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INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, som e 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 quality 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 had 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 corner and continuing from left to right in equal sections with small overlaps. Photographs 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. ProQuest Information and teaming 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 ® UMI THE ELECTRONIC “OTHER’: A STUDY OF CALCULATOR-BASED SYMBOLIC MANIPULATION UTILITIES WITH SECONDARY SCHOOL MATHEMATICS STUDENTS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Michael Todd Edwards, B.A., B.S., M.A., M.S., M.Ed. ***** The Ohio State University May 18,2001 Dissertation Committee: Approved by Douglas T. Owens, Chair Bert K. Waits Advisor College of Education Suzanne K. Damarin UMI Number; 3011050 UMI' UMI Microform 3011050 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Copyright by Michael Todd Edwards 2001 All Rights Reserved ABSTRACT This study investigated two groups of suburban high school students using computer algebra systems (CAS) with graphing calculators. The CAS group (n = 25) used symbolic manipulation utilities systematically throughout the school year. The non- CAS (n - 22) group were taught using graphing calculators without CAS. Group performances on a year-end Advanced Algebra Final Examination were analyzed, using scores on two pretests as separate covariates. An analysis of covariance (ANCOVA) revealed that non-CAS students significantly outperformed CAS students (p < 0.000). Additional applications of ANCOVA revealed that low-performing non- CAS students significantly outperformed their CAS counterparts, (p = 0.029). Middle- performing non-CAS students significantly outperformed their CAS counterparts (p = 0.033). Differences in the performances of high-performing students were not statistically significant (p = 0.463). Additionally, chapter test and quiz means were examined. Of the nineteen tests and quizzes administered, separate analyses of variance (ANOVA) revealed no statistically significant differences. Non-CAS students outperformed CAS students on all tests and quizzes until the fifteenth week - suggesting that CAS posed challenges for students at the beginning of the year. On specific test and quiz items, CAS students employed symbolic methods more frequently than non-CAS counterparts (51 percent to 42 ii percent). CAS students used tables 14.1 percent of the time, compared to 19.4 percent for non-CAS. The CAS group used graphs 32 percent of the time, compared to 31.3 percent for non-CAS. Several conclusions can be drawn from student essays, short writing items, and a teacher journal. First, CAS students’ perception that CAS would be useless in subsequent coursework interfered with their mathematical performance. Secondly, CAS students indicated that by-hand manipulation helped them understand algebraic concepts more clearly than CAS. Awkwardness of calculator output and the calculator’s tendency to perform "too many steps,” contributed to students’ preference for by-hand methods. Focusing student attention on the theory of equation solving is the real promise of s}Tnbolic manipulation with CAS. CAS enables students to choose transformations to apply to equations without worrying about arithmetic errors. However, until CAS utilities exist allowing students to choose transformations for themselves, the use of CAS as a means of teaching equation solving to secondary school students is not recommended. Ill To my loving wife, Jennifer, and my daughter, Cassady, for their patience, guidance, and love. I would have never survived this process without their support. To my parents, Charles and Francine, for instilling within me a passion for learning. IV ACKNOWLEDGMENTS A pivotal moment in my career as a mathematics student and educator occurred when I was only eight years old. For my birthday, I received two gifts that dramatically influenced the course of my education and professional life. My parents bought me a Texas Instruments TI-1600 pocket calculator, and my Aunt Georgia bought me a small tape recorder. For hours, I lay on the floor of my bedroom, intently pressing buttons on the calculator. While carefully examining the results of various computations on the calculator’s red LED screen, 1 recorded my discoveries onto cassette tape. Each evening I played edited versions of the tape-recorded lectures back to my parents, sharing my findings eagerly with them. Although my early calculator activities were purely recreational — a diversion from cartoons and kickball — they developed within me a strong interest in the interplay of mathematics, technology, and learning that continues to be my passion to this day. The activities also fostered within me a curiosity regarding research and calculators that has nourished me as I continue work with calculator-based research in mathematics education. As I grew older, I became fascinated with sports — particularly baseball — both as a player and as a hobbyist. I spent countless hours poring over statistics of my favorite players — comparing batting averages, home runs, and earned run averages of the all- time greats. In a thirst for more statistics and more colorful player profiles, I soon began producing my own baseball cards. Using fictitious athletes and teams, I drew action "photos" on the front of heavy cardboard cutouts while composing extensive statistics on the back. With the assistance of my calculator, I compiled statistics not commonly found on baseball cards — on-base percentages, slugging percentages, and homerun-to-hit ratios. After creating teams of fictitious players, I developed statistics- based card games that allowed my athletes to compete against each other. The creation of these card games provided me with invaluable practice for the cross-curricular lesson planning 1 would find myself engaged in two decades later and provided me with a view of mathematics as a creative enterprise with numerous connections to art, writing, and atheletics. This document is dedicated to my parents — Charles and Francene Edwards — for their tireless support of my academic pursuits; to my wife — Jennifer — for her patience regarding my calculator tirades; and to my daughter Cassady — the best CAS with which I’ve ever worked. A special thanks also goes to all of my math teachers, both good and bad, for helping me build my own view of what mathematics instruction should be. In particular, thanks to Dr. Dermis Davenport — professor of Mathematics at Miami University — for turning me onto higher level mathematics and Dr. Bert Waits — professor emeritus at Ohio University — for introducing me to the “power of visualization.” I am forever grateful to Bert and Dr. Ed Laughbaum for their help with VI this study. Last, but not least, thanks to Dr. Douglas T. Owens — my Ph.D. adviser at Ohio State University — for his kind words of encouragement and his tireless editing of my dissertation drafts. VII VITA 1990 .............................B.A. and B.S. in Mathematics, Miami University, Oxford, Ohio 1991-199 2 ...................Computer Programmer, Cincinnati Bell Information Systems, Cincinnati, Ohio 1992-199 5 ...................Graduate Teaching Associate, Department of Mathematics, Ohio University, Athens, Ohio 1995 ............................ M.S. Mathematics, Ohio University, Athens, Ohio 1995 ............................ M.Ed. Education with Secondary Certification, Ohio University, Athens, Ohio 1995-2000 ...................Mathematics and Computer Science Teacher, Upper Arlington High School, Upper Arlington, Ohio 2000-2001.................. Mathematics Teacher, Linworth Alternative School, Worthington, Ohio FIELDS OF STUDY Major Field: Education via TABLE OF CONTENTS ABSTRACT................................................................................................... ,.......................... ii ACKNOWLEDGMENTS........................................................................................................v VITA .......................................................................................................................................viii LIST OF TABLES.................................................................................................................xiii

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