YOUNG CHILDREN CONCEPTUALIZE RELATIONSHIPS AMONG POSITIVE AND NEGATIVE NUMBERS AND ZERO A dissertation submitted to the Kent State University College and Graduate School of Education, Health, and Human Services in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Peggy D. Manchester May 2011 © Copyright, 2011 Peggy D. Manchester All Rights Reserved ii A dissertation written by Peggy D. Manchester B. S., Youngstown State University, 1978 M. Ed., Kent State University, 1996 Ph.D, Kent State University, 2011 Approved by ___________________________, Co-director, Doctoral Dissertation Committee Michael Mikusa ___________________________, Co-director, Doctoral Dissertation Committee Anne Reynolds ___________________________, Member, Doctoral Dissertation Committee David Dees Accepted by ___________________________, Director, School of Teaching, Learning and Alexa L Sandmann Curriculum Studies ___________________________, Dean, College and Graduate School of Education, Daniel Mahony Health, and Human Services iii MANCHESTER, PEGGY D., Ph.D. May 2011 Curriculum and Instruction YOUNG CHILDREN CONCEPTUALIZE THE RELATIONSHIPS AMONG POSITIVE AND NEGATIVE NUMBERS AND ZERO (191 pp.) Co-Directors of Dissertation: Michael Mikusa, Ph.D. Anne Reynolds, Ph.D. 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. iv 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. v 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 . ..

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