group, and counting objects in the set by merely reciting the sing-song counting sequence as they randomly point to objects. Children in stage two use one-to-one correspondence but do not maintain a lasting equivalence of number. Piaget argues that the

level of correspondence between the numerals and the objects is still purely

verbal, and the child has not yet acquired the notions necessary for the

construction of number itself, i.e. permanence and equivalence of sets irrespective

of the distribution of the elements of which they are composed. (1952/1964,

p. 47)

Children in stage three use both exact correspondence and lasting equivalence of number.

Through his research with young children aged four to seven, Piaget (1964) determined that young children learn number through a developmental sequence of

16 one-to-one correspondence, seriation, ordination, and finally cardination. Young children initially develop one-to-one correspondence in a sense of comparison of quantities either globally, as density or size of set, or in a matching context. Seriation refers to the ordering of objects by a quantifiable characteristic. Ordination expands on the concept of seriation by distinguishing the number of order by placement with same units. An

“ordinal number is a series whose terms, following one another according to the relations of the order that determine their respective positions.” (Piaget, 1952/1964, p. 157)

Cardinality of number is perhaps the most complex because it involves logico- mathematical reasoning with which a child constructs a relationship between the physical and social, or conventional, knowledge of objects in regard to number and may be able to represent this knowledge with symbols or signs (Piaget, 1952/1964). “A cardinal number is a class whose elements are conceived as ‘units’ that are equivalent, and yet distinct in that they can be seriated, and therefore ordered.” (Piaget, 1952/1964, p. 157) A study of number representation by kindergarten children conducted by Sinclair, Siegrist, and

Sinclair (1983) supported Piaget’s theory of progressive stages of learning numbers. In the study, children represented objects in five different ways from global representation to cardinal value representation. The children first represented a quantity with tally marks and then with representative images of the object (not necessarily quantitatively representative). In one-to-one correspondence, the children initially made representation with invented or conventional symbols and then with numerals, such as 1 2 3 or 3 3 3.

Children with a sense of cardinal value represented with the numeral alone or depiction of the object in both qualitative and quantitative respects, e.g. drawing of three crayons.

Piaget (1950) argues that children construct knowledge by adapting their

cognitive structure with the construction of three kinds of knowledge; physical, social,

and logico-mathematical. Physical knowledge of an object is an external reality and is

gained empirically by observation (e.g. the color or weight of an object). Social knowledge is conventional knowledge as evidence in the numeral 2 meaning the set composed of one object and another object. Logico-mathematical knowledge consists of relationships established through reflective abstraction as the child assimilates or tests previous knowledge against a new idea that is evoking a conflict or perturbation and then accommodates or makes changes to his or her base of knowledge or schema (Piaget,

1967/71).

Clements and Sarama (2007) report that findings from recent research suggest that knowledge of number and quantification develops earlier than Piaget proposed and even claim that infants demonstrate understanding of cardinality. Piaget’s concept of cardination is challenged by the concept of subitizing. Subitizing is the process of making a quick perceptual determination of numerosity of a group of objects without counting the objects. Preschoolers were found to be able to subitize or determine the number of objects in a small group without counting (LeCorre, Van de Walle, Brannon,

& Carey, 2006). Clements and Sarama (2007) summarize that early numerical knowledge includes subitizing, learning names and order of number words, enumerating objects (saying the word that correlates with the object in the set), and cardinality.

Zero

The concept of zero is difficult for children to comprehend for a number of

reasons including its definition, usage, and representation. Borba (1995) describes zero

as an ambiguous number defined as the absence of number and as an origin of number.

Children described zero as “nothing”, “none”, “worthless”, or simply referred to zero as

O (oh) as a letter of the alphabet (Catterall, 2008; Hughes, 1986). When children

encountered a negative number situation in which zero was to serve as an origin, or initial

point, the children ignored the zero, said zero indicating a terminal point, or tried to

change the problem to elude the situation (Van Den Brink, 1984). Sometimes the number

zero exists and sometimes it can be ignored depending on the context (e.g. zero

passengers on a bus makes sense but a zero Cuisenaire rod does not ) (Van Den Brink,

1984).

The numeral zero is used as a symbol of quantity and as a place-holder in the base-ten number system (Hughes, 1986). While children have little difficulty grasping the idea that the numeral zero is a symbol that designates no quantity, using zero as an indicator of place value is not as straight-forward. How can a zero added to the right of the number one create the number ten, even though the zero added means nothing? In a study of young children engaged in a classroom activity of modeling phone numbers,

Harrison, Scown, & Marshall (1993) observed a child who left a space where the numeral zero should have been and explained that “zero means nothing, so I’ve put nothing there.” (p. 7)

Children are not certain how to represent the number zero. Hughes (1986)

observed children who represented zero with the conventional notation, drew an empty

box, or simply left the space empty. He also noted that when children were asked to

arrange magnetic numbers in a sequence of one to nine, some of the children placed the

number zero in front of the one while others placed the zero after the nine stating that it

was ten or needed a one to be ten. Additionally, many children encounter difficulties

when addition and subtraction with zero is introduced (Van de Walle, 2001). Addition

and subtraction with zero contradicts children’s intuition or experience that addition

creates a larger number and subtraction makes a smaller number. Van de Walle (2001)

suggests that ample opportunities to model addition and subtraction with zero should be

presented to young children.

Constructivism

Theories of learning by Piaget (1950), Dewey (1916/1944), Bruner (1960), and

Vygotsky (1978) in which children learn as they construct meaning from individual experiences developed the constructivist paradigm. Constructivism is a theoretical approach to teaching and learning rather than a specific pedagogy and describes

“knowing” and “how one comes to know” (Fosnot & Perry, 2005, p. ix). This

constructivist paradigm is paramount in the mathematics education reform movement

espoused by NCTM (2000). Constructivism embodies Piaget’s (1950) argument that the

aim of education should be to develop autonomy in a child; the education process should

develop a child’s desire and ability to think for himself. Fosnot describes learning from

the constructivist perspective as

a process of struggling with the conflict between existing personal models of the

world and discrepant new insights, constructing new representations and models

of reality as a human meaning-making venture with culturally developed tools

and symbols, and further negotiating with meaning through cooperative social

activity, discourse, and debate in communities of practice. (2005, p. ix)

Davis stated that this approach gives children a clearer view of “the nature of mathematics as something invented by human beings in the course of giving sensible responses to sensible challenges, and not as some kind of revealed truth that one could know only because one had been told.” (2003, p. 631)

Clements and Battista (1990) describe constructivism in the context of mathematics by basic tenets addressing active learning, adaptation, subjectivity, social interaction, and disposition of mathematics knowledge. In constructivism, knowledge is actively constructed through individual experience of each child rather than passively received from a knowing source. Children do not discover new mathematical ideas but invent ways of thinking about a new mathematics concept. Knowledge is constructed as the child cognitively adapts new experiences in mathematics to an existing physical and mental experience base of mathematics concepts (Stenger, Keller, Arnon, Dubinsky, &

Vidakovic, 2008). Because each child’s experiential base of knowledge differs from the others, constructing knowledge is subjective and “no one true reality exists, only individual interpretations of the world. Their interpretations are shaped by experience and social interactions.” (Clements & Battista, 1990, p. 34) (This tenet is controversial

among radical, social, and cognitive constructivists.) Knowledge acquisition is a social

process. Clements and Battista argue,

Mathematical ideas and truths, both in use and in meaning, are cooperatively

established by the members of a culture. Thus, the constructivist classroom is

seen as a culture in which students are involved not only in discovery and

invention but in a social discourse involving explanation, negotiation, sharing,

and evaluation. (1990, p. 34)

As a fifth tenet of constructivism, Clements and Battista (1990) state “When a teacher demands that students use set mathematical methods, the sense-making activity of students is seriously curtailed.” (p. 34) Such demands promote rote memorization of mathematics methods and rules and devalue the aspect of sense-making of mathematics.

Meaning-making of Number

In 2000 the National Council of Teachers of Mathematics recommended to start teaching the concept of negative numbers in fifth through eighth grade, introducing negative numbers as an extension of the positive number system. Murray (1985) suggests that children should be introduced to negative numbers earlier than they are now because they are able to understand numbers, numbers patterns, and logic at an earlier age than is considered now. He argues that for young children to extend the number system to include negative numbers may be no more difficult than extending to rational numbers, provided negative numbers were carefully introduced with only addition of signed numbers and no rules or algorithms would be introduced. Results from a study by

Hativa & Cohen (1995) suggest that some negative number concepts such as mental

comparisons, representation on a number line, and distance between integers could be

taught to all fourth grade students. Galbraith (1974) argues that signed numbers and

addition of signed numbers can be introduced at the “top of primary” grades or early in

the secondary grades, but subtraction, multiplication and division of integers should be

delayed until the formal operational stage of development.

The aforementioned research raises the question as to when negative numbers

should be introduced to children. Bruner argues that “any subject can be taught

effectively in some intellectually honest form to any child at any stage of development.”

(1966, p. 113) From a psychological perspective, Kagan & Baird (2004) state that the

ability to activate past representations to interpret current situations accelerates between

the ages of five and eight and plateaus from age eight to puberty. Following meta-

analysis of research, Streefland (1996) questions whether instruction on negative

numbers should possibly be started with 9- to 10-year-olds before they master the natural numbers in a formal context. While the majority of research on negative numbers deals with participants whose ages range from 12-16 (Bruno & Martinon, 1999; Chiu, 1994;

Gallardo, 2002; Vlassis, 2004; and others), some recent research deals with younger children in the first through third grades (Behrhend & Mohs, 2006; Goldin and

Shteingold, 2001). Results from an exploratory study with four- to seven-year-old children suggest that they develop action-based intuitions of negative numbers before they receive formal instruction (Davidson, 1987; Davidson, 1992). Through enactive representation, children can construct their own concepts intuitively rather than through external imposition of rules (Bruner, 1960). Behrend & Mohs (2006) found that young

children are capable of intuitively developing algebraic concepts such as negative

numbers. Following interviews with first and second grade participants with no previous

negative number instruction, Goldin and Shteingold (2001) suggest that children begin to

develop informal representations of negative numbers earlier than is believed, and that

school curriculum might consider earlier introduction of negative numbers before they

become obstacles in learning. Young children come to school with informal

mathematical ideas that have been developed independent of formal instruction; this

knowledge has been acquired through social transmission from older siblings and parents

(Moser, 1992). Young children experience situations that deal with negative numbers in

the context of temperature, scoring of games, and football gains and losses. They may be

familiar with negative integers used to describe physical situations such as −5 for five

points in the hole in scoring of games, −7 denoting seven degrees below zero, −4

meaning a loss of four yards in a football game, or −100 indicating one hundred dollars in

debt (Van Engen & Grouws, 1975).

As young children develop a sense of number with traditional instruction, they

count and manipulate concrete objects to model addition and subtraction, and then

compute with written numerals. During this process prohibitive intuitions, such as not being able to subtract a larger number from a smaller number (Davidson, 1987) or that

multiplying or adding make larger, while subtraction makes smaller (Peled & Carraher,

2007), may develop. These imbedded intuitions are contradicted with the introduction of

negative numbers (Davidson, 1987; Peled & Carraher, 2007). Children must go through

a process of assimilation and accommodation (Piaget, 1952/1964) to internalize negative

numbers within their schema of natural numbers (Gelman & Gallistel, 1978; Vlassis,

2004). Not only do students encounter conflicts with negative numbers producing

solutions that are inconsistent with their previously developed schema of whole numbers,

but children can no longer represent negative numbers with tangible objects. This

transition from concrete to formal mathematics requires the children to use a higher order

of thinking in abstraction (Klein, 1925).

To develop the concept of negative numbers, children should be given ample opportunities to experience situations in which they encounter negatives (Van Den Brink,

1984). For young children, three to eight years old, activities should be presented that can be solved intuitively without reference to a set of rules or without focusing on formal operations (Van Engen & Grouws, 1975). Caution must be taken to design problem situations in which the concept to be learned is justified by the activity and effective interaction of the student (Thomaidis, 1993). Mukhopadhyay (1997) suggests that children are able to make sense of negative number situations when they are encouraged to use a story-telling approach. Davis (1985) argues that to promote algebraic thinking in

elementary children, ages seven to twelve years, a wide array of experiences from which

young children can build mental representations for algebraic ideas, including negative

numbers, should be provided. He cautions that to promote conceptual development these

activities should be worthwhile and powerful and not just fun.

Role of Representation

As children are introduced to the abstract concept of negative numbers, they

experience two major difficulties, mental representation and symbolic notation. Children

need to be engaged in activities that cause perturbations allowing them to construct their

own understanding through discovery. Piaget stated, “To understand is to discover, or

reconstruct by discovery, and such conditions must be complied with if in the future

individuals are to be formed who are capable of production and creativity and not simply

repetition.” (1973, p. 20) Children are determined to understand and are willing to create

their own ways of understanding (Davis, 1992).

The National Council of Teachers of Mathematics (NCTM) (2000) describes

representation as both a process and a product. As a process, the child transfers an

internal or mental image of a concept to an external or visual or tactile representation in the attempt to gain understanding of a mathematical concept. As a product, an external

representation such as a symbol or drawing is created. On the contrary, Borba & Nunes

(1989) suggest that young children may develop more of an understanding of the concept

of negative numbers if they do not have to make external representations. NCTM states,

Representations should be treated as essential elements in supporting students’

understanding of mathematical concepts and relationships; in communicating

mathematical approaches, arguments, and understandings to one’s self and to

others; in recognizing connections among related mathematical concepts; and in

applying mathematics to realistic problem situations through modeling.” (2000,

p. 67)

When children encounter new mathematical situations, they are likely to invent

their own ways of understanding (Erlanger, 1973). If mathematics is a way of thinking

that involves mental representation, then each child must be permitted to invent his own

mental representation of new concepts (Davis, 1992). “Children need to invent to

understand and to assimilate information.” (Brizuela, 2004, p. 35) Although she did not

include negative numbers in her research, Kamii argues that “procedures children

invent are rooted in the depth of their intuition and their natural ways of thinking

…children develop a better cognitive foundation as well as confidence.” (1982, p. 14)

Piaget regarded invention as central to knowledge construction and argued

The problem we must solve, in order to explain cognitive development, is that of

invention and not mere copying. And neither stimulus-response generalization

nor the introduction of transformational responses can explain novelty or

invention. By contrast, the concepts of assimilation and accommodation and of

operational structures (which are created, not merely discovered, as a result of the

subject’s activities), are oriented towards this inventive construction which

characterizes all living thought. (1970, pp. 713-714)

Brizuela further argues,

Genuine optimism would consist of believing in the child’s capacity for invention.

Remember also that each time that one prematurely teaches a child something he

could have discovered for himself, that child is kept from inventing it and

consequently from understanding it completely. (2004, p. 715)

When instructional programs do not promote developmental understanding, even young children invent their own ways of understanding often in terms of misunderstood or incomplete application of rules (Davis, 1992). Such a misunderstanding was evident in the case study of Benny, a young boy, who appeared to do well in mathematics as

27 indicated by his test scores. However, discussions with Benny indicated that, although he had been taught the rules of operations with fractions and decimals, he had misinterpreted and internalized them to fit his own understanding (Erlwanger, 1973).

Notation

Since children are not able to represent negative numbers with tangible objects, they must make some kind of mental or written representation. Research indicates that variations of the number line seem to satisfy that need. Peled, Mukhopadhyay, & Resnick

(1988) and Borba (1995) report that students with some introduction to negative numbers found a number line representation appropriate, although Borba’s sixth grade participants, like Descartes, preferred a divided number line which presented positive and negative numbers as two disjoint sets of numbers. However, Goldin and Shteingold

(2001) argue that “the ability to assign meaning and to use negative numbers in an ordinal external [conventional] representational context of activity neither implies nor is implied by the presence of these abilities in a cardinal external representational context.”

(p. 19)

When children are asked to invent their own representation of negative numbers, they, like early mathematicians, are challenged to invent symbolic notation for the numbers. Historically, negative numbers have been denoted by raised minus signs

(Davis, 2003), numbers enclosed in circles, numbers preceded by the word “moins” meaning minus, or numbers preceded by the letter m (Beery et al., 2004). Logically invented symbolism for negative numbers is a natural development; if conceptual development of negative numbers is the intention, then creative notation with which the

child can develop operations before encountering the algebraic symbols and syntax of

equations makes sense. This development is much like inventive spelling for children

learning to express their ideas in writing. Children create, use, and adapt notations to

communicate mathematically with others and to serve as external memory aids for

themselves; they create new symbols that might not be elements of the traditional number

system, but the notation allows them to continue to develop the concept without symbolic

constraints and progressively adapt to conventional notation (Lee, Karmiloff, & Smith,

1996). Piaget (1951) found that children use two types of numerical representation –

symbols and signs. Symbols may actually resemble the objects or signify the quantity of

the objects being represented; they may make representative drawings or tally marks.

The source of signs is social knowledge that includes the spoken words, such as three, or written numerals, such as 3. According to Piaget (1951), symbols are inventions of the child, and signs are not. In a study of kindergarten children, Brizuela (2004) found that children invented their own notations of number based on their previous knowledge of number and language. A child, Paula, wrote a two-digit number with what she called a

“capital number” because she knew that a two-digit number was different than a one-digit

number and that letters differed by capitalization. Another child, George, wrote dummy

numbers that vaguely resembled numbers because he knew they had to be different but he

was not knowledgeable of the conventional notation (Brizuela, 2004). Brizuela argues

… Children’s invented notations are of utmost importance in children’s learning

and development of notations; second, conventional notations play an important

role in children’s invented notations and provide a support for their

development… they are subordinate to inventions and to the assimilatory aspect

of thought. (2004, p. 27)

If the concept of negative numbers is presented in a non-threatening manner, children may be able to intuitively understand the concept prior to introduction to the rules of operation for integers. This approach allows children to make their own meaning of negative numbers and invent their own symbolic notation for negatives.

One of the major notation difficulties that young children, as well as older students or adults, encounter with negative numbers is the use of the minus sign (Vlassis,

2004). The minus sign has three major functions, unary in which the minus sign makes the number negative, binary for which the minus sign indicates the operation of subtraction, and symmetric for which the minus sign means to take the opposite of

(Gallardo & Rojano, 1994; McAuley, J., 1990). Vlassis argues that “difficulties with the minus sign can be related to the fact that students could not subordinate their use of mathematical symbolism to the need for efficient communication” (2004, p. 482). In a study of 13-14 year olds with prior introduction to negative numbers, Vlassis (2004) found that none of the children thought that the minus sign could be unary and binary.

The children stated that the minus sign had to indicate subtraction and could not be attached to the following number because the indication of operation would be eliminated. Vlassis (2004) also found that eighth grade students thought that the minus sign indicated action to be performed on the first term because they were taught that 6−4 means 6 take away 4 with the action on the 6. Vlassis (2004) found that students who learn the rules prior to understanding the concept have more difficulty relating to the

30 concept. In a study of 12-year-old prealgebra students, Herscovics and Linchevski

(1994) found a tendency of detachment from the minus sign regardless of the placement in the polynomial. They conjectured that students who had learned order of operations erroneously performed addition before subtraction in any context (e.g. [16−16+32]).

Since counting does not result in negative numbers, children do not necessarily conceive of subtraction as adding a negative number. Furthermore, subtracting a negative number contradicts the belief (as in natural numbers) that addition increases numerosity and subtraction decreases numerosity (Gelman & Gallistel, 1978). Murray (1965) reported that eighth and ninth graders overgeneralize the commutative property and complicate integer addition and subtraction; the students explained that “2 – 5 is the same as 5 – 2”

(p. 150).

In an expression such as 7 − 4, the minus sign can be interpreted as seven minus four as a simple subtraction or as seven plus a negative four. This duplicity of the meaning of the minus sign is a constant point of contention for students as they begin studying algebra (Van de Walle, 2001). As children encounter an expression such as

4 − 7 for which they can not use simple subtraction nor find empirical reference or physical representation, they either contend that a larger number can not be subtracted from a smaller number, or disregard order and answer 3, or they simply say the result is zero.

The advent of the calculator that delegates separate keys for minus and negative entries was expected to impact the teaching and learning of the negative number concept

(Van de Walle, 2001). Donoghue (2003) contended that the accessibility of calculators

31 would take the sign problem “out of the question” (p. 389), but the findings the research presented does not substantiate that claim.

Play and Learning Mathematics

Within the behaviorist paradigm, learning mathematics is a passive experience for young children who may be subjected to periods of completing worksheets and practicing rote memorization of mathematics facts. The constructivist approach to learning is supported by play-based learning in which the learners construct knowledge from personal experience (Caswell, 2005). Bruner, Jolly, & Sylva (1976), Dienes (1973),

Dewey (1928), Vygotsky (1976), and Piaget (1951) argue that learning, including mathematics learning, is attainable through simple acts of play. Piaget argues, “Play is the child’s natural approach to knowledge.” (cited in Harding, 1973, p. iii), and young children naturally satisfy their inquisitive nature in playful settings as they learn to talk, reason, and make mathematical and scientific deductions through hands-on, minds-on experimentation and exploration. Piaget states “Play is a particularly powerful form of activity that fosters the social life and constructive activity of the child.” (cited in Kamii

& DeVries, 1980, vii) Vygotsky argues that “…play creates the zone of proximal development of the child. In play a child is always above the average age, above his daily behavior… ” (1976, p. 552). While much of Vygotsky’s research on play concerned preschool children, he noted that, in play, school age children create relationships between “semantic and visible fields” (1976, p. 554) by extending real situations to situations in thought. Since the recent mathematics education reform movement in the United States encourages educators to aid children in establishing

connections between mathematics and the real world (NCTM, 2000), doing mathematics

and playing may be logical partners in education. A study of middle school children by

Caswell (2005) also indicated that play activities in the mathematics classroom engage

the child, enhance mathematical understanding, promote application of previous

constructed mathematical knowledge, and allow the child to advance from concrete

manipulation to abstract mathematical thinking at his level of understanding.

Piaget (1951) argues that play allows the child to assimilate the newly-found knowledge to create an equilibrium with the child’s previous experiential learning.

“Unlike objective thought, which seeks to adapt itself to the requirements of external reality, imaginative play is a symbolic transposition which subjects things to the child’s activity without rules or limitations.” (Piaget, 1951, p. 87) As the child begins socialization within play, he engages in play that requires rules and adaptation to an external reality (Piaget, 1951).

Dienes (1973) defined six stages of learning that are initiated through play. The first stage of learning involves play which is a spontaneous adaptation of the child to his environment. The second stage involves structured play where activities become more restricted by self-imposed or socially-imposed rules. Relationships between concrete and abstract concepts are created in the third, or abstraction stage of learning. Concept discrimination in which representation is made of the knowledge that has been learned is defined as the fourth stage. The fifth stage ties the invented or learned symbolism

(language) to more conventional representation. Within the sixth or formalization stage axioms, theorems and proofs are established. Dienes created models, including the

Dienes rods, that promoted playful activities in mathematics which facilitated

advancement among the stages of learning.

Games and Learning Mathematics

Within the context of play, games for education were first developed as simulation games in the early 1960s (Coleman, 1976) when social scientists discovered that games could function as learning or teaching tools in the classroom (Boocock &

Schild, 1968). Most of these early educational games were reality or strategic type games patterned after war games (Abt, 1968). Researchers have indicated that games are an effective means of enhancing mathematical learning. Studies with young children

(Peters, 1998) and older children (Adaramola & Alamina, 2008) indicated that children taught with mathematical games outperformed children taught by traditional methods.

Piaget (1951), Bruner (1976), and Dienes (1973) argue that playing games in the mathematics classroom conforms to the principles of constructivism eliciting active participation, subjective interpretation, adaptation to individual schema, and, in the case of group games, social interaction. Dewey (1928) argued that games should be an integral part of the school curriculum, not simply a release from school work.

Engagement in educational games allowed children to learn actively rather than passively, aligning with Dewey’s pedagogical philosophy (Boocock & Schild, 1968).

Boocock and Schild credit Dewey with defining principles of argument for educational games,

…the active and simultaneous participation of all students in an educational game,

with the teacher in the role of aid rather than judge; the internal rather than

external locus of rewards, and the motivation, in a game; and the linking of the

student to the outside world through the simulated environment…(1968, p. 57).

Abt (1968), Burns (2003), Cavanaugh (2008), and Ernest (1986) present rationales for using games to teach mathematics; these rationales include reinforcement or practice of skills, development of problem solving strategies, development of a mathematical concept, and engagement that may instill a positive attitude towards mathematics. Furthermore, Ainley (1990) states that playing games promotes decision- making, conjecturing, generalizing, and judging of actions and may aid the child in developing confidence in his own thinking. Abt (1968), Ernest (1968), Peters (1998), and Wakefield (1998) add that playing games provides motivation for children who are less than enthusiastic to complete the same mathematics on worksheets. Ainley (1990) and Kamii (2000) emphasize that ideally the motivation to play games should be intrinsically rather than extrinsically generated. However, Ponce (2007) questions the difference between learning the rules for playing a game to gain proficiency in a concept such as adding or subtracting negative numbers and learning the rules of operations.

Kamii (2000) states that games are superior to worksheets for learning because games, a natural activity for children, are intrinsically motivating and allow for immediate feedback and communication over misunderstandings. Players of games, not the teachers, determine correctness and incorrectness. Games can be adapted to different levels of development of players. Unlike playing games, writing answers on worksheets may interfere with free thought as the child may become engrossed in discrimination of numeral or letter writing (e.g. 5 or S) (Kamii, 2000). Children are more likely to

establish numerical relationships with number generators for games than for isolated

problems on worksheets. Children who are permitted to choose which game to play may

work harder because the activity is appealing. Lastly, children develop sociomorally by

playing games during which they must interact with other players and find amicable and

fair resolutions to conflicts (Kamii, 2000).

The teachers must present opportunities for children to engage in problem solving

that allows for “repeated but meaningful mental engagement” (Wakefield, 1998, p. 269).

A game should be designed so that the skill or concept to be learned is a necessary element for playing the game (Boocock & Schild, 1968). Ainley views educational games as a “way of providing the mathematical equivalent of children’s books and comics.” (1990, p. 86) She argues that many text problems apply to the adult world and not the child’s; educational games allow the child to engage in fantasy or story play where the outcome matters to the child. Ernest (1986) states that games should be selected to meet certain objectives and become a vital element in success and achievement in mathematics education. He notes that although a well-designed educational game may create an environment conducive to learning, learning is not guaranteed. Bright, Harvey, and Wheeler (1985) emphasize that educational games should be more than drill-and-practice activities; educational games should address the highest of Bloom’s (1956) taxonomic levels appropriate for both the mathematics concept and the target population.

When educational games are incorporated in classroom instruction, the role of the teacher changes from source of knowledge to facilitator of learning. Teachers can remain

involved in the learning process by using this opportunity for observation and

assessment. Teachers can play the games with individuals or groups of children to assess

their levels of understanding of mathematics concepts, posing questions or making

observations of play to elicit insight into children’s thinking (Ainley, 1990; Kamii, 2000).

Onslow (1990) argues that children may need guidance to develop an understanding when playing a mathematical game; teachers should question the children to help them overcome conceptual obstacles. Onslow states

A game can provide the embodiment for a meaningful discussion to assist

students in accommodating new concepts. During the discussion, the students’

attention must be clearly focused on the conceptual difficulties by bringing the

conflict to the surface and providing the climate for students to form

generalization and construct more encompassing frameworks based on experience

and the awareness of the limitations of inferior primitive structures. (1990, p. 591)

Children are not as intimidated by a mistake made while playing a game as by a

mistake made on written work, because they are more self-judgmental in play

(Wakefield, 1998). Furthermore, children who engage in discussion while playing a

mathematical game are likely to accept their errors and handle them constructively

(Onslow, 1990).

Although a plethora of mathematics games for young children (Kamii & DeVries,

1980; Kamii, 2000; Harding, 1973; Trivett, 1974; and Wakefield, 1998) and older

children (Fischer, 2004) are available for perusal and use in the classroom, the teacher

should select games with discretion. Earlier games were created for the objective of

practice of skills. Although Kamii includes some of the same games in her selection, she

has analyzed and presented them in context of conceptual development (Kamii, 1982;

Kamii & DeVries, 1980). Studies with young children (Peters, 1998) and older children

(Adaramola & Alamina, 2008) indicated that children taught with mathematical games

outperformed children taught by traditional methods.

While mathematics games are a valuable asset in the early childhood classroom,

they are not a panacea for mathematics education reform. Abt (1968) presents limitations

of the use of educational games. First, teacher attitudes towards the games may

determine whether children take the games seriously or just for fun. Teachers may set

too strict or too numerous restrictions on the games making play undesirable for children.

The intellectual validity of the game may determine its appeal and value to the children; the game should present an appropriate situation for the child. And finally the game should be attractive to the child in the sense of integration with other concepts being learned in the classroom.

Summary

The way children learn about negative numbers in the traditional mathematics

education curriculum is loosely analogous to the way early mathematicians worked with

negative numbers. Children learn to apply a set of rules of operations on integers without

adequate conceptualization of the relationship among positive and negative numbers and

zero.

Although much of the research on learning negative number focuses on children

aged twelve through fifteen and particular methods of instruction for operations with

negative number, more recent research has indicated that younger children have an

inquisitive sense and natural intuition about negative numbers. While variations of

models of operation with negative numbers abound for children in grades six and above,

few models for conceptualization of negative numbers by younger children are presented

in the literature. Most of the models for younger children rely on game-like activities.

Children learn through experience, and many of the experiences of a child are

through playful situations. Ponce (2007) states that playing mathematical games

promotes the essential NCTM (2000) principles of engagement, representation,

communication, and connections for mathematics education. Therefore, games or other

playful activities that promote mathematical concepts may be a valuable addition to the

child’s learning experiences in or out of the classroom.

Negative Number Instruction

Introduction

Since instruction for operations on positive and negative numbers could not

follow modeling with concrete manipulatives, a number of models of instruction to assist

children in conceptual development of negative number operations were developed. The

concept of negative numbers is abstract and difficult for children to grasp, so negative

numbers should be introduced to children using an appropriate manipulative (Thompson

& Dreyfus, 1988). In his studies, Thomaidis (1993) found widespread use of concrete models in the teaching methods for negative numbers. The ambiguity of the minus sign

in negative numbers poses considerable difficulty for children. Therefore, Janvier (1985)

argues that one of the pedagogical objectives of models for integers should be to remove

that ambiguity of notation.

The cognitive goal of modeling in mathematics is to aid the child in developing

“an intuitive acceptance of the concept” (Fishbein, 1987, p. 100). Fishbein (1987) argues

that two intuitive obstacles affect how children make sense of negative numbers. First,

the concept of negative numbers is intuitively contradictory to the concept of positive numbers defined as quantifiable entities. Secondly, negative numbers are a “by-product of mathematical calculations and not the symbolic expression of existing properties.”

(Fishbein, 1987, p. 101) To overcome these obstacles, modeling activities allow children to develop an ability to understand data presented, use intuitive knowledge, and to distinguish between personal and task knowledge (Peled, Mukhopadhyay, & Resnick,

1989). Modeling activities for arithmetic of positive and negative numbers are presented with the ordinal context represented as variations of the number line and the cardinal context represented as some quantifiable collection of objects, often as a variation of the charged particle model.

Models of Integers

As early Chinese mathematicians attempted to make sense of negative numbers by manipulating colored rods to facilitate computations (Beery et al., 2004; Gallardo &

Rojano, 1990), mathematics educators have created concrete models for which the operations with negative numbers could be manipulated. Early models of positive numbers were simply collections of objects of nature, such as rocks and sticks, which could be counted and manipulated for addition, subtraction, multiplication, and division

(Kohn, 1978). However, these models became problematic when applied to negative numbers as mathematicians could not quantify a negative set (Peled, Mukhopadhyay, &

Resnick, 1989), and, according to Galbraith (1974), no concrete substantial model of subtracting negative numbers exists. Schwartz, Kohn, & Resnick, (1993/1994) categorized models as cancellation models, e.g. debts, elevators representing both positions and actions, time (limited in usage and most difficult for young children), temperature (not applicable for binary operations), and formal algebraic operations. But

Featherstone (2000) argues that negative numbers are difficult to align with metaphors of everyday life to which young children can relate.

Janvier (1985) argues that the main purpose of models is transferability from concrete manipulations to abstract thinking. “The power of the model is in the conceptual simplification.” (Ost, 1987, p. 369) Galbraith (1974) argues that children at

Piaget’s concrete level of understanding are more dependent on models of operations than children at the formal operational level. Ost (1987) describes models as tools of inquiry that may be used to develop higher order thinking skills, such as explaining, interpreting, predicting, and analyzing. Nunes and Bryant (1996) contend that “when we understand the logic of a number system, we can generate that which we have never heard before.” (p. 45)

The concept of negative numbers has been introduced in a myriad of ways that include exploring repeated subtraction on the calculator, physically walking forward and backward in reference to a reference point (Naylor, 2006), playing games with cards for which the red and black playing cards represent negative and positive numbers

41 respectively (Ponce, 2007), and creating extended number lines and vector representation

(Carson & Day, 2006). Two models for operations with negative and positive numbers appear to be predominant in current textbooks – the number line model and the collection model which includes colored counters and positive and negative charges (Cathcart et al.,

2003; Janvier, 1985; & Van de Walle, 2001). Goldin and Shteingold (2001) conjecture

“that a fully developed concept or signed number involves internal representations both for the number line and signed cardinality interpretations, whereas partially developed concepts may function in one system but not another.” (p. 9) Both models are fairly easily manipulated, but they have limitations that will be discussed later.

Number Line Model

The number line used in schools today became a popular model for integer addition and subtraction in the 1960’s and 1970’s during the post-Sputnik era (Hoff,

2002; Galbraith, 1975), unlike Descarte’s early number line which was regarded as two discrete half lines of positive and negative numbers (Janvier, 1985). Cathcart et al.

(2003) suggest that early childhood classrooms should have an integer number line visible so the children can familiarize themselves with it and practice labeling points, ordering positive and negative numbers, and denoting direction from zero. With the number line, children model addition and subtraction of positive and negative as forward and backward movement along the number line. To add on a number line, a child must begin on a point of the first addend and then move the distance and direction (right or left for positive or negative respectively) as indicated by the second addend, and the terminal point is the sum (Van de Walle, 200; Cathcart et al., 2003). But, to facilitate subtraction

of a negative number on a number line, the child must comprehend the relationship

between addition and subtraction as focus is on the missing addend and the “final answer

reflects the number of spaces moved and the direction of the move.” (Cathcart et al.,

2005) Sfard (2007) argues that the number line used with this connotation is the best

model for children as it provides a visual representation of negative numbers and

operations with negative numbers. Furthermore, Carson & Day (1995) argue that

abstract concepts such as negative number operations should be taught with a geometrical

grounding including vector operations, or directed arrow representations.

While most researchers used the conventional number line labeled with positive

and negative numbers, Carraher, Schliemann, and Earnest (2006) introduced a variation

of the number line model for research with 8- to 10-year-old children. This number line model emphasizes the relationship of positive and negative numbers in regard to a reference point N rather than zero. The value of the operation is expressed in terms such as N + 1, N +2, N, N – 1, and N – 2 and so on as values are added and subtracted from N.

This approach stresses relationship of number rather than placement or notation of number.

Recent research from a psychological perspective has added to the knowledge base of number line representation of positive and negative numbers and zero. Ganor-

Stern & Tzelgov (2008), Tzelgov, Ganor-Stern, & Mayman-Schreiber (2009), and Shaki

& Petrusic (2005) suggest that older children and adults process negative numbers on a

number line using “components representation” of negative numbers in which the

magnitude of the number and the polarity are considered separately and then integrated

for placement on the number line. Shaki & Petrusic (2005) contend that negative

numbers on the number line are not based on numerosity and are referenced in context of

positive numbers. But Fischer & Roitmann (2005) argue that negative numbers are not

processed as opposites of positives, and participants of studies associated negative

numbers on the left and positive numbers on the right of the mental number line

(Fischer, 2003; Prather & Boroditsky (2003).

Collection or Charged Particle Model

Goldin and Shteingold (2001) claim that the collection model originated from the

argument that “negative whole numbers” can be represented cardinally by denoting the

expression – n as a set of objects “tagged” as being less than zero. In that sense, as a

result of several studies with 12- and 13-year olds, Gallardo (2002) argues that integers

(although she referred to them as whole numbers) should be modeled with discrete

objects that are opposite in nature (e.g. black and white, protons and electrons). Battista

(1983) argues that the model of integers as a collection of positive and negative charges

has a number of advantages over the number line model. First, this model allows the

child to manipulate concrete objects. Secondly, the charged particle model allows the child to model each positive or negative number in multiple ways much like the child is encouraged to do with relationships of natural numbers. Thompson (1988) argues that such modeling reinforces the concept of addition which becomes instrumental in integer subtraction. Battista (1983) and Kohn (1978) agree that the positive and negative charge model is more complete than other models because all four operations can be represented.

Battista (1983), Cathcart et al. (2004), and Van de Walle (2001) describe the operations

with charged particles as follows. For addition, positive chips can be added with positive

chips with the result of a greater amount of positive chips; negative chips can be added

with negative chips to get a greater amount of negative chips. When positive chips and

negative chips are combined, a cancelling process occurs leaving a quantity of single-

signed chips. For subtraction, the first integer (minuend) is represented with a set of chips of the given quantity and charge, then a collection of charges that represent the

second integer (subtrahend) is removed. The remaining chips with a single charge are the

difference. However, if there are not enough chips of the charge to be removed, then a

set or sets of neutral (one positive and one negative) charges must be introduced for

removal. In this charged particle model, multiplication is represented as repeated addition

using the commutative property when necessary. Division is a combination of the

multiplication and subtraction models with reference to the inverseness of the operations

of multiplication and division for sign derivation of the more complicated problems.

Williams and Linchevski (1997) found in their study of sixth-grade children conducting

an activity using a die and an abacus with yellow and blue beads (a modification of the

charged particle model) that the children developed three strategies critical to using the

collection model of integer operations – comparison (of number of beads), cancellation

(of number on die), and compensation or taking from either quantity.

A major limitation of the charged particle model is that to model subtraction and

division the introduction of additional sets of positive and negative charges or chips is

necessary. This cancelling process presents a limitation of this model because students

must understand that the sum of opposites results in zero, or that a positive charge and a

45 negative charge create a neutral charge (Kohn, 1978). A minor limitation could be the color of the chips used for this model. While Battista (1983) used a red chip to represent a negative charge and a white chip to represent a positive charge, Kohn (1978) used red chips for positives and white chips for negatives. Since a negative amount is commonly referred to as being “in the red”, children may take that connotation literally and become confused with the colored chip representation they encounter in the classroom (Gallardo,

2002).

Conversely, Janvier (1985) argued that a model which related integers to concrete items via mental images was more effective than the colored chip model. In a study of

13- to 14-year-olds and 15- to 16-year-olds, Janvier (1985) studied the effects of using the colored chip model or a mental image model of a hot air balloon whose altitude could be determined by adding or subtracting weights that represented positive and negative numbers. The results of the study indicated that the children who used the mental image model were more proficient in the arithmetic of negative numbers than those who used the colored chip model. However, Janvier (1985) questions whether subtraction of integers should be introduced with a model such as this.

In 1991 Davis, Clements, and Battista proposed that children need to develop thinking within an assimilation paradigm using mental images based on familiar objects or familiar situations. “The ‘signed numbers’ (positive, negative, and zero) arise in reference-point situations, in which the starting point is arbitrary…” (p. 24). A common example is measuring Fahrenheit and Celsius temperature in reference to zero. A familiar example in research of negative numbers is Davis’ (1967) Pebbles in a Bag

46 model in which pebbles are accounted for in reference to how many more or less than the initial amount. This approach addresses the cardinality of number in regard to a reference point much like the number line or ordinal approach presented by Carraher,

Schliemann, and Earnest (2006). However, this reference point model is limited to addition and subtraction.

Summary

Although all the number line and charged particle models of operations on integers have limitations, they are efficient methods of assisting children in making sense of the operations with negative numbers. Models of operations with negative numbers should be presented prior to the introduction of the rules of operations. While some educators believe that models should be temporary instructional tools and should be abandoned as soon as the concept is understood, others argue to keep the models readily available in case the learner needs to redevelop the concept (Janvier, 1985). Since children are traditionally not introduced to modeling of abstract mathematics concepts, such as negative numbers, until secondary school, their interpretation process can be undeveloped or eliminated (Peled et al., 1989).

History of Negative Numbers

Introduction

A study of negative numbers would be remiss without viewing their development from a historical perspective. Throughout the history of mathematics, mathematicians perceived the concept of negative numbers as a point of contention. Current students continue to struggle with some of the same conceptual understandings of negative

47 numbers. Unlike positive real numbers that represent visual or tangible objects, negative numbers represent intangible objects. Early mathematicians used negative numbers in solving equations but argued for centuries that negative numbers could not exist because they could not be represented with physical objects (Beery et al., 2004). Many students still learn computations with negative numbers much the same as the early mathematicians did by simply memorizing a set of rules for operations without understanding the concept.

Third to Eleventh Centuries

Although negative numbers may have been used before this time, Beery et al.

(2004) cite the first written evidence of the use of negative numbers by Chinese mathematicians dates back to the third century version of Nine Chapters on the

Mathematical Art by Liu Hui which included sets of problems with general rules for adding and subtracting positive and negative numbers. Mathematicians encountered negative numbers as solutions to problems rather than as quantifying entities, but did not accept these negative numbers as viable solutions or part of the set of integers (Beery et al., 2004; Katz & Michalowicz, 2005; Struik, 1987).

Since early mathematicians worked problems by manipulation of representative objects or by written words rather than by the means of mathematical equations, the idea of working with negative numbers did not cause the Chinese much difficulty as they made calculations with red counting rods to represent positive coefficients and black counting rods to represent negative coefficients (Katz & Michalowicz, 2005; Mattessich,

2000). If different colored rods were not available, then the Chinese used a symbolic

numerical notation of placing a diagonal rod across the ones digit of the number

represented (Beery et al., 2004).

Also in the third century, Greek mathematician Diophantus used negative

numbers in computations and possibly used a minus sign in calculations, but did not

accept negative numbers as solutions to equations (Beery et al., 2004; Katz &

Michalowicz, 2005). Since his work was influenced by Greek geometry that dealt

primarily with measurement, Diophantus would not accept negative solutions and

referred to them as “absurd” (Beery et al., 2004).

In the sixth and seventh centuries the people of India used mathematics primarily

for purposes of astrology and astronomy and for commercial applications with positive

and negative numbers being used in the context of assets and debts (Beery et al., 2004).

Cajori (1917) credits the Indians as the first to recognize the existence of negative

quantities. Indian astronomer and mathematician Brahmagupta (598-668) was one of the

first Indians to state rules for computations with negative numbers as prose to facilitate

memorization and recognized negative quantities in the sense of positives as possessions

and negatives as debts (Beery et al., 2004; Berlinghoff, & Gouvea, 2004; Bradley, 2006;

Katz & Michalowicz, 2005). Brahmagupta is considered to be the earliest mathematican to work with negative numbers and zero in arithmetic operations (Bradley, 2006).

In the late eighth century Arab mathematician Mohammad Musa al-Khwarizmi

(about 790 – 840) also stated rules of operations for positive and negative numbers but

did not accept negative numbers as viable solutions to equations since much of his work

was based on geometric applications for which negative length did not make sense

(Beery et al., 2004). Burton (1988) states that Al-Khwarizmi influenced Western

European mathematicians who likewise did not use negative numbers when they found solutions through discourse rather than symbolic notation (Berlinghoff & Gouvea, 2004).

Twelfth to Sixteenth Centuries

Throughout the twelfth to the sixteenth centuries, mathematicians continued

attempts to make sense of the negative numbers encountered in computations, but focus

on rules of operations with negative and positive numbers rather than conceptual

understanding prevailed (Beery et al., 2004). Indian astronomer Bhaskara II (1114-1185)

stated a more condensed verse for operations on negative and positive numbers than

Brahmagupta, and his verse appears similar to the rules stated in current algebra

textbooks (Datta & Singh, 1961). Like earlier mathematicians, Bhaskara disapproved of

negative solutions to quadratic equations, because he and his contemporaries dealt with

problems involving tangible, not intangible, objects (Beery et al., 2004; Katz &

Michalowicz, 2005; Kline, 1972).

Europeans were slow to accept negative numbers due to the nearly sacred role

that natural numbers served in ancient Greece in which a number was always something

positive; even the universe was deemed reducible to whole positive numbers or their

ratios (Mattessich, 1998). After the Renaissance, Europeans became more dependent on

quantitative numbers and needed to use numbers other than whole numbers (Kline,

1972). As major economic changes, including increased trade, took place in Europe in

the fourteenth century, an increased need for professional mathematicians who could

teach practical mathematics to merchants developed (Beery et al., 2004). With this

impetus for mathematical calculations rather than solutions written in prose, a need for

symbolic mathematics notation arose (Beery et al., 2004). Throughout the twelfth to the

sixteenth centuries, mathematicians including Fibonnacci, Chuquet, and Widman

introduced symbolic notation for equations. Twelfth century mathematician, Fibonnacci

(Leonardo of Pisa) (1170 – 1250) is credited with making advancements in algebraic

symbolism, although not until the sixteenth century did our present numerals finally

replace the Roman numerals (Beery et al., 2004). However, an isolated negative number

was not used in an equation (e.g. x + 3 = −5), for the first time until 1880 in a French algebra book written in 1484 by French physician, Nicolas Chuquet (1445 – 1488) (Beery et al., 2004). Johann Widman (1462-1498), German mathematician, published

Merchantile Arithmetic in 1489 and used the current symbols for addition (+) and subtraction (−) explaining that the plus sign denoted a surplus and the minus sign denoted a deficiency rather than addition and subtraction (Beery et al., 2004; Eves, 1964; Katz &

Michalowicz, 2005).

Another sixteenth century German mathematician, Michael Stifel (1487-1567) continued the debate on the conceptual meaning of negative numbers as he accepted negative coefficients in equations but not negative solutions. He argued that negative solutions were absurd numbers because they carried the connotation of a value less than nothing (Cajori, 1917; Eves, 1964; Katz & Michalowicz, 2005). As mathematicians continued work in algebra in the sixteenth century, they began to encounter and argue the existence of negative numbers in context of complex or imaginary numbers. Italian mathematician, Girolama Cardano (1501-1576), also referred to as Jerome Cardan,

encountered imaginary or complex numbers in his work and argued that complex

numbers were needed to obtain rational solutions to quadratic equations (Beery et al.,

2004; Eves, 1964; Burton, 1988; Katz & Michalowicz, 2005; Struik, 1987). Cardano was

the first to notice negative roots but called them “fictitious” (Burton, 1988; Kline, 1972).

Simon Stevin (1548/49 – 1620), Belgium mathematician, published l’Arithmetique in

1585 using the + and − symbolism and accepted negative numbers, although not complex

numbers, as viable solutions to equations (Beery et al., 2004). Robert Recorde (1510-

1558), English mathematician, and Francois Viete (1540-1603), French mathematician,

made further advances in symbolism for algebra. Recorde stated the rules for operations

using positive and negative numbers in Whetstone of Witte in 1557 and is credited for

introducing the symbol (=) for equal facilitating symbolic rather than verbal equations.

Viete, whose significant contribution to mathematics was to use letters to represent

parameters, also used the + and − symbolism in his work, but was still reluctant to accept

negative numbers as roots, restricting him from developing the fundamental theorem of

algebra (Beery et al., 2004; Eves, 1964; Katz & Michalowicz, 2005).

Seventeenth to Twentieth Centuries

Seventeenth through twentieth century mathematicians continued negative number use in calculations but did not agree on their acceptance ranging from total disregard to partial or reluctant acceptance to full support of their existence. The major point of contention was the conceptual meaning of negative numbers, especially in regard

to the interpretation of negative numbers as a quantity less than zero (Beery et al., 2004).

Sixteenth and seventeenth century mathematicians began to focus on the linear rather

than the quantitative context as they attempted to define negative numbers. Janvier

(1985) states that Descartes was the first to represent negative numbers on a number line

which consisted of two separate half-lines –positive and negative numbers. French

mathematician, Albert Girard, (1595-1632) also used a geometric context for negative numbers on a number line with positives to the right and the negatives to the left of zero

(Katz & Michalowicz, 2005). “The minus solution is explicated in geometry by retrograding; the minus goes backward where the plus advances”. (as cited in Katz, 1998, p. 446) In his algebra text, Universal Arithmetic: A Treatise of Arithmetical

Composition, Sir Isaac Newton (1642-1727) described positive and negative numbers as

Quantities are either Affirmative, or greater than nothing, or Negative, or less than

nothing. Thus in human Affairs, Possessions or Stock may be call’d affirmative

Goods, and Debts negative ones. And so in local Motion, Progression may be

call’d affirmative Motion, and Regression negative Motion, because the first

augments, and the other diminishes the Length of the Way made. And after the

same manner in Geometry, if a Line drawn any certain Way be reckon’d for

Affirmative, then a Line drawn the contrary Way may be taken for Negative…A

negative Quantity is denoted by the sign −; and the Sign + is prefix’d to an

affirmative one…. (as cited in Whiteside, 1967, p. 7)

Colin Maclaurin (1698–1746) of Scotland accepted negative quantities in mathematics and described them in terms of practical applications, such as

excess and deficit, money owed to and by a person, the right-hand and left-hand

directions along a horizontal line, and the elevation above and the depression

below the horizon. He allowed the subtraction of a greater quantity from a lesser

quantity of the same kind, provided it made physical sense. So although one

could subtract a greater height from a smaller one to get a negative height, one

could not subtract a greater quantity of matter from a lesser one. (as cited in Katz

& Michalowicz, 2005, p. 611)

By the eighteenth century, mathematicians had begun to use negative numbers in their work because they were practical, but they were still unsure of the meaning of negative numbers (Katz & Michalowicz, 2005). Throughout most of the eighteenth century, people justified negative numbers analogously with interpretations such as debt, but nineteenth century mathematicians desired more rigorous foundations and shifted the focus to the laws of operations with negative numbers using symbolic notation (Katz &

Michalowicz, 2005). By the end of the nineteenth century, empirical applications for negative numbers had been conceived in fields of accounting, geography, thermodynamics, and electricity (Mattessich, 1998).

Maclaurin and Leonard Euler (1707-1783) considered algebra as a generalized

arithmetic, but in his Elements of Algebra (1770) Euler argued

Arithmetic treats of numbers in particular, and is the science of numbers properly

so called; but this science extends only to certain methods of calculations, which

occur in common practice. Algebra, on the contrary, comprehends in general all

the cases that can exist in the doctrine and calculation of numbers. (as cited in

Beery et al., 2004, p. 28)

He explained the process of subtracting a negative value to that of adding a positive value

by the analogy, “to cancel a debt signifies the same as giving a gift.” (cited in S. Smith,

1996, p. 37)

George Peacock, (1791-1858) attempted to quiet the debate over negative numbers by dividing algebra into arithmetical algebra that used only non-negative numbers and symbolic algebra that used the operations of arithmetic allowing negative numbers (Beery et al., 2004; Katz & Michalowicz, 2005). Peacock did, however, work towards the acceptance of negative numbers as purely symbolic algebra (Beery et al.,

2004). In a similar perspective, Sir William Rowan Hamilton (1805 – 1865) devised an algebraic system, quaterninons, in which “terms of axioms, or rules of operation, and entirely independently of the meaning of the objects in the system.” (Beery et al., 2004,

p. 32)

However, all mathematicians did not agree on the acceptance of negative numbers. British mathematician and logician, Augustus de Morgan (1806 – 1871) adamantly argued in his 1831 teaching manual, On the Study and Difficulties of

Mathematics,

Above all, he must reject the definition still sometimes given of the quantity –a,

that it is less than nothing. It is astonishing that the human intellect should ever

have tolerated such an absurdity as the idea of a quantity less than nothing; above

all, that the notion should have outlived the belief in judicial astronomy and the

existence of witches, either of which is ten thousand times more possible. (p. 72)

De Morgan reasoned that if a person found a negative number as a solution, then the person had erred in setting up the equation (Featherstone, 2000).

German mathematician, Hankel (1839-1873) has been credited with settling the negative number debate by establishing a “principle of permanence of formal laws”

(Streefland, 1996, p. 60) that founded integers in a theoretical sense as a number system to which the mathematical laws applied. Hankel argued that by the distributive law (5-

3)(4-2) = 5(4) + 5(−2) + −3(4) + −3(−2) and since 5(4) = 20 and 5(−2) = - 10 and −3(4)

=−12, then −3(−2) = + 6 to satisfy the multiplication of 2 (2) = 4 (Streefland, 1996).

Beery et al. (2004) credit Karl Weierstrass (1815 – 1897), Richard Dedekind (1831 –

1916), and Georg Cantor (1845 – 1918) with constructing a logical foundation for the

entire real number system including negative numbers and the mathematical symbol for

absolute value making rules for working with positive and negative numbers simpler to

write and making negative numbers more acceptable in mathematics.

While the algebra books written by Hui, Chuquet, Widman, Stevin, Recorde,

Newton, Euler, de Morgan, and other early mathematicians were written for

mathematicians, algebra textbooks of the colonial period into the 1820’s were written for

the student and mostly presented rules and tables to be memorized and applied to given

problems (Donoghue, 2003). Authors of the mathematics textbooks of the early 1900s

were still struggling with the presentation of negative numbers. In 1909 in First Year

Mathematics for Secondary School, Myers (as cited in Donoghue, 2003) introduced positive and negative operations through examples of leverages with balance beams, weights and pulleys and geometry problems. In his Complete School Algebra, Hawkes

argued that “subtraction of 5 from 4 is … regarded as impossible. Arithmetically

speaking, such a subtraction cannot be performed… The fact that such a subtraction is

incomplete is indicated by writing a minus sign before the result.” (Hawkes, Luby, &

Touton, 1912, pp. 18-19) In later algebra textbooks authors, attempting to address both

cardinality and ordinality aspects, introduced negative numbers in real life contexts or as

measurements of direction and magnitude (Donoghue, 2003). Both magnitude and

direction are represented in textbooks today.

Due to discrepancies among mathematics history presentations, this brief historical review of negative numbers is not intended to be all inclusive. Many early mathematicians who made significant contributions to the evolution of negative numbers have been noted, although credit for some contributions differs among researchers. The point of argument is that for two thousand years mathematicians were said to have “used” negative numbers in problem solving but argued their existence. Most of the authors of the histories reviewed stated that early mathematicians used the rules for operations on negative numbers in their work but neglected to explain (or conjecture) when, where, why, or by whom these rules were initially established.

Summary

This history emphasizes the difficulties of working with and understanding the

concept of negative numbers and presents a valuable perspective on the teaching and

learning of mathematics involving negative numbers. This study will be conducted in a

format loosely analogical to the history of negative numbers. Granted, early

mathematicians had mathematical background unavailable to the children, but the

57 children, like mathematicians, will attempt to make sense of negative numbers that cause a conflict or perturbation with their previous knowledge of number and to conceptualize the relationships among positive and negative numbers and zero. The children will be encouraged to create notations for negative numbers much like the early mathematicians had to create notations for negative numbers when problem solving evolved from written discourse to algebraic notation.

Chapter Two Summary

Piaget, Dewey, and Bruner were proponents of the active learning principle in the constructivist approach to learning and teaching. Children naturally learn in an environment in which they constantly make sense of their actions and reactions with the outside world; they construct knowledge when they are active rather than passive participants in the learning process. Children begin formal schooling with a base of informal knowledge which may have been gained through everyday activities such as play. Appropriately created games may promote learning in the mathematics classroom.

Mathematics games can be motivational and educational. However, unless mathematics games are well-developed and supported by the teacher, they may be no more than motivational worksheets.

As children encounter new mathematical concepts, they must adapt this knowledge to their pre-existing base of knowledge. Researchers report that children have an intuition about negative numbers and are willing to invent their own representations for these numbers. Young children have exhibited interest in the concept of negative numbers and may be able to make sense of negative numbers at a younger age than

58 currently recommended. The following study may support a change in perspective of early childhood mathematics education.

CHAPTER III

METHODOLOGY

Introduction

The methodological paradigm for this study is qualitative, and the research will be conducted within a phenomenological approach. This chapter includes descriptions of qualitative research, the research design including descriptions of participants, data collection procedures, analysis of data, and limitations of this study.

Qualitative Research

Until the early 1930’s research in education in the United States was

predominately quantitative, except for anthropological studies using research methods of

the qualitative nature to study education. However, following the Depression, research

became more focused on affective issues of the nature and extent of the problems,

including gender relations, associated with post-Depression situations (Bogdan & Biklen,

1998). In the 1960’s as national attention focused on problems in the education system, researchers began to utilize qualitative research methodology to examine cognitive rather than behavioral aspects of the students and teachers (Bogdan & Biklen, 1998). Lichtman

(2009) states that during this period interest in qualitative research increased as the educational research field opened to women and “people of color” (p. xv), and in the

1990’s “scholars, publishers, journals, and government agencies” (p. xv) began to acknowledge qualitative research as acceptable methodology. Since then qualitative

research has been taught at universities worldwide and accepted by professional associations and government agencies (Lichtman, 2009).

Lichtman (2009) and Bogdan & Biklen (1998) describe qualitative research as a

process of investigating the human experience and then providing in-depth descriptions

that inform the knowledge base of the studied topic from alternative perspectives and

previously marginalized voices. Qualitative research promotes inductive thinking from

which conclusions or grounded theory may emerge (Bogdan & Biklen, 1998; Lichtman,

2009). Qualitative research is interpretive analysis with the researcher functioning as the

instrument, collecting and interpreting the data (Jones, 2007). In the analysis of data for a

qualitative study, “Interpretation always involves making subjective, inferential

judgements regarding the meanings that participants ascribe to particular expressions,

actions, or objects.” (Hatch, 1995) Qualitative research takes place in a setting that is

natural for the experience, provides analysis of descriptive data rather than quantification

of data, focuses on process rather than product, provides inductive analysis of data

searching for emergent theory, and focuses on meaning from the perspective of the

participant (Bogdan & Biklen, 1998). The more natural setting allows the researcher to

access the processes that reflect how the child understands and constructs meaning

(Hatch, 1995; Lichtman, 2009).

Qualitative research can be conducted from the phenomenological, ethnographic,

case study, grounded theory, feminist approach, narrative analysis, biographical or

autobiographical, hermeneutics, or a combination of these approaches (Bloomberg &

Volpe, 2008; Lichtman, 2009). While the phenomenological approach focuses on the

lived experience of the participant, the ethnographic approach emphasizes human

interaction within a particular culture. A case study focuses on a particular person,

happening, or setting. Grounded theory generates theory from data or modifies or

extends existing theory, and the feminist approach to research gives voice to

marginalized populations. Narrative analysis and biographical or autobiographical

research present the story of the phenomenon in the voice of the participant, and hermeneutics presents an interpretation of text (Bloomberg & Volpe, 2008; Lichtman,

2009).

For this study, the research will be conducted with a phenomenological approach utilizing interpretive methods to examine the experiences of the children being studied

(Jones, 2007; Van Manem, 1990). Phenomenologists focus on the subjective aspects of the behaviors of the participants (Bogdan & Biklen, 1998). Lichtman (2009) argues that phenomenology is both a philosophy and an approach. As an approach to research, phenomenology utilizes processes of reduction and bracketing; reduction facilitates the probe for the essence, or deep understanding, of a phenomenon, and bracketing occurs when the researcher sets his or her own beliefs aside in order to make objective analysis of data (Bloomberg & Volpe, 2008). In this study, the researcher will interview and observe the children as they encounter situations evoking negative number concepts. The data collected will provide the researcher with evidence of levels of understanding of negative number concepts; collectively the data will reflect the essence of the phenomenon of negative number concepts. Bracketing will occur as the researcher

62 disregards experiences with adult learners and reflects and analyzes the data from young children.

Since the purpose of this research is to collect data on how young children make sense of the relationships among positive and negative numbers and zero rather than to test a particular treatment on operational performance, the qualitative research methodology is appropriate. The ontological perspective of the study will be within the constructivist paradigm for which constructed realities exist (Lichtman, 2009).

Constructivists strive to understand how the world operates by studying the world through the perspective of those participating in it (Hatch, 1995). Constructivist researchers observe and interview participants to “capture how participants make sense of their own constructed realities.” (Jones, 2007, p. 123) Often, this data is related in rich narratives in the voices of the participants as the voices of the individuals involved give

“depth and breadth to understandings of experience” (Hatch & Barclay, 2006). For this study, data will be collected by the researcher through observation, interviews, and personal documents in the form of scrap paper notes and responses to interview questions in an informal setting during which the young children will be participating in game-like activities. While engaged in the activities, the children will encounter the concept of negative numbers in relation to positive numbers and zero. Each child will be accountable for the numerosity of his collection and encouraged to invent a meaningful representation for negative numbers.

Research Design

Participants

The participants for the study will be a convenient sample of children aged four

through eight years from rural and urban Midwest school districts. Four to six children of

each age will be studied, observed and interviewed. The children in the study will

include representation by gender. The children will have little or no formal instruction of

negative numbers; however, they may have encountered negative numbers such as in

scoring of games or in temperature readings or other life experiences.

Data Collection Procedures

Data for this study will be collected through game-like activities with the

children. The activities will address both cardinality and ordinality of number through

collection-type activities and modified number line activities. Activities will be

presented first as ordinality and then cardinality in accordance with Piaget’s stages of

learning number. During the activities the children will encounter the task of making

sense of enumerations of deficient quantities with respect to zero within a gain or loss

context and within a directional number context. The activities of ordinality will direct

the children to move playing pieces left or right in respect to a starting point (“home”)

and to move vertically (to a treehouse or cave) with respect to a reference level (“the

stone”). These activities are similar to Davis’ (1967) and Davidson’s (1987) “Mailman”

games for which children represented positive and negative quantities as forward and

backward movement along a modified (street-like) number line. Davidson argued that these movements represented iterated actions in opposite directions generated by positive

and negative numbers. The activities of cardinality will direct the children to gain or lose

pennies and to catch or release manipulatives. These activities are similar to the “Hippo”

game for which children put food in or took food out of a bowl which supports

Davidson’s (1987) argument that non-positive actions on concrete objects exist in the

form of decrementing or dividing a collection and null actions, such as adding zero

objects or restoring the set of objects.

For the first activity, the children will be given verbal instructions to move a token horizontally (left or right) on a “street” of houses distinguished by color. (see

Appendix A) Each child will set up the houses to the right and/or left of the designated

“home” house. The home space will serve as the referent or zero. The children will be asked to analyze the aspects of distance and direction in regard to the referent.

The second task will be structured like the first task except that the movement will be vertical rather than horizontal. The playing board will be designed with a predetermined placement of spaces on which to move. The children will start on the

“stone”, referent point, and be verbally instructed to move down a series of steps towards the cave or up towards the treehouse. (see Appendix B) The children will be asked to analyze the aspects of distance and direction in regards to the origin.

During the first collection task the children will catch and release bears, frogs, or pigs (counting manipulatives) according to verbal instructions. The verbal instructions will include commands such as “catch 2 bears” and “release 1 bear”. (see Appendix C)

The directions will be set in a predetermined pattern so that all the children will

65 experience the same sequence of actions. The children will encounter situations of controversy when they do not have enough bears in their collection to release.

The second collection task will involve earning and paying pennies. The directions for this task will be given verbally in a predetermined sequence. (see

Appendix D) Once again the children will encounter being deficit of pennies with which to pay. The children will be asked to account for the pennies that they have earned and that they have in their possession at a particular time.

The children will be encouraged to interpret each situation, discuss, represent, model, and/or make symbolic notation. Paper, pencils, markers, and crayons will be accessible to each child during the play of the game. For all activities, the instructions for movement or collection will be limited to the quantities of one, two, three, four, or five units.

Each observation and interview session will last for approximately sixty minutes.

All sessions will be video recorded for later viewing. All video recordings will be destroyed upon completion of this research. The names of the children will not be presented in the study. Each child will be a presented only as initials with a subscript of age in years (e.g. PM5).

Analysis of Data

Data will be analyzed from the notes and partial transcriptions of the videos, researcher’s journal and field notes, and personal documents of children’s work. All data may be coded by setting or context, perspectives of the children, process, activity, event, strategy, social structure, and method (Bogdan & Biklen, 1998) and other emergent

codes. Codes will be analyzed and grouped into categories from which key concepts will

be identified (Lichtman, 2009). Analysis will also be presented as anecdotal records

from the field notes and rich narrative in the children’s voices.

Pilot Study

A pilot study was conducted with two children, a five-year-old girl and an eight- year-old boy. Both children participated in all four tasks. As a result of the pilot study, phrasing of the questions was modified from “you go to a house (or step on the ladder) that is two houses away from your house” to “you move two houses (or step on the ladder) from your house” to avoid the misinterpretation of skipping two units rather than moving to the second unit. The five-year-old consistently made this misinterpretation with her movements along the modified number line (street or ladder).

In the pilot study, the children were encouraged to create markings by each house to distinguish one house from another. The researcher intentionally did not ask the children to number the houses in an attempt to allow the children to invent their own notations for the negative numbers. Both children made random markings not associated with number ordering. For this study, the children will be encouraged to number the houses in reference from the “home” house or “stone” by the ladder.

Both children said that there was no difference between going two houses to the left or going two houses to the right. In the study, questions that encourage reference to direction as well as magnitude of number will be included. For each activity, tasks to address combinations of positive numbers only, cancellation to establish an initial

reference location or zero, combinations that result in negative numbers, and then

combinations of positive and negative numbers and zero will be presented.

Issues of Trustworthiness

In qualitative research potential biases, issues of validity and reliability, which

may be present in studies are defined as credibility, transferability, dependability, and

confirmability (Bloomberg & Volpe, 2008; Lichtman, 2009). Credibility involves the

extent to which the findings actually describe the reality of the experience from the

perspective of the person or group of persons being studied and can be ensured by the use

of triangulation, member-check, and peer evaluations (Bogden & Biklen, 1998;

Bloomberg & Volpe, 2008). Transferability refers to the ability to transfer findings

across different contexts (Lichtman, 2009) or the fit “between research context and other contexts judged by the reader” (Bloomberg & Volpe, 2008, p. 78). Bogdan and Biklen

(1998) describe dependability as the fit between the data and the actual occurrences in the setting in the study. Dependability can be controlled by the use of “thick description” in the voice of the participant. Confirmability evaluates whether the findings of the research are resultant of researcher bias and subjectivity (Bloomberg & Volpe, 2008). Since young children are the participants in this study, member checking is not an appropriate method of evaluation; therefore, issues of trustworthiness will be controlled by triangulation of data collection – observation, interviews, and review of children’s work and peer evaluation by another mathematics education professor (Bloomberg & Volpe,

2008; Bogdan & Biklen, 1998).

Current Study

Following collection of the data, the videos, children’s work, and field notes were reviewed by the researcher. Videos were transcribed; detailed notes were compiled from the voluminous transcripts. Consistent behaviors of the children were identified and noted. The notes and transcriptions were then coded by prevalent mathematical concepts recognized, developmental differences exhibited, and interactive behaviors of each child for each individual age group. The codes used for this study were: cardinal numbers, ordinal numbers, subitization, rote memorization, zero as referent, zero as quantity, cancellation/additive inverse, positive number relationships, positive and negative number relationships, negative number relationships, deficit in collections, magnitude on the number line, direction on the number line, representation - horizontal number line, representation – vertical number line, representation – collection (pigs, frogs, bears), and representation – collection (pennies). The theme of narrative or story-telling emerged.

Following the coding of data, data from each age group were analyzed and synthesized.

Then the data from the six age groups were synthesized and analyzed. Categories of relationships among positive and negative numbers and zero, zero as a quantifiable set, zero as a referent, contextual situations as interactive stories, representation of intangible numbers, and rote learning were established.

Peer evaluation was completed by a mathematics professor, a preschool teacher and an intervention specialist for children aged six through nine. The mathematics professor provided a algebra-based mathematics perspective, and the two early childhood educators provided perspectives of young children’s learning behaviors. All three of the

69 evaluators reviewed the detailed notes, transcripts, copies of the children’s written work, the presentation of the data in Chapter Four and analysis of data in Chapter Five. The early childhood educators requested permission to view portions of the videos to clarify data involving mathematical concepts, in particular cardinality, ordinality, subitization, and additive inverse, with which they were not currently familiar. The mathematics professor questioned whether children so young could actually conceptualize abstract algebraic concepts. The researcher argued that if the purpose of the study was to determine whether and how young children conceptualize an abstract algebraic concept like negative numbers, then reference to such abstract concepts should be made with appropriate mathematics terminology.

The early childhood teachers, both of whom teach children including those who are cognitively and behaviorally atypical, expressed skepticism of the abilities of the youngest children. They requested to view some of the videos. After reviewing portions of the videos, they acknowledged the possibilities of the insights and intuitions of the young children. Following further discussion on the abilities of the children and review of copies of the original work of the children – the modified number lines and the accounting records, all three of the evaluators agreed with the researcher’s analysis.

Limitations

Conducting a qualitative study with young children presents a number of issues in regard to reliability of data collected during interviews and observations. Although

Hatch (1990) studied preschool and kindergarten children, he implied that the issues encountered could be transferable to studies with slightly older children. He identified

four problems that could affect the quality of the interviews with young children - the

adult-child problem, the right-answer problem, the pre-operational thought problem, and the self-as-social-object problem (Hatch, 1990). The adult-child problem addresses the fact that the child knows the expectations of interaction with adults and of the state of the relationships between adult and child. The child may not understand his or her role in the interview and simply strive to obey and respect the adult/researcher. The right-answer problem exists when the child thinks that his or her responsibility in the interview is to give the researcher the expected, assumedly correct, answer inhibiting reflection and presentation of the child’s perspective. This problem may be more evident in slightly older children who have already internalized the right-answer emphasis in teaching

(Hatch, 1990). In response to this problem, researchers have the arduous charge to respond with nondescript expressions and non-judgmental reactions to the input of the child. Hatch (1990) argues that the pre-operational thought problem suggests that young children are in Piaget’s pre-operational stage of cognitive development in which they exhibit characteristics of egocentrism, complexive thinking, and centering. Interviews with young children in this stage of development may yield irrelevant data as children focus on a single aspect of self, situation, or object. The last issue is the self-as-social- object problem which addresses the inability of young children to think of themselves as separate entities and reflect and analyze their own behavior or thoughts.

The first two problems pose potential limitations to this study as the researcher will not be a familiar adult to the children. The second two problems may limit the

71 children’s ability to reflect and respond especially if they have had little experience in reflecting and responding to open-ended questions.

Another limitation of this study is previous informal introduction to negative numbers. Most children within this range of age are at least aware of the notation of temperatures below zero. Some of the older children, in particular, may have experienced playing games for which a tally score is recorded with positive and negative numbers.

Summary

Researchers using qualitative research methods rely on their participants, in this study young children, rather than numerical statistics from a separate instrument to provide the data for analysis. In qualitative research, the researcher is the instrument; the researcher processes and interprets the data. Findings are presented as thick descriptions or rich narratives that reflect the voice of the participant. Reliability and validity of qualitative research studies are measured by means of assessing the reasonableness of the interpretation of the researcher rather than reducing to numerical coefficients.

These characteristics of qualitative research will be utilized in this study of young children as they attempt to make sense of negative number concepts. The children will be observed and interviewed as they participate in game-like activities encouraging them to construct relationships among positive and negative numbers and zero. Interpretive analysis of the data using qualitative methods will determine the findings that follow.

CHAPTER IV

RESULTS

Introduction

This study was conducted in a format loosely analogous to the historical evolution of negative numbers. The lengthy review of the literature on the history of negative numbers set the tone for this study. Rather than introducing the children to traditional representation of negative number and then asking them to add and subtract with positive and negative numbers using a novel or traditional operational model, the children were presented with contextual activities from which the concept of negative numbers could emerge. And, like early mathematicians, the children who demonstrated some conceptual understanding of negative numbers were challenged to make written or iconic representations of the negative numbers that they encountered.

This chapter presents a description of the data collected from the interviews conducted with twenty - four children between the ages of four and eight. In the presentation of the results of this study, each child is identified by initials and a subscript of his or her age. The sample consisted of the following children:

Four-year-olds: JB4, PS4, CM4, and BS4 Five-year-olds: IS5, HB5, RC5, SS5, and ES5 Six-year-olds: LS6, KM6, HC6, FC6, and CM6 Seven-year-olds: AS7, ES7, LM7, SH7, and MS7 Eight-year-olds: KS8, NB8, CH8, AH8, and KG8

Some of the data collected is most effectively presented as dialog between the child and the researcher. Each child was interviewed individually; all interviews were

video-recorded. The data is presented in relation to the following two research questions

posed by the researcher:

1. Do young children make sense of the relationships among positive and negative

numbers and zero encountered in contextual activities? If so, how?

2. What representations, expressed verbally and/or expressed as pictures, tally

marks, numerals, etc., do young children use to designate numbers in recording

results of task-related activities?

Research data contained in this chapter includes two sources: interviews with the children and written work of the children during the interviews. Data addressing the first research question was collected primarily through one-to-one interviews with each child using interactive game-like activities. Data addressing the second research question was collected from written work done by the children during the interviews as well as explanations given throughout the interviews.

The researcher conducted an interview with each child in his or her own home with the exception of a seven-year-old boy whose interview was conducted in the office of the researcher at the request of the child’s mother. The researcher provided all the materials used in the interactive interviews. Manipulatives included plastic bears, rubber pigs, and plastic frogs. Each child was asked to select bears, pigs, or frogs to use in the activities. In one activity, they used pennies. The number line activities were conducted with modified vertical and horizontal number lines. The vertical number line was a board game on which each child was directed by the researcher to move the shoes (game token) on the ladder toward a treehouse or toward a cave. The horizontal number line consisted

74 of cut-out houses and a poster board street/road. The researcher also provided a paper mat in the shape of a piggy bank for one of the collection activities. Pencils, markers and a notebook with which each child could record any representations or calculations were also provided.

The children used the above materials for six interview activities to demonstrate understanding of cardinal numbers and ordinal numbers, to demonstrate conceptualization of the relationships among positive and contextually negative numbers and zero on modified horizontal and vertical number lines, and to demonstrate conceptualization of the relationships among positive and negative numbers and zero in collection-type activities with the aforementioned manipulatives. As the children engaged in the activities, they were encouraged to make representations of numbers for the modified number line tasks and for the contextual situation task of collections of manipulatives and/or pennies.

Interview Activities

The first activity addressed the children’s understanding of cardinal numbers and established a baseline of each child’s number sense. This baseline guided the researcher in setting up the tasks with a manageable number of manipulatives or movements for the child. For the cardinal number activity the children were asked to count out a set of ten manipulatives - bears, pigs, or frogs. Because of the diverse age span of the children, the number of manipulatives was limited to no more than ten. Then the researcher presented each child with varied sets of manipulatives and asked the child to enumerate the set.

The child was permitted to determine the number of manipulatives in the set by any

method (e.g. subitization, counting each manipulative in the set, or guessing). The child was then encouraged to explain his or her enumeration.

The second activity addressed the children’s understanding of ordinal numbers.

The purpose of this activity was to assess the child’s knowledge of the series of ordinal number names and the association between cardinality and ordinality of number. The number extent for the modified number line tasks were tempered by the results of this activity. Another goal of this activity was to facilitate the construction of a working number line on which the children could model addition and subtraction of positive and negative numbers and zero.

For this activity the child was asked to arrange the set of chosen five or six manipulatives in a row. When the arrangement was completed, the researcher asked the child to point to the first, second, fourth, fifth, and third manipulative. The order of selection was intentionally not sequential; the third manipulative in order was deliberately skipped to determine whether the child was making a counting-ordinal association or rotely pointing to each successive manipulative. The children were encouraged to explain why they pointed to a particular manipulative.

For the first number line activity the children worked with a horizontal modified number line. Each child set up a street or road containing his or her “home” house as the referent house and three to five other houses situated to the right or left of the referent house. The children were permitted to place the additional houses along one side of the street at their discretion. The only stipulation was to have at least one house on either side of the “home”. When the street was set, the researcher instructed the children to

move the shoes (game token) from house to house in reference to positive or negative

(opposite) numbers and zero. The tasks explored concepts that included addition and

subtraction of positive and negative numbers and zero, cancelling or additive inverses,

and counting on a continuous number line that crossed the referent point or zero. The

children in the study were asked to determine the number of movements from one

location to another on both the vertical and horizontal number lines. Magnitude and

direction were also addressed through questions posed to the children (e.g. If your friend asked you to come to his house that was two houses from yours, would you know which direction to go to get to his house? Does it make a difference?). Children who demonstrated competency in writing numerals and a general understanding of the questions of movement were encouraged to assign numbers to the houses in reference to their “home” house.

The second number line activity was similar to the first number line activity but was based on a vertical modified number line. The number line was presented as a board game on which the children moved their shoes (game token) from the stone, or referent point, upward toward the treehouse or downward toward the cave as directed by the researcher. As in the previous activity, concepts of addition and subtraction, additive inverses, referent point, magnitude, and direction were addressed in the interviews.

Again, the children were encouraged to create a number line to which the researcher could reference further questioning.

The first collection activity dealt with the concept of cardinality of set and deficit situations. Using his or her preferred manipulative, each child was asked to play the role

of a zookeeper for bears, pig farmer, or frog collector and gain or lose quantities of the

chosen manipulatives as directed by the researcher. The children were presented with

situations of composition and decomposition of set, cancellation or additive inverses, and

contextual situations of deficit.

The second collection activity was similar to the first collection activity except all

of the children used pennies as manipulatives. Once again, the children received pennies

to add to the set in the paper piggy bank or were directed to spend or lose pennies which

were removed from the piggy bank. The children were presented with situations of

composition and decomposition of set, cancellation or additive inverses, and contextual

situations of deficit. The children were encouraged to keep a record of the amount of

money, or absence of money, in the piggy bank.

The preceding activities addressed the first research question of conceptualization

of relationships among positive and negative numbers and zero. The number line

activities were designed to explore how children organize numbers in a linear context and

then to observe how the children develop relationships among the numbers. The

collection activities were designed to explore how children account for objects as they

compose and decompose sets and encounter deficit situations. The second research

question concerning representation was addressed by tasks in which the children were

encouraged to number the houses and steps or spaces and to record the quantities of set.

Children’s explanations and behaviors also contributed to the data collected.

Data Addressing Conceptualization of Relationships of Number

The preceding activities address whether and to what extent a child conceptualizes the cardinality and ordinality of set, number relationships with zero, number relationships with positive and negative numbers, and number relationships with a referent. The children in this study demonstrated multiple levels of understanding of these concepts. Some of the concepts were universally understood, while other concepts were understood in varying levels of complexity or not understood at all.

Cardinality of Set

While cardinality of set is a complex conceptual principle of mathematics, young children may simply apply the rule of last-word response to answer the question of how many (Fuson, Pergament, Lyons, & Hall, 1985). In an attempt to determine whether the children had developed a sense of cardinality of set or rotely counted objects in a set, the researcher presented each child with a set of manipulatives and asked the child to determine how many manipulatives were in the set. All of the children appeared to subitize sets with a small amount of manipulatives. Dehaene (1997) argues that subitizing occurs when “all objects in the visual field are processed simultaneously” (p.

169). Conversely, Gelman and Gallistel (1978) argue that subitizing is actually counting unconsciously. An observer may experience difficulty distinquishing between subitizing and quick mental counting.

All of the children provided evidence of subitizing sets of one, two, or three manipulatives. When asked to explain how they knew the number of manipulatives in the set so quickly, several of the children explained their number sense in terms of fast

counting or visualization of sets and subsets. PS4 explained that he knew how many were

in the set “because I counted two” and that he could “count without talking”. ES7

explained that she just “thought two” and then named sets of three and four as “two plus

one” and “two plus two” respectively. CH8 explained his sense of cardinality of set as “I

just look and say”. SH7 said that he knew there were four manipulatives in the set because he saw a subset of two and another subset of two, and that he knew that two plus two is four. He also explained the set of three as a subset of two and one more. NB8

determined the number of manipulatives in sets of four, six, and eight using a strategy

similar to that of SH7; she visualized groups of two and four and combined those subsets

for the totals of six and eight. AH8 explained how she determined that there were five

pigs in the set, “I counted without counting” and “I just count in my mind”; her

explanations do not define whether she subitized or simply made a quick mental count.

Some of the younger or less outspoken children were unable to explain their

strategies for determining the cardinality of the set. Since many children this age are not

often asked to explain their thinking or verify their answers, some of the children in this

study merely shrugged their shoulders, said that they didn’t know, stated that they knew

because they had been in school for years or learned the concept before, or rationalized

that they knew the answer simply because they were “smart”.

All of the children in the study determined the cardinality of a given set by

subitizing and/or mental counting. Distinction between subitizing and mental counting is

difficult to discern from the explanations provided by the children. For this study, the

researcher interpreted a rapid response of quantity without reference to counting as

subitization.

Ordinality of Set

Unlike cardinality of set that requires the child to determine a quantity of the set

and then associate a numeral with that cardinality, ordinality of set requires the child to

understand seriation of equal units as well as cardinality of set. Each child was asked to arrange five or six manipulatives in a row. Seventeen children lined the manipulatives from left to right, five lined them from right to left, and two children lined them up vertically, one top to bottom and the other from bottom to top. Data on the preference of orientation was not gathered for this study. The researcher observed that the children used this same orientation when they identified the ordinal placement of each manipulative. The researcher then asked each child to point to the first, second, fourth, fifth, and third manipulative in the row deliberately in that order. By skipping the third manipulative the researcher observed whether the children determined order through a connection with counting or cardinality or assigning the ordinal number name in a rote

pattern. All of the children aged four, five, and six pointed to the first and second

manipulative successfully. When they were next asked to point to the fourth

manipulative in the row, they pointed to the third manipulative which was adjacent to the

second manipulative that they had previously identified. They demonstrated knowledge

of the rote sequence of ordinal names and attempted to assign the next ordinal name to

the next manipulative in the row. Regardless of the position of each manipulative within

the row, the four-, five-and six-year-olds pointed to the third for the fourth manipulative,

the fourth for the fifth manipulative, and the fifth for the third manipulative. PS4 argued that his identification of order was correct “because I said it”, implying that he had rotely learned the series of ordinal number words without conceptual understanding of position in a set. Once again, very few of the younger children were able to explain their ordinal number and position associations.

The seven- and eight-year-olds, however, demonstrated some association between cardinal and ordinal numbers. LM7 verified the fifth position by one-to-one counting of the five manipulatives. SH7 determined four manipulatives as “two plus two is four” and explained that the fourth manipulative was actually the fourth “because if this is the second (pointing to the second) I can skip and go to the fourth”. When AS7 was asked whether identifying the first and second house had anything to do with counting, she confidently replied, “No”. NB8 confidently pointed and counted, “one, two, three, four”, when asked how she identified the fourth bear. Another eight-year-old, CH8, offered a different insight to the question. (Note: In all of the dialogs that follow, the researcher is represented as R.)

R: Will you point to the first [of six] bear? (CH8 points to the bear on the right.)

How do you know that’s the first?

CH8: Well, there are two, but I just say that I want this one [on right] to be first.

They can be one [points to the bear on the far left] or one [points to the bear on

the far right].

R: Will you point to the second bear? (CH8 points to the second bear from the

right. Will you point to the fourth bear? (CH8 starts to point to the third bear,

smiles, and then points to the fourth from the right.) How do you know that’s the

fourth?

CH8: Two, four. (CH points to the group of first and second and then to the group

of third and fourth.)

AH8 also presented an association of order with counting as she explained that to

determine the second pig she had to “count in my mind”. She continued to identify the

fourth pig, “because if we have two, you just need to skip one and go to the next one” and

the fifth pig, “because if you just told me the fourth one, you just go one more.” AH8

used the combination of counting on and skip counting strategies to justify her

determination of ordinal number.

The younger children in the study, aged four, five and six demonstrated uncertain

understanding of ordinal numbers as they correctly identified the first and second

manipulative in the row but incorrectly identified the third, fourth, and fifth

manipulatives. They made no connection between the position of the manipulative and the ordinal designation beyond the first and second positions. Most of the seven- and eight-year-old children justified their assignment of the ordinal number to each position in the row of manipulatives using counting principles. Some of these older children faltered at identifying the third manipulative as the fourth, but corrected their ordinal association after studying the row of manipulatives.

Zero as a Referent

The conceptualization of zero was presented within the activities in two contexts, absence of movement on the modified number lines and absence of quantity in the

collection activities. The number line context of zero was not recognized by all the

children, but the children from all of the age groups associated the number zero with a

context of absence of quantity and expressed zero as “none” or “nothing” or simply made

a circle with their thumbs and index fingers.

Fourteen of the twenty-four children associated zero with the “home” house on

the horizontal number line or the stone on the vertical number line, but most of their

explanations dealt with absence of movement rather than zero as a referent for the other

positions on the modified number lines. LM7 assigned zero to the “home” “because that’s the first number”. When asked what number he would put on his house, CH8 wrote

“0” and hinted of a referent point as he explained, “because that you automatically lived

there and there’s one, two, three, four” [points to the other four houses and numbers to

the right]. AH8 also implied that zero was a referent when she numbered the ladder for the treehouse task. She counted the steps from the stone to the cave and then numbered from the bottom, “7, 6, 5, 4, 3, 2, 1”. Then she numbered the stone “0” and explained,

“because that’s where you need to go”. She continued to number the steps from the stone up to the treehouse, “1, 2, 3, 4, 5, 6, 7, 8”.

Even five-year-old IS5 knew that if he was on the stone then he was “none” or

“zero” steps away from the stone, but he was not able to apply that concept to other spaces or steps on the vertical modified number line. KM6 also expressed that zero

should be assigned to the stone “because I did not move any” but continued to number

the spaces to five in a single (upward) direction and then downward from zero beginning with six and numbering to twelve (see Table 1). Another six-year-old, LS6, had difficulty

discerning whether to count either of the end houses and/or the houses between. She

argued that there were zero houses to go from one house to the adjacent house on her

number line because there were no houses between.

The majority of the children associated the stone or “home” with zero, but they were inconsistent with using the zero as a referent particularly in regards to the houses to

the left or steps/spaces below. The children in this study demonstrated little knowledge

of number line organization by constructing either disjoint number lines or unidirectional

number lines.

Zero as a Quantity

When encountered with a deficit situation in the collection activities, the children

responded in three ways: recognize the deficit amount but retain the previous count,

recognize the deficit amount but respond with “zero”, or recognize the deficit and attempt

to represent that deficit amount. Children in all age groups realized that they needed

more manipulatives and determined how many they needed to complete a set since they

did not have enough. Lacking symbolic representation for the deficit, some children

stated that they could not give away the requested amount and retained the previous

amount in their record. Other children knew they did not have enough manipulatives to

meet the demand, knew how many manipulatives they were deficit, but recorded zero as

the number in the collection. HC7 demonstrated her understanding of deficit referring to

zero in the following exchange. Prior to this dialog, HC7 had seven pigs and had

previously explained that the number for “none” is zero.

R: If I bought seven pigs, how many would you have?

HC7: None.

R: If I bought eight pigs, how many would you have?

HC7: None.

R: If I bought nine pigs, how many would you have?

HC7: None.

R: So no matter how many pigs I bought, you would always have none?

HC7 shakes her head in agreement.

HC7’s understanding of deficit was always represented as zero as the least quantity

possible.

Only two children recognized the deficit and made a record of that amount in

words or symbols. SH7 represented the deficit as “n 2” accompanied by two invented

iconic representations of frowning-faced pennies (see Appendix H). When NB8

determined that she was in a deficit situation, she recorded zero and also wrote “owe 2

pigs” beside the zero. She expressed the deficit in words instead of invented symbolic or

iconic representation (see Appendix I).

While the children failed to recognize zero as a referent on the modified number lines, they readily identified zero as a number of quantity when involved in collection activities. When encountered with a deficit situational context, most of the children determined zero to be the least quantity that they could represent. Their knowledge of number prohibited them from recognizing or representing a number of lesser value than zero. Only two of the older children proceeded to represent the deficit with words or

invented symbols.

Relationships Among Positive and Negative Numbers and Zero

Children in this study demonstrated conceptualization of relationships of numbers

through activities involving only positive numbers, positive and negative numbers, and

only negative numbers. They also explored more advanced mathematical concepts of

additive inverse and addition and subtraction of negative numbers within the context of

age- and ability-appropriate activities.

Relationships Among Positive Numbers

All of the children were able to add and subtract positive numbers especially in

the collection activities. When they received or relinquished pigs, frogs, bears, or

pennies, the children accurately determined the cardinality of the set by counting all,

counting on, skip counting (especially by twos) or subitizing. Five-year-old IS5 verified

his count of four, “Um, because like this, um (removes one frog from set). If this one was

gone, then there would only be three. But there’s one (puts frog back in set) one, two,

three, four frogs.” In another activity, he relied on skip counting by two to determine that

he had four frogs. KM6 knew that she had five pigs because “that’s four because there’s two down there and two down there. Like two plus two is four and I don’t have to count

them. But if I add one more, there would be five.” NB8 explained her determination of

four moves in one scenario, “If I’m here (home) and only had to move three to move here

(Three houses left of “home” on the house number line, referred to as L3), it’s only one

more. So it equals just three plus one.” In another scenario, she subitized two houses and

then skip-counted “two plus two” to account for four moves.

CH8 applied principles of addition of positive numbers to determine the number of moves on one side of his ladder number line. Even though his number line went from

top to bottom without designation of the stone as zero, he explained that to go from step

four to retrieve his bear on step nine, he would have to move nine [error noted] steps

“because five plus four equals nine”. He also explained, without error, that to move from step two to step ten he would have to move eight steps because “two plus eight equal ten”.

The children had no difficulty determining relationships among only positive numbers especially in the collection activities. They added and subtracted using previous knowledge of number relationships or using counting strategies that included one-to-one

counting, skip counting – especially by twos, and counting on. The children did not

present the same relationships on the number lines. They relied more on counting to

ascertain the number of moves or combinations of moves.

Relationships Among Positive and Negative Numbers

In the study the researcher presented the children with number line scenarios

regarding relationships between positive and negative number (e.g. having to move back

across the zero to secure a forgotten pig, bear, or frog, requiring that they determine

movement from one side of the referent to the other). The major fault with this activity

was that the children did not recognize the numbers to the left of zero or below zero as

negative numbers and did not consider relationships as such. In essence, some of the

situations encountered in this activity explored the algebraic concept of subtracting a

negative number. Asking a child to determine the distance or number of moves from one

side of the referent to the other (e.g. move from the third house on the right of zero to the

second house on the left of zero) is akin to asking a child to subtract negative two from

positive three. The children found this task challenging in regard to determining whether

to include the referent, “home” or stone, in the count or to exclude the referent because it

did not belong to either set (representatively positive or negative) above/below or

right/left of the referent.

When presented with the scenario of moving from two steps above the stone to

the forgotten toy on five steps below the stone, ES7 and IS5 relied on counting each move

including the stone, or zero, to determine seven and six moves respectively. In a similar

situation, SS6, who exhibited considerable difficulty determining the number of moves

between the stone and steps on the ladder, successfully counted from two steps above to three steps below including the stone to determine five moves.

NB8 had a slightly different approach to addition of positive and negative number in context. The researcher posed, “If you are at 2 left and want to go to 1 right [NB8 had

already numbered the houses (see Table 1).] How many houses do you have to move?”

Without counting, NB8 responded, “Three. Two plus one.” NB8 had determined the

number of houses to the left of “home” and how many houses to the right of “home” and

added instead of traversing the number line house-by-house. NB8 repeatedly verified her

answers with mental addition rather than counting on or counting all.

CH8 had difficulty with the modified number lines, especially when determining which houses or steps to include in his count. In a scenario of moving from “home” to the second house to the right of home (R2), the researcher and CH8 discussed his strategy

CH8: Well, counting this house (R2) two, but I say one.

R: Why do you say one?

CH8: Because you don’t count this one (home) and you don’t count the one you

land on. (CH8 points to R2)

R: You don’t count the space you land on when you play a game?

CH8: Oh, you do.

R: So how many houses did you move?

CH8: Two

R: Two houses from your house, right? (CH8 nods in assent.) Now move to the

purple house (R3). (CH8 moves the shoes.)

CH8: Zero

R: Zero houses?

CH8: Oh, one.

In another scenario, CH8 determined that to go from right 3 to left 3 (verbally identified

as right or left) he would have to make seven moves because “Three plus three is six and

you count the one you land on, so that equals seven.” He was inconsistent with this application on the vertical number line; but in later activities CH8 successfully counted

out the movement including the stone.

The errors that KM6 made when she attempted to determine the number of steps

between a location above and a location below the stone on her vertical number line

resembled errors made by young learners of linear measurement, i.e. reading only the last

number on the measuring instrument corresponding to the end of the object being

measured. Children working with linear measurement often focus on the numbers on the

measuring tool rather than the iteration of units of measurement (Kamii, 1995). When

asked to determine how many moves from step 8 to step 3, KM6 stated, “I don’t have to

count these because I have numbers. … Eight.” KM6 then counted down five steps, but

did not correct her error (see Table 2).

Although the number line activities provided more opportunities than the collection activities for the children to experience opposite effects of movement, the children did not refer to negative numbers. They used unidirectional numbering or two positive number lines with opposite directions. Since only one child created and utilized a notation system similar to operations with positive and negative numbers, the intended study of relationships between positive and negative numbers was not addressed in depth.

The majority of the children were more concerned with whether or not to count the beginning and/or the ending house or step/space. When faced with determining whether to include “home” or the stone in the count when moving from one side of the referent to the other, some of the children became more perplexed. The confusion may have resulted from using spaces and/or steps for counting. Their difficulties were compounded when they attempted to use their numbering on the number lines.

Relationships Among Negative Numbers

Only one child, NB8, was able to conceptualize addition of negative numbers.

Although she used words to account for the deficit rather than traditional negative number notation, she demonstrated understanding of the concept of adding two negative

numbers in context. In the conversation between NB8 and the researcher, NB8 captured the essence of adding two negative numbers.

R: What if Mom told you that she would give you two pennies so you could buy

it now when it is on sale? Would that be a good deal?

NB8: Yeh. So now I have zero and owe Mom two cents.

NB8 records “owe Mom 2¢”

R: Now you want to buy something else you like for three pennies and Mom says

again that she’ll help you out.

NB8: Five.

NB8 writes 5 over the 2 in “owe Mom 2¢”

(see Appendix I.)

NB8 surpassed all the other children with her conceptualization of negative numbers in context. Not only did she invent a conceptual representation of deficit using words rather than symbols, but she demonstrated her understanding of calculations with positive and negative numbers or only negative numbers using her own representation.

Relationship Among Opposite Numbers

A foundational level of the concept of additive inverse, or adding opposites, was presented to the children in both number line and collection contexts. Since the children were not familiar with the notation for opposite numbers, they experienced the concept of additive inverse at the level of cancellations of movements or actions. In the street and ladder number lines, the children were asked to move their shoes a designated number of houses from the “home” house and then move back the same number of spaces or to

move up the ladder a designated number of spaces/steps and then move down that same

number of spaces/steps. The researcher then asked the children to identify the resultant

position after the sequence of moves. All of the children aged six, seven, and eight,

determined that the succession of moves landed them on the space or house from which

they started, but all did not verbalize an association with zero. They recognized that

movement in one direction and then the same amount of movement in the opposite

direction, on both the vertical and horizontal number lines, resulted in no movement from

the starting location, stone or “home”. After successive equal moves along the vertical

and horizontal number lines in one direction and then in the other direction, the children

simply stated that they were back at the “home” house or on the stone but did not refer to

zero.

In the collection contexts, the children were given a set of manipulatives or

pennies and then asked to give away the same number of the set. The researcher then

asked the children to determine the number of manipulatives or pennies that were in the

set now. All of the children stated that the result of their actions was “none” or “zero”,

unlike in the horizontal and vertical number line activities when the children did not

intuitively assign the number zero to the starting location. The children associated the result of successive opposite movements on the number lines with the origin, but they associated the result of successive opposite actions of composing and decomposing objects with zero.

JB4 successfully responded that he would have to move the same number of

houses from the “home” and the same number of houses back to reach the “home” house.

He determined the correct number of counter movements needed to return to the starting

point for options up to ten without counting the spaces forward or backward to make his

determination. The other four-year-olds were inconsistent in their preliminary

understanding of the additive inverse concept. A distinct possibility exists that these children may have successfully determined the resultant position after successive

opposite moves or simply recognized the pattern of the researcher’s repeated questions

(e.g. “If you move three houses from ‘home’ and then three houses back, where will your

shoes be?”) and responded “home” repeatedly. Their inconsistency of answers may have

resulted from confusing the context of two questions: Where will you land? How many

moves will you make?

The five-year-old children demonstrated more understanding of the additive

inverse concept in the collection activities than in the modified number line activities.

IS5 demonstrated basic understanding of the additive inverse as he moved forward and

backward on the horizontal modified number line and when he collected and removed

manipulatives from his sets in the collection activities. SS6 and ES6 did not recognize

that forward and backward movements resulted in no change on the number line, but they

demonstrated an understanding of additive inverse when they collected and lost frogs and

pennies. ES6 argued that if she took three steps to the treehouse, then she only needed to

take one step to get back to the stone because she did not have to count the three moves

again. The younger children had particular difficulty with determining movement as

directed. When asked to move three steps or houses, for example, they frequently made

one move of three steps instead of three movements of one step. IS5, HH5, and RC5 kept

track of how many times, or turns as in playing a game, they had been asked to move

their shoes and responded with that count rather than the count of moves between houses.

The children’s confusion with the terminology used in the questioning (i.e. “moves” or

“houses away from”) may have affected their responses.

All of the six-year-olds recognized the additive inverse concept in one or more

activities. The six-year-olds recognized that opposite movements of the same magnitude

on the number lines resulted in no movement from the origin. They also recognized the

additive inverse concept in the collection of manipulatives or pennies and were able to

make a determination of counter actions without counting. HC6 had a slightly different

approach. When the researcher asked her how many of her three pigs would have to run

away to have none, HC6 declared, “All of them.” She did not respond with a number, but

fully demonstrated understanding of the concept.

The children, aged seven, struggled with the additive inverse concept especially in

reference to the modified number lines. LM7 determined the inverse movement with the aid of counting. SH7 required several attempts before he could determine the inverse movement without counting. ES7 successfully recognized inverse movements for the

horizontal number line but did not make the association for the vertical number line.

The oldest children in the study, aged eight, succeeded in recognizing the additive

inverse, although some of the children had difficulty modeling the concept. They did not

require counting for their determination of inverse operations. AH8 recognized the result

of the inverse actions on the modified number line after some difficulty with modeling.

R: Start at home and move to the white house (R3). (AH8 moves to R3.) If you

move three houses back, where would you be?

AH8 moves shoes to R2, then R1, hesitates over home, and then drops back to R1.

AH8: The red house (R1).

R: Move to the purple house (R2). (AH8 moves shoes to R2.) How many moves?

AH8: Two

R: And you move two houses back. Where would you be?

AH8: (AH8 moves to home). My house.

R: If you move to the yellow house (R4), how many moves?

AH8: Four

R: How many houses would you have to move to get back to your house?

AH8: Four. If you move up you get four. (AH8 moves shoes to R4). If you move

back you get four. (AH8 moves shoes back to home)

But when asked the result of having three pennies and giving away three pennies in a

collection activity, AH8 confidently said, “zero” emphasizing her response by forming a zero with her thumb and forefinger. NB8 argued the concept of additive inverse,

“Because if you move down four, you’re gonna have to move four to get back.”

In the study, all of the children demonstrated some level of understanding of

inverse actions. They appeared more confident of the concept of additive inverse, or

cancelling, in the collection activities than in the number line activities. Some of the

children successfully modeled the cancelling process; others recognized the result of

cancelling but were unable to associate proper numbers with the process. The younger

96 children’s inability to justify their answers raises some doubt as to whether they understood inverse actions or responded to the pattern of questioning.

Data Addressing Representation of Number

During the modified number line activities the researcher encouraged the children to create representations of number on a number line as determined by magnitude and direction associated with the placement of the houses or steps/ spaces. In the collection activities the researcher encouraged the children to record quantity of set in regard to the number of pennies in the piggy bank or manipulatives in the set. Assigning numbers to the number lines was less troublesome than the recording of quantities of set because the children used familiar positive numbers regardless of direction. The numbers that the children assigned to the number lines did not reflect negative number connotations, just opposite actions (e.g. NB8’s number line with “2 left 1 left 0 1 rite 2 rite”). However, when recording the quantity of set, the children encountered numbers of deficit with unknown representation. The children did not instinctively number their number lines or record the quantities of sets, but cooperated when the researcher asked them to do so.

Number Line Representation

The children were assigned two tasks involving modified number lines – a street representing a horizontal number line, and a ladder representing a vertical number line.

The children were asked to number the houses to the right and the left of the “home” house in reference from that house. Three of the five-year-olds and one of the six-year- olds were unable to complete the task of representation on either the horizontal or vertical number line due to confusion and frustration. Three five-year-olds counted the number

of times they had been asked to move rather than the number of houses moved. Although

CM4 created a number line that included his real house number and random other

numbers, he relied on rote counting and not one-to-one counting to determine the moves.

Table 1 presents the children’s representation of order on the horizontal number line.

Table 1

Presentation of Children’s Representation of Horizontal Number Line

Child Number Line

JB4 2 1 5 1 2 3 4

PR4 3 1 2 0 1 3 4

CM4 5 0 5 5 1 1 2 5 0

BS4 3 1 2 01 2 4

SS5 8 7 3 1 0 2 4 5 6

ES5 6 5 H 1 2 3 4

KM6 9 8 7 61 0 1 2 3 4 5

LS6 9 8 7 6 5 4 3 2 1 1

HC6 8 7 9 1 2 3 4 5 6

FC6 4 3 2 1 1 2 3 4

AS7 4 5 6 0 1 2 3

ER7 4 3 2 1 0 1 2 3 4

LM7 4 3 2 1 0 1 2 3 4

SH7 4 3 2 1 0 1 2 3 4 5 L R MS7 4 3 2 1 0 1 2 3 4

KS8 3 10 12 15 81 13 11 9 8

NB8 left 2 left 1 left 0 1 rite 2 rite 3 rite

CH8 4 3 2 1 0 1 2 3 4 (verbally indicated right or left)

AH8 9 8 7 H 1 2 3 4 5 6

KG8 4 3 2 1 0 1 2 3 4 Note: The numerals assigned to the “home” house are indicated in bold font. A bold-font H, H, represents a “home” house that was not assigned a numeral. Children who experienced

difficulties comprehending the task and were unable to complete a horizontal number line were not included in this table. KM6 wrote 1 and then crossed it out and wrote 6.

JB4, BS4, ES5, KM6, FC6, and CM6 and all of the seven- and eight-year-olds,

except KS8, successfully numbered the houses to the right of “home” with respect to placement from “home”. Although JB4, and ER7, SH7, MS7, LM7, and KG8 successfully

numbered the houses to the left with reference to the “home” house, they focused on

magnitude and not direction maintaining that there was no difference between going to

house “2” on the left or house “2” on the right. Only NB8, CH8, and AH8 made note of

direction on their number lines. The other children simply applied the numbers in a rote

counting sequence or randomly to the indicated houses. The exception, CM4, failed to

conceptualize any connection with placement from the referent house and simply wrote

his own house number and other random numerals under the houses. Although KS8

demonstrated a number sense in the other activities, he assigned “9” to the “home” house because “that’s the first number I thought of”. He continued to assign numbers randomly to the other houses.

NB8 experienced considerable difficulty numbering the houses on the horizontal

number line. She knew that there was a difference in moving two houses to the right or

two houses to the left of her “home” and was able to distinguish between the two when

“left” and “right” or house colors were used as reference. She numbered her “home”

house as “0” and the houses to the right of “home” with numerals “1, 2, 3, 4”. When NB8

was asked to number the houses to the left of her “home” house, she looked at the street

pensively, looking at the right most end of the street, then at the researcher, and then at

the leftmost end of the street. She then pointed to the houses third-left (L3), second-left

(L2), and first-left (L1) from “home” and said, “five, six, and seven” respectively.

Working with NB8’s number line of “5 6 7 0 1 2 3 4”, the researcher asked, “So if this

[house] is number five (pointed to L3) and I ask you to move to the yellow house [L3], will you make five moves?” NB8 shook her head and said “No”. She then decided to number the houses with two “1”’s and argued that the researcher would have to tell her right or left. Following further discussion, NB8 decided that the color of the house would also be a reliable reference. Although her color reference was functional, the researcher further explored her thinking.

R: But what if all the houses fell down?

Researcher removes all the houses leaving only the number line that NB8 had

created.

R: Now, we want to build the houses again. You know at 0 you will build your

house, right? The other people come back and say, “We used to live in number

one house”. How would the builder know which location that is?

NB8: The builder doesn’t know which to go to.

R: So is there a way you could make the numbers different so the builder would

know where to build the house?

NB8: Put this one [L3] zero, [L2] one, [L1] two, [H] three, [R1] four, [R2] five,

[R3] six, [R4] seven.

R: But what if we build a house here [Points to space left of L3]

NB8 picks up the house and moves it to the far right.

R: So you don’t want that house on the left? This end always has to be zero?

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NB8 nodded.

NB8: Or I could do this.

NB8 picks up the far right house and places it far left on street and moves each

house one space to the right.

R: How did that help?

NB8: Because this one [new far left] is gonna be zero, and this one is one, two,

three, four, five, six, seven, and eight. [NB8 had a very satisfied look on her face.]

R: Now what number is your house?

NB8: Now my house is going to be five. [NB8 had a smug look.]

R: That makes sense to you? You always start with zero?

NB8 nods in assent.

Through the discussion above, NB8 attempted several different numberings of the houses.

She seemed satisfied with her number line that resembled the traditional number line,

although the numbering of her “home” house did not satisfy the original reference to a starting point.

The children had considerable more difficulty creating a vertical number line using the stone (representing ground level) as the referent. Only fourteen of the children in the study were able to complete a vertical number line. Of those fourteen children,

NB8 and AH8, ES5, and BS4 provided evidence that they may have developed a

preliminary understanding of magnitude and direction. All four of these children labeled

the number line upward from one to six or eight and downward from one to six or eight

from the stone [zero], but only two eight-year-olds NB8 and AH8 reasoned that there was

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a difference between going up or down and that the numbers should be denoted as such.

NB8 marked the numbers below the stone as “1 down, 2 down, 3 down, 4 down, 5 down,

6 down”. When asked whether she needed to write 1 up and 2 up, she argued, “No, because we already have down. You can say either one, two, three, four, or five or one

down.” At the conclusion of the interview, the researcher asked NB8 if she had learned

anything. NB8 said that she had learned that two house numbers can not be the same number. The researcher reminded her that she had had difficulty with those numbers and

asked her if she wanted to go back and look at them again. NB8 agreed although she still

felt she didn’t know what to do with those numbers. The researcher brought out the

street and continued,

R: Do you remember what you did on the treehouse activity?

NB8: Down

R: Can you do something like that here [on the street]?

NB8: Left or right.

R: So how do you want to mark them now?

NB8: One right, two right, three right, three left, two left, one left.

R: So if I ask you to move to house one right…

NB8: I know where to go.

R: If you are at two left and want to go to one right, how many houses do you

have to move?

NB8: Three. Two plus one.

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At first, NB8 was better able to visualize the vertical number line than the

horizontal number line. Later, she transferred her understanding of the numbering of the vertical number line to the horizontal number line, marked it appropriately and accurately

found the distance between positions from the left of zero to the right of zero.

Although she was working with a vertical number line, AH8 designated the

numbers above zero as “right” and the numbers below zero as “left”. She also drew a

line below the “1” on her number line to help define right and left. When asked whether zero would be right or left, AH8 responded, “Right, because I drew the line here. I should

have put it there” pointing under the “0”. The other children numbered in sequence from

the top of the ladder or from the bottom of the ladder or starting from 1 to the space or

rung above the stone to the top of the ladder and then continuing the number sequence

from the space or rung below the stone to the bottom of the ladder, as presented in

Table 2.

Table 2

Presentation of Children’s Representation of Vertical Number Line

Child BS4 SS5 ES5 KM6 HC6 LS6 AS7 SH7 CH8 NB8 AH8 KG8 KS8

8 R 7 i 6 12 13 6 g 8 6 5 5 5 5 11 10 12 5 5 h 7 5 4 4 4 5 4 10 9 11 4 4 t 6 4 3 3 3 4 3 9 8 10 3 3 8 5 3 2 2 2 3 2 8 7 9 2 2 7 4 2 1 1 1 2 1 7 6 8 1 1 6 3 S 0 0 0 1 1 6 5 7 0 0 5 S 1 1 1 6 7 1 5 4 6 1Down 1 4 2 2 6 2 7 8 2 4 3 5 2Down 2 3 1 3 7 3 8 9 3 3 2 4 3Down 3 2 4 8 4 9 10 4 2 1 3 4Down 4 L 1 5 9 5 10 11 5 1 2 5Down 5 e 0 6 6 11 12 1 6Down 6 f 12 7 t 8

Note: The stone (starting space) is indicated in bold font. A bold-font S, S, represents the stone that was not assigned a numeral. Children who did not complete a vertical number line are not included in this table.

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In this study the children numbered both vertical and horizontal number lines with

positive numbers regardless of the orientation from zero or referent position. While most

of the children maintained that there was no difference between movements to the right

or left or up or down, three of the older children used descriptive works to define

opposites. Even the few children who realized that direction did matter did not consider

negative numbers. Instead they used descriptive words such as “rite”, “left”, “up”, or

“down” to distinguish between numbers on opposite sides of the referent.

Representation of Deficit

The children were asked to record the quantity of pigs, frogs, bears, or pennies

that they had following each action of gaining or losing. Only two children, SH7 and NB8 used their own written representations for a deficit situation. One child, CH8, stated that

he had “minus one” when he needed one to complete a set, but did not attempt to make a

written or pictorial representation. The other children used only positive integers and

zero to record the amounts. They either left the set intact and recorded the quantity they

had prior to encountering a deficit situation or removed the number of manipulatives they

had, albeit not enough to satisfy the demand, and simply recorded zero.

During a collection task NB8 took on the persona of a pig farmer and managed a

collection of pigs. NB8 and the researcher engaged in a conversation about how to record the number of pigs in her collection when she was unable to meet the demand. At the beginning of this discussion NB8 had three pigs in her set.

R: A farmer needs five pigs.

NB8: I can give you the three, but once I get two more I’ll give ‘em to him.

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NB8 removes the three pigs from her set.

R: Now what number will you write in the book?

NB8 writes “0”.

R: How will you remember that you owe the farmer two pigs?

NB8 writes “owe 2 pigs”. NB receives six more pigs from the researcher.

NB8: I owed two so now I give two to the farmer. He has his five pigs.

R: And you have?

NB8 writes “4” and scribbles out “owes 2 pigs”

R: And you crossed that out?

NB8: That way if I have two, I don’t always keep owing them.

(see Appendix I.)

In another scenario in which she had three pigs, NB8 continued to use her written words to account for the pigs that she owed.

R: Can you give me four pigs?

NB8: Give you three. When I get a pig, I’ll give it to you.

NB8 writes “0” and “owe 1 pig”.

R: Now someone wants to buy 5 pigs from you.

NB8 shakes head.

R: I have the money right now for you.

NB8: But I don’t have any pigs.

R: Is there any way we could work out a deal?

NB8: Give me the money and then I’ll give the pigs to you.

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NB8 crosses out the 1 and writes 5, crosses out the 5 and writes 6.

R: Why did you do that?

NB8: Because I have to owe the farmer six, I mean five, and I owe that one guy

one.

R: So now you owe six?

NB8 nods in assent.

R: Now you get four pigs.

NB8: I’ll give the person their one because they were the shorter, and now cross

out 6 [and writes 5]. But I give him his three, so it’s two.

R: How did you know that?

NB8: Three plus two is five.

NB8 carried her representation of deficit amounts with written words to the task of collecting pennies. She started this discussion with one penny which she has recorded as “1” in her tally column.

R: Now you find something you really like and it costs three pennies. What are

you going to do?

NB8: Look for pennies (Grins.)

R: How many more pennies do you need?

NB8: Two

R: What if Mom told you that she would give you two pennies so you could buy

it while it is on sale. Would that be a good deal?

NB8: Yeh. So now I have zero and owe Mom two cents.

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NB8 writes “0” and “owe Mom 2 ¢”

R: And now you find something else you like for three pennies and Mom said

again “I’ll help you out”.

NB8: Five.

NB8 writes “5” over the “2”.

R: But how many do you have in your piggy bank?

NB8: Zero.Right there.

NB8 points to “0”.

R: Now you find four pennies.

NB8: I’ll give them all to Mom. Now I just need to owe her one.

NB8 crosses out “5” and writes “1”.

R: Now you get two pennies.

NB8: Put one in piggy bank and one to Mom.

NB8 crosses out the “owe Mom 1 ¢” and writes “1” in the column registering

pennies in the bank. NB8 makes a very satisfied face

In these three scenarios, within context of the activity NB8 correctly added and subtracted positive and negative numbers not as numbers but conceptually using her own invented representation of negative numbers.

SH7 also invented his own representation for negative numbers; he used a variety of symbols including a combination of the letter n and numbers and drawings rather than words as NB8 used. SH7 successfully composed and decomposed his set of frogs and referred to “negative two” when he did not have enough frogs to meet the demand.

108

However, at this point in the activities, SH7 did not attempt to represent that number.

Later, after a series of questions about gaining and losing pennies, SH7 had three pennies

in his collection. The researcher asked SH7 to record the amount of pennies in the piggy bank. SH7 wrote a large 3 followed by a backwards cent sign and the image of three

pennies.

SH7: I’m not a very good Abe Lincoln drawer.

R: Now you get this many pennies [four pennies]. How many do you have now?

SH7: (writes “7”).I don’t have thirty-seven cents. I don’t have seventy-three

cents. (crosses out the three) I have x-seven cents.

R: Now let’s say you buy something for five cents.

SH7: Can I buy all those frogs? No, wait, how about the colored one?

R: How many pennies do you have now?

SH7: Two. [writes “2” and backward cent sign]

R: Now you want to buy another frog for three cents.

SH7: I don’t have enough money for that.

R: You don’t have enough money for that? Can you keep track of that money?

SH7: Yeh.

R: What would it be?

SH7: Two cents. Wait, negative one.

R: How would you write that?

SH7 writes “N1” and draws a frown face.

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SH7: If there was a kind of penny that was like saying you had negative one

penny, that’s how it would look.

R: So you are spending these [two pennies] and you have negative one in your piggy bank now. Right?

SH7: Yep.

R: Now if you get four pennies. How many pennies do you have now? You had

negative one. How many do you have now?

SH7: That’s a happy face [drawn over the frown face] and I just made a c

[backward] over the one and I changed the N to a four.

R: Now you are going to spend two of those pennies.

SH7: That’s like where did my money go?

R: You have how many pennies now?

SH7 writes “2” and a backward cent sign.

R: Now you want to spend four pennies.

SH7 writes “n 4”and draws a frown face.

SH7: Has big ears there.

R: So if you are spending four pennies, we’ll take these and you now have a

negative four?

SH7: Yep

R: If you have a negative four now and you want to spend three more pennies,

what would you have?

SH7 writes “0”and draws a large frowning face.

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R: That’s a pretty scary penny. What’s your number? Negative?

SH7: Zero.

R: Negative zero

SH7 nods in agreement.

(see Appendix H)

CH8 expressed that he knew that when he had four pennies he needed one more penny to meet the demand of five pennies. He also stated that his tally was “minus one” and attempted to justify that verbal representation by addition and subtraction calculations with the following explanation.

R: Five minus four is one?

CH8: Yeh

R: Does that mean you have one penny in your bank?

CH8: No. I mean (pause) no. It’s a minus. (He referred to subtraction in the

notebook)

R: So if Mom came into the room and asked you how many pennies you have.

CH8: I’d say ‘zero, Mom’.

R: But you have written one.

CH8: That’s a minus one. I only need one more.

R: So what’s a minus one mean?

CH8: That means that you take… (He looked at his work.) Actually that’s a four.

Five minus one equals four and four minus four equals zero.

R: Does this one mean how many pennies you have in your bank?

111

CH8 shakes his head.

CH8: It means how much money you needed.

R: Looking at your work would your mom know that you need one penny?

CH8: (He shakes his head.) I would have to say, “Mom, I don’t have any

money.”

(see Appendix J.)

These three children invented representation for the deficit amounts. However,

only one child used her representations for calculations. The other two presented some

evidence of informal introduction with negative number notation, but they were not

consistent with the notations and were not able to make calculations with their notations.

Making Sense of Number in the Context of a Story

Many of the children in this study appeared to have never encountered situations

of deficit. When presented with a situation of deficit, e.g. having three pennies and

wanting to spend four pennies, the children all grinned, smiled, or even laughed as if they

had encountered a trick of impossible dimensions. Their reactions to the situation varied

from offering no response to problem-solving to eliminate the seemingly impossible

situation. Quite a few of the children offered alternative solutions. When ES7 had two

pennies and needed four, her solution was “Ask a friend to give you two more pennies”.

JB4 simply resolved the problem by “buy something for two pennies” rather than for four pennies. IS5 declared that he could “get more [pigs] at the store” or “get some [pennies]

from my wallet” rather than have to face a deficit. AH8 resolved that if she needs six

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pennies and only has four “You need to go back home or go in your purse and find some

more [pennies].” or “Gonna have to get it next week.”

When CH8 was faced with the situation of the researcher wanting to buy three

bears when he only had two bears, he continued the story,

CH8: Sorry only have two… I can’t sell you anything. I only have two left and I

can’t give you three because I only have two. But if I ask one of the zookeepers

for one back, I’ll get you the money soon.

R: So you need one more [bear] back to give the zookeeper three bears?

CH8: Um humh

R: What would happen if you gave me all the bears you have?

CH8: Well you….it depends how much you had. Like if it was fifty bucks and I

only had two. Actually if it was sixty bucks, you could only pay me thirty,

because I only have two bears.

R: If I had the money for three bears would you take the money for three?

CH8: I can’t. I don’t have enough. Maybe next week, I’ll get a shipment of bears,

and I’ll save the money ‘til then, and I’ll give you some.

The discussion continues between CH8 and the researcher. When CH8 doesn’t have

enough bears to meet the demand, he offers, “Sorry. Maybe next week.” and “I’m getting

a new shipment after lunch.”

AS7 was unable to determine what she could do when she needed three pennies to

buy something when she only had two. However, after a few pensive moments, AS7

related this story,

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Well, actually I have like a story to tell you. Well once I was at Walmart in the

toy section, and I see this Fur Real Kitty. And I really, really, really want that.

But I don’t have enough money. I have to save up for it. Now I have to wait, and

I don’t have enough money to buy it. That’s how I did it. It was still there when I

came back … I have a new idea. So I’m just going to have to save. But if you

had allowance day, and it was the day after the day that you really really wanted.

And you got one penny for cleaning your room. And … on that day you got the

penny and really, really, really, really wanted to go to the store now and see if it

was still there. And you came in, and it was still there, you would be so happy.

Throughout the study, the researcher presented the directives for many of the

activities to the children in a contextual, story-like format. The children were asked to

move from one house to a friend’s house with the enhancing narrative of having to ask

Mom for permission. They were involved in the story of leaving a toy (manipulative) at

one house or step while they moved onward. The researcher posed the question as how

many moves were required to go back and get the toy rather than how many moves from

one given space to another. The stories for the collection activities included scenarios of

pigs or frogs or bears running away or buying and selling pigs or frogs or bears for farms

or stores or zoos. The children became involved and animated, acting out the directives.

Only one child, CH8, briefly wrote the mathematics problem in the traditional vertical

alignment format to determine his answer. The stories presented by the researcher elicited stories created by the children.

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A few of the children attempted to make sense of the situation by creating their

own stories with similar circumstances or stories that did not put them in deficit

situations. Several children processed the situations by anticipating and stating the

researcher’s next question. JS4 had just finished a discussion in which he was two pigs

short of the demand. The researcher gave him seven pigs. JS4 commenced to count as he

touched each pig, “One, two, three, four, five, six, seven. And the two person came and

said, ‘You have more pigs.’ And I’ll say, ‘Yes’ And [he’ll say] ‘I have two pigs’. And

I’ll say, ‘Yes’.” Another child, KM6, created lengthy stories and marveled at her own ability to communicate mathematically,

What happens if someone comes and they wanted four [pigs] and I was out of

pigs and someone else came and they wanted some pigs? If you left a mom pig

out and it had babies, you would have more pigs. Your next door neighbor has

some pigs and you could borrow some from them…. You could go to your

neighbor and get one more. If you had a mom pig and she waited one more week

and the mom had one more baby and she came back and wanted three more.… I

never talk this much when I learn. I never do…

The children became engaged in the activities with the researcher’s story lines. In response to problems for which they became perplexed by encountering a deficit, the children resorted to problem-solving strategies. The children attempted to resolve the situation by presenting a sensible answer. Answers varied from a finite “can’t do it” to alternative solutions to extensions of the problem in a context of which they could make

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sense. Making up their own stories allowed the children to extend and/or solidify their thinking.

Summary of Findings

In this study the children demonstrated an understanding of cardinality of set through one-to-one counting or subitization. The children verified their determination of cardinality by counting all or skip counting. All of the children, except two eight-year-

olds, assigned ordinal number names to manipulatives in a row beyond first and second

without regard to the actual placement in the row. All of the children successfully

identified the first and second manipulative in the row, but four-, five-, and six-year-olds

pointed to each successive adjacent manipulative as the fourth, fifth and third

manipulative. These younger children failed to connect the cardinal number name of the

manipulative with the ordinal number name. Their explanations inferred a rote learning

of the ordinal number name sequence.

The children demonstrated varied levels of understanding of additive inverses, or

cancellation, through movements in one direction and then in the opposite direction or

actions of opposite effects. The children defined zero as nothing or none and

demonstrated more understanding of zero as a quantity in collection activities than as a

referent point in number line activities. The children demonstrated little knowledge of

the organization of the number line, particularly regarding zero and negative numbers.

The children also demonstrated varying levels of understanding of the

relationships of positive and negative numbers and zero in a situation of deficit or

negative number context. All of the children demonstrated understanding of relationships

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between only positive numbers, but did not present evidence of understanding

relationships between positive and negative numbers or only negative numbers. Negative

numbers did not exist, symbolically or conceptually, in their individual number systems, so the children attempted to develop relationships with positive numbers only, even in the case of deficit situations. At a very basic level of understanding, the children determined that quantifying a deficit was not possible within the positive number system with which they were familiar. All of the children in the study, however, could determine the number of manipulatives needed to satisfy the demand but were unable to express it as a number less than zero. A few of the older children demonstrated a more developed level of understanding by using verbal or written representation of the deficit quantity. Only one of the older children could add or subtract contextually negative numbers using her system of representation.

The children did not present the same number relationships on the number lines.

They focused more on counting to determine the number of moves or combinations of moves. The children’s arguments that direction of movement did not matter further substantiate the conclusion that they functioned only in a realm of positive numbers.

The children expressed more concern over including the beginning or end house or step/space than direction of movement. Assigning numbers to the houses or spaces/steps did not facilitate development of understanding of the effects of both magnitude and direction. Since only three children indicated that direction of movement on the number line made a difference, the intended study of relationships between positive and negative numbers on the number lines was not pursued in this study.

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The children were reluctant to explore written and pictorial representations of

numbers, particularly negative numbers with which they were not familiar. They relied

on the set of counting numbers and zero to represent quantities of set. Only two of the

older children ventured to represent a negative number in a deficit situation. The other

children simply referred to the deficit amount as zero.

When the children encountered an unfamiliar deficit situation, some of the

children attempted to make sense of the situation through problem-solving or creative story-telling. Although their stories did not always answer the initial question, the children made sense of the situation within the limits of their experiential knowledge base, and they seemed satisfied with their solutions to the problem.

Throughout this study the children demonstrated a very basic foundational understanding of algebraic principles. Fundamentally, the children in this study demonstrated a sense of deficit but were not driven to use negative numbers to address the situations presented to them. The children provided evidence of a preliminary awareness of algebraic relationships of numbers, additive inverse in particular. The children also attempted to make sense of this novel concept of deficit or negative number context by relating to their personal experiences and imbedding the problems they encountered in story-telling.

CHAPTER V

ANALYSIS

Introduction

This chapter contains analysis of the findings of the study. An overview of the results precedes analysis of data on how children conceptualize the relationships of numbers including only positives, negatives and positives and zero, and only negatives.

Children’s representation of negative numbers is analyzed. A discussion of the role of story-telling in solving problems with a context of negative numbers is also presented.

The implications of the findings of this study are presented in this chapter.

Limitations of this study and suggestions for future research are also indicated.

Overview of Results

The findings from this study affirm that young children, ages four through eight, do conceptualize relationships among positive and negative numbers and zero. The children demonstrated varied levels of understanding of the relationships from basic counting of objects to algebraic concepts such as additive inverse and operations with integers, although negative integer notation was not introduced. In the positive number sense, the children had no difficulty composing and decomposing sets of manipulatives, but they demonstrated less understanding of addition and subtraction on a number line.

As for the concept of zero, all of the children declared a working definition for zero as

“none” or “nothing” implying lack of quantity in a collection or lack of movement in the number line activities. The children exhibited some understanding of the concept of

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additive inverse but did not create notation that represented opposites.

The findings of this study also suggest that young children use verbal and numeral

representation more than pictorial expression to designate numbers, negative numbers in

particular, in recording results of task-related activities. The children verbalized a mental

image of deficit, that is, knowledge of how many more manipulatives they needed to

create a set of a given cardinality, but most of them made no attempt to create a written or

pictorial representation of the number. Two of the children in the study, a seven-year-old

boy and an eight-year-old girl, made pictorial or written representation of the deficit

number. His representation was with drawings, and hers was with words. Of all the

children in the study, only the eight-year-old girl invented representation that allowed her to add and subtract in a negative number context.

Results of the Study

Relationships Among Positive and Negative Numbers and Zero

The results of this study confirm that children between the ages of four and eight experience little or no difficulty with the concepts of addition or subtraction of positive numbers in collection-type activities. Given a set of manipulatives the children successfully determined the quantity in the new set when manipulatives were added and when manipulatives were removed. The children determined the quantity of the new set by counting all, counting on, skip-counting, or subitizing. However, the children experienced greater difficulty with the concepts of addition and subtraction in modified number line activities. Children in all age groups did not indicate an association between addition and subtraction and the movement along the number lines. They effectively

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determined the number of steps, spaces, or houses from the origins (“home” house or

stone) or from the previous location, but they were inconsistent with including the

beginning and/or the ending step, space, or house. Some children included the beginning

or ending location in the count, and some children only counted the steps, spaces, or

houses between the beginning and ending location. Even when the directions were

worded with utmost care to eliminate the confusion between how many moves and how

many houses between, some children still focused on the number of steps/spaces/houses

between the two endpoints.

Only one child, NB8, actually added and subtracted positive and negative numbers without traditional negative number notation. When she already had a deficit of manipulatives, pigs or pennies, and lost more, she accurately determined the accumulated

deficit value. When she had a deficit of manipulatives and gained some, she successfully calculated the sum of the deficit amount, or negative amount, and the positive amount

regardless of whether the result remained negative or became positive in context. Her

written word representations were logical in context and fully functional for her in

calculations. Her mental processing of the negative number context was similar to the

thinking of early mathematicians who used the written word to solve problems (Beery et

al., 2004).

All of the other children were cognizant of the deficit amount when they were

asked to relinquish more manipulatives than they possessed but were unable to record

that amount or to mentally retain that deficit amount and combine it with quantities found

in successive gains or losses. Although one seven-year-old recorded the deficit with

121 symbols and icons, he did not retain that amount in further compositions of sets. Another eight-year-old stated that he had “minus one” pennies when he had four pennies and needed five pennies but became confused when he tried to justify the “minus one” with written calculations. All three of these older children demonstrated a more advanced conceptualization of negative number context than the rest of the children in the study.

If these children, and others with similar negative number conceptualization levels, were given ample opportunities to develop their understanding of the negative number concept beyond their foundational level of understanding, they would undoubtedly be better prepared to complete operations including negative numbers when they encounter negative numbers in the traditional mathematics curriculum.

If children are sensitive to conceptual principles of arithmetic and can use these

principles, at least occasionally, then clearly we must seek to understand how

knowledge of these principles emerges and interacts with the development of

mathematical procedures … (Klein & Bisanz, 2000, p. 113)

Children who learn at a young age that subtraction of eight from three is not possible must later unlearn that knowledge to learn that the subtraction is possible with the use of negative numbers.

Zero as a Quantifiable Set

The children in this study identified zero as a set containing “nothing” or “none” objects when they were engaged in the collection-type activities. On the modified number line, some of the children used zero to represent the position from which no

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movement was made. In both cases the children used zero to represent a number

representing nothingness within the context of the activity.

The findings of this study indicate that the children had some understanding of the

concept of additive inverse, a more advanced algebraic principle, in both the collection

and modified number line context. In the number line activities and the collection

activities, the children recognized that opposite actions of the same magnitude, e.g.

moving left two spaces and then right two spaces, moving up three steps and then down

three steps, receiving and then losing the same number of manipulatives, resulted in zero

as a set with “none” or a position of no movement. Studies by Nunes, Bryant, Hallett,

Bell, & Evans (2009) and Bryant, Christie, and Rendu (1999) also presented findings that

children of age five, six, seven and eight were able to conceptualize the additive inverse

principle in collection activities and word problems. Baroody & Lai (2007) argue that

children of kindergarten age, ages five and six, may be developmentally ready to learn

inverses. Findings from previous research suggest that even three- and four-year-olds

(like the four-year-olds in this study) may develop a partial understanding of the inversion principle (Bryant, Christie, and Rendu, 1999; Klein & Bisanz, 2000; Sherman

& Bisanz, 2007; and Vilette, 2002).

While in algebraic terms the children in this study were conceptualizing the additive inverse principle, other researchers may argue that the children were utilizing the inversion principle (Sherman & Bisanz, 2007) or reversibility (Piaget, 1952/1964). The

inversion principle is defined as “a + b – b must equal a.” (Sherman. & Bisanz, 2007,

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p. 333). According to Piaget (1952/1964) reversibility is the process of cancelling the

effect of a transformation with an opposite transformation on an object. Piaget

determined that reversibility develops in children of ages six and seven. While the

researchers above have conducted studies on cancelling transformations with objects and

with numbers, Vilette (2002) suggests that children may base the transformations on

objects and not number. The children in this study also based their reasoning on

movements rather than numbers. The children naturally based their reasoning on

movements along a number line or decomposition of sets of objects because they lacked a

workable notation for opposites (i.e. negative numbers). Like early mathematicians, the children in this study made sense of the situation without algebraic notation. Both the children and the mathematicians worked through the problems of deficit utilizing mental images, written word, or manipulation of tactile objects.

In the current study, the observations of the children were not definitive whether some of the children, especially the younger ones, were simply responding to the repetitive nature of the questioning about inverses. The researcher is uncertain whether the children simply recognized the pattern of the actions or words and responded accordingly. Klein & Bisanz (2000) suggest that the concept of inversion (or additive inverse) may arise from the give and take in which children participate when playing with others.

When faced with a deficit situation in the collection activities, some of the children gave up as many manipulatives as they had in the set, although that quantity was not enough to meet the demand. They then recorded zero for the quantity of the set.

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Although they knew the deficit, they did not account for the number of manipulatives

needed to complete the set in demand. They inferred that zero was the finite end, or

beginning, of the number system. In a similar study with children aged seven through

nine, Mukhopadhyay, Resnick, & Schauble (1990) found that young children knew they

did not have enough objects and knew how many more objects they needed but were

unable to make calculations with the deficit quantities. Like a fifth grader, Beth, in a study by Mukhopadhyay (1997), HC6 and others in this study resolved the conflict of deficit by taking away all the manipulatives and declaring a total of zero. The thinking of the children is analogous to the thinking of early mathematicians in this aspect.

Mathematicians argued for hundreds of years that a number less than zero can not exist.

Both groups thought in a positive number sense and struggled to, or even refused to,

comprehend numbers that contradicted that concrete representation. Ending, or

beginning, the number system with zero simply made sense to the young children and

mathematicians.

However, NB8 carried this strategy of recording zero to another dimension. When

she had three pigs and needed five, she gave up the three pigs and recorded “0”. But she

did not stop there. NB8 went on to record “owe 2 pigs”. At first glance, NB8 seemed to

have made a mathematical error by recording “0”. Algebraically she should have

recorded −2. However, her reasoning was sound. She did have zero pigs in the set in a

tangible sense, yet she accounted for the abstract concept of deficit of pigs by writing the

words “owe 2 pigs”. NB8’s responses further substantiate children’s thinking that the

number system has a finite end with zero. Like many of the other children, NB8

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terminated her counting at zero, but she further accounted for the deficit in a positive

sense. To “owe 2 pigs” implies that she needs two pigs with pigs being accounted for as a

positive tactile quantity.

The children who recorded zero for the deficit number are at a more preliminary

stage of conceptual development of negative numbers than NB8. They did not express a

need to account for the deficit quantity. If the younger children continue to develop their

thinking in a similar manner to NB8, they only need to develop a mental, written, or symbolic representation for a deficit or negative number concept.

As researchers and educators and parents, we must remember that children do not

usually function in a negative number or deficit context. Adults routinely manage money

with negative number concepts in balancing a checkbook, performing accounting

procedures, securing loans, or buying with credit. All of these examples are adult-

oriented, rather than child-oriented, applications of negative numbers. “In the case of

negative numbers, it is not entirely clear what everyday experiences could serve as the

basis for the development of relevant concepts.” (Mukhopadhyay, 1997, p. 36) As the

children in this study demonstrated, a child who does not have enough money or objects

will typically wait until he or she gets that deficit amount or secures the needed amount

(perceived as a positive) from another source. If the child was able to secure the needed

money or object from another source, he or she did not express the need to replace the

borrowed amount. Even mathematicians until the sixth century were not involved with

accounting procedures that warranted a representation for deficit amounts. Much of their

problem solving was based on context and not notation.

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Zero as a Referent

The findings of this study offer little evidence of young children conceptualizing zero as a referent point on a number line. At the age of seven or eight, the children in the study represented the starting point, either the “home” or stone, as zero. The children were unsuccessful at integrating direction and magnitude on the number line. On the modified number lines, some of the younger children assigned counting numbers of magnitude in both directions from the zero and argued that direction did not matter, e.g. one move to the left of home was the same as one move to the right of home. Although they did recognize the “home” or stone as an origin or starting position, some children did not provide evidence of perceiving the number lines as continuous dimensions. Like some early mathematicians, the children encountering the abstract concepts of zero and negative numbers perceived the number lines as a composite of disjoint sets of numbers

(Beery et al, 2004; Mukhopadhyay, 1997). Discontinuous number lines consist of two or three segments: numbers to the right of zero, numbers to the left of zero, and zero. (Zero may be included with the right segment.) Further studies may indicate that these children were at a preliminary stage of development of zero as a referent and were unable to account for opposite actions or direction of number.

Some of the children numbered the modified number lines with zero as a starting point and continued numbering in a rote counting sequence to the right on the street and upward on the ladder. When they reached the end of the number lines, the children then moved back to the “home” house or stone and continued the numbering sequence to the left or downward on the horizontal or vertical number lines respectively (e.g. numbering

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the stone as zero and the spaces going up as 1, 2, 3, 4, 5) and then continued numbering

the spaces below the stone as 6, 7, 8, 9, 10, 11, and 12. They were unable to justify their

numbering; they could not explain why space numbered 7 was only two spaces away

from zero. These children applied traditional counting sequences to the positions on the

number lines without regard to the referent.

Only one seven-year-old and two eight-year-old children numbered the horizontal number lines with zero as a starting point and then successive counting numbers going in both directions, left and right. On the vertical number line only two eight-year-old children numbered the referent as zero and the other spaces in the counting sequence up and down. While these children demonstrated some understanding of the sequence of numbers on a number line, they did not use negative numbers to indicate direction.

However, they did use descriptive words to indicate difference of direction from zero

(e.g. NB8 writing “rite 1” or “left” and “1 Down”).

Many of the children made no association of the modified number lines with traditional number lines involving magnitude and direction. They appeared, instead, to view these modified number lines as linear measuring tools rather than instruments of record of movement in a game-like activity. Children who created modified number lines with one continuous unidirectional sequence of counting numbers exhibited applications of linear measurement similar to a ruler with which they most likely were familiar. These children disregarded the context of the tasks in which they had previously engaged. The referent point, “home” or stone, carried little mathematical significance of origin as they assigned numbers to the houses or steps/spaces. More often in the horizontal number

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lines than in the vertical number lines, the referent position was labeled or referred to as

zero. The number lines resembled rulers; and, like reading a ruler, the children referred

to the referent points as five, six, or eight. Such actions are similar to the way young children frequently read the positional number on the measuring tool regardless of reference of the endpoint (Kamii, 1995; Gravemeijer, 1994).

In another unpublished study with young children of the same age, this researcher

found that the children had little trouble measuring with a non-standard unit measuring tool that was not labeled with numbers or endpoints. These children successfully measured lengths on the tools from any point of reference on the tool. The critical difference in understanding was that they were focusing on units of iteration rather than numbers or endpoints. Misconceptions of linear measurement may affect how young children perceive a number line that includes positive and negative integers and zero.

Additionally, the way in which the number line is taught may provoke negative attitudes toward using the number line and may be detrimental to the development of integer operations later in children’s mathematics education (van den Heuvel-Panhuizen, 2008).

The findings from this current study raise the question of how and when to teach linear measurement.

Surprisingly, the children in this study did not associate the vertical number lines with a very common application of negative number – the thermometer. Not one child spoke of this association. No child mentioned the temperature of negative two as being two degrees below zero. The researcher was reminded by some preservice teachers that today children read temperatures in digital notation rather than from a vertical

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thermometer. With this fact in mind, educators may consider the effectiveness of

introducing negative number concepts using the number line model, especially a vertical

number line like a thermometer.

Like the children in this study, Turkish children (age seven to twelve) found

negative numbers too abstract to conceptualize as real-life models and failed to relate negative numbers to a number line context (Altiparmak & Ozdogan, 2009). Failure to make connections led the children to memorize rules of operations. More effective number line models to consider for use with integer operations may be the empty number line (Bobis & Bobis, 2005; Klein, Beishuizen, & Treffers, 1998; Freudenthal, 1973; van den Heuvel-Panhuizen, 2008) and a number line with the referent N rather than zero

(Carraher, Schiemann, and Earnest, 2006). The empty number line model was originally created as a tool for teaching calculations with numbers up to 1000, but could be adapted to integer addition and subtraction. The number line with referent N focuses on the count or movement in either direction from an arbitrary point and is effective for developing concepts of negative numbers.

Contextual Situations as Interactive Stories

The findings from this study affirmed that learners of mathematics, including young children as well as early mathematicians, need to learn and do mathematics in contextual situations or story-like contexts. “A narrative, thus, portrays an individual thought process... and provides an opportunity in knowing young children’s perspectives, their naïve theories of negative numbers.” (Mukhopadhyay, 1997, p. 48) “Constructing stories in the mind… is one of the most fundamental means of making meaning, as such

130 it is an activity that pervades all aspects of learning.” (Wells, 1986, p. 194) Children of all ages in the study demonstrated the need to learn mathematics in context rather than as manipulation of mathematical symbols. “A context in which the learners express their ideas and discuss the matter comfortably eliminates the concerns about mathematics.”

(Altiparmak & Ozdogan, 2009, p. 45) When the children associated the tasks with real- life situations or stories for which they had an experiential reference, they demonstrated more depth of conceptual understanding of negative number or deficit context.

“Mathematics is ultimately a tool applied in context. Making sense of the context, the story, is essential to making sense of the mathematics.” (Swanson, 2010) Educators should not assume that young children can associate negative and positive numbers with debts and assets; young children may not have experience or knowledge of debts and assets (Mukhopadhyay, 1997). Children must be encouraged to construct their own associations of deficit in simple everyday situations. The findings of this study are consistent with findings from Mukhopadhyay, Resnick, and Schauble’s study (1990) of children aged seven through twelve and unschooled boys from India aged ten through thirteen. These children demonstrated more understanding of negative and positive number calculations when the problems were presented in the context of a story or a familiar social situation. Mukhopadhyay et al. explain,

Since cognition or the knowledge developed is correlated to the situations that

promote and nurture sense-making for a given concept, one finds the study of

situations an essential element of understanding and knowledge development. In

a problem solving situation, the problems are attempted and solved, ideas are

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formed, and concepts are developed via the use of language and conversations,

either with self or with others present in the context. (1990, p. 37)

Although not by design of this study, almost all of the mathematics done in this study was done mentally by the children. Only one of the children asked to write out the mathematics problems. Instead, the children related similar personal experiences with money or collections or presented alternative solutions to avoid the dilemma of deficit.

Several of the children became quite animated with the pigs, frogs, and bears and still

remained on task. In these problem-solving situations, the children used language as in story-telling or communication as in acting out as a “tool for rich investigations”

(Mukhopadhyay, 1997). Taking ownership of the problem helped the children define their understanding or present misunderstandings. “A dialogue between a child and adult helps the child find a voice of his or her own and establish cognitive empowerment, an absolute necessity of one’s intellectual development.” (Mukhopadhyay, 1997, p. 48)

The children were more likely to relate to a story during the collection activities than activities with the modified number lines, and they related more stories in the money context than in the collection context. Most children between the ages of four and eight have some experiences with money, thus, the children in this study related real experiences with money (e.g. having to wait to buy a desired toy until enough money was saved or earned). An eighth grade boy in one of Mukhopadhyay’s (1997) studies presented a sound argument for using a money context in stories, “you cannot owe anything other than money.” (p. 44). However, in the number line activities, the children’s apparent unfamiliarity with the number line thwarted their creativity and

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limited many of the children’s stories to replications of the scenarios presented by the

researcher.

Representation of Intangible Numbers

The results of this study suggest that children younger than seven or eight are able

to determine a deficit amount as a mental image but are unable or unwilling to make a

written or pictorial representation of that amount. Furthermore, the young children did

not attempt to retain or record the mental images of deficit for use in their calculations of

composition of sets. Only two children in this study attempted to invent a notation for the deficit or negative number. In the collection activities, the children’s reactions (e.g. laughing at the prospect of having to give away more than they had) suggested that they knew that a number representing a deficit amount is different than a number representing a tangible positive quantity. In the number line activities, the majority of the children showed little evidence of recognizing a difference in moving left or right to a house or up or down to a step/space and responded with positive numbers only.

The findings of this study indicate that young children conceptualize negative numbers in a positive number and tangible context. They knew the deficit amounts in the collection activities, but they perceived them as positive numbers. They knew that they needed, for example, one more bear. One bear is a positive tangible object. To alleviate the deficit situation, some of the children introduced more manipulatives to the set by transferring the needed manipulatives from the supply pile, and others made up stories in which they secured the needed manipulatives. In essence, these children worked in an abstract negative number context by applying tangible positive number principles.

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A few of the children tried to work with the deficit in a more negative context.

SH7 was somewhat familiar with the negative number vocabulary, because he had been

informally introduced to the wording prior to participation in this study. He attempted to

transfer that wording to representative notation. He invented an alpha-numeric symbol

for negative two; he also created iconic representations for negative numbers. His alpha-

numeric symbol, “n 4”, closely resembled the symbol, “m 4”, recorded by ancient

mathematicians (Beery et al., 2004). His iconic representation was emotionally based; he

drew a “Lincoln” penny with a frowning face since he explained that not having enough

money is a sad situation. Although SH7 used the vocabulary for negative numbers and

provided evidence of some sense of deficit with his frowning face, he was unable to use

that vocabulary or notation in computations. He exemplified the potential detriment of

introducing a child to mathematical terminology and concepts prior to the development of

conceptual understanding of the mathematical concept.

NB8 demonstrated the most advanced conceptualization of the relationships among negative and positive numbers and zero. Not only did she invent a written word

representation for the deficit or negative concept, but she successfully used her representations in application of addition and subtraction within context. Her representation of negative numbers, although not traditional symbolism, was logical and functional. If NB8 continues to use her representation of deficit, she will undoubtedly

find it cumbersome in traditional algorithms and strive to construct notation that is more

conducive to algebraic manipulation.

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Rote Learning

In this study, the results of the task in which children were asked to identify

manipulatitives by their ordinal position and of activities in which the children were

asked to create a number line demonstrated the emphasis, or possibly over-emphasis, on rote learning of both cardinal and ordinal number sequences. At a very young age children, even younger than the children in this study, have their first counting experience as a simple recitation of an “unbreakable” chain of words with no contextual meaning

(Fuson & Hall, 1983; Wynn, 1989). These same children who reportedly know how to count from one to ten are frequently unable to use these counting words to demonstrate one-to-one counting of objects. “There is no evidence that children represent the

(necessary) stable-order of the counting words any differently than the (arbitrary and

nonessential) ordering of the letters of the alphabet.” (Wynn, 1989, p. 32) Rote recitation

of the number words does not guarantee that a child can count in a cardinal sense (Van de

Walle, 2001) or recognize positions in an ordinal sense.

Cardinality means that “the child’s response refers to the numerosity of the whole

set of elements presented,” (Berjemo & Lago, 1990, p. 233), but cardinality is not an

innate concept and must be developed through a child’s individual number sense

development (Belman, Meck, & Meike, 1986; Bermejo, 1996; Bemejo, Morales, de

Osuna, 2004; Fuson, 1988). Cardinality and counting are not dependent on each other.

Cardinality should not be considered as a principle of counting, but counting is a means

for determining cardinality; counting is the procedure and cardinality is the result

(Bemejo, Morales, de Osuna, 2004). Rogers (2008) and Wynn (1989) argue that

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three- and four- year-olds, like the four-year-olds in this study, do not have a strong

conceptual foundation of number on which to build cardinal number sense and representation.

The younger children in this study, except one, had advanced beyond the stage of

randomly pointing to objects while rotely counting and counted sets of manipulatives

using one-to-one counting strategies or skip-counting strategies. Only one four-year-old,

CM4, said that there were three bears in the set but recited, “one, two, three, four, five,

six, seven, eight, nine, ten” as he randomly pointed to the bears. He determined the

cardinality of the set of three without counting, possibly by subitizing, but his rote

counting inferred that he had not developed a sense of cardinality of number. Young

children of this age do not necessarily relate numerosity and cardinality (Frye, Braisby,

Lowe, Mancudas, & Nicholls, 1989). Findings of this study suggest that the children

used the last-word-counted strategy for determining cardinality.

In this study, the children, with the exception cited above, provided evidence of a

sense of cardinality for one, two, or three objects. They determined the number of

manipulatives in the given set and affirmed that numerosity by one-to-one counting.

The children encountered more difficulties working with ordinal numbers than

cardinal numbers possibly as the result of a misunderstanding and /or misuse of ordinal

numbers. As with cardinal number names, children learn to recite the sequence of ordinal

number names, “first, second, third” and so on without association to the cardinal number

represented in order of place within a set. Children first experience ordinal numbers in the context of comparison of size and not placement within a row (Bruce & Threlfall,

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2004; Piaget, 1952/1964). Ordinal number abilities in this sense develop as early as in

infancy (Brainerd, 1978; Brannon, 2002; Kingma & Koops, 1981; Piaget, 1952/1964). In

the present study, the researcher asked the children to point to the manipulative

representing a certain ordinal number, and many of the children, especially the four- and five-year-olds, demonstrated the ordinal number association error by simply pointing to the next adjacent manipulative. Bruce and Threlfall (2004) describe the “consecutive ordinal sequencing” (p. 22) as the process of assigning ordinal number names to consecutive positions in a row irrespective of the actual ordinal position. They continue to argue that early experiences with ordinal numbers are limited and mostly involve consecutive positions starting with the first position and that “order is established as a holistic process, without matching specific words to particular positions.” (Bruce &

Threlfall, 2004, p. 20) Many of the children demonstrated no association of place in the row with the ordinal number beyond first and second; they simply pointed to the

manipulative behind or beside the previously identified manipulative. These children in

the study provided little evidence of association between the cardinal number name for a

particular manipulative and its ordinal number name. All of the four-, five-, and six-year- olds and a few of the seven- and eight-year-olds mistakenly identified the third manipulative in the row as the fourth when asked to point to the “first”, “second”, and

“fourth” manipulative in the row. The older children in the study, however, had established some cardinal and ordinal number association and were more successful in identifying the manipulative that associated with the ordinal number stated. They also were able to verify their actions in context of cardinality. Some of these older children

137 started to point to the third manipulative as the fourth, but hesitated and then pointed to the fourth.

All of the children in this study demonstrated more understanding in the tasks involving cardinality than ordinality. They successfully determined the numerosity of a set of manipulatives and assigned a cardinal number value to the set. However, further research is necessary to definitively determine whether the children had developed an authentic conceptualization of cardinality. Development of a sense of cardinality could be explored further by asking the children to determine the quantity of the set by counting from two or by counting backwards. If the children still determined the numerosity of the set by the cardinal number rather than the last word counted, then their sense of cardinality would be more verifiable (Bermejo & Lago, 1990).

This pattern of understanding of cardinality before ordinality in this study contradicts early research in which children developed an understanding of ordinality prior to an understanding of cardinality (Brainerd, 1978; Kingma & Koops, 1981; Piaget,

1952/1964). Such research, however, focused on the concept of ordinality in regard to

“sequences of numerosity” (Brannon, 2002, p.238) or comparisons of characteristics of objects or sets of objects (Brainerd, 1978; Kingma & Koops, 1981; Piaget, 1952/1964).

More recent researchers argue that ordinal number development follows cardinal development (Bruce & Threlfall, 2004; Michie, 1985). Bruce & Threlfall (2004) credit social experiences at home and in preschool for development of both cardinal and ordinal number sense and further argue that adult attention to cardinal and ordinal numbers affects the order of development.

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This contradiction of ordinality and cardinality development stems from the

working definition of ordinality. The ordinal number sense in this study developed from

the definition of ordinal numbers as names of objects within a set as determined by their

relative placement rather than characteristics of size or value. Piaget’s (1952/1964)

studies of ordinality progressed from ordering by comparison of characteristics of set or

object to ordering by placement within a set. Piaget’s preliminary study of ordinality did not necessarily involve number or cardinality.

Summary of Analysis

In regard to the first research question of this study, the findings affirm that young

children between the ages of four and eight do conceptualize relationships among

positive and negative numbers and zero. While the children demonstrated conceptual

understanding primarily with positive numbers and zero, they presented varied levels of

understanding of the concept of negative numbers. The children demonstrated

understanding of cardinality, ordinality, and counting principles, composition and

decomposition of sets, as well as a preliminary understanding of the concept of additive

inverse.

The children in this study provided evidence of understanding cardinality of number more than ordinality. These children demonstrated a rote learning of the sequence of ordinal numbers without substantial prior association with cardinal numbers.

The children’s difficulties with ordinal numbers transferred to their difficulties with understanding the modified number lines.

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In a positive number sense, the children in this study demonstrated little difficulty in composing and decomposing sets with concrete objects in the collection activities.

Relationships among only positive numbers were not problematic for the children in the study. When the children attempted to apply only positive numbers to the number lines, they encountered pertubations. The children assigned numbers to the number lines using their understanding of positive numbers sequencing and cardinality with disregard to direction. They could not recognize nor explain the difference between moving one space left and one space right of the referent. Even the children who distinguished a directional difference of number used words connected with positive numbers, e.g. “1 Down”.

While their conceptualization of the number line surpassed most of the other children in the study, these children used representation analogous to interpreting “– 2” as the opposite of positive two rather than negative two.

Relationships with zero were studied in two aspects – zero as a quantity and zero as a referent. The children had little difficulty recognizing zero as a lack of quantity in the collection activities and described that amount as “none” or “nothing”. They also recognized that a series of opposite movements resulted in no movement in the number line activities. Although they did not use traditional notation for opposites, the children provided evidence of preliminary conceptual understanding of the additive inverse principle. Knowledge of traditional notation for opposites may play a critical role in the development of the additive inverse concept. The findings of this study did not suggest that the children conceptualized zero as a referent in the number line activities. Only three of the children recognized that direction of movement made a difference. The

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others, when asked to make two moves, argued that the direction was insignificant, e.g.

moving left two houses or moving right two houses was essentially the same, a movement of two houses. Opposite movements or actions made sense to the children, but opposites as numbers were not recognized nor sought. As they constructed the vertical and horizontal number lines, many of the children failed to assign any significance to zero as a point of origin or referent.

Since most of the children did not create working notations for negative numbers, they were unable to construct relationships among only negative numbers. The children conceptualized negative numbers in context as positive numbers or zero, as in lacking a positive quantity or having a quantity represented by an empty set. Even the one child who demonstrated some foundational understanding of operations with negative numbers represented the deficit amount in reference to a positive quantity.

In reference to the second research question, the children used verbal or numeral representation more than pictorial or iconic representation. Most of the children represented only positive numbers and zero and did not acknowledge the existence of negative numbers. Only two of the children attempted to represent negative numbers through written words, iconic representation, or alpha-numeric notation. All of the children verbalized a mental image of the deficit but were reluctant to make a visual representation.

Only one eight-year-old child successfully demonstrated conceptual understanding using her own written representation for negative numbers. For her number lines, she stated that she only had to use notation to differentiate the numbers

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below the stone or left of the home on the number lines. She argued that familiar

counting numbers without any designation of position were sufficient for the numbers

above the stone or right of the home. For the collection activities she maintained records

of quantities as positive numbers and deficits as words and (positive) numbers.

Although the other children did not create working notation for negative number

contexts and thus demonstrated difficulty in constructing relationships among positive

and negative numbers or two negative numbers, they attempted to make sense of the

negative number situations by telling stories. Through their stories, the children problem-

solved to alleviate or eliminate the negative number context. More children told stories for the collection activities than for the number line activities suggesting that activities with concrete objects carry more mathematical meaning to young children.

The children in this study experienced a mathematical journey similar to the

historical journey by mathematicians during the evolution of negative numbers. Like

early mathematicians, some of the children totally disregarded the need for negative

numbers and argued the finiteness of zero. Mathematicians, historically, and young

children, presently, made similar arguments about the existence of negative numbers.

Both the mathematicians and the young children experienced a quest for meaningful and functional negative number notation, and both early mathematicians and these young children constructed meaning of negative numbers through stories or narratives.

Limitations of the Study

This study was qualitative in nature exploring whether and to what extent young children conceptualize relationships among positive and negative numbers and zero. The

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study focused on how children made sense of deficit situations. Therefore, the dialogs

between the researcher and the children and the explanations given by the children were

paramount to the counting and enumeration procedures demonstrated throughout the interviews. Within this qualitative study, social aspects, interaction time, age disparity, and sample characteristics created some limitations.

The qualitative nature of this study compelled the researcher to collect data from

not only observations but also from discussions in which the children shared their

insights and intuitions about the concepts. Many of the children were uncomfortable with

or unaccustomed to explaining their thinking. The children argued that they “just knew”

or shrugged their shoulders and/or stated that they didn’t know what prompted their

answer. Therefore, the self-as-social-object problem (Hatch, 1990) created a certain

limitation with this study.

The models for the modified numbers lines may also have limited the children’s

responses and elicited misunderstandings. The “home” house and the stone were

intended to represent the referent or zero. Many of the children of all age groups

demonstrated confusion as to whether to include the “home” or stone. The children did not visualize the discrete objects and numerical representation of distance.

Another limitation of the study was the short amount of time spent with each

child. Each session lasted approximately one hour. The tasks were diverse enough and

interesting enough to keep most of the children engaged for this period of time, but

evidence of boredom and fatigue transpired in some children prompting the researcher to

curtail those interviews. More frequent and shorter sessions may have facilitated more

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engagement from the children and more in-depth questioning by the researcher. Two

separate sessions would have provided for isolated experiences with number line

connotations and collection experiences.

Another limitation of the study was the disparity of age of the children. Some of

the tasks were too complex for the youngest children and too mundane for the oldest

children. Several of the youngest children were unable or unwilling to make

representations for tasks involving modified number lines or recording quantities in

collection activities. While, in this study, the wide age range of development of number

sense presented a perspective of growth, a similar study with a more homogeneous age

group may yield more in-depth insights into children’s conceptualizations of number,

negative numbers in particular.

The study was limited also by the use of a convenient sampling of children who volunteered to participate. Although gender representation was addressed by having at least two of each gender in each age group, ethnic, racial, cultural, and educational representation was not addressed in selecting children for the study. Educational factors including mathematics aptitude and the source of education (i.e. public or home schooling) were noticed by the researcher but were not included in the selection process nor in the analysis of data. All of the children were Caucasion, but socio-economic status of families was not determined.

Implications of the Study

The results of this study clearly indicate that children as young as four years can develop an understanding of the concept of deficit in a collection situation. Even the

144 youngest child in the study determined how many more manipulatives were needed when he did not have enough to complete a set. Understanding the concept of deficit is a precursor to the understanding of the abstract concept of negative numbers. Teachers and parents and other influential people in a child’s life should seize every opportunity for the child to experience the concept of deficit. Experiences need not be extensive. Children encounter situations of deficit in their daily lives at home and in school (e.g. not having enough money to buy a snack or toy). Slow and careful exposure to the negative number concept will better prepare the child for operations with negative numbers later in the mathematics education curriculum. Children, like the early mathematicians, need time to process this new and abstract concept.

While children are developing an understanding of the negative number concept, they also need to understand the number zero in two contexts, quantity and reference point. Although zero appears to be a simple number, zero presents numerous contradictions in usage. Adding or multiplying by zero does not make a larger number.

Dividing by zero is undefined. Sometimes we write the zero; sometimes we eliminate it.

If zero is the enumeration of a quantity as the children argued, then is zero the finite end of a number system? Children wrestled with this idea. Modeling on the number lines is comparable to determining linear measurement for which children experience similar difficulties. The zero on the number line is similar to zero in linear measurement. The children were perplexed as to whether to include the starting and /or ending houses/spaces or to simply count the houses/spaces between. Children who explore linear measurement with unnumbered non-standard unit measuring tools focus on unit

145 iteration rather than number labels. Using unnumbered measuring tools also aids the children in establishing a referent unit or zero (NCTM, 2000). Connecting the number line with a linear measuring instrument raises the question of how and when to teach linear measurement.

Findings from the study indicate that young children develop cardinality of number before ordinality of number. The children experienced more difficulty with the tasks involving modified number lines; counting discrete objects like houses did not register to the children as counting on a number line. Thus, instruction of negative number operations should begin with collection-type activities rather than number line activities. In the traditional mathematics curriculum, negative numbers are often introduced via a number line. Learners, both children and adults, are expected to focus on the notation and placement of negative numbers on a number line without conceptual development of negative numbers. Educators should structure a curriculum in which investigation of the concept of negative numbers in a collection context precedes the introduction of negative numbers in the number line context.

By the age of seven or eight years, children should be encouraged to try to represent the numbers encountered in situations of deficit. The children in the study constructed meaning of abstract mathematical concepts with a mental or verbal representation prior to using traditional notation. If the children are better able to deal with the deficit amount within the context of a story or situation, then their own written or drawn notations should be acceptable. As children construct knowledge of new and/or abstract concepts, they should be encouraged to invent their own personally-meaningful

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representation of the concept. Such mathematical representation is not unlike children

expressing themselves with invented spelling. Children who work with their own

invented representation may more easily focus on the concepts than struggle with

negative number notation which is indistinguishable from subtraction notation. Teachers

should embrace each child’s pictorial, iconic, or verbal representation although it may

deviate from the traditional notation. If mathematicians struggled for hundreds of years

to create an appropriate and functional notation for negative numbers, then children

should be afforded an appropriate time for notation development as well.

Young children should be encouraged to express themselves mathematically and

present mathematical concepts in the context of familiar real-life situations. Explaining mathematics concepts through narratives or story-telling strengthens the child’s understanding of the concept. Early mathematicians also worked with problems of deficit situations in prose long before they settled on negative number notation for calculations (Beery et al., 2004). Arguably, those mathematicians solved more complex problems than these children, but the approach of developing contextual understanding before procedural understanding with algebraic notation is common to both groups.

The above implications challenge educators to consider the pedagogy of mathematics education for young children. Presentation of new concepts as explorations or story problems prior to introduction of algorithms allows the child to construct his or her own knowledge of the concept. After foundational knowledge of a new concept is constructed the child is more able to transform that knowledge to use of a more traditional notation or algorithm.

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Current textbooks do not support this curriculum. Problem-solving usually

follows introduction of a concept or procedure. Textbooks publishers should consider

presenting problems involving concepts with which the children are not familiar prior to

process. Children need to develop an understanding, however flawed, before attempting

to apply a process. Granted, the curriculum need not necessarily be driven by the

organization of the textbook. So the issue is one of the pedagogy of mathematics

education rather than textbooks or curriculum.

In response to a call for introductory problems for abstract algebraic concepts, textbook publishers may include problems that challenge children to think beyond the concepts presented in the traditional mathematics curriculum. Inclusion of real-life situations (e.g. a child not having enough money to buy a bottle of juice for lunch) coerces the child to consider the options and then determine how to handle the situation.

However, the challenge problems may pose an even bigger threat to conceptual development of negative numbers as teachers ignore, misinterpret, or attempt to “teach” the concepts rather than encourage exploration and conceptual development.

To summarize, the findings from this study are not convincing that the concept of negative numbers or representation of negative numbers should be formally introduced to children of ages four through eight. Furthermore, with all the other mathematical concepts that young children must understand, the question becomes whether adding the concept of negative numbers to the preschool through second grade curriculum is a justifiable demand on learning time. However, children within this age group will benefit from experiencing negative number concepts in contextual situations appropriate for

148 young children. They should be given ample opportunities to investigate perturbations to their established number sense and encouraged to express those discrepancies through verbal, pictorial, iconic, or other means.

Recommendations for Future Research

More research should be conducted on how young children conceptualize the relationships among positive and negative numbers and zero with emphasis on negative numbers and zero. The research should focus on how children explain encounters with deficit amounts rather than how children adapt to certain models of operations on integers or procedures dealing with negative numbers and zero. The research should be conducted with groups of children within small age bands to allow for more depth in the study.

Further research is needed on the relationship of linear measurement and integer number lines. Since the number line is one of the models on which operations of integers is structured, children should have a sound understanding of the linear measurement prior to learning negative numbers with a number line model. The children in this study had much difficulty discerning whether to count the starting or ending space in the modified number line activities. Such difficulties may have resulted from misunderstandings or lack of understanding about linear measurement.

More research should be conducted on how playing games or conducting activities that include situations of deficit affect the development of the concept of negative numbers. The children in this study appeared to respond well to collection type

149 activities that involved deficits or the concept of negative numbers suggesting that activities involving positive and negative quantities may be beneficial.

Overall, further research with young children should focus on their developing conceptualization of the negative number concept rather than introducing negative number operations with certain models of integer operations. Research with emphasis on real-life experiences or game-like activities that create contextual negative number encounters should be conducted. Children need some conceptual understanding of an abstract concept prior to attempting calculations with traditional notation.

APPENDICES

APPENDIX A

Activity One: Streets with Houses Modified Horizontal Number Line

Appendix A

Activity One: Street with Houses Modified Horizontal Number Line

For this activity the children will move a playing piece along the number line modified as a street. The children will be asked to select one cut-out house to be their “home” and locate it on a strip of poster board designated as a street. The children will line up five or more houses of different colors and shapes to the right and/or left of their house. Questions similar to the following will be posed to each child. Additional questions may be asked to address each individual child’s interest, insight, and conceptualization of the relationships among the positive and negative numbers and zero.

Task One: Moving on the Right Side of the Initial Reference Point Q1: Please put your marker on your house. Now move to the blue (second right) house. How many houses did you move from your house? How do did you decide that? Q2: Now move your marker from the blue house to the green (fifth right) house. How many moves are you from the blue house? From your house? Please explain your numbers. Q3: Now from the green house move to the brown (third right) house. How many moves are you from the green house? From the blue house? From your house? Are you sure? Q4: From the brown house, move to the white (first right) house. How many moves are you from your house? From the green house? Can you check that number? Q5: If you start at the white house, how many houses would you have to move to go to the yellow (fourth right) house? To the green house? To your house? Can you explain how you decided on that number?

Task Two: Moving Playing Piece to Address Cancellation (Adding to make Zero) Q1: Please move your marker to your house. Now move your token to the blue (second right) house. How many houses did you move from your house? Q2: From the blue house, now move to the yellow (fourth right) house. How many houses would you have to move to go to the blue house? To your house? Move to your house. Please explain your answers. Q3: From your house, move to the white (first right) house. How many houses did you move? How many houses would you have to move to go to your house? Can you explain how you got that number? Move to your house. Q4: From your house, move to the green (fifth right) house. How many houses did you move? How many houses would you have to move to your house? Are you sure? Q5: Can you give me another example of moving your marker from your house to another house and ending at your house? Can you show me how you would make those moves?

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Task Three: Numbering the Houses Q1: To help us talk about the houses, could you give numbers to each house? What number do you want to put on the white house? Blue house? Brown house? Yellow house? And the green house? Why? Q2: What number would you put on your house? Why? Q3: Let’s suppose another house is built on the other side of your house. Please move to that house. What number would you put on that house? And the house beside that one?

Task Four: (Assuming the houses are numbered with the child’s notation) Combining Positive and Negative Numbers and Zero Q1: Place your marker on your house. Please move your piece to the brown house. How many houses did you move? Now move to the red (second left) house. How many houses did you move from the brown house? Can you show me how you got your answer? Q2: Move to the (fourth left) house. How many houses did you move from the gray (first left) house? How many houses are you from your home? Can you make sure? Q3: Please move back to your home. Now move three houses from your house. (No direction given.) Q4: (After child moves marker). How did you know to go that way? (If the child does not move the marker, ask him or her to explain why.) Q5: Place your token on the gray house. Now move your marker to the white house. How many houses did you move? Can you count those houses?

Questions will continue with respect to the child’s notation and conceptualization of the number line.

Questions may include: Q1: If the red house is two moves from your house and the blue house is two moves from your house, how do you know which house to move to if I ask you to move two houses from your house? Q2: If the child numbered his or her house as 0, how many houses do you move from the blue house (second right) to the red house (second left)? Why do you count your house? Q3: Will you please give me directions to get from the yellow house (fourth right) to the red house (second left)? From the brown house (third right) to the green house (first right)?

APPENDIX B

Activity Two: Ladder from Cave to Treehouse Modified Vertical Number Line

Appendix B

Activity Two: Ladder from Cave to Treehouse Modified Vertical Number Line

For this activity the children will move markers along the number line modified as a ladder. The children will be asked to place their markers on the stone located at ground level. The ladder will have six steps up to the treehouse and six steps down to the cave. The following questions will be posed to each child. Additional questions may be asked to address each individual child’s interest, insight, and conceptualization of the relationships among the positive and negative numbers and zero.

Task One: Moving Up From the Initial Reference Point Q1: Please put your marker on the ladder at ground level. Now move three steps toward the treehouse. How many steps/spaces did you move from the stone? Will you count them please? Q2: From where you are now, move your marker three more steps toward the treehouse. How many steps are you from your last step? From the stone? Can you show me how you got those numbers? What step are you on now? Q3: From the step your marker is on now, move four steps toward the ground. How many steps from your last step? From the treehouse? From the ground? Q4: If you start at step four, how many steps would you have to move to go to step six? To step three? To the ground?

Task Two: Moving Playing Piece to Address Cancellation (Adding to make Zero) Q1: Please move your marker to the step at the ground. Now move your marker two moves toward the treehouse. How many steps did you move from the step at the ground? Q2: From this step (two), now move to this step (four). How many steps would you have to move to go to this step (one)? To the step at the ground? Please explain your answers. Move to the step at the ground. Q3: From the step at the ground, move to this step (one). How many steps did you move? How many steps would you have to move to go to the step on the ground? Will you please count your steps? Move to the step at the ground. Q4: From the step at the ground, move to this step (four). How many steps did you move? How many steps would you have to move to the step at the ground? How do you know? Q5: Can you give me another example of moving your marker from the step at the ground to another step and then ending at the step at the ground? Will you show me what you would do?

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Task Three: Numbering the Steps or Spaces Q1: To help us talk about the steps on the ladder, could you give numbers to each step/space? What numbers will you use on the steps/spaces towards the treehouse? Why did you choose to use those numbers? Go ahead and write those numbers on the poster board strip that is beside the ladder. Q2: What number would you put on the step/space at the ground? Why? Q3: Now let’s look at the steps/spaces of the ladder going towards the cave. What number would you put on the step that is one step/space closer to the cave? And the next step/space closer to the cave? Why did you decide on those numbers?

Task Four: (Assuming the steps/spaces are numbered with the child’s notation) Combining Positive and Negative Numbers and Zero Q1: Place your marker on the stone. Please move your marker two moves closer to the cave. How many moves did you make? Now move three moves closer to the cave. How many moves did you make the stone? From this step (two below the stone)? Can you count out your answers? Q2: From that step/space, move to this step (six). How many moves did you make from this step (one)? How many steps/spaces are you from the stone? Are you sure? Q3: Please move back to the stone. Now move three steps from the stone. (No direction given.) Q4: (After child moves token). How did you know to go that way? Q5: From that step/stone move to this (opposite) step/stone. How many moves did you make? Count them out please.

Questions will continue with respect to the child’s notation and conceptualization of the number line.

Q1: Why do we have to use different numbers for the steps/spaces above and below the ground? Q2: Can you explain how you numbered the steps/spaces for the previous activity and the steps for this activity? Does it matter? Q3: If we wanted to extend the ladder what number would you put on the next step/space after the treehouse? What number would you put on the next step/space after the cave? Why would you use those numbers?

APPENDIX C

Activity Three: Collecting Bears, Pigs, or Frogs

Appendix C

Activity Three: Collecting Bears, Pigs, or Frogs

For this activity each child will be asked to account for a collection of bears, pigs, or frogs as he or she is instructed to get a number of bears or lose a number of bears. Each child will be asked to count out a set of ten bears. The child will take bears, pigs, or frogs from the set and put bears, pigs, or frogs back in the set. The following questions will be posed to all children. Additional questions may be asked to address each individual child’s interest, insight, and conceptualization of the relationships among the positive and negative numbers and zero.

Task One: Composing and Decomposing a Set of Counters For simplicity, the following questions are written with bears only although the child may be working with pigs or frogs. Q1: How many bears are in the pile? How many bears do you have? Q2: You may take three bears from the set. Now how many bears do you have? How do you know? Q3: Take one more bear from the pile. Now how many bears do you have? How do you know? Q4: Now put two bears back in the pile. How many bears do you have? How do you know? Q5: Put three bears back in the pile.

If the child says that he or she has zero, questions like these will follow. Q6a: Why are there zero bears? Q7a: If you get four more bears, now how many bears do you have? Are you sure? Q8a: If other children were collecting bears, could it be fair for you to say you have zero? How is that fair? Similar questions will follow.

If the child indicates that he or she needs more bears to put back the three bears, questions like these may follow. Q6b: You say that you need one more bear to put back three bears. Is there any way that you could keep track of that number? Why did you choose that way to remember that number? Q7b: Now take four bears from the pile. How many bears do you have now? Count them. Do you think about the one bear that you needed before? How does that affect the number of bears in your pile? Similar questions will follow.

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Task Two: Accounting for the Number of Bears in the Set Q1: For this activity will you keep a written account of how many bears you have? Write the number of bears you have now. (Zero). Q2: Get three bears from the pile. Now how many bears do you have? Are you sure? Q3: Now get two more bears from the pile. How many bears do you have now? How do you know? Can you write how many bears you have now? Q4: Please put four bears back in the pile. How many bears do you have now? How do you know that? Can you write how many bears you have now? Q5: Please put two bears back in the pile. How many bears do you have now? Are you sure? Can you write how many bears you have now? Does that make sense to you? Q6: Now take three bears from the pile. Now how many bears do you have? Is that a fair count? Please write the number of bears you have now.

Questioning will continue with more questions of a similar nature. The children will be observed for consistency or adaptation of their sense-making of the deficit quantity.

APPENDIX D

Activity Four: Collecting Pennies

Appendix D Activity Four: Collecting Pennies

For this activity the children will be asked to account for a collection of pennies as they are instructed to get a number of pennies or give back a number of pennies. Each child will be asked to count out a set of ten pennies. The child will take pennies from the set and put pennies back in the set.

The following questions will be posed to each child. Additional questions may be asked to address each individual child’s interest, insight, and conceptualization of the relationships among the positive and negative numbers and zero.

Task One: Composing and Decomposing a Set of Pennies Q1: Please count the number of pennies in the pile. How many pennies do you have? Q2: You may take four pennies from the set. Now how many pennies do you have? Can you count them? Q3: Take one more penny from the pile. Now how many pennies do you have? Can you count them? Q4: Now put three pennies back in the pile. How many pennies do you have? Count them please? Q5: Put three pennies back in the pile.

If the child says that he or she has zero, questions like these will follow. Q6a: Why are there zero pennies? Q7a: If you get four more pennies, now how many pennies do you have? Are you sure? Q8a: If other children were collecting pennies, could it be fair for you to say you have zero? How is that fair? Similar questions will follow.

If the child indicates that he or she needs more pennies to put back the three pennies, questions like these will follow. Q6b: You say that you need one more penny to put back three pennies. Is there any way that you could keep track of that number? Why did you choose that way to remember that number? Q7b: Now take two pennies from the pile. How many pennies do you have now? Count them. Should you think about the one penny that you needed before? How does that affect the number of pennies in your pile? Similar questions will follow.

Task Two: Written Representation for the Number of Pennies in the Set

Q1: For this activity will you keep a written account of how many pennies you have? Write the number of pennies you have now. (Zero). 161

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Q2: Get three pennies from the pile. Now how many pennies do you have? Count them. Q3: Now get two more pennies from the pile. How many pennies do you have now? How do you know? Can you write how many pennies you have now? Q4: Please put four pennies back in the pile. How many pennies do you have now? How do you know that? Can you write how many pennies you have now? Q5: Please put two pennies back in the pile. How many pennies do you have now? Are you sure? Can you write how many pennies you have now? Does that make sense to you? Q6: Now take three pennies from the pile. Now how many pennies do you have? Is that a fair count? Please write the number of pennies you have now.

APPENDIX E

Consent Form for Parent of Minor

Appendix E

Consent Form for Parent of Minor

Young Children Conceptualize the Relationships Among Positive and Negative Numbers and Zero

Your child is being invited to participate in a research study. This consent form will provide you with information on the research project, what your child will need to do, and the associated risks and benefits of the research. Your child’s participation is voluntary. Please read this form carefully. It is important that you ask questions and fully understand the research in order to make an informed decision. You will receive a copy of this document and the assent form from your child.

This study is being conducted to explore how young children work with positive and negative numbers through games. Since the children in this study are younger than the children who traditionally work with negative numbers in the current mathematics curriculum, the information gathered from this study may provide insight for teachers, administrators and local, state, and national education policy makers into mathematics curriculum to facilitate students’ learning of algebraic principles involving operations with negative numbers.

Your child will be asked to participate in game-like activities that involve making sense of number relationships. Your child will be asked to account for the gain or loss of bear-shaped counters and to account for movement along modified number lines. Your child will be interviewed during the activity and be asked to make written representation of his or her understanding. Your child will be asked to meet for two half-hour or less sessions. Sessions with your child will be video-recorded. Your child’s written work will be kept for analysis. Following the completion of the data collection, the video- recordings will be transcribed for use in data analysis. Upon completion of this study, all video-recordings and written work will be destroyed.

Your child may not benefit directly from this study. Your child may incidentally benefit by the insight into number relationships that he or she may gain from participation in this study.

There are no anticipated risks to your child beyond those encountered in everyday life.

Anonymity will be maintained though the use of initials of the child with a subscript of the year, e.g. PM5, for each child in transcripts and narratives included in the written presentation of the study. Consent forms and assent forms will be stored in a

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locked file in Office 50 on the Kent State University Regional Campus at Salem. The study-related information from your child will be kept confidential within the limits of the law.

Taking part in this research study is entirely up to you and your child. You or your child may choose not to participate in the study. You or your child may decide to discontinue your child’s participation at any time, and no one will hold that decision against either of you.

If you have any questions or concerns about this research, you may contact Peggy Manchester at 330.337.4264. This project has been approved by the Kent State University Institutional Review Board. If you have any questions about your child’s rights as a research participant or Kent State University’s rules for research, you may call Dr. John L.West, Vice President and Dean, Division of Research and Graduate Studies, at 330.672.2704.

You may keep a copy of this consent form.

Sincerely,

Peggy Manchester, Mathematics Instructor

I have read this consent form and have had the opportunity to have my questions answered to my satisfaction. I voluntarily agree to allow my child to participate in this study.

______Parent’s Signature Date

APPENDIX F

Assent Form for Minor

Appendix F

Assent Form for Minor

Young Children Conceptualize the Relationships among Positive and Negative Numbers and Zero

1. Hi, [child's name].

2. My name is Peggy Manchester, and I am trying to learn more about how young children think about numbers.

3. I would like you to participate in some activities and share your thoughts in words and writing. I will be video-recording you as you take part in the activities and will collect any written work that you do so that I may look at it later.

4. Do you want to do this?

5. Do you have any questions before we start?

6. If you want to stop at any time, just tell me.

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APPENDIX G

Audio/Videotape/Photograph Consent Form

Appendix G

Audio/Videotape/Photograph Consent Form

I agree to video taping of my child at______on______Signature Date

I have been told that I have the right to see the video tapes of my child before they are used. I have decided that I: ____want to see the tapes ____do not want to see the tapes

Sign now below if you do not want to see the tapes.

If you want to see the tapes, you will be asked to sign after seeing them. Peggy Manchester and other researchers approved by Kent State University may/may not use the tapes made of my child. The original tapes or copies may be used for: _____this research project _____teacher education _____presentation at professional meetings

______Signature Date Address:

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APPENDIX H

SH7 Record and Representation

Appendix H

SH7 Record and Representation

171

APPENDIX I

NB8 Record and Representation

Appendix I

NB8 Record and Representation

173

APPENDIX J

CH8 Record and Representation

Appendix J

CH8 Record and Representation

175

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