The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes Anthony M. Miele Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2014 © 2014 Anthony M. Miele All Rights Reserved ABSTRACT The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes Anthony M. Miele The purpose of this study was to determine how the study of number theory might affect high school students’ metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. The study utilized questionnaire and/or interview responses of seven high school students from New York City and 33 high school students from Dalian, China. The questionnaire components served to measure and compare the students’ metacognitive functioning, mathematical curiosity, and mathematics attitudes before and after they worked on a number theory problem set included with the questionnaire. Interviews with 13 of these students also helped to reveal any changes in their metacognitive tendencies and/or mathematics attitudes or curiosity levels after the students had worked on said number theory problems. The investigator sought to involve very motivated as well as less motivated mathematics students in the study. The participation of a large group of Chinese students enabled the investigator to obtain a diverse set of data elements, and also added an international flavor to the research. All but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics after the students had worked on the assigned number theory problems. The results of the study thus have important implications for the value of number theory coursework by high school students, with respect to the students’ metacognitive processes as well as their feelings about mathematics as an academic discipline. TABLE OF CONTENTS Page CHAPTER I – INTRODUCTION ............................................................................................. 1 Need for the Study ......................................................................................................... 1 Purpose of the Study ...................................................................................................... 2 Procedures of the Study ................................................................................................. 3 CHAPTER II – LITERATURE REVIEW ................................................................................ 7 Attitudes Towards Mathematics .................................................................................... 7 Number Theory in High School ..................................................................................... 16 Metacognition ................................................................................................................ 25 CHAPTER III – METHODOLOGY ......................................................................................... 37 CHAPTER IV – FINDINGS AND RESULTS ......................................................................... 46 Preliminary Trial Summary ........................................................................................... 46 Preliminary Trial Interviews—Common Theme ............................................... 48 Preliminary Trial Details.................................................................................... 48 Student Q1 ............................................................................................. 48 Student Q2 ............................................................................................. 51 Student Q3 ............................................................................................. 53 Main Study Summary .................................................................................................... 56 Main Study Interviews—Common Themes ...................................................... 58 Main Study Details ............................................................................................ 59 Student Q4 ............................................................................................. 59 Student Q5 ............................................................................................. 61 i Page CHAPTER IV (continued) Student Q6 ............................................................................................. 64 Student Q7 ............................................................................................. 66 Student Q8 ............................................................................................. 69 Student Q9 ............................................................................................. 71 Student Q10 ........................................................................................... 73 Student Q11 ........................................................................................... 75 Student Q12 ........................................................................................... 78 Student Q13 ........................................................................................... 80 Student Q14 ........................................................................................... 82 Student Q15 ........................................................................................... 83 Student Q16 ........................................................................................... 85 Student Q17 ........................................................................................... 87 Student Q18 ........................................................................................... 89 Student Q19 ........................................................................................... 90 Student Q20 ........................................................................................... 92 Student Q21 ........................................................................................... 94 Student Q22 ........................................................................................... 96 Student Q23 ........................................................................................... 98 Student Q24 ........................................................................................... 100 Student Q25 ........................................................................................... 101 Student Q26 ........................................................................................... 103 ii Page CHAPTER IV (continued) Student Q27 ........................................................................................... 105 Student Q28 ........................................................................................... 106 Student Q29 ........................................................................................... 108 Student Q30 ........................................................................................... 110 Student Q31 ........................................................................................... 111 Student Q32 ........................................................................................... 113 Student Q33 ........................................................................................... 115 Student Q34 ........................................................................................... 116 Student Q35 ........................................................................................... 118 Student Q36 ........................................................................................... 120 Student Q37 ........................................................................................... 122 Student Q38 ........................................................................................... 124 Student Q39 ........................................................................................... 125 Student Q40 ........................................................................................... 127 CHAPTER V – SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS .................. 130 Summary ........................................................................................................................ 130 Conclusions .................................................................................................................... 131 Recommendations .......................................................................................................... 135 REFERENCES .......................................................................................................................... 138 Internet Resources .......................................................................................................... 145 iii Page APPENDICES Appendix A – Questionnaire Parts A, B, and C ........................................................................ 146 Appendix B – Interview Script .................................................................................................. 153 Appendix C – Certificate of Participation Templates and Consent for Reference Form .......... 156 Appendix D – Institutional Review Board Required Forms ...................................................... 159 Appendix E – Mandarin Translations