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University Microfilms 300 North Zmb Road Ann Arbor, Michigan 40106 a Xerox Education Company 73-11,544 INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. University Microfilms 300 North Zmb Road Ann Arbor, Michigan 40106 A Xerox Education Company 73-11,544 MITCHELL, Bruce Alex, 1937- THE EFFECT OF A TEACHER-DEVELOPED UNIT IN HYPERBOLIC GECMETRY ON STRUCTURAL OBJECTIVES IN TENTH GRADE GECMETRY. The Ohio State University, Ph.D., 1972 Education, general University Microfilms, AXEROX Company , Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. THE EFFECT OF A TEACHER-DEVELOPED UNIT IN HYPERBOLIC GEOMETRY ON STRUCTURAL OBJECTIVES IN TENTH GRADE GEOMETRY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Bruce Alex Mitchell, B.S., M.S * * * * * The Ohio State University 1972 Approved by A civ is or lece of Education PLEASE NOTE: Some pages may have Indistinct print. Filmed as received. University Microfilms, A Xerox Education Company AC KNOWLEDGMENTS I wish to acknowledge those people who helped make the completion of this research project possible. To Dr. F. Joe Crosswhite, my major adviser, and to Dr. Alan Osborne for their time investment in me and their many helpful suggestions, my deep appreciation. A thank you to Ted Scheick of the mathematics department for his help, encouragement, and friendship. Thanks to Dr. Dan Eustice of the mathematics department for his help in the instrument development stage. I also thank Dr. Peter Anderson and Dr. Jae Lee for their assistance. To Ray Tata, Hike Myers, and Gary Smith, of the Brookhaven mathematics department, thank you so much for your help. A special thanks to my parents and the Hemingtons for the help and encouragement they have always given me. To my wife Judy and children, Kathe, Bob, and Jim, who gave love, understanding, patience and help, I express my deepest appreciation. The combination of people and events that effected my life during the last three years contributed to what I consider the best three years of my life. VITA March 26, 19 37 * Born - Chicago, Illinois 1959 . B.S. in Education, Northern Illinois University, DeKalb, Illinois 1959-1962. Mathematics Teacher, Zion-Benton Township High School, Zion, Illinois 1962-1963. M.S. in Mathematics, University of Tennessee, Knoxville, Tennessee 1963-1966. Mathematics Teacher, Niles Township High School - West Division, Skokie, Illinois 1966-1967 Participant, NSF Academic Year and Summer Institute, University of Wisconsin, Madison, Wisconsin 1967-1969. Mathematics Instructor,.Lake Forest College, Lake Forest, Illinois 1969-1971. Teaching Associate, Department of Mathematics, The Ohio State Uni­ versity, Columbus, Ohio 1971-1972, Teaching Associate, Department of Science and Mathematics Education, The Ohio State University, Columbus, Ohio FIELDS OF STUDY Major Field: Mathematics Education Professor F. Joe Crosswhite Professor Alan Osborne Professor Harold C. Trimble i i i Minor Fields: Mathematics Professor John T. Scheick Teacher Education Professor Herbert Coon iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS . i i VITA............. i l l LIST OF TABLES v Chapter I. THE PROBLEM Introduction and Background of the Problem Rationale for the Study ................. Method and Procedure .................... Instrumentation Hypotheses............. 8 Analyses of Data. 8 Delimitations. 9 An Overviev.’ of the Report g II REVIEW OF THE RELATED LITERATURE .... 11 Goals of Geometry Related to This St.rdy . 11 Dissatisfaction VJith Student Fulfillment of the Goals of Geometry Related to This Study .............................. 19 Recommendations for the Geometry Course . 2 3 Non-Euclidean Geometry in the Iligh School Feasibility of Teaching Hyperbolic Non- Euclidean Geometry to High School Geomctr Students.................................. 32 Summary..................................... 34 III RESEARCH PROCEDURE. 36 Sample and Design 36 Treatment . 39 Instrumentation . 41 Analysis of Data. 44 v Chapter Page IV. THE SEMINAR .................................. 46 Background.............................. 46 Preparing the Teachers................ 50 Developing the Unit................... 5 3 Discussing Teaching Strategies ..... 54 V. ANALYSIS OI-* D A T A .................... 59 Initial Difference Hypotheses ............. 59 Analyses Related to Main Hypotheses . 64 Analyses of Posttest ........................ 67 Chi-Square Analysis ........................ 72 Item Analysis of Posttest: Results . 76 Achievement Test in Hyperbolic Geometry . 8 3 VI. SUMMARY AND CONCLUSIONS................ 86 Summary................................. 86 Conclusions .......................... 88 Discussion............................. 89 Recommendations for Further Research . 90 APPENDIX A ...................................................... 94 B ...................................................... 103 C ...................................................... 114 D ................................ 137 E ...................................................... 139 BIBLIOGRAPHY ............................................ 347 vi LIST OF TABLES Table Page 1. Results of the Posttest Pilot ....... 43 2. Distribution of Pre-Student Teachers and Student Teachers. ..... 47 3. Summary of Results of the Elementary Geometry Facts Test— September 1 9 7 1 ........................ 60 4. Summary of the Results of the t-Tests for the September Geometry Facts Test ................. 61 5. Summary of Results of the Pretest..................... 62 6. Summary of the Results of the t-Tests Used on the Pretest ..................................... 6 3 7. Summary of t-Test R e s u l t s ................ 65 8. Blocking on the P r e t e s t ...............................66 9. Summary of Results of 2 X 3 Analysis of Covariance.............................................67 10. Summary of Posttest Results. 68 11. Chi-Square Item by Item Analysis ..................... 70 12. Relative Difficulty and Discrimination Indices of Posttest I t e m s .................................. 77 13. Response Pattern on Posttest .................... 79 14. Results of an Achievement Tost in hyperbolic Non-Euclidean Geometry............................... 84 15. Item by Item Mean Scores on Hyperbolic Geometry Achievement Test................ 85 vii CHAPTER I THE PROBLEM Introduction and Background of the Problem Each year thousands of high school sophomores com­ plete a course in Plane Geometry. The literature suggests that the understanding of mathematical structure, inference patterns, and patterns of proof are commonly accepted goals for this course. The degree to which these goals are achieved appears to be unknown. To some, the attainment of these goals is questionable. As Brumfiel states: Students of 1954 who studied an old fashioned hodge­ podge geometry had no conception of geometric structure. Students of 1971 who have studied a tight axiomatic treatment have no conception of geometric structure (4, 5). The content that will enable students to emerge with an appreciation of mathematical structure and the ability to recognize valid inference and proof patterns has been a controversial issue among mathematicians and mathematics educators. A source of fuel for this controversy has been supplied by attempts to utilize recent developments in geometry (20) . The affine approach to Euclidean geometry (24), the transformational approach (13), the vector 1 2 approach (46), and the inclusion of non-Euclidean geometry (30), were among the suggestions for the geometry course that reflected these developments. It was the purpose of this research to develop a unit in hyperbolic non-Euclidean geometry as a means of achieving goals related to mathematical structure and inference and proof patterns. Specifically, the
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