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Xerox University Microfilms INFORMATION TO USERS This material 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. 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Xerox University Microfilms 300 North Zaab Road Ann Arbor, Michigan 40100 77-2383 DAMARIN, Suzanne Kidd, 1941- AN INQUIRY INTO THE USE OF LOGIC IN MATHEMATICAL CONTEXTS BY PRESERVICE ELEMENTARY TEACHERS. The Ohio State University, Ph.D., 1976 Education, mathematics Xerox University MicrofilmsAnn , Arbor, Michigan 48106 © Copyright by Suzanne Kidd Damarin 1976 AN INQUIRY INTO THE USE OF LOGIC IN MATHEMATICAL CONTEXTS BY PRESERVICE ELEMENTARY TEACHERS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Suzanne Kidd Damarin, A.B., M.A. *■* * * The Ohio State University 1976 Reading Committee: Approved By Richard J. Shumway Joseph Ferrar Alan Osborne Arthur White Advisor Faculty of Science and Mathematics Education ACKNOWLEDGMENTS Many individuals have contributed to the success of this research. More than 300 elementary education students served as research subjects. They were quite cooperative and several of them volunteered their thoughts and concerns about the tests. Professors Joan Leitzel, T. Ralley, P. Rolfe, J. Schultz and L. Stull generously permitted the testing of these students during class time. The Undergraduate Commit­ tee of the Department of Mathematics and several of the professors named above raised probing questions concerning the nature and purpose of the research, thereby helping to shape it. Each member of the reading committee had a unique influence on the development of the dissertation. Professor Joseph Ferrar raised questions which led to the clarifica­ tion of relationships among several parts of the manuscript. Discussions with Dr. Alan Osborne concerning the relation­ ships between this research and other studies and points of view were quite helpful, as was his critical reading of the manuscript. Dr. Arthur White*s keen scientific intuition and concern with experimental validity influenced the research design and development from the outset. Dr. Richard J. Shumway served conscientiously and well in the many roles of advisor. Concerning himself with all aspects of the dissertation he raised many perti­ nent questions especially with regard to the implications of the research. His spirit of inquiry and keen insight will be a lasting resource. VITA February 19, 1941. Born - Easton, Pennsylvania 1962 ................... A.B., Wilson College, Chambers- burg, Pennsylvania 1964 ...................M.A., Bryn Mawr College, Bryn Mawr, Pennsylvania FIELD OF STUDY Mathematics Education v TABLE OF CONTENTS Page ACKNOWLEDGMENTS..................................... iii VITA ............................................. v LIST OF TABLES .................................... viii LIST OF FIGURES................................... xi Chapter I. STATEMENT OF THE PROBLEM................. 1 II. REVIEW OF THE LITERATURE................... 14 Studies in the Tradition of Experi­ mental Psychology................... Studies and Findings in the Genetic Tradition............................. 26 Studies in the Tradition of Educa­ tional Research.......................33 Synthesis of Three Traditions ........ 37 III. THE CONSTRUCTION OF A MODEL 41 A Formal Relationship Between Mathe­ matical and Logical Statements. The Operation of the Expanded Model . 50 Testing the Model I: Delineation of Contexts ........................... 56 Testing the Model II: Research Hypotheses............................. 57 IV. PROCEDURES FOR TESTING THE MODEL.......... 68 Segment 1. Finding Sortings for Statements............................. 74 Segment 2. Comparing Content D o m a i n s............................... 76 Segment 3. Selecting Statements for Sortings. .............. 77 Segment 4. Finding Examples fcr which Statements Differ ............ 82 vi Page Segment 5. Bringing Six Tasks Together.............................. 89 Segment 6. The Class Room Logic T e s t s ................................ 91 V. R E S U L T S ............................... 94 Segment 1. Finding Sortings for Statements........................... 96 Segment 2. Comparing Content D o m a i n s ................................100 Segment 3. Selecting Statements for Sortings................................102 Segment 4. Finding Examples for which Statements Differ ............. 121 Segment 5. Bringing Six Tasks Together................................133 Segment 6. The Classroom Logic T e s t s ..................................150 VI. CONCLUSIONS.............................. 155 APPENDIX A. A Preliminary Investigation: Interpretations of Statements ........ 167 B. Instruments...............................179 BIBLIOGRAPHY........................................ 259 vii LIST OF TABLES Page 1. Presentations of equivalent information in 3 m o d e s ..................................... 11 2. Principles used in inference tests.......... 19 3. Equivalence of two sample statements........ 46 4. Standard statements................. 69 5. Brief descriptions of research segments .... 70 6. Pairing of tests for segment 3 ............... 80 7. Classification of pairs available for test LM . 86 8. Classification of pairs available for test ML . 87 9. Test administration data..................... 94 10. An overview of results as related to hypoth­ eses......................................... 95 11. Basic test statistics for segment 1 tests . 97 12. Basic test statistics for segment 2 tests . 101 13. Mean percentages correct for subsets of MA. 102 14. Basic test data for segment 3 tests .......... 105 15. Correlations between test scores............. 106 16. Repeated measures ANOVA for ordered tests . 106 17. Basic test data for segment 3 subscales .... 107 18. ANOVAs subscales of four tests {segment 3) . H I 19. Repeated measures ANOVAs. for L and M test f o r m s ........................................H 2 20. Analysis of Variance: Comparing L with M . 115 viii Page 21. Subscale means for L and M ........................116 22. Distribution of errors in segment 3 ............. 120 23. Speededness data for segment 4 tests............. 122 24. Basic test data for segment 4 t e s t s ........ 123 25. Summary of ANOVA for statement types........ 124 26. Basic test data for segment 4 subscales .... 125 27. Summary of ANOVA: LM vs. LL (segment 4). 127 28. Summary of ANOVA: ML vs. MM (segment 4) . 129 29. Comparison of true key and conjunctive key scores............................................ 132 30. Basic test data for segment 5 t e s t s ............. 134 31. Differences between means for segment 5 t e s t s ............................................ 135 32. Means and reliability estimates for test L (segment 5 ) ......................................137 33. Means and reliabilities when scored for conjunctive interpretation....................... 138 34. Means and intercorrelations for selected subscales of test M (segment 5 ) ................. 140 35. Drawing power of options in L*M (segment 5) . 141 36. Distribution of errors on test M*L (segment 5 ) ......................................142 37. Correlations among scale scores ............... 144 38. Correlations corrected for attenuation (segment 5 ) ............................... 146 39. Multiple correlations among tests (segment 5 ) ......................................147 40. Tetrachoric correlations (high vs. low group)............................................ 148 ix Page 41. Reliabilities of three-test units ............. 149 42. Basic test data for segment 6 t e s t s ..............150 43. Errors on correct and incorrect arguments.........................................151 44. Data for hypothetical tests .....................
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