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Information to Users INFORMATION TO USERS This manuscript has been reproduced from the microfilm 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 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. Each original is also photographed in one exposure and is included in reduced form at the back of the book. 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. UMI University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9123572 Art and mathematics: Enhancing achievement through curricular design Winter, June Frances M., Ph.D. The American University, 1991 Copyri^t ©1991 by Winter, June Prances M. All rights reserved. UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 ART AND MATHEMATICS; ENHANCING ACHIEVEMENT THROUGH CURRICULAR DESIGN by June Frances M. Winter submitted to the Faculty of the College of Arts and Sciences of The American University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Education Signatur^^of Committee: Chair:. ean of the College Date 1991 The American University i\\A Washington, D.C. 20016 THE IHBRICAN UNIVERSITY LIBRARY © COPYRIGHT by JUNE F. M. WINTER 1991 ALL RIGHTS RESERVED To Tom and To Professor Mary W. Gray ART AND MATHEMATICS: ENHANCING ACHIEVEMENT THROUGH CURRICULAR DESIGN BY June Frances M. Winter ABSTRACT Glaring deficiencies in the mathematical achievement of American youth, as evidenced by national and international studies, indicate that the demand for mathematically literate persons will continue to exceed the supply unless wide-ranging educational changes are instituted. Educators must develop innovative and creative intervention programs to aid students in growing mathematically. To this end, we explore an alternative approach which would supplement current teaching practices with a curriculum interrelating the studies of mathematics and art. Students in grades five through eight are targeted for the project. Developing the curriculum model, we review interdisciplinary, cultural, and motivational considerations in the first three chapters. Recent research is cited from the fields of physics, education, psychology, sociology, and medicine. These studies lend 11 support to the premise that the teaching of mathematics through art is not only feasible but educationally sound. In chapter four, we investigate some of the many connections between art and mathematics, then focus on the concepts of ratio, proportion, and symmetry. This material can be used by teachers to extend their mathematical understanding and to relate the discipline to the realm of art. Additionally, the chapter can serve as a source of reference material helpful in creating and planning additional student learning experiences. The activities chapter incorporates premises of the Standards (National Council of Teachers of Mathematics, 1989) with an interdisciplinary, inter- cultural, visual and tactile constructivist teaching approach. Both cognitive and affective objectives are included, thus recognizing the importance of developing all facets of a student's capabilities. The goal is to encourage reluctant learners to comprehend the contributions of mathematics to world culture, thus stimulating pupil interest in further mathematical study. Appropriate components are included to enhance student experience with reading and writing in the language of mathematics. Activities facilitate the growth of mathematical communication. Connections among XXI mathematics, other school subjects, culture, and practical applications in everyday life are reinforced. IV ACKNOWLEDGEMENTS A work of this scope and nature could not be completed without the aid of an entire community. I should like to thank my dissertation committee members. Professors David S. Crosby, Mary W. Gray (chair), and Myra Sadker of The American University for their guidance and constructive criticism. To Marymount University, I express gratitude for both financial and moral support. In particular, I thank Provost Alice S. Mandanis, Professor Alice Turner Schafer of Mathematics, and Artist-in-Residence Andrew Monje. Marymount University research librarians Mridula Ratnam and Kim Dessalegn facilitated inter-library loans of research materials. I am grateful to Daphne Cox of The American University for superb administrative support. I appreciate Mr. Abe Rybeck's permission to reproduce seven linear-band patterns from Peter S. Stevens, Handbook of Regular Patterns; An Introduction to Symmetry in Two Dimensions (Cambridge, Massachusetts: M.I.T. Press, 1980). Last, but certainly not least, many thanks to my family— Tom, Mother, Father, Tom III, Catherine, and Stuart— for their unflagging love, devotion, patience, and enthusiastic encouragement. VI TABLE OF CONTENTS ABSTRACT .......................................... ii ACKNOWLEDGEMENTS ................................ V LIST OF ILLUSTRATIONS.............................. ix Chapter 1. THE PROBLEM AND ITS SETTING................ 1 Introduction ............................ 1 Statement of Purpose .................... 6 2. WHY DEVELOP CURRICULUM MATERIALS FOR GRADES FIVE THROUGH EIGHT? ........................ 8 Introduction ............................ 8 Responsibilities, Time, and Training of Teachers with a Focus on Elementary School Personnel .................... 8 Characteristics of Middle-School L e a r n e r s ............................ 15 Conclusion.............................. 17 3. ESTABLISHING A FRAMEWORK .................. 19 Why Use Art to Teach Mathematics? .... 19 The Need for a New A p p r o a c h ............ 22 Interdisciplinary, Cultural, and Motivational Considerations .......... 23 Brain Research Considerations .......... 33 Brain Theory, Mathematics Achievement, Sex, and Spatial A b i l i t y ............ 38 V l l Chapter Page Implications of the Right-Brain/Left- Brain Construct for Teaching Mathematics...................... 48 Learning Style Considerations .......... 52 Additional Insights .................... 60 4. MATHEMATICAL PERSPECTIVES ON A R T ....... 65 Introduction .................. ..... 65 Ratio and Proportion................. 71 Symmetry............................. 112 5. ILLUSTRATIVE ACTIVITIES FOR A CURRICULUM INTERRELATING THE STUDIES OF MATHEMATICS AND ART FOR GRADES FIVE THROUGH EIGHT .... 159 Introduction ............................ 159 Activity O n e ............................ 162 Activity T w o ............................ 169 Activity Three .......................... 185 Activity F o u r .......................... 193 Activity F i v e .......................... 203 6. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . 217 Summary and Conclusions............. 217 Recommendations ........................ 219 SELECTED BIBLIOGRAPHY ............................. 222 V l l l LIST OF ILLUSTRATIONS Figure Page 1. Constructing the Golden Section Point of a Line Segment.......................... 79 2. Given a Square, Construct a Golden R e c t a n g l e .............................. 86 3. Construct the Root Rectangles: Two, Three, Four, and F i v e .......................... 98 4. A Special Property of the Root Two R e c t a n g l e .............................. 101 5. The Root Three Rectangle as Two Equilateral Triangles Placed Base to B a s e .......... 103 6. The Regular Hexagon and the Root-Three R e c t a n g l e .............................. 104 7. Repeating Regular Hexagons ................ 105 8. Translation as a Composition of Reflections............................ 121 9. Rotation as a Composition of Reflections . 124 10. Symmetries of the Equilateral Triangle . 131 11. Cayley T a b l e .............................. 133 12. Group 2 M o t i f .............................. 141 13. Group 2 m m ................................. 141 14. Group 3 M o t i f .............................. 143 15. Group 3m M o t i f ............................ 143 16. Group 3m Motif with Three Mirrors........... 145 IX 17. Linear Band Pattern Type " t " .............. 148 18. Linear Band Pattern Type "tg" ............ 149 19. Linear Band Pattern Type "t m " .............. 151 20. Linear Band Pattern Type "m t " .............. 153 21. Linear Band Pattern Type "t 2 " .............. 154 22. Linear Band Pattern Type "t2mg"............ 156 23. Linear Band Pattern Type "t2mm"............ 157 24. Reference T r i a n g l e ........................ 171 25. Symmetries of Equilateral Triangle ....... 172 26. Remaining Symmetries of an Equilateral Triangle...............................
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