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INFORMATION to USERS the Most Advanced Technology Has Been INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the original text directly firom tiie copy submitted. Thus, some dissertation copies are in iypewriter face, while others may be from a computer printer. 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 copyrighted material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each oversize page is available as one exposure on a standard 35 Tmn slide or as a 17" x 23" black and white photographic print for an additional charge. Photographs included in the original manuscript have been reproduced xerographically in this copy. 35 mm slides or 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. Accessing the World'sUMI Information since 1938 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA Order Nxaniber 8824468 Characterizing levels of understanding of functions and their graphs Browning, Christine Annette Cheadle, Ph.D. The Ohio State University, 1988 Copyright ©1988 by Browning, Christine Annette Cheadle. All rights reserved. UMI SOON. Zeeb Rd. Ann Aibor, MI 48106 PLEASE NOTE: in all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark V 1. Glossy photographs or pages. 2. Colored illustrations, paper or print _____ 3. Photographs with dark background ____ 4. Illustrations are poor copy _____ 5. Pages with black marks, not original copy. 6. Print shows through as there is text on both sides of page. 7. Indistinct, broken or small print on several pages Y 8. Print exceeds margin requirements _____ 9. Tightly bound copy with print lost in spine. 10. Computer printout pages with indistinct print. 11. Page(s) __________ lacking when material received, and not available from school or author. 12. Page(s) __________seem to be missing in numbering only as text follows. 13. Two pages numbered . Text follows. 14. Curling and wrinkled pages _____ 15. Dissertation contains pages with print at a slant, filmed as received ________ 16. Other______________________________________________________________ UMI CHARACTERIZING LEVELS OF UNDERSTANDING OF FUNCTIONS AND THEIR GRAPHS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Christine Cheadle Browning, B.S., M.A. ***** The Ohio State Univerâty 1988 Dissertation Committee: Approved by A. Osbome, Chairman F. Demana j M. Kasten Adviser College of Education J. Verducci Copyright by Christine Cheadle Browning 1988 To my family, David, Joshua, and David 11 ACKNOWLEDGEMENTS I express sincere appreciation to Dr. Alan Osbome for his patience,understanding, and encouragement throughout the many years. Many thanks go to the other members of my advisory committee, Drs. Peggy Kasten, L. Alayne Parson, and Joe Verducci, for their guidance and encouragement Thanks go to Drs. Frank Demana and Bert Waits, my "employers." Their guidance, suggestions, and humor were greatly appreciated; I count it a privilege to have worked with them. The technical assistance of Barbara Aronson and Jïïl Baumer is gratefully acknowledged. The encouragements and prayers of my Mends, Pam S tamer, Jane Miller, Marsha Parra, Amy Edwards, Liz Busey, Mary Maxwell, and Chuck YonderEmbse, throughout the years were appreciated. I am thankful for my Mends at Berean Bible Church, for their prayers and support To my parents, I express sincere thanks for your support and encouragement ; I am grateful to have parents such as you two. To my husband, David, I offer my sincere thanks for your faith in me and your helpful suggestions throughout my research. To my children, Joshua and David, I thank you for letting Mommy go to school when you wanted her to stay at home. And thank you Lord, for none of this would have been possible without You. Ill VTTA May 5, 1958 .......................................................Bom - Anniston, Alabama 1980 ....................................................................B.S., The Ohio State University, Columbus, Ohio 1980-1983 .......................................................... Classroom Teacher, Bloom Carroll Schools, Carroll, Ohio 1983-1985 .......................................................... Graduate Teaching Assistant, The Department of Mathematics, The Ohio State University, Columbus, Ohio 1985-198 6 ......................................................... Administrative Assistant, The Ohio Board of Regents, Columbus, Ohio 1986-1988 ..........................................................Graduate Research Assistant, The Department of Mathematics, The Ohio State University, Columbus, Ohio FIELDS OF STUDY Major Fiel± Mathematics Education Studies in Precalculus curriculum development with Dr. Bert Waits, Dr. Frank Demana, Dr. Alan Osbome, and Dr. Greg Foley. IV TABLE OF CONTENTS ACKNOWLEDGEMENTS .................................................................................................i VITA ................................................................................................................................... iü LIST OF TABLES ..............................................................................................................vü LIST OF FIGURES ................................................................................................. viü CHAPTER PAGE I. INTRODUCTION ...................................................................................1 1.1 Motivation for Litensrfîed Study of Graphing ........................ 2 1.2 Graphing in the Curriculum .............................................. 3 1.3 An Assessment Instrument .............................................. 6 1.4 Research Technique ........................................................... 8 1.5 Research Questions ................................................................... 9 1.6 Summary..................................................................................10 II. REVIEW OF THE LITERATURE..................................................... 11 2.1 Introduction ..............................................................................11 2.2 Mathematical Coimections and Graphical Learning. ............ 12 2.3 Levels of Understanding .......................................................... 15 2.4 The Concepts in Secondary Mathematics and Science (CSMS) Itogramme ...............................................................19 2.5 Computers and Graphing .........................................................23 2.6 Visual Information Processing and Display Design _____ .25 2.7 Summary ................................................................................. 30 m . METHODS AND PROCEDURES............................................................. 32 3.1 Introduction .......................................................... :................. 32 3.2 Population .......................................................................... 32 3.3 Instrument Construction and Implementation ......................... 33 3.4 Analysis...................................................................................37 3.5 Interview Protocols .................................................................. 40 3.6 Procedures ................................................................................41 IV. ARE THERE LEVELS?............................................................................. 42 4.1 Introduction ............................................................................. 42 4.2 Clustering of Items into Levels ....................................... .42 4.3 Pretest Clusters ........................................................ .43 4.4 Cluster IDeraichy.................................................................... .46 4.5 Pre- and Posttest Comparisons .........................................49 4.6 Hypotheses and Results .................................................... 53 4.7 Summary ............................................................................... .54 V. GRAPHING LEVELS OF UNDERSTANDING............................... .56 5.1 Introduction ........................................................................... 56 5.2 Forming the Clusters ................................................................56 5.3 Characteri2dng the Clusters ...................................................... 57 5.4 Characterizing the Students ...................................................... 65 5.5 Summary .................................................................................69 VI. DISCUSSION AND RECOMMENDATIONS................................... .71 6.1 Introduction ........................................................................... 71 6.2 Group similiaiities and differences ..........................................71 6.3 Van Hiele and instruction ........................................................ .74 6.4 Implications for the "Graphing Curriculum" ...........................76 6.5 Future Research......................................................................
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