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Information to Users INFORMATION TO USERS This manuscript has been reproduced from the micro&hn master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quali^ of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material bad to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photogrtq>hs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 DEVELOPMENT OF ALGEBRAIC REASONING IN CHILDREN AND ADOLESCENTS: CULTURAL, CURRICULAR, AND AGE-RELATED EFFECTS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Anne Krislov Morris, B.A., M.S., M.A. ***** The Ohio State University 1995 Dissertation Committee: Approved by Alan Osbome Vladimir Sloutsky 7<f/L . Sigrid Wagner s I o - f y Advisors College of Education OHI Number: 9544648 Copyright 1995 by Morris, Anne Krislov All rights reserved. OMI Microform 9544648 Copyright 1995, by DMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Copyright by Anne Krislov Morris 1995 TO MY DAUGHTERS—JENNIFER, GENIE, AND GWEN ACKNOWLEDGMENTS I express sincere gratitude to Dr. Vladimir Sloutsky and Dr. Alan Osbome for their assistance and intellectual guidance throughout the research study. I am also indebted to Sigrid Wagner for her suggestions and comments. This research was supported in part by Johann Jacobs Foundation of Switzerland, a Graduate Student Alumni Research Award and Presidential Fellowship from Ohio State University, and the Center for Cognitive Science at Ohio State University. Their support is gratefully acknowledged. Thanks are also due to Sergei Baklochinsky, Melanie Scully, and Elizabeth Squires for their assistance during the project. The greatest thanks are due to my children, Jennifer, Genie, and Gwen: Thanks for your patience, your senses of humor, and your acceptance of imperfection. m VITA October 6,1955 Bom - New York, New York August, 1976 B. A., Psychology; University of Minnesota, Minneapolis March, 1988 M.S., Applied Mathematics; University of California, Davis June, 1990 M.A., Teaching of Mathematics; University of California, Davis PUBLICATIONS Morris, A. K. & Silk, W. K. (1992). Use of a flexible logistic function to describe axial growth of plants. Bulletin of Mathematical Biology, 54 (6), 1069-1081. Morris, A. (1994). Meaningful instruction in fractions: Implementing a theory in a low achieving mathematics classroom. Focus on Learning Problems in Mathematics, 77 (3^ Morris, A. K., & Sloutsky, V. M. (1995, March). Cognitive tools of higher-order thinking: A cross-cultural analysis of children's use of algebraic symbols. Paper presented at the Biennial Meeting of the Society for Research in Child Development, Indianapolis, Indiana. Morris, A. K., & Sloutsky, V. M. (1995, October). Development of algebraic reasoning in children and adolescents: A cross-cultural and cross-curricular perspective. Research report presented at the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, Ohio. Sloutsky, V. M., & Morris, A. K. (1995, August). Understanding of logical necessity: Developmental antecedents and cognitive consequences. Paper presented at the Annual Meeting of the American Psychological Association, New York, N.Y. FIELDS OF STUDY Major Field: Education Studies in Applied Mathematics, Mathematics Education IV TABLE OF CONTENTS ACKNOWLEDGMENTS.......................................................................................................iii VITA.......................................................................................................................................... iv LIST OF TABLES...................................................................................................................vü LIST OF FIGURES........................................................................................................ viii CHAPTER I. INTRODUCTION..........................................................................................1 Goals of the study .......................................................................................................1 Introduction ................................................................................................................. 1 Theoretical framework .......................................................................................... — 4 Research questions ..................................................................................................... 12 Significance ................................................................................................................. 13 CHAPTER II. THEORETICAL ANALYSIS: CULTURE AND CURRICULUM AS CONTRIBUTING FACTORS TO ALGEBRAIC REASONING ........................................................................................................................ 14 Characteristics of a culture that may affect mathematical and logical reasoning ......................................................................................................................14 Characteristics of a curriculum (curricular variables) that may affect the development of algebraic reasoning .......................................................................... 34 CHAPTER m . METHODOLOGY AND INSTRUMENTATION................................. 75 D e sig n ......................................................................................................................... 75 Subjects........................................................................................................................ 75 Measurement and Instruments ...................................................................................83 P ro c ed u re s .................................................................................................................89 Data analysis ................................................................................................................90 CHAPTER IV. RESULTS AND DISCUSSION....................................................... 92 Algebraic reasoning ....................................................................................................92 Algebraic structure ...............................................................................................125 Algebraic letter interpretation......................................................................................147 CHAPTER V. CONCLUSIONS AND GENERAL DISCUSSION................................158 Cultural, curricular, age, and contextual effects ......................................................158 V Testing the models of cognitive performance and algebraic reasoning ................... 161 Future studies ................................................................................................................170 APPENDIX A. WRITTEN TEST INSTRUMENT (ENGLISH VERSION)...............171 APPENDIX B. WRITTEN TEST INSTRUMENT (RUSSIAN VERSION)...............181 APPENDIX C. INTERVIEW GUIDE (ENGLISH VERSION)....................................... 192 APPENDIX D. INTERVIEW GUIDE (RUSSIAN VERSION).......................................200 APPENDIX E. CODING CATALOGUE.................................................................210 APPENDIX P. PERCENTAGES FOR PROBLEM SOLUTIONS...................................224 LIST OF REFERENCES................................................................................................ 228 VI LIST OF TABLES Table 1. Experimental curricular models for developing algebraic reasoning ..................35 Table 2. Experimental curricular models for developing concepts of algebraic stru c tu re .................................................................................................................................. 36 Table 3. Experimental curricular models for developing algebraic letter c o n c e p ts .................................................................................................................................. 37 Table 4. Sample.......................................................................................................................76 Table 5. Sample described in terms of two age groups, 10-14 years and 14- 16 years ....................................................................................................................................
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