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The Comparison of Physical/Virtual Manipulative on Fifth-Grade Students’ Understanding of Adding Fractions

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The Comparison of Physical/Virtual Manipulative on Fifth-Grade Students’ Understanding of Adding Fractions Running Head: COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE The Comparison of Physical/Virtual Manipulative on Fifth-Grade Students’ Understanding of Adding Fractions A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Education in the Department of Curriculum and Instruction of the College of Education by Sami Alshehri M.A. Umm Al-Qura University July 2008 Committee Chair: M. Sally, Ed.D. COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE ABSTRACT The purpose of this quantitative study was to compare two types of manipulatives in order to see their effects upon understanding of adding fractions for three comparable groups of fifth grade students. A total of 163 students who demonstrated low mathematical performance participated in the project in order to learn the addition of fractions by using physical and virtual manipulatives for the experimental groups and the normal mathematic curriculum for the control group. The intervention occurred during a two-week time frame in six public elementary schools in Abha, Saudi Arabia where students used fraction bars for both physical and virtual manipulatives in order to build conceptual understanding of adding fractions properly. Instructions were provided to all the participants directly regarding what the participants were to do each day of the 2-week experiment. Pre-and post-tests, an attitude survey, and a preference survey were the instruments that were used to collect data during the study. A repeated measures design with a cross over treatment was used for comparing the effects of the two modes of treatments, virtual and physical manipulatives, compared to a control group for the understanding of adding fractions for the three groups of students. Overall findings revealed that fractions performance differed significantly as a function of use of manipulative, F(4, 320) = 506.49, p < .001, η2 = .86. Also, findings revealed that fractions performance was significantly better after students were exposed to either virtual or physical manipulatives. In addition, results indicated that fractions performance was significantly better after students were exposed to both types of manipulatives, F(1, 161) = 1452.59, p < .001, η2 = .90. The change in the final scores indicate that using fraction bars as a manipulative tool can be helpful in teaching the concept of adding fractions because students build a better conceptual understanding of the concept of fractions. ii COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE iii COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE DEDICATION To my wife Asma, your support throughout these many years has been resolute. Your understanding has been unwavering, and your encouragement has been infinite. Thank you for being there for me, especially in times of despair. To my parents Musbah & Aishah, from any early age you instilled in me values that have molded and shaped my life. These values to date dedicate my aspirations, my achievements, and my resilience in persevering relentlessly through obstacles. From you I have learned to work hard and dedicate myself to accomplishing my life goals. Without such values I could not have accomplished this task. For that and more, I thank you. To my children Yazeed, Yazan, Waleed, and Wisam, you are my joy and inspiration. Over the course of journey, as I watched you grow into young adults and childhood, your own accomplishments encouraged my efforts in realizing this goal. Thank you for your unconditional love and encouragement. You made the journey worthwhile and I love you dearly. To my siblings, thank you for your inquiries as to my progress, your expressions of love, and your constant encouragement. I hold a deep appreciation for you love and caring. I could not have done this without all of my family’s help. iv COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE ACKNOWLEDGMENTS My Chair: Dr. Moomaw, your constant encouragement and guidance throughout the dissertation process ensured the completion of this task. Your patience and support have helped me navigate this long journey. This was aided in no small way by your timely responses to email queries, coursework, in addition to weekly meetings. Thank you for being my intellectual mentor who believes in my ability more than anyone else and had encouraged me and provided me with opportunities to grow as a researcher. Also, I would like to express my deepest gratitude to the members of my Dissertation Committee, Dr. Gregson, Dr. Pelikan, and Dr. Zydney. It was with their encouragements, insights, and expertise that I was able to complete my doctoral dissertation. Thank you for being supportive throughout this entire process. Thank you for the hours and hours you happily and graciously read my study to make sure my writing was superb! There are no words to express the appreciation that I feel toward you both. Lastly, I would like to acknowledge and thank members of the University of Cincinnati faculty and staff. v COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE TABLE OF CONTENTS ABSTRACT ………………………………………………………………………………. ii DEDICATION ……………………………………………………………………………. iv ACKNOLEDGMENTS …………………………………………………………………... v LIST OF TABLES ………………………………………………………………………... x LIST OF FIGURES ………………………………………………………………………. xi CHAPTER I: INTRODUCTION …………………………………………………………. 1 Background and problem statement……………………………………………………. 3 Definition of terms …………………………………………………………………….. 16 Purpose of study ……………………………………………………………………….. 17 Objectives ……………………………………………………………………………… 17 Research questions …………………………………………………………………….. 18 Research hypotheses …………………………………………………………………... 19 Significance of the study ………………………………………………………………. 20 CHAPTER II: LITERATURE REVIEW ………..………………………………………... 23 Introduction ………………………………………………………………………….… 23 Theoretical Framework ……………………………………………………………….. 24 Literature Review ……………………………………………………………………... 27 Mathematics in Elementary School ………………………………………………. 27 Teaching mathematics in elementary school ………………………………….. 38 Students’ attitudes toward mathematics ………………………………………. 31 Students and mathematics difficulties ………………………………………… 33 Summary ……………………………………………………………………… 35 Fractions ………………………………………………………………………….. 36 Understanding of fractions ……………………………………………………. 36 Difficulties with fractions ……………………………………………………... 39 Fraction equivalence ………………………………………………………….. 46 Addition of fractions ………………………………………………………….. 50 Conceptual Understanding ……………………………………………………….. 55 Importance of conceptual understanding in mathematics …………………….. 55 Developing a conceptual understanding of fractions …………………………. 56 Using Manipulatives ……………………………………………………………… 60 Support for manipulatives in the classroom …………………………………… 60 Manipulatives help students learn mathematical concepts ……………………. 63 Research Studies on Manipulative Effectiveness ……………………………... 67 General improvement of mathematics abilities …………………………... 68 Improving conceptual understanding …………………………………….. 72 vi COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE Manipulatives for fraction instruction ……………………………………. 74 Combining physical and virtual manipulatives …………………………... 78 Summary of the studies …………………………………………………... 80 Physical Manipulatives vs. Virtual Manipulatives ……………………………. 81 Ease of use ………………………………………………………………... 82 Guided instruction and instant feedback ………………………………….. 83 Built-in constraints and amplifications …………………………………… 85 Linking representations …………………………………………………… 86 Distracters ………………………………………………………………… 87 Unique affordances ……………………………………………………….. 88 Students' Attitudes Toward Manipulatives ……………………………………. 89 How manipulatives improve students’ attitudes towards math …………... 91 Students with Math Difficulties ………………………………………………. 92 Intervention …………………………………………………………………… 95 Conclusion ……………………………………………………………………………. 100 CHAPTER III: METHODOLOGY ……………………………………………………… 104 Introduction …………………………………………………………………………… 104 Research design ……………………………………………………………………….. 104 Research procedures …………………………………………………………………... 106 Internal validity ………………………………………………………………………... 111 External validity ……………………………………………………………………….. 113 Confounding variables ………………………………………………………………… 115 Description of the independent variable ………………………………………………. 115 Description of control group lessons ………………………………………………….. 119 Training for Instruction ………………………………………………………………... 122 Sampling and participants ……………………………………………………………... 123 Instrumentations ………………………………………………………………………. 130 Data collection ………………………………………………………………………… 133 Data analysis …………………………………………………………………………... 134 Assumptions …………………………………………………………………………… 135 CHAPTER IV: RESULTS ……………………………………………………………….. 137 Introduction ……………………………………………………………………………. 137 Methods of analyzing data …………………………………………………………….. 138 Descriptive statistics …………………………………………………………………... 139 Description of the sample ………………………………………………………… 139 Description of study variables ……………………………………………………. 139 Results of the hypotheses tests ………………………………………………………... 152 vii COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE CHAPTER V: Discussion, Implications, Limitations and Recommendations …..………. 164 Introduction …………………………………………………………………………… 164 Interpretation of findings ……………………………………………………………… 165 Theoretical implications ………………………………………………………………. 169 Practical implications …………………………………………………………………. 170 Limitations ……………………………………………………………………………. 171 Future studies …………………………………………………………………………. 173 REFERENCES …………………………………………………………………………... 176 APPENDICES ……..…………………………………………………………………….. 225 Appendix A: Cover Letter
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