The primary purpose of this study was to investigate experimentally the following questions: (1) What is the relative effectiveness of the vector approach and the traditional approach to teaching certain topics in plane analytic geometry to high school students in terms of achievement and understanding? (2) Does teaching these topics in plane analytic geometry via vectors produce more transfer to other topics in analytic geometry (mainly solid analytic geometry) than teaching these topics via the traditional approach? / Typescript. / "December, 1968." / "Submitted to the Department of Mathematics Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

by Law Huk-yuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 235-243). / ACKNOWLEDGEMENTS --- p.ii / ABSTRACT --- p.iii / LIST OF TABLES --- p.x / LIST OF FIGURES --- p.xii / Chapter CHAPTER 1 --- INTRODUCTION --- p.1 / Chapter 1.1 --- Background of the Study --- p.1 / Chapter 1.2 --- Purpose and Significance of the Study --- p.2 / Chapter CHAPTER 2 --- REVIEW OF LITERATURE --- p.5 / Chapter 2.1 --- Research in Mathematics Education : A Cognitive Science Perspective --- p.7 / Chapter 2.1.1 --- Issues in mathematics education --- p.8 / Chapter 2.1.2 --- The cognitive science perspective --- p.13 / Chapter 2.1.2a --- Task-based interviews --- p.16 / Chapter 2.1.2b --- Information-processing framework --- p.18 / Chapter 2.1.2c --- The knowledge structure --- p.21 / Chapter 2.1.2d --- The nature of concepts --- p.23 / Chapter 2.1.3 --- The psychological studies of mathematical concepts --- p.25 / Chapter 2.2 --- Mental Models and Conceptual Knowledge --- p.29 / Chapter 2.3 --- Expert-Novice Discrepancies in Knowledge Representation --- p.32 / Chapter 2.4 --- Assessing Mathematical Understanding --- p.38 / Chapter 2.4.1 --- Assessing knowledge structure --- p.39 / Chapter 2.4.2 --- Test validation --- p.41 / Chapter 2.5 --- Summary --- p.42 / Chapter CHAPTER 3 --- RESEARCH METHOD --- p.44 / Chapter 3.1 --- Research Questions --- p.44 / Chapter 3.2 --- Subjects --- p.45 / Chapter 3.3 --- Design and Procedures --- p.46 / Chapter 3.3.1 --- Phase 1. : Initial testing for conceptual knowledge --- p.46 / Chapter 3.3.2 --- Phase 2 : Task-based interviewing --- p.47 / Chapter 3.3.3 --- Phase 3: Revised testing for conceptual knowledge --- p.48 / Chapter 3.3.4 --- Model for assessing conceptual understanding --- p.49 / Chapter 3.4 --- Data Analysis --- p.51 / Chapter 3.5 --- Time Frame --- p.52 / Chapter CHAPTER 4 --- RESULTS AND DISCUSSION --- p.53 / Chapter 4.1 --- Overview --- p.53 / Chapter 4.2 --- Conceptual Knowledge in Coordinate Geometry --- p.54 / Chapter 4.3 --- Phase-one Analysis : Selection of Expert and Novice Students --- p.57 / Chapter 4.3.1 --- The scoring of initial test --- p.58 / Chapter 4.3.2 --- A profile of expert students and novice students --- p.60 / Chapter 4.3.3 --- A preliminary discussion ： Expert students vs. novice students --- p.61 / Chapter 4.4 --- Phase-two Analysis --- p.63 / Chapter 4.4.1 --- Constructing students' knowledge representations --- p.64 / Chapter 4.4.1a --- Mental representation of <PARALLEL LINES> --- p.66 / Chapter 4.4.1b --- Mental representation of <SLOPES> --- p.78 / Chapter 4.4.1c --- Mental representation of <INTERCEPTS> --- p.82 / Chapter 4.4.1d --- Mental representation of <POINT COORDINATES> --- p.85 / Chapter 4.4.1e --- Knowledge representations --- p.92 / Chapter 4.4.2 --- Comparison of expert mental representat ion and novice mental representation --- p.101 / Chapter 4.4.3 --- Generating and testing hypotheses --- p.105 / Chapter 4.5 --- Phase-three Analysis --- p.123 / Chapter 4.5.1 --- Scoring of the revised test --- p.123 / Chapter 4.5.2 --- Test reliability and validation --- p.135 / Chapter 4.6 --- Summary --- p.139 / Chapter CHAPTER 5 --- "CONCLUSIONS, IMPLICATIONS AND SUGGESTIONS FOR FUTURE RESEARCH" --- p.142 / Chapter 5.1 --- Conclusions --- p.142 / Chapter 5.2 --- Limitations --- p.144 / Chapter 5.3 --- Implications --- p.145 / Chapter 5.4 --- Suggestions for Future Research --- p.147 / APPENDICES / Appendix 1. Initial test for conceptual knowledge --- p.149 / Appendix 2. Scoring record of initial test --- p.152 / Appendix 3. Questions for the first interview --- p.153 / Appendix 4. Subjects' protocols of the first interview (the Chinese version) --- p.157 / Appendix 5 . Subjects ' protocols of the first interview (the Knglish- trans1ated version ) --- p.176 / Appendix 6. Questions for the second interview --- p.202 / Appendix 7. Record of subjects' responses to the questions of second interview --- p.208 / Appendix 8. Revised quiz for conceptual knowledge --- p.210 / Appendix 9. Item and score distribution of the two-halves of the revised test --- p.232 / Appendix 10A. Scoring record of Test A ( the first half- test ) --- p.233 / Appendix 10B. Scoring record of Test B (the second half- test ) --- p.234 / REFERENCES --- p.235