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...... 173 27. Grid for Cayley T a b l e ...... 174 28. Scale Version of Manipulative...... 178 29. Construct a Perpendicular to a Line Through a Given Point, X, on the Line . . 204 30. Construction of a Regular P e n t a g o n ...... 205 CHAPTER 1 THE PROBLEM AND ITS SETTING

Introduction: The Current Status of Mathematics Education in the United States The United States is in a crisis. From the Wall Street Journal to the neighborhood newspaper, front-page articles lament the lack of a sufficient number of American mathematically competent college graduates to meet the country's burgeoning needs. As reliance on sophisticated technology and complex problem-solving increases, more occupations require workers who are mathematically literate. "Over 75 percent of all jobs require proficiency in simple algebra and geometry either as a prerequisite to a training program or as a part of a licensure examination" (National Research Council 1989, 4). The demand for mathematicians, statisticians, engineers, and scientists is at an all-time high. In addition, professionals in business, the social sciences, allied health fields, and education must have advanced levels of reasoning skills and mathematical ability. However, universities throughout the country report that the percentage of United States citizens in their graduate programs in mathematics and science is small. Where are the Americans who will teach the young and perform research after the visiting mathematics/science majors have returned home and the current contingent of faculty members have retired? Results of national and international studies attest to the fact that mathematics education in the United States is in dire need of réévaluation and change. Japanese and American students participated in two mathematical studies. The First International Mathematics Study (FIMS) took place in 1964, whereas the second (SIMS) occurred in 1981-1982. In both studies, Japanese scores at each cognitive level were appreciably higher than American scores. In addition, Japanese scores forged ahead from the first study to the second, while American student achievement was relatively stagnant (Horvath

1987). Results of the SIMS indicated that mathematical performance of the top 5 percent of American students was matched by the top 50 percent of Japanese students (National Research Council 1989). The National Assessment of Educational Progress, mandated by Congress, has assessed the mathematical performance of American students since 1972. The study has encompassed the skills, attitudes, and course-taking trends of 9-, 13-, and 17-year-olds. Although trends in the studies may appear to be positive— course enrollments and average performance are rising— these findings are not cause for great rejoicing * Improvements in proficiency were centered in lower-level skills. * Discrepancies between prescribed and actual performance rose as time passed, especially for Blacks and Hispanics. * Academic achievement for Blacks and Hispanics lagged consistently and substantially behind that of Whites.

* White students made only minor improvements in achievement between 1978 and 1986. * Performance gaps between females and males have not appreciably narrowed. * Even in the 1986 assessment, a majority of 17- year-olds reported no advanced coursewc : in mathematics (Dossey et al. 1988).

* Although the percentage of Black males, ages 18

to 24, completing high school has increased since 1974, the rates for both Black and Hispanic males still fall far behind those of White males of comparable age (National Center for Education Statistics 1988). Other studies attest to the mathematical deficiencies of Native American students. Ute pupil 4 mathematics performance on standardized tests lags behind that of non-Native Americans. This situation is mirrored in studies of Indian students from other tribes. Performance deficits appear as early as second or third grade and become more pronounced as higher grade levels are reached (Leap 1988). Thus, Native Americans are frequently led to careers that are not financially rewarding. Census projection indicates that, by the year 2020, the population of school-age children (ages 5-17) will be 48 percent ethnic minority (Duckett 1988). In view of the current sub-standard mathematics performance of non-Asian minority students, future American mathematical proficiency will be appreciably impaired if appropriate intervention measures are not undertaken immediately.

What are some of the aspects of mathematics instruction which contribute to the problem? One is the frequent emphasis on a technique-oriented curriculum (Bishop 1988). This is a curriculum which stresses practice and rote learning. It is one, metaphorically, of endlessly playing musical scales without the joy of participating in a Beethoven concerto. It is a curriculum which presents mathematics as a bag of tricks. It frequently does not develop understanding or meaning. The 5 National Assessment of Educational Progress reported that over 50 percent of students in grades seven and eleven viewed mathematics as "mainly memorization"; the same proportion stated that they do not expect to work in an area requiring mathematical ability (Dossey et al. 1988). What do these statements mean? They imply that many students view mathematics as a hodgepodge of disconnected rules which provide few, if any, links to prior knowledge. To them, mathematics is, at best, a discipline remote from real-world applications (Comiti, Payne, Green, and McIntosh 1984). Another facet of the problem is that of impersonal learning. Emphasizing subject knowledge and technical performance, mathematics education frequently ignores the individuality of the student and the social and cultural aspects of the learning experience (D'Ambrosio 1986; Suina 1988). For those who fail in the attempt to learn mathematics, there is a despair, a sense of boredom, a frustration. For many pupils, both those who do not succeed and those whose progress is marginal at best, there is the we11-documented mathematics anxiety, characterized by a pervading fear— and awe— of mathematics. (See Gough, 1954; Morris, 1981; Hembree, 1990.) This fear not only inhibits students from 6 succeeding in subsequent mathematical encounters, but it may also cause math avoidance. As Sir Hermann Bondi stated, "The negative attitude toward mathematics, unhappily so common, even among otherwise highly educated people, is surely the greatest measure of our failure and a real danger to our society" (as quoted by Skemp 1987, 161).

The mathematics education dilemma is academic, economic, social, and cultural. Any effective solution will require prolonged planning, national consensus, funding, and dialogue. All this involves enormous expenditures of time and money. In the interim, educators themselves must attempt innovative and creative intervention programs to aid students... in growing mathematically.

Statement of Purpose

The purpose of this dissertation is to establish a rationale and to present a model for an alternative mathematics curriculum designed for students in grades

five through eight. The curriculum focuses on the many interrelationships between mathematics and art. The goal is to encourage reluctant learners of mathematics to comprehend the contributions of mathematics to world culture, thus stimulating pupil interest in further mathematical study. Relevant insights from cognitive 7 research and motivational theory will be integrated to create and construct engaging projects and group activities. Appropriate components will be included to enhance student experience with reading and writing in the language of mathematics. Activities will facilitate the growth of effective mathematical communication. One of the primary aims of the work will be reinforcement of the connections among mathematics, other school subjects, culture, and practical applications in everyday life. As Lynn Arthur Steen states, "Students' experiences in school must bring them to believe that mathematics has value for them so they will have the incentive to continue studying mathematics" (Steen 1989, 18). CHAPTER 2 WHY DEVELOP CURRICULUM MATERIALS FOR GRADES FIVE THROUGH EIGHT?

Introduction Demands made upon the elementary school teacher are staggering. With the increasing number of single­ parent households and the concomitant impoverishment of children on one hand, and the growth of two-wage-earner families on the other, teachers are feeling the burden of becoming "all things to all people": teacher, parent- figure, role model, source of emotional and psychological support, advocate, disciplinarian, and mentor. In view of this difficult situation and the need for improvement in mathematics education (see Chapter One), innovative aids to enhance curriculum implementation are required.

Responsibilities. Time, and Training of Teachers With A Focus on Elementary School Personnel Mathematics Anxiety In 1976, Levine made the startling and disturbing statement that many elementary school teachers, most of whom were female, were terrified of mathematics; they felt that "a mathematician must either be a genius or a

8 9 lunatic" (Burton 1979, 265). Since that time, a number of studies have explored the relationship between mathematics anxiety and the elementary teacher. A recent study of college freshmen indicated that elementary education majors scored higher on the Mathematics Anxiety Rating Scale (MARS) than any other group tested— except for the participants in a math- anxiety workshop (Kelly and Tomhave 1985). In his study of mathematics anxiety level by college courses and majors, Hembree found that the highest levels occurred in students preparing to teach elementary school (Hembree 1990). Some evidence exists to indicate that minority- group teachers are more likely to admit to math-aversion than teachers from the majority (Cocking and Chipman 1988). Hembree found that Hispanic college students displayed higher levels of mathematics anxiety than Whites (Hembree 1990).

Members of the Fifth International Congress on Mathematical Education reiterated concerns that teachers' negative attitudes toward mathematics would produce a deleterious effect on their pupils (Dorfler and others 1984, 113). Although Wood claims that research is inconclusive as to whether or not math anxiety is transmitted from teacher to student, it is important to note that a significant minority of elementary school 10 teachers dislike or fear mathematics (Wood 1988). How can teachers with an aversion to mathematics be expected to inculcate a love of the subject in their pupils? The preceding research findings indicate that mathematics curriculum materials in the form of background materials and activity plans may be of considerable assistance to elementary school teachers who feel uncomfortable with the discipline.

Preparation The concept of the self-contained classroom, with a teacher who teaches many subjects, cannot assume that he or she should be a specialist in all of them. This dilemma has led to proposals in some quarters that mathematics in the elementary school, beginning with the fourth grade, be taught by teachers who are prepared as specialists in mathematics. (Hill 1987, 10) Since the demand for mathematics teachers continues practically unabated (National Center for

Education Statistics, 1988), and school funding remains a primary economic problem, it is unrealistic to think that Hill's suggestion will be implemented, nationwide, in the immediate future. Yet the need for quality education in mathematics on all levels remains. Intermediate measures must be taken immediately. To compound the dilemma, several studies have indicated that preparation of elementary school teachers in mathematics should be improved. One research project 11 exploring prospective elementary teachers' understanding of division revealed that knowledge of division was fragmented and incomplete (Ball 1990). Another study (Graeber, Tirosh, and Glover 1989) also produced alarming information: In working multiplication and division problems, 39 percent of the preservice teachers gave erroneous answers to approximately 31 percent of the questions posed. An investigation by Martin and Harel (1989) researched the views held by preservice elementary teachers concerning mathematical proof. Since mathematical proof receives little emphasis in elementary classrooms, the students' concept of proof is transmitted primarily through the teacher. Thus, if the teacher considers proof by example to be adequate, this misconception will be held by the students. Difficulties in secondary school geometry and higher mathematics courses will result. The study by Martin and Harel revealed that more than 50 percent of the 101 preservice- elementary-teacher subjects accepted an "inductive argument" as a valid mathematical proof. By "inductive

argument," the researchers meant a generalized conviction arising from relevant individual experiences: an "argument" which is not necessarily logically correct. 12 Studies by Gibney and Ginther (1988) performed over the past eighteen years focused on the level of mathematical competence of the elementary teacher. The researchers concluded that mathematical understandings of elementary teachers being prepared at the preservice stages in the 1980s were lower than those of a similar group tested in the 1970s. The researchers cited four additional studies which corroborated the observed declines in academic competence. In Everybodv Counts; A Report to the Nation on the Future of Mathematics Education, the authors lament: Too often, elementary teachers take only one course in mathematics, approaching it with trepidation and leaving it with relief, such experiences leave many elementary teachers totally unprepared to inspire children with confidence in their own mathematical abilities. What is worse, experienced elementary teachers often move up to middle grades (because of imbalance in enrollments) without learning any more mathematics. (National Research Council 1989, 64) The observed deficiencies in mathematical understandings indicate a need for improved preparation of preservice and inservice elementary school teachers. If the errors and imperfect understandings are not corrected, teachers will perpetuate misconceptions in their pupils. The National Council of Teachers of Mathematics (1982) recognizes that mathematics programs, particularly in grades seven and eight, are intended to function in two capacities: (1) as a review of concepts and skills 13 encountered in grades one through six, and (2) as an introduction to new areas of mathematics which prepare for the secondary school curriculum. The Council states. These classroom teachers are specialists in the rather clear-cut mathematics programs of the elementary school or the senior high school but not in the mathematics program of the middle grades. They are not prepared to handle both a different treatment of familiar content and a treatment of content intrinsically satisfying to mathematicians. (National Council of Teachers of Mathematics 1982, 4) The materials developed in this paper can be used by teachers both to extend their mathematical understanding and to promote their awareness of the importance of integrating art and science. Senechal (1990) quotes Rudolph Arnheim: The lack of visual training in the sciences and technology on the one hand and the artist's neglect of, or even the contempt for, the beautiful and vital task of making the world of facts visible to the enquiring mind, strikes me . . . as a much more serious ailment of our civilization than the "cultural divide" to which C. P. Snow drew so much public attention some time ago. He complained that scientists do not read good literature and writers know nothing about science. Perhaps this is so, but the complaint is superficial. . . . Snow's suggestion that "the clashing point" of science and art "ought to produce creative chances" seems to ignore the fundamental kinship of the two. (Arnheim as quoted by Senechal 1990, 171) Additionally, the activities section can serve as a source for readily available reference material in creating and planning interesting and meaningful student learning experiences. 14 Time A recent study of 1,509 elementary school teachers underscored the lack of time and assistance which adversely affects the teaching of mathematics. Working with students of all ability levels, teachers generally instructed without the aid of another adult. Few participated in team teaching or taught in a departmental structure. Seventy-eight percent of the sample primarily utilized whole-class instructional organization. Small group instruction was eschewed for reasons of: (1) too little time for planning purposes, and (2) impracti- calities of supervising multiple groups (Good, Grouws, and Mason 1990). Students were thus deprived of opportunities to exchange information, work cooperatively to solve problems, discuss observed patterns, and construct hypotheses.

Recent statistics underscore the lack of time available for elementary school teachers: * The mean number of students an elementary school

teacher works with on a typical day is sixty (The Carnegie Foundation 1988, 19). * 37 percent of all elementary teachers queried have 10 or more preparations each week (The Carnegie Foundation 1988, 15). * 60 percent of all teachers interviewed have less 15 than one hour of formally scheduled preparation time on a typical school day (The Carnegie Foundation 1988, 17). Lack of time is also cited as the reason that a large proportion of teachers have not observed another person instructing a class since their pupil-teaching days. Also, in contrast to Japan where teachers' meetings focus on instructional and curricular matters, American counterparts deal primarily with administrative concerns (Hill 1987). Again, time pressures impede teacher growth and effectiveness.

Other Problems Other acute problems identified by surveying elementary school teachers were: different ability levels in the same class; lack of concrete materials; and too many students per classroom (VanDevender 1988).

Characteristics of Middle-School Learners

A comprehensive report by the Carnegie Council on

Adolescent Development (1989) cogently delineates salient characteristics and needs of the adolescent learner which directly bear upon the substance of this dissertation. The early adolescent years— ages ten to fifteen— constitute a "Turning Point" for many young people. 16 Attitudes are formed and decisions made which profoundly affect students' futures. During early adolescence, an expanded urge for exploration emerges. If not channeled in positive directions, students become extremely vulnerable to self­ destructive tendencies. Experimentation with drugs and sexual practices can lead to both health problems and a diminished capacity for economic self-sufficiency. Cognitive growth manifests itself in the ability to approach complex problems with an increased capacity for abstract thought. However, as the Carnegie Council (1989) states, the organization and curricula of many middle-grade schools are antithetical to the intellectual and emotional needs of its student body. Caught in a vortex of changing demands, the engagement of many youth in learning diminishes, and their rates of alienation, substance abuse, absenteeism, and dropping out of school begin to rise. (p. 8) Thus, researchers find that over one-quarter of American middle-school students can be considered "at- risk" pupils for whom school failure is compounded by negative behavior patterns (Carnegie Council 1989, 8). Risks are particularly high for the poor and members of racial or ethnic minorities. One manifestation is in higher rates of retention in grade (p. 25). Another is the high drop-out rate. 17 An estimated 59 percent of Hispanic dropouts leave school before completing the 10th grade. Data from one school system, Washington, D.C., . . . show that more than half of all dropouts leave before completing the lOth grade. Ninety-two percent of all students in the Washington, D.C. school system are Black. (Carnegie Council 1989, 27) The Carnegie report further states that young adolescents need to develop meaningful self-esteem, inquisitive habits of thought, and an identification with a valued group (p. 12). In this vein, the Carnegie Council advocates an interdisciplinary approach to facilitate inquiry, association, and synthesis across disciplines. Fragmentation of knowledge is thus minimized. Interdisciplinary themes can also enhance young peoples's recognition that the United States is now a part of a global economy and an interdependent society of nations, and that the composition of our own population is racially and ethnically increasingly diverse. (Carnegie Council 1989, 48)

Conclusion How can the use of art in conjunction with mathematics ameliorate the situation?

* An interdisciplinary approach promotes cohesiveness of knowledge. Knowledge and skills obtained in one area can be reinforced in the other. * By including art from various cultures, the

curriculum fosters a heightened sense of the 18 contribution made by women and men of all racial and ethnic groups to humanity's common body of knowledge. * The small-group learning activities advocated in Chapter Five promote peer interaction and aid in satisfying the need for meaningful group membership. * The integration of art and mathematics can maximize efficient utilization of classroom time.

* Motivation to learn may be sparked by appealing to the visual-spatial strengths of students. * Use of activity plans diminishes teacher preparation time and allows for more attention to individual differences in student ability.

* Inclusion of appropriate background material (Chapter Four) assists elementary school teachers in expanding their background in mathematics.

* Both the analytical and intuitive modes of thought can be activated in pupils, thus promoting collaboration between left-and-right hemispheric thinking. * Spatial-visualization skills, so important in mathematics, can be enhanced. CHAPTER 3 ESTABLISHING A FRAMEWORK

Why Use Art to Teach Mathematics? A review of the current literature in mathematics education reveals a permeating cry for innovative approaches. The crisis in education, manifest in catastrophic dropout rates, widespread failure to teach the basics, and looming teacher shortages, is largely a problem of the cities and of poverty. But there is also a crisis of responsiveness, of imagination. Almost all children expect school to be mostly unresponsive, disconnected from life, and unrelated to their deepest desires and concerns. Life in school is likely to be teacher-centered, textbook dominated, restrictive, impersonal, and rigid. (Ayers 1989, 72)) This scathing condemnation is reinforced by the findings of the National Assessment of Educational Progress which indicate that traditional methods of teaching— lecture, teacher presentation, and reliance on textbooks— still predominate. Innovative forms of instruction remain rare (Dossey 1988).

A literature search reveals that using art to enhance mathematics learning is indeed unusual. In this paper, we shall explore the advantages of utilizing art to

19 20 stimulate and promote mathematics learning in grades five through eight. The need for a fresh approach to the teaching of mathematics is underscored by Skemp in a striking metaphor: We would not think it sensible to teach music as a pencil and paper exercise, in which children are taught to put marks on paper according to certain rules of musical notation, without ever performing music, or interacting with others in making music together. . . . For most of us, mathematics, like music, needs to be expressed in physical actions and human interactions before its symbols can evoke the silent patterns of mathematical ideas (like musical notes), simultaneous relationships (like harmonies) and expositions or proofs (like melodies). (Skemp 1987, 207) For too many years, mathematics has been taught, particularly on the elementary and middle-school levels, as a disparate jumble of unconnected rules and procedures without coherence or applicability. Intimations of the discipline's inherent beauty and power have frequently been ignored or eschewed. This approach to the teaching of mathematics is akin to writing music by blindly following set rules of counterpoint. There is a need to hear the music, to enjoy its sound, to relate it to other music, from classical masterpieces to the touching pleasures of folk ballads. For teaching to be successful— in music, art, or mathematics— connections must be made within the discipline, between disciplines, and in relation to daily life. 21 Disconnected, fragmented, uninspired, and unimaginative approaches to the teaching of mathematics contribute to the appalling prevalence of "mathematics anxiety" and the sobering fact that half of the American 17-year-olds cannot handle junior high school-level mathematics problems (Jackson 1990, 666). Participants in a recent convocation of the Mathematical Sciences Education Board stressed that changes in curricula may help address some of these problems. Rote learning and remediation must give way to a student-centered approach. "Mathematics must be taught in such a way as to relate to the lives of the students and to their cultural backgrounds. . . . People have to see themselves in what they do" (Jackson 1990, 667). The aesthetic element is an integral component of one's daily life. Art, in varying forms— a painting, a beautiful car, a graceful bridge, a striking piece of jewelry, or an attractive textile— is around us everywhere. By using works of art to introduce mathematical topics, the teacher can enhance students' opportunities to learn. How? That is the subject of the remainder of this paper. For the moment, however, let us examine one facet of the premise. The visual aspect of art is obvious. Bishop (1989), in his review of research on visualization in mathematics education, states: 22 For more than 100 years, mathematics educators have been interested in the visual and figurai representation of mathematical ideas both in the work of individuals and in the process of teaching about those ideas. . . . Mathematics is a subject which is concerned with objectivizing and representing abstractions from reality, and many of those representations appear to be visual, (i.e., they have their roots in visually sensed experiences). . . . We would therefore agree with, and go beyond, Gaulin's statement (1985) that "The main contention of this presentation is that in mathematics education, much more emphasis should be put on various types of plane representations of three-dimensional shapes and relations, both in curricula . . . and in research and development. (Bishop 1989, 13-14) Chapters Four and Five of this dissertation correlate visual aspects of art to mathematics. A more "playful" approach to mathematics learning can be taken, establishing low-risk learning situations and fostering the construction of knowledge by utilizing visual, tactile, and kinesthetic components. For example, in Chapter Five, the data-gathering and analysis activity is introduced by paintings of leisure time pursuits.

Activities of this nature can promote group discussion and cooperation as cultural links are strengthened and mathematics-learning enhanced.

The Need for a New Approach In a recent issue of UME Trends. Garfunckel underscores the necessity of immediately establishing innovative programs for students already in the K-12 educational system. One goal is to spark and maintain 23 Student interest in mathematics. These new programs may not be perfected, but, nevertheless, should be put into operation quickly. Using the story of The Flood and the Ark, Garfunckel creates a visual metaphor. People are already drowning. Serviceable rafts must be built posthaste; later, an ideal, sturdy ark can be constructed to accommodate all (Garfunckel 1990). Utilization of art to motivate the study of mathematics creates one of these "rafts" which may well, in the future, be incorporated into the ultimate ark. In view of the plight of mathematics education, Skemp concurs by stating, "We need all the ideas we can get in arriving at new discoveries, and should not be too fussy about their sources" (Skemp 1987, 133). Nielsen adds, "You've got to do what you can in the classrooms today, with the teachers and students you have. . . . Don't wait for the perfect program. There are things you can do right now" (as quoted by Jackson 1990, 667).

Interdisciplinary. Cultural, and Motivational Cons iderations Let us now explore the bonds between mathematics and art in terms of the relatedness of knowledge, culture, and personalization of learning. Jerome Bruner gives us an insight into successful teaching approaches: 24 We live in a "multi-bonded pluriverse" in which, if everything is not related to everything else, at least everything is related to something. The only possible way in which individual knowledge can keep proportional pace with the surge of available knowledge is through a grasp of the relatedness of knowledge. (Bruner 1960, 617) If students, from an early age, begin to view mathematics as a discipline connected to other academic endeavors and meshed tightly to the realities of everyday life, they may become lifetime learners and grow academically. The 1986 National Assessment of Educational Progress supports this position. The NAEP surveyed student attitudes, beliefs, and values concerning mathematics. A statistical analysis of student perceptions of mathematics and subject proficiency showed a positive correlation. Those who reported they grasped the relevance of mathematics to daily life also exhibited higher levels of mathematical proficiency (Dossey 1988). The National Council of Teachers of Mathematics published its long-awaited Curriculum and Evaluation Standards for School Mathematics in 1989. One of the goals promulgated by the Standards is "Learning to value mathematics." This goal is approached by providing the learners with numerous and varied experiences focusing on the cultural, historical, and scientific evolution of mathematics. Relevant applications of mathematics also promote the affective goal of "valuing" while 25 simultaneously providing motivation for the study of mathematics. This approach of relevancy and connection is emphasized in Chapter Four, where we explore the interrelationships between art and mathematics via the study of ratio, proportion, and symmetry. For the moment, however, let us note that an enterprising teacher can find art objects from many different cultures illustrating mathematical concepts. Concentrating on the mathematical functions of counting, measuring, locating and designing, one can utilize: * Asante brass weights used in Africa to measure gold dust * African gold jewelry used as an indication ("measure") of a person's social status

* Gold and silver antique coins from Rome, Greece, and the Mid-East utilized as measuring and counting aids for ancient commerce * Indian and Persian astrolabes employed to locate position * LeNotre's landscape plans of Versailles as one illustration of a design for locating objects in a spatial milieu * Gaudi's church of La Sagrada Familia in Barcelona, the Mayan temples at Chichen Itza in 26 Mexico, Simon Rodia's Watts Towers in Los Angeles, and the Museum of Modern Art's Italian red-aluminum sports car as examples of the designing function * Britain's Stonehenge, connected to all the named functions and related to the sciences through physics, astronomy, mechanics, and meteorology Another goal affirmed by the Standards is "Becoming confident in one's own ability." student utilization of mathematics in everyday problems is stressed. "To decorate a ceramic pot with a regular pattern is doing mathematics. School mathematics must endow all students with a realization that doing mathematics is a common human activity" (National Council of Teachers of Mathematics 1989, 6). Another use of art in mathematics learning is evidenced in a project by Okamori and Yokochi (Japan). Children made decorative objects for their own personal use; these became prized possessions. Younger children made stamps: each one a shape and its mirror reflection. Children in the middle grades made book covers with a kaleidoscopic design. Thus, by manipulation of three-dimensional objects and creation of a valued decorative piece, the students explored mirror transformations (Carss 1984). 27 The Association for Supervision and Curriculum Development also recommends utilizing problems related to the real world (May 1990). Implementation of the curriculum thrust proposed in this paper may enable students to grasp the usefulness of mathematics and perceive its flow through architecture, sculpture, municipal planning, automobile design, and other aspects of daily life. Incorporation of a real-world context into the curriculum promulgates a ". . . richness of discussion [that] is astonishing" (The Association for Supervision and Curriculum Development 1990, 5). Studies of Native American children revealed that teachers' understanding of students' culture, languages, and values influenced effectiveness in conveying academic subject matter and its relevance (Cheek 1984; Cocking and Mestre 1988). Researchers in California investigated the adaptation of Southeast Asian refugee children to the public schools. They found that higher grade point averages correlated positively with maintaining close social and cultural ties with the parent ethnic group (Divorky 1988). The Mathematical Sciences Education Board laments curriculum neglect of mathematics' cultural contributions to society (National Research Council, 1990). There is a definite need to incorporate social and cultural threads into the tapestry of mathematics 28 education. Thoughts and emotions are part of reality and thus affect all learners. As D'Ambrosio states. The individual is not alone, but is part of a society. Reality is also social. The interplay of the environmental, of the abstract, and of the social is a key issue in mathematics education, . . . unfortunately often disregarded. (D'Ambrosio 1986, 3) By a judicious use of art in mathematics learning, the teacher can personalize instruction, strengthen cultural values, and foster student self-esteem. For example, the Standards stress the importance of pattern in the K-8 curriculum. Visual pattern development promotes the growth of skills in perceiving numerical patterns and relationships. Pattern recognition, extension, and formulation thus help develop problem-solving skills (Carss, 1986). Art objects from varying cultures provide an appropriate introduction to mathematical pattern studies. For example: * Native American pottery * Native American beading on wearing apparel * Mexican bedcoverings and related textiles

* Amish quilts * Turkish, Persian and Indian rugs and wall hangings * African textiles * African baskets

* South American embroidery 29 * European wall paper * Irish "Aran Isle" sweaters and blankets The above are but a few concrete items which could be brought into the classroom or featured on field trips. These art objects provide important links between society, the individual, and subject matter (here, mathematics), the three primary components of curriculum design advocated by Tyler (1964). Student activities should evolve, say the Standards. from problem situations and should involve active, as well as passive, components (National Council of Teachers of Mathematics 1989, 9). Let us consider the following problem situation: appropriate decoration of a funerary cloth to reflect the social, political, and religious status of a nineteenth-century African tribal chief. A particular work of art, an African Adinkra cloth, could be featured to introduce the mathematical concepts of symmetry and utilization of symbolism. Cultural components in this activity can capture student interest and enhance African-American self esteem. Zaslavsky (1990), in an address to the History in Mathematics Education Conference, summarized some benefits of introducing multicultural, interdisciplinary perspectives into the mathematics curriculum by stating: * Students become aware of the role of mathematics in all societies. They realize that mathematical 30 practices arose out of a people's real needs and interests. * Students learn to appreciate the contributions of cultures different from their own, and to take pride in their own heritage. * By linking the study of mathematics with history, language, fine arts and other subjects, all the disciplines take on more meaning. * The infusion into the curriculum of the cultural heritage of people of color builds their self-esteem and encourages them to become more interested in mathematics. As one eleven-year boy wrote in his evaluation of a classroom activity based on African culture: "As you probably know, I feel very strongly and am in deed thrust [sic] with my black people, and the math has made me feel better." There is little to be added to this heart-felt comment! (Zaslavsky, 1990, 7) Jenifer (1990) reiterates Zaslavsky's thrust in an article written for the Washington Post: Whether in elementary school or a university campus, students in this nation need to know about the heritage and contributions of people of African descent and of Asian descent and of Latin American descent as well as people of European descent. An education that views the European heritage as central, and, by assumption, superior, and that views non- European heritages and [sic] peripheral, and by assumption, inferior, is a deficient education. For it fails to prepare our youth for the reality of a world that is increasingly interdependent and a nation that will increasingly be composed of non-white peoples or, the preferred term these days, "people of color." (Jenifer, 1990)

The curriculum should provide for personalization of knowledge. Bruner maintains that learning should get to the child's feelings, fantasies, and values. Students, Bruner believes, should discover "kinship and likeness in what at first seemed bizarre, exotic and repellent" (Bruner 1966, 160). The dramatic qualities of art can be 31 Utilized in our nation's schools to personalize . the dilemmas of the culture, its aspirations, its conflicts, and its terrors" (Bruner 1966, 162). How can this aid mathematics learning? Let us review some examples:

* The drama of a bronze sculptural group of Benin warriors can be utilized in an introduction to symmetry. The patterns appearing on the soldiers' robes have cultural, aesthetic, historical, and mathematical implications. * The searing drama of Picasso's Guernica under­ scores the horrors of war and the need for international cooperation and dialogue. The painting can be used as an entré into the concept of mathematical abstraction and symbolism. (For instance, the bull stands for the triumph of malevolence, brutality, and darkness. The dying horse stands for the suffering of the innocent and their futile

protestations. These are symbols, taking the place of a concept. Similarly, rr is a symbol standing for the ratio of the circumference of any circle to its diameter. The square root of -1 is represented by the symbol, i. If students grasp the significance of a symbol in art, they 32 may not be terror-stricken by the symbols of mathematics.) * For eighth graders who have had numerous experiences with one, two, and three-dimensional figures in an informal manner, (as recommended by the Standards), an arresting encounter with George Braques's cubistic Man With A Guitar or Salvador Dali's surrealistic Corpus Hypercubis could further motivate spatial exploration. In the Braque painting, one encounters a seminal approach to the age-old problem of representing a three-dimensional object on a two-dimensional canvas. The Dali work examines the possibility of representing four dimensions on a two- dimensional surface by means of the hypercube. Also included in the goals set by the Standards is "Learning to Communicate Mathematically." The National

Council of Teachers of Mathematics holds that this aim is most appropriately approached by problem situations that promote student reading, writing, and discussion using the language of mathematics. "As students communicate their ideas, they learn to clarify, refine, and consolidate their thinking" (National Council of Teachers of

Mathematics 1989, 6). An activity introduced by utilizing the fascinating and mysterious Escher drawings and leading 33 to tiling theories can be adapted to many grade levels. This activity gives ample opportunity for group exploration, communication, the making of conjectures, and refinement of theories. (See Chapter Five.) To recapitulate, in this section we have examined some of the perceived benefits of our innovative approach to the teaching of mathematics. Interdisciplinary ties can be strengthened. The role of mathematics in different cultures can be emphasized, enhancing interpersonal understanding and building self esteem. Relevant applications of mathematics to everyday life can be examined. Pattern skills can be developed. The curriculum can be adapted to provide for personalization of knowledge. And students can be given the opportunity to communicate ideas concerning mathematics.

Brain Research Considerations An area of concentrated medical, scientific, and educational interest is that of cerebral hemispheric asymmetry. That is, left brain vs. right brain orientation of learners. Evidence concerning specific hemispheric/locale involvement in the cognitive/affective function has been accumulating for over a hundred years. And research continues. It is surmised that the two hemispheres perceive different kinds of stimuli in varying manners: 34 The right hemisphere is primarily sensory, visual, and spatial in nature. The left hemisphere is more verbal and symbolic in orientation (Glennon 1981; Johnson, 1982; Skemp, 1987). The right hemisphere processes information in a synthetic, intuitive, holistic manner, while the left hemisphere processes in an analytic, logical, and linear mode (Glennon, 1981).

In 1972, David Galin and Robert Ornstein of the Langley Porter Neuropsychiatrie Institute in San Francisco investigated the posited dichotomy in hemispheric involvement by means of the electroencephalogram (EEG). In one of their papers, they stated, "In a subject performing a verbal or a spatial task, we expected to find electrophysical signs of differences in activity between the appropriate and inappropriate hemispheres" (as quoted by Springer and Deutsch 1989, 105). Part of their findings concerned the amount of alpha production during participation in different types of tasks. (Alpha activity, electrical in nature, is predominant in the EEG when the subject is resting.) The researchers postulated that the left hemisphere should show relatively less alpha when subjects are involved in a verbal endeavor than when they are grappling with a spatial-orientation task 35 (Springer and Deutsch 1989, 106). Confirming the hypothesis, Glennon states, A large number of studies of electroencephalographic data have been done for the purpose of comparing asymmetries in brain wave patterns when verbal or spatial tasks are being performed. . . . When verbal and arithmetic tasks are being processed, the alpha wave production in the right hemisphere is greater than in the left hemisphere; that is, the right hemisphere is "idling." And when the tasks are non­ verbal, spatial in nature, the alpha production in the left hemisphere is greater. (Glennon 1981, 6) Current research in cognitive neuroscience has utilized sophisticated equipment and approaches such as PET (positron emission tomography). Imaging studies performed at the Malinkrodt Institute of Radiology at Washington University in St. Louis support the network theories of neuroscience and cognition. That is, the ability to perform rudimentary aspects of cognitive functions is localized in "nodes." A "network" connects these nodes and coordinates their capabilities to enable cognitive functions— simple to complex— to occur (Posner et al. 1988). "The task itself is not performed by any single area of the brain, but the operations that underlie the performance are strictly localized" (Posner et al. 1988, 1627). The imaging data gathered by these researchers supports posited localization of operations performed on visual, phonological, and semantic codes. They found, for example, the following relationships:

(1) The ventral occipital lobe to visual word form 36 (2) The parietal lobe to active selection and visual search (3) The lateral left frontal lobe to a semantic network for coding word associations. The research performed by Posner, Petersen, Fox and Raichle (1988) has added to the extant body of scientific knowledge obtained from the study of split- brain and brain-damaged patients. Their findings reinforce a cautionary statement made by Springer and Deutsch (1989), who maintain that a strict left/right hemisphere dichotomy is simplistic: For one thing, most of the processes neatly labeled as visual perception, speech production, voluntary movement, or memory are really the result of many complex cerebral interactions. Whether they are diffusely spread over large areas of the brain or are limited to particular regions appears to be determined by which function we are studying, how precisely we are defining it, and how successfully we are able to limit our tests to what we assume they are testing. (Springer and Deutsch 1989, 23) Thus, in this paper, let us use the terms "right- brain child" and "left-brain child" as metaphorical in nature. We choose to accept a modified cerebral hemispheric asymmetry as a theoretical construct. By that we mean:

(1) Children use both hemispheres with their special attributes in learning and reacting to their environment. The difference lies in preference and 37 dominance. Certainly, the typical child has functioning left and right brain hemispheres; The distinction is that (1) each will be more sensitive to different stimuli than the other; (2) given the same stimulus, one may process the information in a different manner than the other; and (3) the types of responses that are typical of the individuals may differ. (Cresswell, Gifford, and Huffman 1988, 119) For example, in attempting to work a proof in geometry, a "left-brained" student may approach the task by reviewing theorems and definitions or referring to the proof of a related problem. The "right-brained" student may begin with carefully drawn diagrams of both the problem and associated theorems. (2) Complete knowledge necessitates an integration of hemispheric perception and processing (Levy, 1982; Crouch-Shinn, 1984; Jones, 1985). Engagement of both hemispheres results in the "... synthesis of the specialized abilities of the left and right into a full, rich, and deep understanding that is different from and more than the biased and limited perspectives of either side of the brain" (Levy 1983, 70). Some authors assert that research findings have extensive applications in the classroom (V. Johnson 1982; S. Johnson 1983; Contreras 1986; Miller 1986; Browne 1986). Others state that theories of hemispheric specialization are oversimplified and urge caution in blind implementation (Kosslyn 1988; 38 Hart 1982; Levy 1982; Caplan, MacPherson, and Tobin 1985). At a minimum, the findings seem to confirm what perceptive classroom teachers observe: Students have different learning styles, attention to which may improve their performance. Before discussing implications of the "right- brain/ left-brain" construct for enhancing mathematics instruction, we shall explore the gender issue as it relates to mathematics achievement and brain-hemisphere theory.

Brain Theorv. Mathematics Achievement. Sex and Spatial Ability Research literature indicates that gender differences in mathematics test performance are not consistently observed until early adolescence (Burton 1979). Generally, female adolescents outperform males in formal mathematical operations; male adolescents show higher performance in less formal and more analytical reasoning tasks. However, by the time the students are seventeen years old (or twelfth graders), male students at every level of course participation score higher than female students in solving two-step word problems (Moore and Smith 1987). In reporting on the fact that males score higher than females on math sections of the Scholastic Aptitude Test (SAT) and College Board 39 Achievement Test and on the math usage portion of the American College Testing Program Examination (ACT), Sadker, Sadker, and Steindam (1989) comment: Girls are the only group who enter school scoring ahead and 12 years later leave school scoring behind. The decline of academic achievement by half our population remains an invisible issue. (Sadker, Sadker, and Steindam 1989, 46) How does one account for these gender discrepancies in mathematical performance? Some postulate that the difference is due to brain development, brain structure, and/or brain "lateralization." Studies indicating gender differences in verbal and spatial skills led to the hypothesis of differential distribution: there are gender differences in the distribution of verbal and spatial functions between the cerebral hemispheres. "A variety of evidence suggests that males tend to be more lateralized for verbal and spatial abilities, whereas women show greater bilateral representation for both types of functions" (Springer and Deutsch 1989, 222). They base their conclusions on clinical studies, dichotic listening, tests of tactual perception, cerebral blood flow, and electrophysiological studies.

A study by Benbow and Stanley (1980) created a great deal of controversy. Data collected from approximately 10,000 males and females prior to 40 differential mathematics-course enrollment pointed to large gender differences in mathematics performance: males excelled. The researchers postulated that . . . [S]ex differences in achievement and attitude toward mathematics result from superior male mathematical ability, which may be related to greater male ability in spatial tasks. This male superiority is probably an expression of a combination of both endogenous and exogenous variables. (Benbow and Stanley 1980, 1264) Stanley later suggested that the gender differential in mathematics achievement might be due to different brain lateralization in males and females which could be genetically programmed (Beckwith 1983). At once the media seized upon the concept of genetic components in perceived mathematical ability/achievement. Educational journals discussed implications of the research for educational policies, suggesting alternative curricula and teaching strategies to accommodate differences in brain lateralization and brain development. (See Beckwith [1983] for a review.) Over the past decade, many researchers rejected the notion that genetic components alone lead to mathematical achievement. They emphasized the importance of social, cultural, and environmental factors (Fennema 1980; Schafer and Gray 1981; Reyes and Stanic 1985; Yee 1986; Pedersen 1986; Syron 1987; Moore and Smith 1987; Sadker, Sadker and Steindam 1989). 41 Focusing on the issue of differential mathematics achievement, spatial ability, and brain lateralization, we note that a large body of research indicates that, in the American culture, males typically outperform females on tasks involving spatial perception (Sherman 1974; Tobias 1978; Fennema 1983). Some researchers connect this superior performance to aspects of posited male brain lateralization. (See Caplan, MacPherson, and Tobin [1985] for a review.) This inference is subject to debate. Researchers . . . present an often overwhelming number of studies that are difficult to interpret because of their limitations. Furthermore, after carefully detailing contradictory results and alternative explanations of findings, reviewers often conclude that trends in the literature support their preferred hypothesis. . . . Is a skill superior if it is unilaterally based? . . . Theorists who propose brain lateralization explanations have tried to address the question of how knowledge of lateralization of function would explain the supposed male superiority in spatial ability. Levy (1970) claimed that unilateral representation for spatial abilities leads to greater performance, whereas Buffery and Gray (1972) claimed that bilateral representation of spatial ability is better. (Caplan, MacPherson, and Tobin 1985, 791-792) More research is needed to answer the question. For purposes of this paper, however, it is important to focus on relationships between spatial abilities and mathematics performance. What is spatial visualization? The term has been defined as ". . . the capacity to perceive and hold in mind the structure and proportions of a form or figure, grasped as a whole" (MacFarlane-Smith 42 1964, 6). It also involves movements (transformations) of the figures and changes in their properties (Fennema 1983). Additionally, The person with relatively high spatial ability has a marked tendency to seek for and recognize regularities and patterns in his experiences. He tends to experience tensions when he becomes aware of a lack of completeness in any of these patterns and he continues to search until he achieves the most satisfactory completion or "closure." (MacFarlane-Smith 1964, 230) There is a consensus that spatial ability plays a significant role in general intellectual ability; however, the nature of spatial ability is varied and needs to be clarified. Some researchers maintain that there is a close connection between mathematical performance and spatial intuition (Guay and McDaniel 1977; Fennema and Sherman 1977; Battista, Grayson, and Talsma 1982). There are mathematicians also who hold that their discipline is inextricably related to spatial ability; According to Bronowski (1974), the total discipline of mathematics can be defined as the language for describing those aspects of the world that can be stated in terms of "configurations." Meserve (1973) believes . . . that "even the most abstract geometrical thinking must retain some link, however attenuated, with spatial intuition." In the Russian literature, mathematics and spatial abilities are regarded as inseparable. (Fennema 1983, 214) Nevertheless, results from empirical studies have been neither clear-cut nor consistent. Some experimenters have concluded that spatial skills and mathematics 43 performance are not significantly correlated. (See Fennema [1983] for a review.) Others consider spatial visualization necessary for all levels of mathematics learning. Still another group hold it is necessary only for mathematicians who specialize in geometry or topology (Tobias 1978). Newcombe, Bandura, and Taylor (1983) and Fennema (1983) discuss studies which conclude that spatial ability and mathematical problem-solving successes are closely allied. More-recent research found that: "Correlational studies show a strong relationship between spatial abilities and scientific and mathematical reasoning, probably reflecting the role of general ability in performance in each area" (Linn and Petersen 1985, 1493). Clearly, it is apparent that additional data and investigation are necessary. Studies in the future should identify and separate each type of spatial ability, if indeed different categories exist. (Linn and Petersen [1985] list three types of spatial ability: spatial perception, mental rotation, and spatial visualization.) Identification and classification may be of assistance in clarifying the relationship between spatial acuity and the learning of mathematics. Reasons for the perceived difference in spatial perception aside, we note that researchers observed gender 44 differences in spatial-ability tasks appearing at the same time that mathematics-achievement discrepancies between males and females became evident (Maccoby and Jacklin 1974; Fennema 1983; Linn and Petersen 1985; Leder 1985). "Male superiority on visual-spatial tasks is fairly consistently found in adolescence and adulthood, but not in childhood" (Maccoby and Jacklin 1974, 351). Marsh, (1989, 193) reports, "Despite the consistency of results favoring boys in mathematics constructs, there is also evidence that such differences may be diminishing." Marsh qualified his statement, however, by stating, Feingold noted . . . that the decline in sex differences in mathematics performances was due primarily to the improved performance of mathematically less able girls. Consistent with this observation, gender differences tended to be larger for selective samples of high ability students such as the self-selected sample of high-ability students who take the Scholastic Aptitude Test as opposed to the representative samples for the PSAT norms. (Marsh 1989, 195) Nevertheless, females are gaining substantially in visual- spatial skills (Rosenthal and Rubin 1982; Gitelson, Petersen, and Tobin-Richards 1982). Do these studies give further credence to the spatial-abilities/mathematics- achievement link? Obviously, additional research is required.

However, if there is even a strong possibility that such a link exists, it is worthwhile to investigate 45 inculcation of more-effective visual-spatial skills as a means of improving mathematics performance. If visual- spatial abilities are related to mathematical achievement, and if visual-spatial skills can be developed by appropriate educational experiences, the argument for connecting the study of mathematics and the study of art will be further strengthened, as art and spatial perception are incontrovertibly associated. Moreover, developing visual-spatial skills in this context may enhance the appeal of mathematics to under-represented groups, such as females. A recent study by Baenninger and Newcombe (1989) identifies pertinent issues, establishes useful preliminary findings, and points out areas where further investigation is necessary. The research focuses on the relationship between spatial-activity participation and spatial ability. It was prompted by the hypothesis that males' differential exposure to spatial experiences contributes to the perceived gender disparity in successful completion of tasks involving spatial perception. This phase of the research was correlational in nature. Baenninger and Newcombe's second concern centered on the use of various types of training programs to foster improvement in spatial-ability skills. The approach in this segment of the research was experimental. 46 In the first phase of the investigation, data gathered from relevant studies were reviewed by the researchers using the meta-analytic procedures employed by Rosenthal and Rubin (1982). Results provided evidence that prior participation in spatial activities— as reported in studies based upon The Spatial Activities Questionnaire devised by Newcombe, Bandura and Taylor, 1983— correlated positively (even if weakly) with higher spatial test scores. Next, spatial training was investigated from the standpoint of gender, duration of training, and type of training administered. Specific training focused on a single spatial measure. General training included various types of spatial measures. Indirect training, as the authors establish, "... had an inherent relationship to spatial ability but was not spatial training per se (e.g., engineering coursework)" (Baenninger and Newcombe 1989, 338). The spatial-training research analysis revealed that spatial ability test performance was improved by each type of spatial training; moreover, the training did not benefit one sex more than another.

To recapitulate, current research indicates that visual-perception skills can be enhanced by training. Also, participation in spatial activities is correlated with spatial ability. 47 Subsequent sections of this paper will explore relationships among mathematics and art. (See Mathematics, Art, Ratio, and Proportion; Mathematics, Art, and symmetry.) Connections between mathematics, architecture, sculpture, painting, the decorative arts, applied arts, and interior design will be described. Illustrations will be viewed through the medium of slides, books, videotapes, museum visits, and objects brought into the classroom. Students will be involved in many of the spatial activities described by Newcombe, Bandura, and Taylor (1983). Students will: * build models * use the compass * create designs for jewelry, cars, ceramics,

clothing, embroidery, wallpaper, and textiles * make and decorate pottery

* sketch plans for buildings, gardens and interior layouts * create paintings and sculpture

Thus while students participate in spatially related activities (here of an artistic nature), their spatial abilities may be enhanced. This, in turn, may promote further mathematical growth, for, if future research can verify the bonds between spatial perception and mathematics postulated by Bronowski, Meserve, and some 48 members of the Russian school, increased competency in one area could promote corresponding growth in the other.

Implications of the Riaht-Brain/Left-Brain Construct for Teaching Mathematics Since individuals process tasks in different ways (Whittrock 1981; Zenhausern 1982), a major implication for education is that accurate assessment of a learner's hemispheric preference and dominance could lead to more appropriate, efficient matches between learner and methods/activities of instruction. "The need for curricula better geared to the abilities of specific groups is clear" (Springer and Deutsch 1989, 225). For example, in learning how to compute the area of a rectangle or circle, the right-brain student may be aroused by viewing a David Smith sculpture. Eleven Books, Three Apples, at the Storm King Art Center in New York. (See photograph in Holt 1971, 93.) The spare, striking metal work stands out against a broad expanse of lush lawn surrounded by large specimen trees. The sculpture is a vertical arrangement of rectangular forms and a circular disc, very bold and arresting. An introduction to the study of squares might commence with viewing Albers' painting. Homage to the Square. (See photograph in Feldman 1971, 436.) Here three superimposed square forms in brilliant colors capture the viewer's eye. "Art's 49 modernist practitioners want to create forms meaningful in themselves, perfect to behold and unsullied by contact with the flawed and mutable world most of us occupy much of the time" (Feldman 1971, 437). After mastering visual and geometric relationships, the mathematics student can then successfully assimilate and understand these geometric forms in a symbolic equation. The Learning Stvles Network Newsletter (1982) cites a study by Markman which investigated the correlation between learner preference, teaching approaches, and test results. Subjects for the study were seventy-two male and female fifth and sixth graders who had been classified physiologically into Right or Left Hemispheric learners. After administering a baseline recall and recognition test of word pairs, the

investigator subjected half of each group to left-brain verbal teaching strategies and half to right-brain imagery approaches. A post test was given. The results were identical for speed, number correct, and number of errors. Rights who had been taught through the verbal strategy achieved significantly less well than the Lefts taught that way. Rights who had been taught through imagery achieved significantly better than the Rights taught "incorrectly," but Lefts taught through imagery revealed no differences from their Left counterparts. The results suggest that it is imperative that Right preferenced individuals be taught through visual imagery via print, pictures, filmstrips, transparencies, etc. Lefts, however, may be better able to adapt to less responsive strategies than Rights. (Learning Stvles Network Newsletter 1982, 1) 50 Right-brain students are considered "neglected" by current school procedures and methods (Elliot 1980; Grow and Johnson 1983). They have difficulty constructing and synthesizing knowledge solely from verbal and symbolic sources (Rubenzer 1982; Crouch-Shinn and Shaughnessy 1984). Their forté is discriminating complex visual and non-verbal auditory patterns (Levy 1982). Thus, there is an implication that right-preferenced learners could benefit from instruction in a more visual mode. This could be accomplished, for example, via photographs, slides, pictures, filmstrips, transparencies, videotapes, computer-assisted design, visits to museums, and use of concrete manipulatives. In any case, it is generally agreed that successful teaching strategies provide the learner with a rich variety of activities, allowing the learner to make choices in information-processing modes (Munsell, Rauen, and Kinjo 1988). Another implication of the right-brain/left-brain construct for the teaching of mathematics relates to the role of affect in learning. (The affective domain is the area of learning involving attitudes, values, and emotions.) Coop and Sigel (1971), in their examination of the concept of cognitive style, state that Since learning appears to involve some combination of existing cognitive skills plus a motivation or tendency to acquire some particular information, it appears feasible to suggest that both the cognitive 51 and affective domains tend to interact in the learning process as well as in the cognitive style construct. (Coop and Sigel 1971, 154-155) Nevertheless, affect and its relation to the teaching of mathematics has been an underrepresented theme, as research on mathematical problem solving has focused on such cognitive issues as knowledge organization, processing, and retrieval. Recently, however, more attention has been given to the part played by affect in the learning sequence (Eisner 1982; McLeod, 1987; Lester and Garofolo 1987; Smith 1988). There is a considerable body of evidence indicating that the right hemisphere plays a major role in the affect areas of emotion, activation, and arousal functions (Springer and Deutsch 1989). Consequently, if the learner can be emotionally aroused and activated, both sides of the brain will participate actively in the educational endeavor. This integration of hemispheric activity leads to a fuller understanding of the material and concepts to be learned (Levy 1982, 180).

Our premise is that the use of art (painting, sculpture, architecture, jewelry) to arouse emotionally and activate learners will facilitate mathematical achievement. How? Art "... elicits memories, creates imaginations, and has meaning that makes contact with our own experiences. It causes us to think, to reason, to 52 feel" (Levy 1983, 68). For example, an exploration into the concept of four-dimensional space could be introduced by selected works of Pablo Picasso, Juan Gris, Georges Braque, Marcel Duchamp, and Max Weber. The complexity, mystery, and almost-bizarre aspects of many of these paintings have fascinated and intrigued viewers for over half a century. A quote from Levy cogently summarizes why we propose to use art to enhance and promote the study of mathematics: I believe that children will learn best if their limits are stretched, if their emotions are engaged, and if they are helped to understand themselves and their own special ways of thinking and seeing the world. . . . Regardless of how the subject matter may be best introduced for a given child, whether through left or right-hemisphere processes, this is only an initial step, a gateway into the whole brain. The synthesis we seek is not merely the sum of understanding of each side. This would yield merely two biased and incomplete representations of reality. . . . How is such a synthesis to be achieved? . . . [P]erhaps it will be possible for teachers with sensitivity to find a way long before scientists can supply specific recommendations. (Levy 1982, 183)

Learning Stvle Considerations

Every cast of mind has its creative activity, which explores the likeness appropriate to it and derives the value by which it must live. (Bronowski 1965, 71) Considering the terms "right-brain" and "left- brain" tendencies as indicating preferred ways for receiving and processing information, we are logically led to the field of learning styles. The concept of learning 53 Style varies in definition from researcher to researcher; however, the comprehensive learning style models of Dunn, Dunn, and Price (1984) and Keefe (1985) encompass factors in addition to those typically addressed in hemispheric research. Learning style includes motivation, intake preferences (eating and/or drinking) while concentrating, chronobiology (time of day most beneficial to productive study), mobility needs, and lighting predilections. The Dunn model, for example, defines the construct in terms of over twenty components related to how learners perceive, process, and retain information (Dunn 1984; Dunn 1988). Keefe describes learning style as a gestalt— not merely a composite of cognitive, affective, and physiological attributes, but a whole which is greater than the sum of its parts (Keefe 1985).

How can one define learning style? Dunn, Beaudry, and Klavas (1989) state it is ". . . a biologically and developmentally imposed set of personal characteristics that make the same teaching method effective for some, and ineffective for others" (p. 50). It is postulated that each learner has a distinctive learning style. This style includes specific senses through which individuals prefer viewing the environment. And it encompasses psychological, cognitive, environmental, and affective aspects. Because the focus 54 of this paper is the utilization of art in teaching mathematics, we shall concentrate first on the sensing mode.

People vary in their penchant for using visual symbols as opposed to verbal symbols: As long ago as the 1880's, Galton (1882-1911) found that people differed greatly in their mental imagery. Some, like himself, had strong visual imagery; others had none at all, and thought mainly in words. This is as true today as it was then, and there are also individuals who have available both, though often with a preference for one or the other modality. (Skemp 1986, 88)

Through observation, teachers can frequently identify certain aspects of an individual student's learning style configuration. A more complete assessment can be made by personal interview or administration of a learning-style inventory instrument. (See Dunn, 1988 for a list.) Educators, however, must be cognizant of the fact that only a few learning-style identification instruments are reliable and valid (Dunn, Beaudry, and Klavas, 1989).

In addition to the study reported by The Learning Stvles Network Newsletter, at least eight additional studies performed since 1980 investigated achievement of students who were taught with instructional resources that both matched and mismatched their preferred means of perception. The perceptual preferences examined were auditory, visual and tactile in nature. Students achieved 55 statistically significant higher test scores when instructed through their preferred modality rather than through a mismatched modality. (See Dunn and Dunn [1987] and Dunn [1988] for specific studies, the majority of which are recent doctoral dissertations.) Dunn reiterates that, although many people can learn through several channels of perception, learning through one is less onerous than another. Children's preferences evolve from psychomotor to visual and then to auditory. Four recent studies (one in mathematics and three in reading) indicate that children's modes of learning are primarily tactile and kinesthetic in the primary grades. As children mature, their ability to process visually increases (Grades 2 and 3). Dunn states that children's ability to remember 75 percent of what they hear in a forty to fifty minute time span typically does not develop prior to the sixth grade. Among underachievers, this ability does not appear until high school, if at all (Dunn 1988). Thus, by integrating

auditory, visual, tactile, and kinesthetic approaches in a timely fashion, teachers can improve the quality and efficiency of instruction.

Learning-style considerations can also be viewed from the personality theories of Carl Jung (1923), which form the basis of the Myers-Briggs Type Indicator, a 56 cognitive-style assessment instrument. This instrument, utilizing four dichotomous scales, posits that apparently random human behavior is consistent with the individual's preferred mode of perception and judgement. Kalsbeek, in his investigation into the Myers-Briggs Type Indicator, states that "... [S]tudents with different preferences will find different kinds of environments rewarding, supportive, challenging, . . . all of which affects choices about persistence" (Kalsbeek 1986, 9-10). To the "sensing" students, art may provide an ideal approach to learning, as these pupils have an affinity for immediate experience perceived through one of the five senses. An extended institutional research effort at St. Louis University underscored the number of "sensing" students at the higher-educational level. It behooves the elementary and middle-school teachers to note that "sensing" pupils exist and their needs must be met. The study— the TRAILS Project (Tracking Retention and Academic Integration by Learning Style)— integrated the students'

Myers-Briggs Type Indicator scores with other data. The database included, for example, high school grade-point average, ACT/SAT scores, university major, and university grade-point average. The researchers sought answers to such questions as: What types of learners enroll in specified fields of concentration? How well do different 57 kinds of learners perform in specific schools or majors? How is student learning style related to retention rates? The study determined that approximately 52 percent of the students assessed in the endeavor fell into either the sensing-introverted or sensing-extraverted Myers-Briggs categories, while the remaining students fit into the "intuition"-grounded learning style type (Kalsbeek 1986). David Kolb's Experiential Learning Theory involves a framework that also can be utilized in effecting improvements in mathematics education. Kolb argues that learning encompasses two processes whereby ideas and skills are "grasped" through concrete experiences or abstract concepts and then "transformed" through reflective observation or active experimentation. "Learning . . . occurs through the active extension and grounding of ideas and experiences in the external world and through internal reflection about the attributes of these experiences and ideas" (Kolb 1984, 52). His phrases mirror aspects of the "constructivist" theories of learning preeminent in today's mathematics education. The constructivists maintain that the learnings, motivations, experiences, and memories that students bring to a learning situation significantly influence their understanding of it. "As a result, we learn what we actively construct from being taught, not only or 58 necessarily what the teacher attempts to tell us" (Wittrock 1981, 12). According to Kolb's theories. At lower levels, individuals habitually tend to use a predominant learning style to acquire basic information. At higher levels, individuals integrate the learning skills of valuing, perceiving, thinking or acting appropriately in differing situations. (Kolb 1984, 1) Certainly, the senses— including the visual and tactile— are conduits through which an individual can perceive "concrete" experiences. For those students who prefer to grasp experience through its concrete form (rather than by abstract conceptualization), the educator must make available many learning activities based on the senses. Thus, by recognizing the importance of learning style, the educator is enabled to motivate and assist students in achieving their academic potential. As James Keefe, director of research for the National Association of Secondary School Principals, stated, The concept of learning styles revives the hope for an authentic individualized education, since it starts with the learner and then proceeds logically to a consideration of the teaching and learning environment. An understanding of the way students learn is the door to educational improvement. (Keefe 1979, 124) Certainly, further research in learning-style theory should be performed. However, in view of the pressing need for student improvement in mathematical skills, attitudes, and understanding, attempts to 59 integrate a variety of sensory modalities into teaching activities are surely warranted. It is quite obvious that teachers who utilize lecturing as their primary approach will not engage the visual, tactile, or kinesthetic learners. As Smith and Renzulli (1984) point out: Every technique has its advantages and disadvantages and will be differentially effective depending on many factors, including the topic being addressed and the students being taught. For this reason, Joyce and Hodges (1966) suggest that "a teacher who can purposefully exhibit a wide range of teaching styles is potentially able to accomplish more than a teacher whose repertoire is relatively limited." . . . Learning styles assessment can help teachers direct their attention to the strategies that are most effective with either individuals or small groups of students. (Smith and Renzulli 1984, 49) Specific learning activities must be developed to reinforce varying modality or styles. Utilization of art as one avenue of approach to mathematics can engage the learners' visual, tactile, and kinesthetic perceptual strengths.

* VISUAL strengths can be tapped by viewing paintings, sculpture, architecture, landscape design, furniture and jewelry design, textiles, pottery, etc. This can be accomplished by means of slides, videotapes, pictures or field trips. * TACTILE strengths can be addressed by using hands to manipulate materials: building models; drawing diagrams; modeling in clay or plastic; painting; creating designs; using geoboards. 60 * KINESTHETIC strengths can be utilized in activities that focus on whole-body movement (for example: taking field trips or engaging in the art of dance).

Additional Insights into the Learning and Teaching of Mathematics You cannot improve the state of education without a model of the learner. Yet the model of the learner is not fixed but various. . . . Perhaps the best choice is not a choice of one, but an appreciation of the variety that is possible. The appreciation of that variety is what makes the practice of education something more than a scripted exercise in cultural rigidity. (Bruner 1985) Bruner asked how human beings translate experience into a model of the world. His research led him to believe that there are three ways. The first is through action. The second is through visual or other sensory organization and use of summarizing images. And the third way is through representation in words or language. These three ways are respectively termed the enactive, the iconic, and the symbolic. Intellectual development proceeds through these three stages until the individual is able to command and utilize all three (Bruner 1966). Bruner put his theory into practice. Working with eight- year-olds, he led them to "discover" mathematically significant principles of factoring by first allowing them to construct large flat squares of wood using components of fixed dimensions. This active manual task facilitated visual summary. And finally, visual organization led to 61 the development of a symbolic notation which delineated the relationship between the entire square and its component parts. Students could grasp, for instance, that (X + 2)2 = x^ + 4x + 4. similarly, the classroom teacher can use art to take pupils from visual and tactile stimuli to the construction of symbolic conceptual learnings, as we shall demonstrate in the Activity Chapter. Skemp, a mathematician and psychologist, maintains that "problems of learning and teaching are psychological problems, and we can expect little improvement in the teaching of mathematics until we know more about how it is learnt" (Skemp 1986, 14). He portrays the learning of mathematics in terms of constructing a cognitive map in which each "dot" represents a concept and each "line" represents a connection between concepts. These conceptual structures, or "schemas," integrate existing knowledge and act as enablers of future learning and understanding. The conceptual nodes, he claims, have their origins in the sensory experience of the external world and motor activity within it (Skemp 1986, 37). Schemas also expand by comparison with the schemas of others. That is, discussion among students is imperative. Finally, the schemas grow from within, through imagination and intuition, internal creative processes. He makes a striking analogy: 62 I have come to think of these properties of schemas as being organic in quality. They begin like seeds that put out roots for gathering nourishment from the soil, and stems and leaves for gathering energy from the sun: in both cases, selectively. By making these into their own substance, they enlarge their roots and leaf structures, and thereby increase what they can take in. Schemas do this, turning experiences into knowledge. (Skemp, 114) How can we help learners to build conceptual structures in mathematics? Skemp maintains that, particularly in their early years, children should be exposed to as many physical embodiments of mathematical concepts as possible. The physical forms correspond more closely to the relevant mathematical concepts than the associated symbolic notation (Skemp 1986, 145). Thus by utilizing the visual and tactile aspects of art, we can facilitate mathematics learning. The constructivist view of how people learn held by Piaget and Kilpatrick also lends credence to coupling the study of mathematics and art. Constructivism embodies

two principles: Learners actively construct knowledge from their own experiences; and learning involves a

cumulative refining process of organizing and relating one's experiences. For example, the student may view mosaics from the Byzantine churches of Ravenna, the St. Louis Cathedral, and The National Shrine of the Immaculate Conception, or see portions of Roman tiled floors in the Metropolitan Museum of Art and the Walters Gallery. This 63 activity would serve as an interdisciplinary introduction to tiling and basic concepts of symmetry. Seeing Dali's surrealistic Persistence of Memory (Canaday 1959, 522) or Braque's analytical-cubist work, Man with a Guitar (Canaday 1959, 475), the learner comes to know that "reality"— form, organization, meaning, structure, affect— can exist in one's mind, with little or no connection to the pre-existing independent physical world. How much easier now to imagine n-dimensional space. Resnick also promulgates a constructive, active view of learning which The National Council of Teachers of Mathematics Standards endorses. Students come to every learning experience with a panoply of previously acquired knowledge; they perceive, filter, and process new information and then construct their own meanings (NCTM 1989). Thus, the Standards state that mathematics instruction must include opportunities for * appropriate project work; * group and individual assignments; * discussion between teachers and students and among students; * practice on mathematical methods; * exposition by the teacher (NCTM 1989,10) In Chapter Five, the art-mathematics learning activity introduced by Escher drawings contains all of the foregoing components plus a visual appeal for the child who is global/sensing/right-brained. All the activities described in Chapter Five attempt to incorporate the 64 premises of the Standards with an interdisciplinary, inter-cultural, visual, and tactile constructivist teaching approach. In summary, this chapter discusses why an innovative approach to the teaching of mathematics through art is not only feasible but educationally sound. Research from the fields of education, psychology, sociology, and medicine lends support to the mathematics- art curriculum. CHAPTER 4 MATHEMATICAL PERSPECTIVES ON ART

Introduction Too many students view mathematics learning as a meaningless, repetitive, boring, and frustrating exercise in learning rules and jumping over hurdles. Yes, students may acknowledge the utility of mathematics in televised space shuttle flights, medical advances, and periodic Wall Street panics, but what does it have to do with their immediate lives? Calculators have their place in the classroom, but students may view their accessibility as another "reason" to shun even the acquisition of basic arithmetic skills. For these students, in particular, it is important that the educational system assist learners in making connections between mathematics and "the real world," the learners real world. As Boles and Newman state. In this world of overspecialization, much of education deals with discrete bits of information rather than large systems. People, therefore, are not trained to find connections. Without connections, value systems are difficult to develop. In the evolution of civilization. Art and Mathematics are disciplines that have been seen as polarities without connections. Yet, in fact, they are the left and right hand of cultural advance: One is the realm of

65 66 metaphor and the other the realm of logic. Our humanness depends on a place for the fusion of fact and fancy, emotion and logic. Their union allows the human spirit freedom. (Boles and Newman 1987, 4) The connections between art and mathematics are many. We shall briefly touch on some of them before discussing a select few in detail.

Linear Perspective and Projective Geometry In the Renaissance, artists turned from the medieval practice of depicting people and objects in a flat, highly stylized manner. The trend moved towards representing the world more realistically. For example, Giotto imbued his figures in the Arena chapel frescoes (1305-1306) with a sense of solidity and weight. As time passed, others also attempted to solve the problem of depicting the three-dimensional world on a two-dimensional canvas. Simultaneously, the discovery of ancient Greek works revived the concept that nature adheres to basic mathematical laws. Out of this combination was born linear perspective, a method of depicting objects on a plane surface in conformity with the way they are perceived, without reference to their absolute shapes or relations (Giedion 1949). The key to this method was found in the principles of projection and section. Kline (1956) states that Renaissance artists had to derive 67 theorems which would specify how a scene would appear on canvas. These theorems are, in a logical sense at least, part of Euclidean Geometry, in that they are deduced from the axioms and theorems of Euclid. . . . The theorems raised questions which proved to be momentous for mathematics. Professional mathematicians took over the investigation of these questions and developed a geometry of great generality and power. Its name is projective geometry. (Kline 1956, 625)

Proportion Proportion enters into human concepts of beauty in design. Hambidge states that "... design of any kind demands proportional and similar figures. Without these, design cannot exist" (Hambidge 1932, 59). Consider the automobile, prized by many as an art object as well as method of transportation. A survey of American advertising literature reveals that proportions for a desirable vehicle have changed over the years: from the box-like form of a 1938 Packard to the sleek, contemporary Jaguar. Study of nineteenth and twentieth century female attire similarly reveals changing tastes in proportion. The wasp-waist and bustle of the 1880s, the board-like androgyne form adopted by American women in the 1920s, the post-World-War-II clothing proportions of Dior's "New Look", and the miniskirt era of the late 1960s all illustrate ties between the world of fashion and the mathematical concept of proportion. (See Laver, 1969.) 68 Geometry and Architecture Throughout the ages, basic geometrical forms have been utilized in architectural design. For example: * The circle appears in the ground plan for an African hemispheric thatched house (Zaslavsky 1973, 157). * Bell towers in Italy, such as The Leaning Tower of Pisa, often exhibit a basic right-circular- cylinder shape. * The triangle appears in the roof trusses of old barns or traditional houses like the Fairbanks house c. 1636, Massachusetts (Feldman 1971, 571). * A ground-plan based on intersecting circles and ovals is used for the Church of the Vierzehnheiligen (1743-1772) in Bavaria (Giedion 1949, 62). * The sphere-like geodesic dome designed by

Buckminster Fuller for the American Pavilion in Montreal (1967) combines aesthetic appeal with economy. It encloses a large amount of space using very little material (Feldman 1987, 396). 69

Four-Dimensional Space In art, as well as mathematics, attention has focused on representing aspects of four-dimensional space. As Giedion states: In 1908 the great mathematician, Hermann Minkowski, first conceived a world in four dimensions, with space and time coming together to form an indivisible continuum. His Space and Time of that year begins with the celebrated statement, "Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." It was just at this time that, in France and in Italy, cubist and futurist painters developed the artistic equivalent of space-time in their search for means of expressing purely contemporaneous feelings. (Giedion 1949, 14) Cubism is characterized by fragmentation and simultaneity. Frequently, objects are dismembered and only key parts retained in the work. Additionally, the subject matter is viewed from more than one vantage point. In this approach to rendering reality, the artist presents an object simultaneously from all sides. The effect can be interpreted as adding a fourth dimension to the art work: time.

Much later, in the 1970s, there appeared a group of artists and mathematicians who were interested in visually representing the spatial fourth dimension. A symposium sponsored by the American Association for the Advancement of Science featured the work of a number of these researchers. Two of them, Thomas Banchoff and

Charles Strauss of the Mathematics Department of Brown 70 University, manipulated four-dimensional objects on a computer screen. "The results of this technological advance are four-dimensional images of an intricacy and accuracy never dreamed of in the early twentieth century" (Henderson 1983, 350). Another American, Tony Robbin, a painter absorbed by the complexities of a space-time continuum, stated, "Our reading of the history of culture has shown us that in the development of new metaphors for space, artists, physicists, and mathematicians are usually in step" (Henderson 1983, 352).

Symmetry The relationship between art and mathematics is clearly seen in the study of symmetry. Principles of symmetry are evident in architecture, jewelry, pottery and textile design, tessellations, and Islamic non- representational art. Symmetry is found in such diverse areas as change ringing. Renaissance typographic ornament, and the three-dimensional packing arrangements utilized in such disparate ways as designing apartment complexes and arranging boxes and cans in a supermarket. (The stability and order of symmetrical arrangements contributed to the appeal of Andy Warhol s rendering of Brillo boxes [Feldman 1971, 342] and Campbell-Soup cans.)

A few of the many connections between art and mathematics have been noted. In this chapter, we shall 71 demonstrate with very specific examples: ratio, proportion, and symmetry.

Ratio and Proportion History In this section, the relationship between mathematics and art will be explored through the medium of ratio and proportion. A ratio is defined to be an indicated quotient of two numbers. An equality between two ratios is a proportion. Any discussion of ratio involves a rudimentary investigation into counting, measuring, and designing. Since elementary school education in the United States typically includes some exposure to the civilizations of ancient Egypt and Greece, we shall commence with the measurement system of Egypt. Living in an agricultural society in close proximity to the Nile, the people of Egypt have had to deal with periodic flooding of their land and concomitant eradication of borders and loss of property. Accurate means of land measurement (geometry) were vital. As the historian Herodotus stated (485-425 B.C.):

They said that this King [Sesostris] divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away something, had to go to him and notify what had happened. He then sent the overseers, who had to measure out by how much the land had become smaller, in order that the owner might pay on what was left, in 72 proportion to the entire tax imposed. In this way, it appears to me, geometry originated which then passed to Hellas. (Pedoe 1976, 15) The ancient Egyptians used a duodecimal mensuration system that was based on human proportions. Their short cubit was equivalent to six handbreadths, while their Royal cubit (pertaining to the Pharaoh) equaled seven handbreadths. The foot was approximately 2/3 of a small cubit (Doczi 1981, 37). (The figure of Hesy-ra from Saqqara, Egypt, c. 2650 B.C. [Janson 1970, 35] is a contemporary depiction of the human body and its proportions.) The short cubit measured just under a half meter (.4236m), while the Royal variant was slightly more than a half meter (.5234m). (It is interesting to note that, building the pyramid of Khufu 4,500 years ago and utilizing the Royal cubit, the Egyptians employed a unit closely related to the present-day meter. The Egyptian royal cubit is equal to one-twelfth the circumference of a circle whose radius is exactly one meter (Grillo i960, 158].)

Egyptian land measurement was accomplished by means of the "compass": a rope length with demarcations in knots, pegged at one end. Ropes with twelve equally spaced knots have been found in pictorial representations of the goddess of construction, Sefkhet (El-Said and Parman 1976). 73 The planning of an architectural site was called "unfurling the net" or establishing a grid. Depictions exist of the Pharaoh placing the peg while rope stretchers commenced their "compass and straight-edge" mensuration (El-Said and Parman 1976). It is believed that the Egyptians constructed a right triangle by holding their knotted rope-length together at the third and eighth knots. Thus many call the 3-4-5 triangle "The Ropestretchers' triangle" or "Egyptian triangle" (Doczi 1981). Historians of mathematics disagree as to whether Egyptians of the Pharaonic period were familiar with the Pythagorean theorem. However, Egyptian construction indicates great practical familiarity with "what works." Later, in the time of the Greeks, measurement was still tied to human proportions, reinforcing Protagoras' statement that "Man is the Measure of all things." A stone relief exists showing a man with his arms outstretched. The distance from fingertip to fingertip was called "the fathom." Above the man's shoulder is a footprint, indicating another fundamental unit of mensuration (Doczi 1981, 36). When the Pythagoreans discovered that the beauties and harmonies of existence could not always be described in terms of a ratio of two whole numbers, measurement and 74 numerical operations became closely associated with geometry. (As we now know, the ratio of the diagonal of a unit square to a side is V2 : 1, an "irrational" number.) Legend has it that the Pythagoreans were so distressed when they discovered this "incommensurability," that they drowned a member who divulged the shocking discovery (Linn 1974). Not content to use the less precise method of the

Babylonians, who approximated 1 / 2 by 1.41, the Greeks decided to work with all numbers from a geometrical standpoint. A length was conveniently selected to represent the number one. Other numbers were then represented in terms of this length. To represent the square root of two, for example, they used a length equal to the diagonal of a unit square. To represent the sum of

V 2 and 1 , they would simply adjoin a unit length to the diagonal of the unit square. Thus, adding an irrational number to a rational one became as commonplace as adding one to one (Kline 1976). This "manipulâtives" approach,

rarely used in the teaching of elementary algebra, might allay the fears that contemporary students, like the ancient Greeks, often experience when first introduced to the irrational numbers.

Since the Egyptians and the Greeks produced so many works of art— and of mathematics— with the simple 75 instruments of straight-edge and compass, it is appropriate that Raphael's School of Athens (Vatican; see Hartt 1969, 461) includes a depiction of Euclid, with compass, bending over a slate covered with geometric constructions. Also fitting is the poet/artist William Blake's rendition of God as geometer (see Janson 1970, 467): The Ancient of Days, Striking the First Circle of the Earth. We have begun our search for the relationships between art and mathematics by examining some examples of early mathematics and its ties to human proportions in measurement, construction, sculpture, and painting. As Bronowski states, I found the act of creation to lie in the discovery of a hidden likeness. The scientist or the artist takes two facts or experiences which are separate; he finds in them a likeness which had not been seen before; and he creates a unity by showing the likeness. (Bronowski 1965, 27)

Different eras held varying opinions concerning the beauty of human proportions. The studies of mathematics and art unite in an overview of this topic. A brief survey of the Ideal might include such contrasting examples as: * The Venus of Willendorf, c. 15000-10000 B.C. (Janson 1970, 21) * Mycerinus and his Queen, Egypt c. 2500 B.C. (Janson 1970, 40) 76 * Cycladic idol, Aiaorgos, c. 2500-1100 B.C.

(Janson 1970, 6 8 ) * Minoan bulldancers from The Toreador Fresco, Crete 1500 B.C. (Janson 1970, 60) * Female Kore, Greece, c. 650 B.C. (Janson 1970, 82) * Botticelli's Birth of Venus, 1480 (Janson 1970,317) * Parmigianino's Madonna of the Long Neck, 1534-40 (Hartt 1969, 524) * Oba monarch and chieftains. Kingdom of Benin, late seventeenth century (Davidson 1971, 110-11) * Matisse's The Blue Nude, 1907 (Canaday 1959, 405) * Wilhelm Lehmbruck's Standing Youth, 1913 (Janson 1970, 519)

Some Greek and Roman writer/artists conceived of ideal proportions in terms of an element of the human anatomy. A now-lost treatise by Polycleitos (fifth century B.C.) reveals his predilection for an ideal figure of height equal to seven heads. Polycleitos' sculpture, the Doryphoros, or Spear Bearer, was analyzed in terms of the head measurement; a repeated use of the special ratio called The Golden Section is indicated (Doczi 1981, 104- 5). The human figure was allegedly employed in 77 formulation of the Doric column. Architects of the time supposedly measured a man's foot, found it was one-sixth of his height, and used this ratio in designing the entire column shaft, including the capital (Pedoe 1976). Studies by Vitruvius, a Roman architect- mathematician of the early Christian era, show a scheme of eight heads for the ideal figure. Vitruvius attempted to incorporate that format into a system unifying the beauty of human form, geometric precepts, and architecture (Bouleau 1963). Closer to our era. Le Corbusier, the modern French architect who dealt with construction in terms of modules, decided to use the standard of six feet (182.88 cm.). Why the use of six feet in elaboration of a standard module? Like many Greek and Roman artists. Le Corbusier viewed ideal proportions in terms of human anatomy. He once wrote, "Have you ever noticed that, in English detective novels, the good-looking men, such as the policemen, are always six feet tall?" (Pedoe 1976, 25). Here is another example of personal preference and human form determining the choice of a standard module for artistic endeavor. Education should help students perceive unity in the various disciplines and discover patterns, not only within a discipline, but between bodies of knowledge: A mathematician is a maker of patterns. . . . His patterns, like the painter's or the poet's, must be 78 beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the final test. (G.H. Hardy, as quoted by Linn 1974, iii)

The Golden Section Just what is The Golden Section referred to previously? If a line segment is divided into two parts so that the shorter portion is to the longer part as the longer part is to the entire segment, the point of division is termed The Golden Section. (See Figure 1.) If we call the longer part of the segment x and the shorter part y, then we can write; y is to X as X is to [x+y] or y/x = x/[x+y] The reciprocal of either side of the above proportion is called the Golden Ratio.

There are many reasons why the Golden Section is so pivotal in a study of art and mathematics. To quote a modern-day architect:

The power of the golden section to create harmony arises from its unique capacity to unite the different parts of a whole so that each preserves its own identity, and yet blends into the greater pattern of a single whole. The golden section's ratio is an irrational, infinite number which can only be approximated, yet such approximations are possible evenwithin the limits of small whole numbers. This recognition filled the ancient Pythagoreans with awe: They sensed in it the secret power of a cosmic order. It gave rise to their belief in the mystical power of numbers. (Doczi 1981, 13) 79 Given: Line Segment AB Construct: G so that AG/GB = AB/AG = $/l 1) Find the midpoint of AB, call it D. 2) Erect a perpendicular to AB at B. 3) Mark off BC so that BC = BD. 4) Connect A to C. 5) On AC, mark off CT so thatCT = BC.

6 ) On AB, mark off AG so that AG = AT. 7) G is the Golden Section of AB.

8 ) We have AG/GB = {(V^/2) p - p/2} / [p - {{p/E - p)/2}]

= [{PO/S - 1 )}/2 ] / {(3p - pv/5)/2}

= p/s - l)/(3 - VS)

$/l

Figure 1. Constructing the Golden Section point of a line segment. 80 In the early days of the twentieth century, it was suggested that the Greek letter Phi ($)— the first letter of Phidias' name— be utilized as a name for the Golden Ratio (Huntley 1970). (Phidias was the Greek sculptor and general construction director of the Parthenon, a magnificent temple honoring the goddess Athena, built between 447 and 432 B.C.) Algebraically, it is easy to compute a numerical approximation. In the proportion given above, replace x with $ and y with 1 and invert. Then

(1) 0 _ [$ + 1 ] 1 $

(2) $ 2 = $ + 1

(3) 0 ^ - 0 - 1 = 0

(4) $ = 1 + 1 - l4J (-1J 2

(5) $ = ( 1 +/5) 2

0 is approximately 1.61803

The seventeenth-century astronomer, Kepler, said of the golden ratio; 81 Geometry has two great treasures: one is the Theorem of Pythagoras; the other the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewell [sic]. (as quoted by Pedoe 1976, 72) The occurrences of the $ ratio in everyday life are numerous, as we shall learn in the next section. One common instance is its approximation in the sizes of printing paper: • commercial size: 8.5" x 11" (11 divided by 8.5 ~ 1.29; and ~ 1.27)

• legal size: 8.5" x 14" (14 divided by 8.5

~ 1 .6 ; and $ ~ 1 .6 )

Looking at $ in another fashion, we note it is simply the ratio of the diagonal to the side of a regular pentagon. The regular pentagon is found throughout the world of nature; some illustrations are the starfish and the apple blossom. The pentagram, or five-pointed star, is derived from the regular pentagon by joining non- consecutive vertices. The pentagram itself illustrates,

in many ways, the Golden Ratio. (See Boles and Newman 1987, 50.)

Throughout the ages, the pentagram has been a recurrent symbol. It was used as a sign in the sixth century B.C. by the Pythagorean fellowship, a group dedicated to a life of moderation and healing. Thus, the five-pointed star came to be regarded as a symbol of 82 health, harmony, and order (Huntley 1970). In the Middle Ages, the pentagram was carved in doors to repel evil spirits (Doczi 1981). As a universal symbol of auspiciousness, it is found in contemporary national flags. For example, flags of the following countries feature the five-pointed star: Chile, Venezuela, Tunisia, Benin, the U.S.S.R., Yugoslavia, New Zealand, Pakistan, Ghana, Yemen, and the U.S.A. (Hoffman 1990). (Also see the Jasper Johns painting. Three Flags in Janson 1970, 555.) In the world of Islam, the pentagonal star has mystical significance; it is often associated with "Allah" and the Prophet Mohammed. In this capacity, it is frequently found positioned at the center of domes or vaults (El-Said and Parman 1976). In Christianity, the pentagram symbolizes the five wounds Christ received at the Crucifixion (Sill 1975). The decagon, or ten-sided polygon, can be derived from the pentagon. It contains the harmonic properties of the regular pentagon and is often used in Islamic tiles, mosaics, and wood carvings. What are some unique mathematical properties of $?

* $ is unique in that it becomes its own

reciprocal when diminished by one: 1 /$ = $ - 1 ~ 0.61803 83 * i generates a geometric sequence of interest. (A geometric sequence is a set of numbers in which the first term is "a" and successive terms are obtained by multiplying the preceding term by a constant: a, ar, ar^, ar\ . . .) By replacing "a" by 1 and "r" by 9, we have the sequence: *** 1, 9, 9^, . *** where n is a positive integer. This sequence can also be written as a summation sequence, one in which each term— after the second— is the sum of the two preceding terms. From equation (2) above: 9^ = 9 + 1

thus 9^ = 9^ [$]= [9^ + 1 ] [$] = + $ and 9* — 9^ [$]= + §] [$] = 4^ + 9^

* * *

^ jn-2 or 4“*^ = 4“*^ + 4“ If one writes the above sequence in the following fashion, another striking relationship, or pattern, becomes evident:

4^ = 14

42 =19+1

$3 = 4 : + 4 ' = [4 + 1] + 4 = 2 4 + 1

4^ = *3 ^ $ 2 = [24 + 1] + [4 + 1] = 3 4 + 2

*5 = 4* + 4 : = [34 + 2] + [24 + 1] = 5 4 + 3 84 4® = 4® + 4® = [54 + 3] + [34 + 2] = 8 4 + 5 etc. Observe that the following sequence occurs twice: (It appears once as successive coefficients of 4; it appears again as the displayed constant term.) *** 1, 1, 2, 2, 5, 8, 13, 21, . . . **** This sequence, inextricably bound to the Golden Ratio, is called the Fibonnacci Sequence. After the fifteenth term, the ratio F„: F^.j is equal to 4 (to five decimal places). The sequence is named after Leonardo, son of Bonnacci, (Fibonnacci), an extremely proficient mathematician of the Middle Ages. Travel in the mercantile business took him to Egypt, Sicily, Greece, and Syria, where he became acquainted with Eastern and Arabian mathematics. Fibonnacci is credited with introducing the Hindu-Arabic numerals and the decimal system, along with the famous sequence, to Europe over 800 years ago (Eves 1983). Jacob Bernoulli of Basel, Switzerland, (1654- 1705), was one of the first to connect the Golden Ratio to the Fibonnacci sequence. Now, how are we to relate the mathematics of the Golden Ratio to art? Let us remember that much intellectual endeavor is an attempt to find connections, patterns, and unity in seeming diversity. When Coleridge tried to define beauty, he returned always to one deep thought: beauty, he said, is "unity 85 in variety." Science is nothing else than the search to discover unity in the wild variety of nature— or more exactly, in the variety of our experience. Poetry, painting, the arts are the same search . . . for unity in variety. Each in its own way looks for likenesses under the variety of human experience. (Bronowski 1965, 16) The Golden Ratio is evidenced in the so called Golden Rectangle. (See Figure 2.) * The ratio of the length of the rectangle to the width is * ~ 1.61803:1.

* The Golden Rectangle contains a square. If we cut out the square and examine the remaining rectangle, called the reciprocal, the dimensions of the latter are in the ratio 1:0.61803 which is equal to $. * The Fibonnacci numbers can be employed to construct a rectangle that approximates a Golden Rectangle, but whose sides are commensurable, say 8:5, 13:8 etc.

The Golden Rectangle occurs frequently in nature as a reference frame— say for a sunfish and dragon fly. (See Boles and Newman 1987.) It also occurs with amazing regularity in varying art forms throughout the ages. Why? Possibly individual predilection. A German psychologist, Gustav Fechner, made an extensive study in 1875 into the aesthetic aspect of i. His tests, crude as they were, indicated that many people prefer a particular rectangle: one whose shape is between that of a square and that of a 86 Given; square ABCD of side "p" Construct: a Golden Rectangle 1) Locate the midpoint, M, of AB.

2) Join M to C. 3) On AB, mark off point X so that MX = MC. 4) Erect a perpendicular to line AX at X. 5) Label the intersection of line DC and the perpendicular to line AX at X: Y.

6 ) Quadrilateral AXYD is a Golden Rectangle. Length / Width = [(p + p/s)/2] / p = (l+Vs)/2 = $/l ~ 1.618033989

P/2

Figure 2. Given a square, construct a Golden Rectangle 87 rectangle in which the ratio of width to length is 0.4. His experiments were repeated by Lalo in 1908 and Thorndike in 1917; the results of these three researchers were surprisingly similar. They indicated a popular preference for rectangles closely approximating the shape of the Golden Rectangle. Huntley gives a tabulation of Fechner's and Lalo's measurements and says: If these figures mean what they seem to mean, they point unambiguously to a popular preference for a rectangular shape closely approximate to that of the golden rectangle. Of Fechner's observers, 75.6 per cent voted for it and of Lalo's, 47.6 per cent. (Huntley 1970, 65) A brief introduction to the use of $ in art can commence with Stonehenge. Let us accept this monumental stone structure, whose construction began prior to 2500 B.C., as architecture, utilizing an art historian's view that architecture is "the art of shaping space to human needs and aspirations" (Janson 1970, 24). This impressive complex was discussed carefully by the astronomers Gerald Hawkins (1965) and Fred Hoyle (1977). Both held that the arrangement of massive, hand-hewn stones functioned as an astronomical tool. Hoyle maintained that its primary use was to predict the occurrence of eclipses. Based on Hawkins's investigations, the architect Doczi (1981) found that: 88 * The Station Stone rectangle in the plan of Stonehenge I consists of two reciprocal Golden Rectangles. (See plan: Hoyle 1977, 37.) * The proportions of the Sarsen archways approximated the 3-4-5 triangle. (See photographs: Hoyle 1977, 5 and 74.) (The ratio 5:3 = 1.666... approximates the Golden Ratio.) The Parthenon, temple to the goddess Athena, was studied exhaustively by the American, Jay Hambidge. His research (1924) indicated that the Golden Rectangle, its reciprocal rectangle (1 x 0.61308), and the root five rectangle were utilized in the temple's ground plan, cornice, entablature, and flank elevations. (A root five rectangle is composed of two Golden Rectangles, overlapping in a square. Its approximate dimensions are 1

X 2.236.) When the triangular pediment of the Parthenon was intact, the facade of the temple could be fit into a Golden Rectangle (Huntley 1970).

Doczi (1981) states that the Golden Rectangle can be found in: * The base plan of Rome's Colosseum (Two Golden Rectangles) * The Triumphal Arch of Constantine (The face-on overall shape is enclosed in two Golden Rectangles.) 89 * Certain aspects of the architecture of Central- America. The temple-pyramids of Mexico were frequently utilized for astronomical purposes, as was Stonehenge. Dimensions of the Pyramid of the Sun, near Teotihuacan, indicate enclosure within two (approximate) Golden Rectangles. (See photographs: Leonard 1978, 44-45.) The influence of this pyramid spread throughout Central America, and Mayan architects adopted its structure. The frame of Chichen Itza's Castillo (tenth and eleventh centuries A.D.) would fit into a series of root five rectangles. Although the popularity of the Golden Section waned— particularly in France— barely 150 years after it was exalted in a book by Pacioli, it was employed successfully by such renowned architects as Palladio, Christopher Wren, and the Adam brothers. Baroque architects in Italy, Spain, and Germany incorporated elements of the Golden Section— via logarithmic spirals—

into their edifices (Ghyka 1977). Studies by Jay Hambidge in the early 1920s, focusing on the occurrence of Golden-Rectangle harmonies in ancient pottery, were continued by Doczi (1981). His scholarship indicated the use of Golden harmonies in pottery from such disparate eras as the Cretan and 90 Mycenaen civilizations of the late Bronze Age, Greece of the sixth century, B.C., the Sung dynasty 960-1279 A.D., and the Zuni and Pueblo people of the New World. His investigations of Pacific-Northwest Native American blanket weaving and basketry hats also demonstrate the $ ratio. He concludes that, in many cultures, folk art frequently reflects the Golden Section regulating principle. In furniture, one finds the $ ratio applied to the Egyptian stool of Tutankhamen and the proportions of many pieces of the Queen Anne and Chippendale styles (Ghyka 1977). Prints and paintings from different ages also frequently manifest use of the Golden Ratio. Among these are the late fifteenth century wood engravings by Durer in his Apocalypse series and many paintings by Veronese,

Tintoretto, Rembrandt, and Vermeer (Bouleau 1963). Closer to our time, Mondrian adopted rectangular motifs in his quest for pure and ultimate beauty. His Broadway Boogie Woogie is analyzed by Bouleau as consisting of horizontals and verticals which exhibit— almost exclusively— the Golden Ratio. Moreover, until recently in France, the standard sizes of canvases sold for artistic use were all based on the Golden Section (Thomas 1969). 91 Triangle, Hexagon, and Square When relating mathematics to art, one needs an understanding of the triangle, an elementary plane figure used in measurement, design, and construction. It is found repeatedly in both nature and man-made structures for the simple reason that it is intrinsically stable. No angle of the triangle can be altered without simultaneously altering the lengths of the sides. Old timber barns, bridges, animal bones, and even the sand dollar utilize the triangle in a buttressing capacity (Williams 1981). In the plane, simple polygons can be subdivided into triangles. In space, the composite forms, tetrahedron, octahedron, and icosahedron, are created via rotation and connections of triangles. In recent times, the dome (derived from the sphere) and the triangle yield in conjunction the geodesic dome, associated with Buckminster Fuller. "A geodesic dome is composed of triangles and triangle-derived forms whose legs fall together to form a series of geodesic lines over the surface of the dome" (Williams 1981, 51). The geodesic dome has achieved an impressive strength-to-weight ratio. (See Williams for extensive sketches.) Continuing our discussion of ratio and proportion, we shall review various types of triangles which play a special part in art: 92 * The 3-4-5 or "Ropestretchers' triangle" was explored earlier. It is the only right triangle whose sides are in arithmetic progression. If the hypotenuse, side of length 5, were hinged and swung perpendicular to the side of length 3, a rectangle of dimensions 3x5 would be formed. With ratio of length to width of 5:3 or

~ 1.66667, its dimensions would approximate those of the

Golden Rectangle, §:1 ~ 1.61803, with an error of less

than 5 percent. The 3-4-5 triangle is useful in demonstrating (not proving) the Pythagorean theorem to elementary school students. This triangle was used by the Persian (Achemenid and Sassanid) architects in creating their elliptic domes (Ghyka 1977). * The Triangle of Price has its derivation from the Pyramid of Khufu (Cheops). (See photograph: Janson 1970, 39.) It is the half-meridian section of the Great Pyramid and widely used in art of the Gothic period (Grillo 1960). Its sides are in the ratio of and

it is the only right triangle whose sides are in geometric progression. * The Harmonic triangle is derived from the regular pentagon. It is a right triangle obtained from the apothem and radius. It can also be viewed as a right triangle with hypotenuse equal to the Royal cubit and one leg equal to the (small) cubit. (With 0.1 meter as the 93 unit, the Royal cubit equals + $ + 1 while the (small) cubit is equivalent to 4^+ 4.) The Harmonic triangle supposedly was used as a standard drafting tool during the Middle Ages (Boles and Newman 1987). * The Triangle of the Pentalpha, or Golden Triangle, is an isosceles triangle obtained by drawing two diagonals from the same vertex of a regular pentagon. The ratio of either of the two congruent sides to the base is 4:1. This triangle is found repeatedly in Islamic art. * The 30-60-90 triangle, or Timaeus triangle, is built with the radius and apothem of a regular hexagon. The ratio of the shorter leg to the hypotenuse is 1:2, while that of the longer leg to the hypotenuse is V3:2. This triangle is found primarily in inorganic— as opposed to living— forms (Boles and Newman, 1987). Two Timaeus triangles, fused along the longer leg, form an equilateral triangle; the latter figures prominently in art as a symbol of religious and philosophical tenets: It is used in the Jewish Star of David. It is the Christian sign of the Trinity. As a triangular halo, it identifies God the Father or the Trinity (Sill 1975). Enclosing a circle, it symbolizes the eternity of the Trinity (Sill, 1975). 94 ® In Indian thought, it represents Shiva and Shaki forming the wheel of Vishnu (Boles and Newman 1987). * In art analysis, the equilateral triangle is often employed in a scheme to clarify compositional format. For example, Annibale Carracci's The Baptism can be viewed as organized around two equilateral triangles; the vertex of one intersects the base of the other at its midpoint. In the over-all rectangular canvas, the lower triangle encloses one featured figure, the Christ, while the upper triangle frames the depiction of God the Father. Such analysis based on geometric forms supports "... the more general proposition that while the higher flights of pictorial invention were intuitive, such imagination was underpinned by quite elaborate designing procedure consciously applied in the studio" (Thomas 1969,

11). The regular hexagon is comprised of six equilateral triangles; it appears frequently in Islamic art. Designers in the world of Islam used the regular hexagon to avoid dealing with irrational numbers. El-Said and Parman (1976) state that, utilizing the ratio of the

height of a regular hexagon to its diameter [v ^ : 2 ], a geometrical system of relating the two dimensions of a 95 hexagonal repeat unit could be achieved without reference to non-rational numbers. Islamic art offers a rich and varied resource for the illustration of numerous geometric shapes and concepts. These appear in tiles, wood inlay, pottery, lacguerwork, textiles, manuscripts, and metalwork— including lanterns, swords, gold and silver objects, book covers, and bronze vessels— to name just a few artistic objects. (See El-Said and Parman 1976; Hayes 1983; Bourgoin 1973; Revault 1973; Fehervari & Safadi 1984.) Muslim theology viewed geometry as an arbitrary convention, an intellectual device intended to lead the viewer to contemplation of a reality separate and distinct from that of the temporal world. In addition, the common Muslim attitude toward art rejected the use of images as symbols of political entities and religion. Consequently, representation of living forms is limited in Islamic art, and geometric devices and designs predominate (Hayes 1983) .

If one views a square as the union of two isosceles right triangles, the five regular solids can be integrated into a discussion of the Golden Ratio and special triangles. Found in Euclid's writings, the five regular solids are often associated with Plato, who tried 96 to relate them to the five basic elements of Greek cosmology: fire: the tetrahedron (4 equilateral triangles)

earth: the cube ( 6 squares)

air: the octahedron ( 8 equilateral triangles)

universe: the dodecahedron( 1 2 {regular} pentagons)

water: the icosahedron ( 2 0 equilateral triangles) * The 12 vertices of the icosahedron can be grouped into three coplanar sets of four. Each coplanar set of four vertices forms a Golden Rectangle. * An icosahedron can be inscribed in a octahedron

in such a manner that each vertex of the 2 0 -sided figure

divides an edge of the 8 -sided figure into the Golden Ratio. * The centroids of the faces of a dodecahedron, when separated into three coplanar groups, also form three Golden Rectangles (Huntley 1970, 33-34).

The Dynamic Rectangles

A family of rectangles— often termed the Dynamic Rectangles— all derived from the square, are often found in various facets of art throughout the ages. Their charm and utility stems from an "harmonious" interrelation between their proportions. These proportions occur not only among individual elements of the group, but within distinct members of the family. When employed in the same 97 composition, they relate to each other in a pleasing fashion; thus they are particularly useful in design, modular composition, and préfabrication (Grillo 1960, 164) . The Dynamic Rectangles fall into two categories: the Root Rectangles and the Golden or 4 Rectangle and its derivatives. (See Figures 2 and 3.) The Root rectangles each have a width of one. Their lengths are, respectively, V2 , V3 , V4 = 2 , Vs, etc. In the 4 Rectangle family, each rectangle has width of one; the lengths take on the values V 4, 4, and 4+1. Associated with each of these rectangles is its Reciprocal, a rectangle similar to the parent and cut from the latter. (Let W = width of parent; L = length of parent; w = width of Reciprocal, and W = length of Reciprocal = width of parent. Then L : W = W : w.) This means that the parent rectangle can be divided and subdivided into similar shapes. The root rectangles particularly lend themselves to design motifs of a repetitive nature, as the length of the parent rectangle is a counting number multiple of the width of the Reciprocal. Edwards (1967) states The foregoing series of rectangles, in which the property of continued proportion and of similarity of shape are [sic] inherent, all have potential dynamic qualities and give a great number of structural skeletal forms which are of great practical use in

composition and design, (p. 8 ) 98 Given: Square ABCD Construct: The Root Rectangles from One to Five. 1) Given square ABCD of side one. On ray AB, Locate X,

such that AXi = AC = V 2 . 2) Construct a perpendicular to line AB at X^. Label its intersection with line DC: Y^. 3) Quadrilateral AX^Y^D is a Root Two Rectangle.

4) On ray AB, locate X^ such that AXj = AY^ = V 3 . 5) Construct a perpendicular to line AB at Xj. Label its intersection with ray DC: Yj.

6 ) Quadrilateral AXgYgD is a Root Three Rectangle.

7) Continue in the above manner: Locate X3 such that AX3

= AYj = V%. Quadrilateral AX3Y3D is a Root Four Rectangle.

Locate X4 such that AX^ = AY3 = VE. Quadrilateral AX4 Y4 is a Root Five Rectangle.

D

Figure 3. Construct the root rectangles : two, three, four, and five. 99 He proceeds to produce an extensive compendium of designs based upon the Dynamic Rectangles. These designs could be utilized in numerous fashions: picture/mirror frames floor/bath tiles wallpaper/wallpaper borders stencils for room trim fabric design architectural ornament stained glass motifs Since the Dynamic Rectangles are derived from the square and often used in conjunction with it, a comment concerning symbolic use of the square in art is appropriate: The 4 elements 4 seasons 4 ages of humankind

4 directions of the compass, are all "... solid, orderly, earthly concepts suggesting the order, stability, and gravity of the square" (Sill 1975, 203).

Since the aspects of a Golden or i Rectangle have been already discussed, let us consider the prominent root rectangles. 100 The Root 2Vo Rectangle

It is particularly important for the graphic artist, as it is the only rectangle in which half of the figure is similar to the whole. (See Figure 4.) Also, due to its properties of retaining the same proportions when bisected, the root-two rectangle is frequently used in design of office furniture and file cabinets and in the process of modular préfabrication. Here, economy of means is achieved by curtailing waste of material in cutting. The peculiarities of the root two rectangle led it to be utilized in a majority of European-village windows throughout the last centuries (Grillo 1960). Two specific examples of the rectangle lie in the area of composition/design. Thomas' (1969) analysis of Piero della Francesca's painting The Flagellation indicates a format of two (approximate) root-two rectangles side-by-side. Hambidge's (1932) analysis of a bronze mirror from ancient Greece revealed an underlying format of two root-two rectangles and a square.

Edwards (1967) chooses to view the octagon as a compound figure consisting of a square, four root-two rectangles and four triangles, where the area of the square is twice that of any of the triangles. In this guise, the root-two rectangle figures prominently in the octagon as employed since early times in Arab, Byzantine, 101 A unique property of the Root Two Rectangle is that half the figure is similar to the whole.

1) Given; AB = VT and AD = 1. 2) Locate the midpoint of AB. Call it M.

3) Then AB/AD = V 2 /I = BC/MB = I/ 0 /2 /2 )

AM B

Figure 4. A special property of the root two rectangle 102 Islamic and Romanesque art (Ghyka 1977). The octagon's use in design of religious buildings can be seen in San Vitale of Ravenna, the Carlovingian cathedral of Aachen, and the Cordoban mosque, to name just a few. Persian and Afghani rugs employ the octagon in overall pattern and detail (Reed, 1972).

The Root Three Rectangle Two equilateral triangles of unit edge placed base to base will determine a root-three rectangle. (The width will be one and the length twice the altitude of the triangle, or 2 x (V^]/2 -V3j see Figure 5.) One example of the use of this rectangle in art is given by Edwards (1967), who demonstrated its repetition in an exhaustive analysis of three different white marble ornaments from ancient Athens and Lykosoura. The regular hexagon is closely connected to a root-three rectangle: If a pair of opposite sides of the regular hexagon is selected, and parallel line segments drawn from the vertices, a root-three rectangle and four 30-60-90- triangles are obtained. (See Figure 6.) Looking at the regular hexagon in another fashion, one can obtain three intersecting root-three rectangles by joining opposite sides of a regular hexagon by parallel line segments from the vertices. This produces a smaller regular hexagon in the center. (See Figure 7.) The 103 Given : A Side of an Equilateral Triangle Construct: A Root Three Rectangle 1) Given side AB of an equilateral triangle. (AB = p) 2) Construct a perpendicular to AB at A: line AF^ Construct a perpendicular to AB at B: line BF; 3) Construct two equilateral triangles using AB as the base of each. 4) From C, drop a perpendicular to line AF,. Extend. From D, drop a perpendicular to line AFj. Extend. 5) Quadrilateral JKLM is a Root Three Rectangle.

P 'fS

> F . 30°

■>R J B K

Figure 5. The root three rectangle as two equilateral triangles placed base to base. 104

Given; A Regular Hexagon Show: Its relationship to the Root-Three Rectangle

30 A

s

Y

Figure 6. The regular hexagon and the root-three rectangle 105

Given; A Regular Hexagon. Construct; A Repeating Pattern of Smaller Regular

Hexagons.

Figure 7. Repeating regular hexagons 106 process can be carried out repeatedly, creating smaller and smaller regular hexagons within the original (Boles and Newman 1987). This property is utilized in certain Islamic decorative motifs (El-Said and Parman 1976).

The Root Four Rectangle This is the 2:1 rectangle which figures so prominently in Japanese architecture. It is the typical size of a tatami mat, a woven-fiber article used in place of rugs to cover the floors. Its use "... ensures that in Japanese architecture, room sizes and also the grouping of rooms into entire buildings will always be in harmonious proportions" (Doczi 1981, 123).

The Root Five Rectangle

The root-five rectangle seen in Figure 3 can also be viewed as two Golden Rectangles which overlap in a square. Many uses of this rectangle in art forms have

already been discussed. To the list, however, we add Hambidge's analysis of an ancient Greek drinking cup from the Boston Museum of Fine Arts. His meticulous studies indicate that ". . . o f hundreds of examples of Greek design examined in American and European museums, about 85 percent show dynamic schemes based upon root five; about 10 percent upon root two; 1 or 2 percent upon root three, and the remainder are either uncertain or are clearly 107 static" (Hambidge 1932, 47). He also points out that five of the major Greek temples were constructed on a root-five base. The root-five rectangle seems to have been accorded a particularly favored position in Greek design after the sixth century B.C.

The Spiral The spiral is the last form we shall discuss in our study of correlations between art and the mathematics of ratio and proportion. The spiral may be defined in terms of the locus of a point P moving along a ray OA (O is the initial point, or pole of the ray) as the ray rotates in the plane about point 0 (Struik, 1987). (See Boles and Newman [1989] for detailed instructions concerning the construction of logarithmic and Archimedean spirals.) Different types of spirals are formed depending upon the fashion in which P moves away from the pole, O. Throughout history, many cultures have employed the spiral in their art. From late Neolithic times through the Christian era, Irish art utilized the spiral. It appears in the burial tombs of Knowth, the La Tene- culture Turoe stone and Petrie Crown, Iron age horse- trapping ornaments, the Book of Kells, and the eleventh- century Shrine of Saint Patrick's Bell, to name a few (O'Brien 1972). Some think the spiral is a symbol of 108 death and rebirth, as a number of depictions show one line coiling in (entombment) as another emerges in an opposite direction (rebirth) (Cook 1979). Spirals appear on 4,000-year-old pottery and 3,000-year-old coins from Crete (Janson 1970). Some scholars think that the spiral may be associated with the Minotaur's maze and, as such, be linked to fertility symbols (Doczi 1981). In ancient Greece, spiral scrolls appeared on pottery and painted frieze fragments. In Ionic and Corinthian capitals, the scroll form is thought to represent the power of waves and winds, primal elements in a society dependent upon the sea (Cook 1979). As a symbol of healing, the spiral takes the shape of intertwined snakes in Hermes-Mercury's wand (bronze sculpture. The Rotunda, National Gallery of Art in Washington, D.C.). As the caduceus, it is an insignia of the medical profession. The Maoris and Vikings alike utilized spiral decoration in their prow ornaments (Feldman 1987). Among Native Americans, the spiral is found in Hopi and Pueblo art (Doczi 1981). Anthropologists report that the spiral, as a symbol of death and rebirth, is employed by native inhabitants all over the Americas. (See Coe [1984], Leonard [1978], and Cook [1979] for Central-American art, especially pottery, goldwork, and temple stone carvings.) 109 Cook (1979) reports that the spiral is found in decorative aspects of the cathedrals of Rouen and Lincoln. Spiral forms appear in such disparate places as sketches by Leonardo da Vinci (Hartt 1969), Whistler's Peacock Room in the Freer Gallery in Washington, D.C., and the spiral minaret of Samarra, Iraq (Feldman 1987). French drawings from the late seventeenth century preserve the shape and form of gold beads created by Akan jewelrysmiths. Discs and spiral beads fashioned by the lost-wax method predominate. Spirals are artfully employed in other Akan pieces such as heavy cast-gold bracelets from the nineteenth century. The spiral, an eternal form, is still seen in contemporary Baule gold beads. Linked as it is to the history of civilization and life forms, the spiral had a particularly interesting function in Baule customs as a type of marriage payment by a suitor to the family of his chosen wife (Garrard 1989). Students can utilize the Golden Ratio to obtain a framework for constructing a logarithmic spiral. This spiral is characterized by; (a) maintaining a constant angle of intersection with the radius vectors, and (b) being associated with a particular geometric sequence. Boles and Newman (1987) explain clearly how to obtain this framework by two different methods. The skeletal form can be created by repeated subdivision of a Golden Rectangle 110 into its reciprocal and gnomen square. (A gnomen is a shape, which when adjoined to a given shape, forms a figure similar to the original.) The resultant spiral is termed a Golden Spiral. Continued subdivision of the Golden Triangle (36-72-72) by means of bisecting successive base angles also produces a defining structure for a logarithmic spiral.

Composition Any discussion of proportion in art must include at least a brief reference to composition. Bouleau (1963, 7, 11), in his treatise on composition, or framework, of paintings emphasizes ". . . a study of the internal construction of works of art.” He says, "A painting is not simply a plane surface; it undertakes the conquest of space, and the different stages of its conquest are bound . . . to be expressed in the composition."

Compositional analysts frequently refer to

artists' preliminary sketches. Initially, painters ”... sketched freely so as to arrive at a grouping of figures which was lucid as illustration, emotionally expressive, interesting as observation, and decoratively striking” (Thomas 1969, 8). Next, the disparate sketches were organized into a coherent whole. Geometrical schemas frequently were employed to establish stability and unity Ill in a work fragmented by vigorous drawing, multiple decorative elements, storytelling devices, symbolism, and the distortions of a chosen perspective. In the case of rectangular canvas formats, the artist may have utilized a grid of vertical and horizontal demarcations of the canvas (datum lines) to organize preliminary sketches. Important aspects of the work could then be placed strategically on the grid. Datum demarcations of the canvas were obtained in diverse manners (Thomas 1969; Bouleau 1963). Some were: * Simple proportion: 1. Golden Section (See Bouleau 1963, 63-79 and 131-34 for a discussion of its use during the Middle Ages and sixteenth-century Venetian school.) 2. Root-Two section

3. Basic whole-number ratios such as 2:3 or 3:5 * Construction: 1. Establishing squares on the width or on half the length

2. Intersection of rectangular picture edges by a) segments at right angles to main diagonal b) fan-like segments originating at 30° or 60° angles to a painting corner c) rotating the width onto the length 112 In other cases, basic geometric forms served as organizing frameworks. The circle, square, triangle, parallelogram, pentagon, hexagon, and octagon all played their roles in unifying a theme. Whether an artist consciously used mathematical schema to unify a composition or subconsciously employed mathematical aids gleaned from studio learning and common practice, the fact remains that artistic composition and mathematics are inextricably bound together.

Conclusion In this section, we have explored a few of the connections between mathematics— in the guise of ratio and proportion— and art. More links exist to be discovered by student and teacher alike:. Shall any gazer see with mortal eyes Or any searcher know with mortal mind— Veil after veil will lift— but there must be Veil after veil behind. (Sir Edwin Arnold as quoted in Huntley 1970, 34)

Symmetry

Introduction We have explored the existing relationships between mathematics and art through means of ratio and proportion. Another link between the two disciplines is discovered in symmetry. 113 Geometric symmetry is apparent in many facets of one's everyday experience. There is symmetry in architectural, interior, and industrial design. A glance into the mirror reveals symmetry in the human body. Symmetrical patterns can be uncovered in critical analyses of Baroque fugues and schematic diagrams of certain Renaissance dances.

Symmetry. Hermann Weyl's classic volume (1952), discusses symmetry from a number of viewpoints. He correlates symmetry to art, citing examples from ancient times to the advent of Islam. Connections among symmetry, nature, crystallography, and quantum mechanics are also noted. In Symmetry Discovered. Rosen (1975) expands the scope of Weyl's work. Rosen attempts to make the mathematical aspects of symmetry comprehensible to those on the pre-graduate level. His exposition of symmetry encompasses a variety of contexts, from botany and art to astronomy and solid-state physics. A Symmetry Festival held at Smith College in 1973 emphasized the multiple manifestations of symmetry in such diverse areas as Renaissance typographic ornament, molecular structure, plant morphology, literary analysis, and change ringing (Senechal and Fleck 1977). 114 In a unique synthesis of many manifestations of symmetry, Godel. Escher and Bach: an Eternal Braid. Hofstadter (1980) relates Bach's fugues to Escher's tessellations in both drawing and woodcut form. Hofstadter's focus on a single conceptual skeleton, that of ". . . two complementary voices carrying the same theme both leftwards and rightwards, and harmonizing with each other," refers to symmetry transformations (Hofstadter 1980, 667). Art historians have frequently referred to symmetry characteristics in describing/criticizing a work of art. Gardner (1959) often analyzes Italian Renaissance painting and architecture in terms of the stabilities, harmonies, balance, and unification created by utilizing the symmetry of square, cube, and equilateral triangle in compositional format.

Arnheim, an art historian-psychologist, often alludes to symmetry when discussing the formal aspects of a work of art. For example, he characterizes symmetry as the "basic law of composition" in Egyptian and early Greek art (Arnheim 1974, 213). Examining a landscape by Hodler, Arnheim states that the artist's strict symmetry turned nature into ornament, creating a "chilly preponderance of order" 115 (Arnheim 1974, 151). In works of painting and sculpture where overall symmetry is appropriate to the subject, he maintains that deviation from a strict format should be included to enliven the work and mitigate "... the fossilizing effect of symmetry" (Arnheim 1974, 119). Washburn, an anthropologist, and Crowe, a mathematician, combined their fields of expertise to produce a book entitled Svmmetries of Culture (1988). The authors' goal was to develop an analytical tool to aid investigations of anthropologists, archaeologists, historians, and others researching human behavior. "Practically every culture in the world is known to decorate at least some portion of its material culture with repeated patterns" (Washburn and Crowe 1988, 29). These patterns can be systematically described in a structural sense. The authors' symmetry classification of motif arrangement— design structure— is presented as a means of relating the graphic system to a comprehensive sociocultural entity.

Symmetry is a cognitive perceptual universal, basic to the processing of all shape information. A culture's symmetries are part of that culture's cognitive organization map, and the classification of symmetries is a meaningful measure of the way members of a particular culture perceive their world. (Washburn and Crowe 1988, 24) 116 What is Symmetry? At a basic level, what is symmetry? Intuitively, one might say that a figure is symmetrical if some operation leaves its appearance unchanged. For example, each of the letters "E" and "H" is symmetrical. If one folds each of these in half, both parts "match exactly." The letter pair "p" and "d" are symmetrical. If one turns the "d" through 180°, it becomes a "p."

A thorough mathematical development of the concept of symmetry can be found in Weyl (1952) and Rosen (1975). Here, it is appropriate to outline basic tenets of linear, planar, and spatial geometric symmetry to facilitate comprehension of the bond between mathematics and art. "Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection" (Weyl 1952, 5). Let us begin by analyzing plane figures in terms of rigid motions. These are distance-preserving transformations of the plane onto itself, and there are exactly four of them. Rigid motions are also termed symmetries or isometries (Washburn and Crowe 1988). 117 Reflection

Reflection is the type of symmetry which frequently comes to mind. It is found daily in nature: the human body exhibits reflection symmetry, where a right appendage has its matching— but reversed— left appendage. The right ear has its counterpart in the left ear, for example. A classical illustration from painting is Michelangelo's Sistine chapel masterpiece. The Creation of Adam (Hartt 1969, 450), where the right hand of God reaches towards the left hand of mankind's prototype. A standard mirror reproduces the object viewed, but reverses spatial order. Using this analogy, one can say that mirror— or reflection— symmetry is present in any form that is divisible into two equivalent, but mirror-image, halves.

A notable and startling exception can be found in the surrealistic painting by Magritte entitled Portraitof Mr. Edward James from the Back (Holt 1971, 81). Here the observer sees Mr. James, frontally viewing himself in a wall mirror. The mirror, instead of revealing Mr. James's face to the onlooker, reflects Mr. James's back, while depicting other real objects in expected reflection form. The power of the painting derives from this startling juxtaposition of reality and a radical departure from 118 everyday experience. Reflection symmetry— and its notable absence— are utilized for artistic effect. Let us continue to examine the rigid motion, reflection. Any plane figure which can be "folded in half" so that one half precisely matches the other admits a reflection or flip. (Example; p|q) The fold line is called the line of reflection or mirror line. Thus, a reflection about a line X is a transformation which maps each point P onto a point P' such that: 1. If P is on the mirror line X, it does not move: P'= P. 2. If P is not on the mirror line, then X is the perpendicular bisector of the line segment joining P and P' (Wheeler 1988, 409). A plane figure which admits a reflection is said to have bilateral symmetry. A reflection is a transformation preserving distance, collinearity of points, betweenness of points, and angle measure. Intuitively expanding this definition to include spatial reflection symmetry, one can grasp another facet of the mathematics-art association. Each of the following is a manifestation of bilateral (reflection, heraldic) symmetry: 119

* The Sumerian silver vase of King Entemena, C. 2700 B.C. (Weyl 1952, 8) * The enameled sphinxes from Darius' palace in Susa (Persia), c. 500 B.C. (Weyl 1952, 11) * Sheet gold ornament. Mistress of the Animals, Rhodes, mid-seventh century (Kuntzsch 1981, 164) * The two-headed eagle, symbol of Czarist Russia and the Austro-Hungarian monarchy (Stevens 1980, 32) * Aubrey Beardsley's drawing, Venus Between Terminal Gods, late nineteenth century (Harris 1967, 85)

* "Sylvia" pendant in gold, agate, rubies and diamonds: a female form interpreted in the popular Art Nouveau "dragonfly" motif by Paul Vevier, Paris, 1900 (Kuntzsch 1981, 243) * Faces in the Bella Coola mask representing the sun. Northwest Coast Native Indian, late- nineteenth to early-twentieth century (Campbell 1988, plate 2)

Translation Loosely considered, a translation or glide can be viewed as a rigid motion in which the motif moves vertically, horizontally, or diagonally, while maintaining 120 its original orientation (Example: B B B B B . . .). Note the repetition through a specified distance and in a specified direction. More precisely, a translation determined by a ray £ and a given distance k is a transformation that maps point Q onto point Q' so that the line segment QQ' is parallel to and in the same direction as ray £. The distance from Q to Q' equals k (Wheeler 1988, 416). A translation can also be interpreted as a composite of two reflections: If r„ represents a reflection about line m and r„ represents a reflection about line n, where m and n are parallel, then the composition of r„ with r„ defines a translation, T. As a composite of two reflections, T is a mapping which preserves distance, angle measure, collinearity of points, and betweenness of points (Wheeler 1988, 416) (see Figure 8).

We can expand this definition intuitively to include spatial translational symmetry. Each of the

following is an illustration of translational (or glide) symmetry: * Persian bowman frieze, c. 500 B.C. (Weyl 1952, 49) * Celtic eighth-century band ornament (Stevens 1980, 98) 121

Line M Line N

Figure 8. Translation as a composition of reflections 122 * Palace of the Doges, Venice, early fifteenth century, arcade of the facade (Weyl 1952, 50) * Old Courthouse, St.Louis, Missouri (1845): repetition of columns in the Greek-Revival facade (Smith 1981, 414) * Bank of America, San Francisco, California (1969): The fifty-two story bank is characterized by repeated angled bays that frame all four sides and define broken-height setbacks at the upper levels without altering the integrity of the basic bay module (Smith 1981, 810-811).

Glide Reflection

A glide reflection can be viewed as a translation (or glide) followed by a reflection about a line parallel to the direction of translation. (Example: b b b . . . P P P The glide reflection is typically found in band ornamentation:

* Mesopotamian "braid" pattern, ninth century, B.C. (Stevens 1980, 107) 123 * Chinese border patterns, Yunnan guild hall, Suifi, Szechwan. These designs are also found in needle-work (Dye 1974, 391). * Imprint of tire tread, twentieth century (Stevens 1980, 112)

Rotation

The symmetry of a rotation (or turn) can be described in terms of reflection. If r„ represents a reflection about line m and r„ represents a reflection about line n where lines m and n intersect in a point P (the center of rotation), then a rotation R about the point P can be defined as a composition of r„ with r„ (Wheeler 1988, 421). (See Figure 9.) As a composition of reflections, the rotation also preserves distance, collinearity of points, betweenness of points, and angle measure.

Alternatively, a rotation may be defined in terms of the center of rotation (rotocenter) P and a given angle of rotation. The rotation maps point R into an image R' in such a way that angle RPR' is of the specified measure and RP = R'P. The direction of rotation (clockwise or counterclockwise) must also be stated (Wheeler 1988, 421). 124

Line M Original Figure

Line N

Figure 9. Rotation as a Composition of Reflections 125 Some examples of rotational symmetry in art are: * Gothic rose windows, with their brilliantly colored stained glass. In particular, note the rose window. West Front, Chartres Cathedral, late twelfth century-first half of thirteenth century (Campbell 1988, plate 7). * Isaac M. Wise Temple, Cincinnati, Ohio, 1866. Rotational symmetry is evident in each of the three round windows of its facade (Smith 1981, 458) . * Necklace of painted tin, Israel, 1982. The rotational symmetry of sixteen identical female figures holding hoops is evident when the piece of jewelry is fastened and positioned on a flat surface (Dormer and Turner 1985, plate 123). To extend one's notion of rotational symmetry to space, Weyl (1952) suggests using an individual motif of length X. This motif is repeated again and again to create a band ornament. Place the band ornament around a circular cylinder, the circumference of which is an integer multiple n of X. One then has a pattern which carries into itself n times. This can be considered a rotation symmetry in which the rotation angle has a measure of 360/n degrees. He continues: The cylinder may be replaced by any surface of cylindrical symmetry, namely by one that is carried 126 into itself by all rotations around a certain axis, for instance, by a vase. (Weyl 1952, 53) An appropriate example can be found in the Athenian eighth-century B.C. Geometric amphora of the Diplyon style, in Gardner (1959), page 131. In architecture, rotational symmetry is evident in: * The Palace Chapel of Charlemagne, Aachen, eighth century A.D. The chapel exhibits octagonal central symmetry (Janson 1970, 201). * The Baptistery, Pisa, eleventh-twelfth century. The building has a central-style plan with six horizontal layers of rotational symmetry, each of a different order (Weyl 1952, 56). * The Leaning Tower of Pisa, eleventh-twelfth century. This bell tower has a central plan with eight horizontal layers of rotational symmetry (Gardner 1959, 236). * The exterior of the dome of St Peter's, Rome, sixteenth century. The dome illustrates two layers of rotational symmetry (Feldman 1971,

577) . * Towers of the Cathedral of St. Basil, Moscow, mid-sixteenth century. Each tower displays multiple layers of rotational symmetry (Feldman 1971, 572). 127 * Interior of the dome of The Capitol, Washington, B.C., late eighteenth-nineteenth century. Here one observes thirteen layers of rotational symmetry (Smith 1981, 221). * Royal Pavilion at Brighton, England, early nineteenth century. Repetition of a lozenge- type window in a large dome yields an additional example of rotational symmetry (Feldman 1971, 572) . Having explored the four rigid motions of the plane, one may find particularly interesting an architectural plan utilizing each isometry. Le Corbusier, a Swiss architect, designed multiple dwelling units for a complex near Bordeaux, France. In the layouts, he started with a basic rectangular unit, then 1. translated the module to orient two units side-by-side; 2 . mirror-reflected the module and joined image to original through the line of reflection;

3. translated the module, then reflected it and joined image to original. (This is a glide reflection.) Creating a second basic, but asymmetrical, module. Le Corbusier rotated it by 180° and then joined image to original in an interlocking format (Stevens 1980, 7). 128 The four rigid motions of a plane have been described and illustrated by examples from the world of art. In recapitulation, then, any rigid motion admitted by a figure is said to be a symmetry of the figure (Washburn and Crowe 1988, 50). Alternatively, a symmetry of a figure is a transformation of that figure for which the image is precisely the original figure (Wheeler 1988, 427) . The concept of symmetry is cogently characterized by Senechal and Fleck who state: The theory of symmetry is a triumph of the human intellect. It is the perception of order in a chaotic universe, the study of the forms that order can take, and the use of that study to give significance to the things we see. In science, as well as in the arts, symmetry is the geometric plan on which the variations of nature and life are drawn. (Senechal and Fleck 1977, 95)

Rigid Motions, Art, and Abstract Mathematics The Curriculum and Evaluation Standards for School Mathematics published by the National Council of Teachers of Mathematics (1989) stresses the study of patterns as an integral part of the mathematics education of all students from kindergarten through secondary school.

Patterns are everywhere. Children who are encouraged to look for patterns and to express them mathematically begin to understand how mathematics applies to the world in which they live. Identifying and working with a wide variety of patterns helps children to develop the ability to classify and organize information. (National Council of Teachers of Mathematics 1989, 60) 129 The widespread occurrence of regular and chaotic pattern behavior makes the study of patterns and functions important. Exploring patterns helps students develop mathematical power and instills in them an appreciation for the beauty of mathematics. (National Council of Teachers of Mathematics 1989, 98) Symmetry and pattern study are inextricably linked. Since a symmetry operation is a rigid motion, one can classify patterns by the kinds of symmetry operations which leave the patterns invariant. Thus, one can begin with a pattern and try to discover the method of its formation. This approach can lead to the exploration of structure; the structure of matter, art, music, and thought (Senechal and Fleck 1977, 12). Alternatively, one can begin with the symmetry motions and try to determine the resulting patterns. This approach is taken by the artist, musician, and scientist. Seemingly disparate, the two approaches are closely related. One is the converse of the other, and they can be abstractly viewed in terms of the mathematical concept of the group. For every pattern of symmetry . . . there is a corresponding symmetry group. . . . Conversely, with an appropriate set of generators and a knowledge of the relations between them, we can generate a group and a corresponding pattern of symmetry. The two dual aspects of symmetry are thus united in the mathematical concept that underlies it. (Senechal and Fleck 1977, 16) A thorough exposition of group theory, inappropriate here, can be found in, e.g., Herstein 130 (1986). However a simple illustration of a group can be found in the symmetries of an equilateral triangle. Consider an equilateral triangle located in a specific spatial position. What actions— motions— can be performed on the triangle, leaving its shape, size, and position in space unaltered? Given an axis of rotation perpendicular to the plane of the triangle and passing through the triangle's centroid, rotations of 120°, 240°, 360° do not change the figure's appearance. In addition, the triangle can be flipped through lines each determined by a vertex and the midpoint of the opposite side. There are three lines of this type, one for each vertex. The rigid motions described above are transformations of the equilateral triangle for which the image (resulting figure) is identical to the original figure. Transformations of this type are called symmetries (Wheeler 1988, 427). To facilitate our characterization of a group, let us name the motions described in the preceding paragraph.

Suppose we call a counter-clockwise rotation of 120° R^, a rotation of 240° Rj, and a rotation of 360° R, or e. Call the flips— or reflections— Fj, and Fj. (See Figure

1 0 .) Now consider a rule for combining any two of the specified transformations. We could create a rule for 131

Figure 10. Symmetries of the equilateral triangle 132 combining motions and R^ by saying, "Let (Rj * Rj) mean we first perform R^^ then take the image (or result) of symmetry R^ and perform Rj on it." We have stated a rule for the composition of the two transformations. Figure 11 is a Cayley table which concisely lists the result when any one of the six symmetries is composed with another. Let us review basic terminology. The result of a transformation on the triangle is called the image of the triangle. If a transformation takes the triangle back to its original position, it is called an identity transformation. Here, the identity transformation is planar rotation by 360°: R, or e. If one transformation cancels or "undoes" the effect of another, the former is called the inverse of the latter. For example, a rotation of 120°, R^, followed by a rotation of 240°, Rj, places the triangle in its original position. We say Rj is the inverse of R^ and vice versa. We may write the process as (R^ * R%) = e =

(«2 * Ri) • We note from the Cayley table (Figure 11) that one symmetry transformation of the triangle followed by a second yields a result which is also a symmetry transformation of the triangle. This is an illustration of closure. For example, a rotation of 240°, Rj, creates 133

* R. F, Fa F3 R, Ra

R. R. Fi F; F3 Ri Ra

F, F, R. Ra R, F3 Fa

Fz Fz R, Re Ra F, F3

F3 F3 Ra R, Re Fa F,

R, R, Fa F3 F, Ra Re

Ra Ra F3 F, Fa Re Ri

Figure 11. Cayley table 134 an image indistinguishable from the original triangle. A flip of the image around a given vertex-to-midpoint axis,

say F3 , results in a new image also indistinguishable from the original triangle. Thus the combined transformation,

Rj followed by F3 , written (Rj * F3 ), is a symmetry

transformation: (Rj * F3 ) = Fg. The combined transformation leaves the size, shape, and spatial position of the triangle invariant. Symmetry transformations are associative. Again, refer to the Cayley table. Let A, B, and C stand for any of the six symmetries of the equilateral triangle. If transformation A is followed by transformation (B * C), the result is the same as if transformation (A * B) were followed by transformation C. In summary, a set of elements, together with a rule of composition— i.e. a rule for combining any two elements of the set— which satisfies the conditions of: * closure * associativity

* existence of an identity. I, such that I * W = W * I = W for each element W of the set and * existence of an inverse for each element of the set is termed an abstract group. 135 The set of transformations of an equilateral triangle discussed above, along with the procedure for combining any two of the transformations, satisfies all the conditions listed in the preceding paragraph. Thus we can say that the set of all symmetry transformations of an equilateral triangle, with the rule for composition, constitutes an abstract group, (Here the term group is used in its mathematical sense (Rosen 1975).) Other examples of abstract groups can be found in the set of integers Z under addition (+); the set of positive real numbers under multiplication (x); and the set Zj of integers mod 2 under addition. The set of all integers under ordinary multiplication do not form a group

because, except for 1 and -1 , the multiplicative inverse

of any integer is not itself an integer; moreover, 0 has no multiplicative inverse. (For additional information, refer to Herstein [1986].)

Thus far, we have examined basic concepts of symmetry and a number of manifestations of symmetry transformations (reflection, translation, glide reflection, and rotation) in the world of art. Finally, we delved into higher mathematics by introducing the relationship between symmetry and the abstract group. Connections between the disciplines of art and mathematics 136 will be further expanded by investigating ornamental symmetry.

Some Aspects of Ornamental Symmetry Schillinger, in The Mathematical Basis of the Arts, emphasizes interdisciplinary relevancies: Any art form is also a derivative of the space-time continuum. Art can be measured and analyzed like any other phenomenon of bur universe. Any analytical result can be represented graphically, i.e., geometrically . . . in terms of space-time relations. . . . Design is geometrically the most obvious art form, since the idea and the realization in an art medium are both accomplished in empirical space. (Schillinger 1976, 365) Schillinger's conviction that interrelationships between mathematics and design exist is also held by Weyl who observes that The Egyptians excelled in the ornamental art four thousand years before the mathematicians discovered in the group concept the proper mathematical instrument for the treatment of ornaments and for the derivation of their possible symmetry classes. (Weyl 1952, 52)

What are these connections between ornament and mathematics? To fully comprehend the relationship, one must first grasp the concept of an n-fold rotational center. Any two-dimensional system admits of a rotation transformation: the entire system is rotated about an axis perpendicular to the plane of the system. The point of intersection of axis and plane is termed the center of rotation (rotocenter). That is: 137 A circle is . . . a system that is symmetric under rotation by any angle about its center. Such a rotation center is called a center of full rotational symmetry. Any system having a regular circlewise repetition is symmetric under rotation, but only by a certain minimum angle and multiples of it. This minimum rotation angle must be 360°/n where n is an integer greater than 1. The corresponding rotation center is called a center of n-fold rotational symmetry. (Note that by putting n = <» we obtain full rotational symmetry.) (Rosen 1975, 41-42) Another approach is to say that a figure has n- fold rotational symmetry if it coincides with itself precisely n times in one full rotation. (This definition accepts 1 as a value for n.) All the regular polygons admit of rotational symmetry (Seymour and Britton 1989, 71) . Two-dimensional symmetry configurations fall into three general classifications. The first class consists of point groups, in which the motif rotates about a fixed point, the rotocenter. Rosen (1975) gives an informative exposition of the cyclic groups C„ and the dihedral groups

D„.) The second class consists of patterns that stretch in a line; they admit translations in only one direction and are called band, strip, or frieze patterns. The only rotations a band pattern can admit are half-turns (180° rotations) (Washburn and Crowe, 1988). A common example of a band-type of ornament can be found in wallpaper borders. There are exactly seven types of repeated, or 138 band, patterns. (See proof by Fisher in Washburn and Crowe, 1988.)

The third class includes repeating patterns that spread uniformly over the plane. Examples are found in wallpaper, upholstery print, and tiled floors. Barlow, in 1897, developed the proof that only one-, two-, three-, four-, and sixfold centers fall into this category (Stevens 1980, 382). Another way of expressing this concept is by saying these patterns admit only of rotations of 60°, 90°, 120°, 180°, or no rotations at all. There are exactly seventeen different kinds of symmetry for a two-dimensional ornament spreading over the plane. (For proof, see Fejes-Toth [1964] or Martin [1982].) (Elaboration of the seventeen kinds of planar patterns, beyond the scope of this paper, can be found in Yale [1968], Stevens [1980], or Washburn and Crowe [1988].) Although this kind of ornamentation existed for thousands of years, conceptual means for completely analyzing it evolved in the nineteenth century with the mathematical abstraction of a group of transformations. On this basis, it was possible to prove that the seventeen symmetries found among decorative patterns of ancient times, particularly in Egypt, categorized the class exhaustively (Weyl 1952). 139 In order further to illustrate the undeniable connection between mathematics and art, we shall show how symmetry enables the artist, historian, archaeologist, and anthropologist to classify symbolic motifs appearing over the ages in diverse forms of art. Establishing links between varying fields of knowledge is essential to intellectual progress. As Schillinger states: The mental growth of humanity . . . may be stated as a tendency to unite seemingly different categories into a complex unity into which previous concepts enter as component parts. The evolution of thought is a process of synthesizing concepts. (Schillinger 1976, 9) Here we shall show how the seven types of borders, or band ornaments, can be classified using symmetry. Following Stevens s (1980) exposition of regular patterns, we shall employ nomenclature compatible with the Hermann- Mauguin system adopted by cry stenographers. It is selected for its concise notational form. First, we shall discuss point groups, as border­ like ornamentation is created from a few specific members of this category. Stevens (1980) places an asymmetric motif in the group l category, as one rotation, a full turn, will cause the pattern to coincide with itself. Consider a question mark "?" and the letter "J” as examples. A group 1 motif frequently found in Oriental rugs is the pine cone or leaf design. (See Reed 1972, 140 plate 26.) This same motif is familiar in the “paisley" format.

Group m members encompass an asymmetric motif reflected in a mirror; the result is a pattern with bilateral symmetry. This type of figure, as with group 1 motifs, coincides with itself after a 360“ rotation. Illustrations of group m motifs are numerous: * Italian drawn work c. 1540 (Fleming and Honour 1979, 244) * Woodcut letter, Paris, 1549 (Fleming and Honour 1979, 814) * Silk coronation mantle, Sicily, 1133 (Fleming and Honour 1979, 734) A group 2 motif appears identical right-side-up and up-side-down, which is the same as saying it looks the same after a rotation of 180°. (See Figure 12.) We could say, for instance, that a face card, like the Queen of Spades, from a standard deck of playing cards, has a two­ fold rotation center. Examples from the field of the decorative arts:

* A type face from the Figgin Foundry, London, c. 1820 (Fleming and Honour 1979, 877) * Pattern on a Yueh ware dish, China, tenth century (Fleming and Honour 1979, 875) 141

Mirror

Figure 12. Group 2 motif Figure 13. Group 2mm 142 A group 2mm pattern results when a line of

reflection is passed through the rotocenter of a group 2 motif. (See Figure 13.) The 2mm design appears the same after rotation of 180°, but displays bilateral symmetry about both horizontal and vertical axes. One could interpret this motif as a single asymmetric ornament reflected in two perpendicular mirrors. Example: * Design on a turned basket, North West Coast California Native American, late nineteenth- early twentieth century (Washburn and Crowe 1988, 113) A group 3 pattern coincides with itself after three rotations. (See Figure 14.) "The group-3 pattern is the epitome of vitality and movement. It hums with rotational action" (Stevens 1980, 52). Examples found in the decorative arts include: * Variations of an ancient Greek symbol, the triskelion: In its archetypal form, the triskelion depicted three flexed legs with a common rotocenter (Stevens 1980, 53).

* The central design on a bronze lachet, Irish, second century, A.D. (Fleming and Honour 1979, 147)

* The design for a Second Empire "Indiscret," a settee formed by three joined armchairs in 143 L

•

Figure 14. Group 3 motif

Mirror J L

Figure 15. Group 3m motif 144 triskelion format, mid nineteenth century France (Fleming and Honour 1979, 395) * The advertising logo of a twentieth century beer: In its three interlocking rings, the motif provides another group 3 illustration, although the same ornament appeared centuries ago as a Christian symbol for the Holy Trinity (Stevens 1980, 53). In architecture, a group 3 motif was utilized by Walter Gropius for The Bauhaus, Dessau, 1926. The overall building plan took a pinwheel form with three hooked arms (Giedion 1949, 422). Our consideration of the point group motif will conclude with the 3m pattern; it is produced by passing a reflection line (or "mirror") through an ornament with a three-fold rotocenter. (See Figure 15.) The resulting figure consists of three bilaterally-symmetric portions, or arms, spaced at 120° intervals. A contemporary symbol utilizing the 3m format is the cautionary sign for biological hazards (Stevens 1980, 55). The 3m pattern can also be viewed as a single asymmetric motif reflected in three intersecting mirrors. (See Figure 16.) Any two adjacent mirrors form a 60° angle. Motifs of even order fall into n or nmm formats. (Refer to Figures 12 and 13.) Those of odd order are 145

j L ✓

Figure 16. Group 3m motif with three mirrors 146 listed n or nm. (Refer to Figures 14 and 15.) Point groups with designations n, nm, and nmm, where n is an integer larger than three, can be inferred from the foregoing discussion. For example: * Group 4 symmetry appears in the rose-and-bud motif of a "Kentucky Rose" quilt, c. 1870 (McCall's 1975, 105).

* Group 6 symmetry is shown in the Magen David, or Star of David, symbol of Judaism (Stevens 1980, 81).

* Group 6 mm symmetry is evident in an alabaster and enamel paten, San Marco, Venice, tenth- eleventh century (Fleming and Honour 1979, 136).

* Group 8 symmetry can be seen in the featured motif of a Pennsylvania "Plume" guilt, c. 1860 (McCall's 1975, 113).

We are now prepared to classify, using symmetry, the band (strip or frieze) patterns which have appeared throughout the history of art. (Notational format will follow that employed by Stevens [1980] for its simplicity of form.) The line groups (borders) are formed by

utilizing motifs from point groups 1 , m, 2 , or 2 mm in conjunction with the rigid planar motions of translation, reflection, and glide reflection. As Washburn and Crowe state in their exhaustive analysis of motifs from varying 147 cultures, "... symmetry classification is concerned . . . with the motions which move the pattern along an axis or around a point. These motions can be thought of as generating the design" (Washburn and Crowe 1988, 55). The simplest linear band ornament is that created by repeated translations of an asymmetric, or group 1, motif. It is designated by the symbol t. Visually, it takes the structural format: b b b b b . . . Examples of the t line group are: * Egyptian design. New Empire Period (Stevens 1980, 97) (See Figure 17.) * Bronze pou (vessel) from the Late Shang Dynasty, China (Washburn and Crowe 1988, 96) * Celtic design: detail of base, Moone Cross, ninth century, Kildare (Mitchell 1977, 151) * Ceramic design, San Idelfonso Pueblo (Washburn and Crowe 1988, 95) The second band ornament repeats translations of the glide reflection of an asymmetric pattern; it is the tg form and assumes the overall shape: b p b p b p . . . Examples of the tg line group are:

* Leaf design from ancient Greece (Stevens 1980, 109) (See Figure 18.) * Celtic band ornament, gold plate over lead, Kildare, c. 700 B.C. (widest of the three 148

o o o

Figure 17. Linear band pattern type "t" 149

Figure 18. Linear bank pattern type "tg" 150 horizontal bands) (Mitchell 1977, 24) * Carved wooden border, Norway (Washburn and Crowe 1988, 124) * Band in decorated barkcloth, Uganda, nineteenth century (Spring 1989, plate 36) * Plan arrangement for apartment houses, twentieth century Europe (Stevens 1980, 115)

If an asymmetric motif is reflected about line t to produce a group m pattern and the latter is reflected about another line fi (where /i is parallel to t ) , the third band ornament is created. Designated tm, it takes on the format: bdbdbdbd. . . * Ancient Egyptian design (Stevens 1980, 119) (See Figure 19.) * Facade of galleried houses in La Coruna, Spain (Padwick and Trevor 1981, 13) * Iroquois beaded moccasin (Washburn and Crowe 1988, 104)

* Band pattern in Sir Bevidere's armor: drawing by Aubrey Beardsley (Harris 1967, 33) The fourth band ornament is created by taking a t line ornament and reflecting it about a line parallel to the direction of translation. This is called the mt classification, and a simple illustration is of the form: 151

w 1 1.V 1 Il II II II H u II II lin

Figure 19. Linear bank pattern type "tm" 152 p p p p . . . b b b b . . . * Inca motif, pre-Columbian (Stevens 1980,127) (See Figure 20.) * Line ornament in Nigerian cloth textile (Spring 1989, plate 3) * Maori rafter pattern (Washburn and Crowe 1988, 99) * Bands on hat of bronze Benin flute player (Zaslavsky 1973, 184) The fifth line ornament is characterized by looking the same right-side-up as upside-down: it has rotational symmetry. Basically, it consists of repeated translations of a group 2 motif. It is the t2 class and can be visualized as: q b g b g b . . . * Eskimo carving (Stevens 1980, 137) (Figure 21.)

* Celtic design of interlaced beasts, each biting its tail. The Book of Dur row, jfol.192 v: carpet

page; mid-seventh century (Mitchell and others 1977, plate 27d)

* Raffia-pile cloth bands, Zaire (two outer bands depicted) (Zaslavsky 1973, 180) 153

US

Figure 20. Linear band pattern type "mt" 154

Figure 21. Linear band pattern type "t2" 155 The sixth band ornament contains all four symmetry operations. It is classified t2mg and may be viewed schematically as; pdbgpdbg. . . * Prehistoric Indian design from Delaware (Stevens 1980, 149) (See Figure 22.) * Ornament on a traveling chopstick case, Japan (Washburn and Crowe 1980, 120) * Arabic wall mosaic. (Washburn and Crowe 1980,

120) The seventh, and final, band ornament is designated t2mm. It can be viewed as successive translations of a 2mm motif. Schematically,it appears as: b d b d . . . p g p g . . . Illustrations of this class are:

* Pompeian mosaic (Stevens 1980, 163) (Figure 23) * Border of Hatchli Bokhara prayer rug (Reed 1972, plate 95)

* Sioux calumet pipe stem (Washburn and Crowe 1980, 114) * Pattern on a wooden door panel from Yoruba, Nigeria (third separating strip from the top) (Zaslavsky 1973, 181) 156 IMEl

Figure 22. Linear band pattern type "t2mg" 157

Figure 23. Linear band pattern type "t2mm" 158

Summary In summary, we have examined certain aspects of symmetry and noted salient links between varying art forms and mathematics. It is hoped that, having made another connection, via symmetry, between the ostensibly separate disciplines of mathematics and art, the reader will be encouraged to continue exploring relationships between symmetry, mathematics, art, and science. It is by making connections and discovering patterns that the individual grows intellectually. In this vein, James Joyce spoke of beholding an object and then putting a frame around it. The object is viewed first as one thing, then another. Then one becomes "... aware of the relationship of part to part, each part to the whole, and the whole to each of its parts. This is the essential. . . the harmonious rhythm of relationships. . . . the epiphany." (Campbell 1988, 220). CHAPTER 5 ILLUSTRATIVE ACTIVITIES FOR A CURRICULUM INTERRELATING THE STUDIES OF MATHEMATICS AND ART FOR GRADES FIVE THROUGH EIGHT

Introduction Having examined the need for improving mathematics education in the United States and studied research findings supporting the feasibility and pedagogical soundness of an alternative mathematics curriculum correlating art and mathematics, we now must detail illustrative activities. As Linn stated, "Neither mathematics nor art is a spectator sport" (Linn 1974, 107) . The activities-planning component of this chapter contains a sample of specific lesson/unit plans. Learning objectives are clearly and carefully delineated, for, as Tyler noted:

Unless the objectives are clearly understood by each teacher, unless he is familiar with the kinds of learning experiences that can be used to attain these objectives, and unless he is able to guide the activities of students so that they will get these experiences, the educational process will not be an effective instrument for promoting the aims of the school. (Tyler 1964, 126)

159 160 Cognitive objectives are stated in behavioral terms in order to facilitate evaluation procedures. Evaluative methods other than pencil-and-paper testing are included in order to obtain a more-complete assessment of student progress. Objectives from the affective domain— the area of learning that involves attitudes, values, and emotions— are included also, although they are more difficult to assess. Discussion concerning the implementation of each activity, possible extensions for individualization and enrichment purposes, and support from research are included. Selection of appropriate activities is crucial to the success of a curriculum. If students are to grow both intellectually and socially, it is imperative for the educator to . . . arrange for the kind of experiences which, while they do not repel the student, but rather engage his activities, are, nevertheless, more than immediately enjoyable since they promote having desirable future experiences. . . . Hence, the central problem . . . is to select the kind of present experiences that live fruitfully and creatively in subsequent experiences. (Dewey 1967, 28)

The activities that follow are purposely rich and varied. They promote the establishment of connections between mathematics, other disciplines, and daily life. They frequently allow for student choice in the processing modes (verbal, visual; analytic, holistic; etc.) utilized in problem solving. They allow students to share learning 161 experiences and communicate mathematical ideas by incorporating small-group cooperative efforts as well as whole-class discussion. The use of manipulatives is stressed to enhance mathematics learning. Illustrative artistic material is drawn from a multiplicity of sources to cultivate appreciation for the pan-cultural nature of mathematics. Selection of art works can be accommodated to include contributions from each student's ethnic background; Zaslavsky (1990) notes that this approach assists students in building build self-esteem. Throughout the activity section, individualization of instruction is considered. Eisner (1982), in his call for new methods of reaching and evaluating wide varieties of human intelligence and capabilities, states: Individualization is regarded as a prime pedagogical virtue, one that is based upon the recognition that children differ in their aptitudes, their interests, their prior experience, and their values, and that school programs and teaching methods should take these differences into account as educational decisions are made. When one examines individualized programs in American schools, one finds that the major variable that is altered for students is time; fast- and slow- learning students are given variable amounts of time to proceed through the same sequence of curriculum activities toward the same performance objectives. Such an approach to individualization alters only one of the several variables that could have been changed. One could alter the goals of educational programs to suit differences in student interest and aptitudes, . . . one could alter the means through which the content was taught, one could alter the form through which students are expected to demonstrate what they learn, [italics added] Individualization, in a subtle way, is related to the ideal of educational equity. Children having different aptitudes need to have an 162 array of educational conditions that optimize their learning in school. (Eisner 1982, 79) The activities that follow encompass two- and three-dimensional geometry, computation, statistics, ratio and proportion, transformations, symmetry, spatial relations, number systems, pattern recognition and analysis, measurement, and even group theory. Mathematics is viewed as communication, problem solving, and reasoning. Connections between the every-day world and mathematics are stressed. In fact, each of the curriculum standards recommended by the National Council of Teachers of Mathematics (1989) for grades five through eight is addressed.

Activitv One Topic: Tessellations Grade Level: Five through Eight

Objectives Students will be able to:

1. describe, recognize, and create a variety of patterns; 2. classify shapes: triangle, quadrilateral, trapezoid, rhombus, parallelogram, pentagon, hexagon; 3. define a polygon; 4. recognize regular polygons; 5. investigate results of combining shapes; 163 6. relate geometric shapes and tessellations to number and measurement ideas; 7. write about their thoughts when viewing a tessellation; 8 . discuss concepts of tessellations with peers and teacher; 9. describe at least two examples of interrelationships between art and mathematics.

Materials Pattern Blocks (These are available from Dale Seymour Publications: 250 blocks/set in six different shapes and colors.) M.C. Escher giftwrap paper (The paper is available from Dale Seymour Publications: sixteen 48 x 69 cm sheets, each with a different print.) Teacher should encase these in plastic to increase longevity. The Mathematics of Islamic Art

(A set of slides and additional explanatory material, this packet can be ordered from The Metropolitan Museum of Art, New York.) Paper and pencil Posterboard (white)/crayons/colored pencils 164 Procedures 1. Divide class into small groups, each at its own table (four to five students per group). 2. Give each group one or two of the Escher gift wrap prints to view. 3. Tell the pupils that they are going to explore some interrelationships between mathematics and art. After looking at the print(s) for a few minutes, students should write down thoughts that occur to them as they view the art work. 4. Explain that what they are looking at is called a "tessellation." Encourage students to share their notions of a "tessellation." Students will discuss their concepts, argue their viewpoints, try toconvince the others, and then form a group definition of the term. One student will write down the definition. 5. Each group will select a member to present its definition to the class. The teacher will clarify student input by posing appropriate questions. The class will revise the information before the instructor summarizes and writes the definition on the board. 6. The teacher will ask pupils to think of examples of tilings (tessellations) that they can see in the classroom or in other aspects of everyday life. (Some examples they might offer: acoustical tiles in ceiling; 165 pattern of vinyl tiles on the classroom floor; pieces of clothing that exhibit tiling patterns; wall paper at home; patchwork guilts; rugs; designs on buildings they pass each day or on their places of worship.) 7. Working individually or in the designated small group, students will create with the Pattern Blocks a design that completely covers the plane: no gaps and no overlaps. 8. The teacher will then name each type of Pattern Block and state that each is a kind of "polygon." 9. Each group will make a written list of the types of blocks that were used in their tessellation. They will try to devise some definition of the terms: polygon, square, rhombus, parallelogram, triangle, pentagon, and hexagon. 10. Teacher will describe a "regular" polygon. Students, in groups, will try to decide what types of regular polygons by themselves will tessellate the plane. 11. As a follow-up, students will view selected slides from the Metropolitan Museum of Art's The Mathematics of Islamic Art. a. Tessellations with hexagons and triangles: Wall panel, Iran, thirteenth century (ceramic); b. Tessellations with squares and rhombuses: Floor detail, Damascus, Syria, 1707 (marble); 166 c. Tilings as architectural decoration: The Friday Mosque, Isfahan, Iran, eleventh and eighteenth centuries.

Evaluation Evaluate individual mastery by having students trace each of the Pattern Block shapes on heavy paper and cut them out, forming stencils. Using the stencils, students draw a design which tessellates. They color the design, using the Escher prints and Islamic slides as inspiration. Then students describe and name each shape used in the construction. The designs are displayed on the bulletin board; students select and vote for their four favorite patterns.

Extension Once students have been introduced to tessellations through Activity One above, Escher art works can be used to explore the fundamentals of symmetry. Using Escher's own commentary on his work, the teacher can illustrate:

* reflection svmmetrv through Symmetry Work §91, bilaterally symmetric beetles (Escher 1989, 26); * translational svmmetrv through Symmetry Work §72, fish motif and sailing vessel and Symmetry Work §105, prancing winged horses (Escher 1989, 27) ; 167 * glide reflection through Symmetry work §96, swans and Symmetry Work §67, warrior on horseback (Escher 1989, 29-30); * rotation through Symmetry Work §70. This work features red, blue, and yellow butterflies; it can be used to demonstrate two-fold, three­ fold, and six-fold rotational symmetry (Escher 1989, 28). Symmetry is of importance to elementary and middle-school pupils, as it leads into the study of transformations in secondary school and into the investigation of groups in more-advanced scholarship. As was mentioned in Chapter Four, symmetry plays a role in such diverse areas as music, anthropological research, crystallography, and design. Nonperiodic tiling using Penrose tiles can be introduced as enrichment for grade six pupils or studied in greater detail at grade levels seven and eight.

Comments * Activity One has many benefits for the pupils. It gives them the opportunity to work with geometric concepts at a concrete level; it utilizes both visual and tactile instructional aids. A study by Raphael and Wahlstrom (1989) investigated the effect of the use of instructional aids on mathematics achievement. Using dual 168 scaling, or correspondence analysis, on data from the Second International Mathematics Study, the researchers noted that the use of instructional aids, in combination with course coverage, was related to higher student achievement levels in work with both plane figures and informal transformations. The authors stressed that, although it was impossible to separate out the effects of teacher experience, content coverage, and use of aids on student achievement, the findings indicated that effective use of instructional aids was associated with successful coverage of the topics studied. * Activity One extends and strengthens the students' vocabulary of geometry; it helps heighten the pupils' spatial sense by exploring geometric figures and their interrelationships. The geometry involved focuses on exploring and using geometric ideas and relationships, as opposed to unthinkingly memorizing definitions and formulas; this approach is advised by the Curriculum and Evaluation Standards for School Mathematics (National

Council of Teachers of Mathematics 1989). * The small-group involvement in Activity One fosters communication of mathematical ideas. "Small groups provide a forum in which students ask questions, discuss ideas, make mistakes, learn to listen to others' ideas, offer constructive criticism, and summarize their 169 discoveries in writing" (NCTM 1989, 79). The activity contributes to a study of patterns, as also recommended by the Standards.

Activitv Two Topic: The Symmetries of an Equilateral Triangle Grade Level: Eight

Objectives: Students will: 1. discover relationships and develop spatial sense by using manipulatives to: a) identify transformations which preserve distance, angle measure, betweenness of points, and collinearity of points in the geometric figure: equilateral triangle; b) draw and label the image of the equilateral triangle when the figure is reflected in the altitudes through the three vertices; c) draw and label the image of the equilateral triangle when the figure is rotated counterclockwise in

the plane through 0° (360°) , 120°, and 240°;

2 . classify these transformations by creating the multiplication table for the group Dj, the symmetries of an equilateral triangle; 170 3. utilize the multiplication table obtained above to gain an insight into some of the distinctions between a group (as evidenced by Dj) and a field (the

rational numbers); 4. discuss concepts of symmetry with peers and teacher; 5. write a paragraph describing an application of symmetry to art; 6. support the thesis that mathematics is pan- cultural .

Materials Slides from the following sources: a) The Mathematics of Islamic Art. Metropolitan Museum of Art, New York City, 1979; b) Bain 1973; c) Mitchell and others 1977; d) Naylor 1975;

e) Spinden 1975; f) Spring 1989. Paper equilateral triangle of side 14 cm. Sheet of paper 8.5" x 11" bearing photocopy of an equilateral triangle of side 14 cm. (See Figure 24.)

A photocopied sheet of twelve equilateral triangles arranged in sets of two (See Figures 25 and 26.) 171

Figure 24. Reference triangle 172

(123) (123) R,

(123) ( ) F

(123)

Figure 25. Symmetries of the equilateral triangle 173

- n

(123)

R

(123) ( )

Figure 26. Remaining symmetries of an equilateral triangle. 174 A photocopied sheet of grid for the group multiplication table (See Figure 27.)

Figure 27. Grid for Cayley Table

Background Students shall have encountered concepts of symmetry as suggested in the Standards (NCTM 1989) and discussed in Chapter Four of this paper.

Procedures 1. Divide class into small groups (four to five students per group).

2. Explain to the class that they are going to explore some interrelationships between mathematics and art. They will view slides illustrating uses of the triangle in design from a number of different cultures. 175 Many of the triangles will be equilateral. (Review definition of an equilateral triangle.) 3. Inform the class that they will explore, in some detail, the symmetries of an equilateral triangle. 4. Show slides. a) Mitchell and others 1977, Plate Two. Here we see a gold lunula from County Westmeath (Ireland). This Early Bronze Age (1800-1600 B. C.) neck ornament is crescent shaped. It is decorated with incised triangular motifs which repeat around the inner and outer rims. b) Bain 1973, Plate E. Pictish knotwork borders from the Gospel of Lindesfarne form two equilateral triangles, one within the other. The equilateral triangle was used as a Christian sign of the Trinity. The equilateral triangle also played a part in symbols of other religions and philosophies. It is used in the Jewish Star of David. In Indian thought, it represents Shiva and Shaki forming the wheel of Vishnu. c) The Mathematics of Islam, slide # 8. This is a

photograph of a molded tile panel. Glazed with turquoise and cobalt from Nishapur, the tiles date from the thirteenth to fourteenth century. Although the Qur'an does not establish a Muslim doctrine on representation of living forms, the Islamic religion does emphatically assert that only God is the Creator and only He is to be 176 worshipped. As a result, representations of living forms played a much more limited part in Islamic art than in other artistic heritages (Grabar 1983). Thus, one finds that Islamic artists frequently fashioned designs of a geometric nature. In the tile panel shown, the pattern includes hexagons and six-pointed stars. (Review the relationship between regular hexagons and equilateral triangles as discussed in Chapter Four.) A design of the type depicted in the slide was economical as well as aesthetically pleasing: only two kinds of molds were required; and a tessellating pattern could easily be devised. (See Activity One.) d) Naylor (1975, 103, Plate C). This depiction of a Northwest Native American basket features a triangular "butterfly wing" motif. The Eastern Woodlands Menominee woven bag (48, Plate C) is decorated with a geometric hexagonal design; it was used for food storage. Pueblo pottery (176, Plate B; 178, Plate A) also utilizes triangular motifs for decorative effects.

e) Spinden (1975, 141). A collection of geometric motifs used in the pottery decorations of Mayan origin includes a triangular repeating pattern. This pattern is usually applied in bands around the neck or rim of a vessel. 177 f) Spring (1989, Plate 36). This slide depicts a decorated barkcloth from Uganda. The design, both stamped and painted freehand onto the cloth, includes a triangular repeating pattern in glide translation format. 5. Distribute paper materials to students. Working individually and in conjunction with other group members, students consider the equilateral triangle of Figure 24. (Print on 8.5" x 11" paper; furnish one copy to each pupil.) This is the reference triangle, an equilateral triangle located in a specific spatial position. Photocopy Figure 24 on a medium-weight paper in contrasting color. (Bright yellow is effective.) Students each cut out the "yellow" triangle and label it as in Figure 28. The "yellow" triangle becomes a manipulative tool. Using the stationary reference triangle and the manipulative, students decide what actions— motions— can be performed with the "yellow" triangle, leaving unaltered its size, shape, and position in space with respect to the reference figure.

6. Ask various group leaders to share their team conclusions with the class. Students should agree that two types of motions could be employed: 178

Verso Recto

Figure 28. Scale version of manipulative 179 a) rotations; Given an axis of rotation perpendicular to the plane of the triangle and passing through the triangle's centroid, rotations of 120* (motion

Rj), 240* (motion R%), and 360* (or 0*) (motion R^ = e) leave the appearance of the triangle unchanged. b) reflections— or flips— about a line passing through a vertex of the triangle and the midpoint of the opposite side also leave the appearance of the triangle unaltered. There are three lines of reflection, one for each vertex. Call a flip about the vertical axis motion Fj. The flip about the altitude emanating from the vertex at the lower left side of the basic triangle is called F%. The flip about the altitude drawn from the vertex at the lower right side of the basic triangle is termed Fj. 7. Inform the students that these motions are called symmetry transformations: they leave the resulting triangle indistinguishable from the original triangle. 8. Using the manipulative, students will perform the rotations and reflections described above, comparing the reference triangle with the manipulative. In this way, students will be able to summarize their findings on the sheets listed as Figures 25 and 26. For example, the rigid motion Fj will leave vertex 2 in place. Vertices l and 3 will interchange. 1 goes into 3 180 2 goes into 2 3 goes into 1 Students will share their results with others in the group. Differences of opinion will be discussed. Group leaders will fill in the instructor's transparency copy of Figures 25 and 26. The transparency will be shown to the entire class by using an overhead projector. 9. Introduce the concept of composition of transformations (without necessarily stressing a definition) by a guided illustration of motion R, followed

by motion F, (denoted Rj * F,) . The rotation R, takes: 1 into 2 2 into 3 3 into 1 If transformation F, is applied to the triangle obtained

through Rj, a flip about the vertical line of reflection is performed. Students use the manipulative to reflect the triangle through the altitude passing through the (new position) vertex 3. They note that 3 remains fixed while 1 and 2 interchange. Thus students observe the result of R, followed by F,r Rj * F, takes vertex

1 into 3 2 into 2 3 into 1 181 Students compare R, * to the six resulting triangles

pictured in Figures 25 and 26. They note that R, * F,

equals F;. On Figure 27, (a Cayley, or Group, Table), they fill in the appropriate space. 10. Encourage students to explore (using the manipulative and the reference triangle): (a) motion e followed by motion Fj (b) motion F, followed by motion e.

Students observe that e * F, = Fj equals F, * e = F, and so on. That is, rigid motion e leaves the vertices of the triangle fixed. Pupils can now fill in column one and row one of their copy of Figure 27. 11. Again, direct students to use the manipulative and the reference triangle to examine combining (composing) two other symmetry transformations. Consulting with other group members, each student fills in the remaining rows and columns of Figure 27, the Cayley Table.

12. Give one group leader a transparency containing the blank grid of Figure 27. The group's version of the Cayley Table is transferred onto the transparency and shown by means of the overhead projector. Differences are discussed and a corrected version established. (See the completed Cayley table in Figure 11.) 182 13. Introduce the terms: closure: What happens if any of the original six transformations is followed by a second? Students note that the result is one of the original six transformations. identity: Is there a transformation that leaves the vertices of each of the six triangles pictured in Figures 25 and 26 unchanged? Students note that motion e plays this role. inverse: Given any of the original six rigid motions, is there another rigid motion listed on Figures 25 and 26 that will "undo" the transformation and return the triangle to its original state pictured on Figure 24, the reference triangle? Using the manipulative or the Cayley Table, students can locate the desired transformation. associativity: Using Figures 25 and 26 (in their completed form) and the manipulative, students can satisfy themselves that, given any three of the original six transformations, call them X, Ï, and Z, then X * (Y * Z) equals (X * Y) * Z. 14. Point out that the set of transformations of the equilateral triangle on Figures 25 and 26, together with the procedure for combining two transformations as discussed in (9), satisfies all the conditions listed in (13). Thus we can say that the set of all symmetry 183 transformations of an equilateral triangle, with the rule for composition, forms an abstract group.

Evaluation At a later class period, students working in groups of three to four will demonstrate mastery by reconstructing completed versions of Figures 25 and 26. They will also complete one row or column of the Cayley Table as specified by the teacher. Each group of students will find at least two illustrations of the use of the triangle in art. Preferably, the art forms selected will reflect group members' ethnic/cultural backgrounds. Pupils will share their choices by bringing the item (or a photograph or sketch) to class and describing it in written or verbal fashion.

Extension Teacher introduces the concept of commutativity by asking students to consider the set of whole numbers under addition and multiplication. It is true that any whole number, say 10, added to any other whole number, say 3, gives the same sum as when the numbers are added in the opposite order? Is 10 + 3 = 3 + 10? What is the case under multiplication? Under subtraction? Under division? Student experience provides correct answers. Next, pupils 184 consider the symmetries of an equilateral triangle. Students perform F, * R, ; they compare to R, * F, and

discuss the outcomes. They note that the property of commutativity is not a requirement in an abstract group.

Comments * The activity assists students to achieve the goal of learning to communicate mathematically, recommended by the Standards; The development of a student's power to use mathematics involves learning the signs, symbols, and terms of mathematics. This is best accomplished in problem situations in which students have an opportunity to read, write, and discuss ideas in which the use of the language of mathematics becomes natural. As students communicate their ideas, they learn to clarify, refine, and consolidate their thinking. (NCTM 1989, 6) * The use of art embodying triangular motifs links the study of mathematics to the fine arts, enriching both disciplines. The inclusion of cultural materials in the activity may help, as Zaslavsky (1990) suggests, various ethnic/cultural groups to build self esteem and become more interested in the study of mathematics. * As recommended by Roberts (1984), students are encouraged to take part in the "doing" of mathematics; there is little passive assimilation of a finished product. The activity provides aids and encourages dialogue so that students can achieve what one educator calls the "heart of the educational process"; that is 185 ”. . . translating experiences into more powerful systems of notation and ordering" (Bruner 1966, 21). Students gain an insight into the power of mathematics to describe complex concepts in concise form. * The activity utilizes manipulatives to help organize, convey, and embody the abstract concept of a group. The use of instructional aids, visual and tactile, assists students in the learning of mathematics (Charbonneau and Steiner 1988; Raphael and Wahlstrom 1989; Sowell 1989). For example, Sowell (1989) analyzed the results of sixty studies which investigated the relationship between mathematics achievement and the use of manipulative materials. Ninety percent of the studies involved students from kindergarten through grade nine. Meta-analysis showed that, for treatments of a school year or longer, mathematics achievement and the use of concrete instructional materials were positively related. (For treatments of less than a school year, results were not statistically significant.)

* The extension leads to broadened student background for higher mathematics.

Activity Three Topic: A Survey of Leisure-Time Activities Grade Levels: Five through Eight 186 Objectives Students will be able to: 1. gather and record data descriptive of student leisure-time activities using individual and group surveys; 2. construct bar graphs, pictographs, and line graphs to display the data; 3. determine an appropriate format for reporting the survey data; 4. compute the arithmetic mean, mode, median, and range of the data (Computations may be obtained by hand or with the aid of a calculator); 5. determine, for the particular situation, whether arithmetic mean, median, or mode should be used in reporting survey results; 6. form hypotheses and strategies, after analyzing survey results and reviewing both school and community resources, to improve recreational facilities for upper- elementary and middle-school students.

Materials Slides (A suggested list follows.) Posterboard, glue, and scissors for each pupil An assortment of newspapers and magazines (For example: Time. Sports Illustrated. People. 187 Connoisseur. European Travel and Life. National Geographic. in addition to students' own magazines) Data recording sheets Graph paper, pencil, calculator, and pencils for data analysis.

Procedures 1. Explain to the class that they are going to continue their study of statistics by means of a project that will combine art, mathematics, and social commitment. They will view slides depicting various leisure-time pursuits/after-school activities. 2. Inform the class that they will conduct a survey of the activities mentioned in (1) above, analyze the data, and share their findings with school and community. 3. Show slides:

a) Leonard 1978, p. 22. Olmec pottery figure of a turbaned man joyously playing a drum (Musical activities);

b) Ragghianti 1968, p.159. Cassat's painting. The Boating Party (Sailing); c) Ragghianti 1968, p.73. Fragonard's oil painting, A Young Girl Reading (Literary pleasures); d) Ragghianti 1968, p.19. Duccio's predella panel. The Calling of the Apostles Peter and Andrew; (The 188 two apostles are fishing, hauling in a net full of sea creatures.) e) Canaday 1959, p. 215. Renoir's painting. Luncheon of the Boating Party; (Young people are dining and enjoying friends.); f) Davidson 1971, p.113. Bronze plaque from Benin, Nigeria depicting two acrobats performing during an annual festival (Gymnastics; attending community festivities);

g) Feldman 1971, p.186. Ben Shan's realistically portrayed work. Handball (Six young men in a crowded city environment are playing ball); h) Ragghianti 1968, p. 86. Degas pastel on cardboard. Ballet Scene (Dancing). 4. Ask students to list their favorite leisure activities. They should estimate what percentage of total free time is apportioned to each activity. 5. Show slide of a collage (a composition made by affixing bits of paper, cloth, fabric, and the like to a background.) See Rubin 1980, p.157 for Picasso's Still Life with Chair Caning. This is a collage of oil, paper, and oilcloth on an oval canvas; it is set off by a border of rope. 189 6. Direct students to create a collage of their favorite activities. (They should use materials available in class supplemented by resources from home.) 7. Ask students, on the following class meeting, to share their collages with classmates. Direct the group to develop a list of favorite leisure activities to be incorporated into a data recording sheet. Duplicate the list and provide one survey sheet for each student in grades five through eight. 8. Plan the data collection as an in-school activity, first obtaining the cooperation of other teachers. Divide the class into pairs of research teams; these teams visit classrooms and conduct the survey. Survey sheets will state: "Please check your four favorite leisure activities." 9. Plan the data analysis as a classroom activity. Present and discuss various ways of summarizing data with graphs— bar graph, line graph, histogram, frequency polygon— and tables.

Stress that visual display of data must always focus on integrity: telling the truth about the data. The Visual Displav of Quantitative Information (Tufte 1983) presents a highly readable discussion of graphical integrity. In a lavishly illustrated chapter, Tufte deals with distortions of scale, perspective, design variation. 190 the use of area to show one-dimensional data, and "lies by omission." After reading Tufte, the teacher could find similar misleading visual displays of data in newspapers and magazines to share with the pupils. Students can be encouraged to locate additional examples of inappropriate displays of statistical data; their findings could be incorporated into a classroom bulletin-board project. Discuss measures of central tendency: the arithmetic mean, median, and mode. Review percent. 10. Divide class into four groups, one for each grade level surveyed. Ask groups to calculate totals and percentages from their data. Group members will decide how to display survey information visually. They will also discuss and decide which measure of central tendency most fairly describes the data. Each group will select a team representative to report survey results to the class. 11. Encourage students to discuss survey results. Based on the data, would the class recommend that additional ball fields or a pre-teen center be built? Should more basketball standards be placed on the school playground? Should park holdings be expanded? Should the public library rearrange floor space to accommodate additional reading rooms? 191 Evaluation Divide class into small groups of five to six pupils. Each group will agree on some aspect of school/student life amenable to a smaller-scale data collecting endeavor. For example, students could poll classmates as to their: two favorite cafeteria dishes; three favorite songs; dream vacation; best-liked color; projected vocation. Each group will devise an appropriate data recording sheet; survey class members; analyze the data; decide how to summarize the data pictorially; and prepare a report which the group leader will make to the class.

Extension * Students utilize computer software packages to facilitate the sorting and organizing of survey data. Computer-generated visual displays provide additional means for effectively communicating the information gathered.

* Students survey school and community recreational facilities, tabulate and analyze the data, and compare resources to needs identified by the leisure- time-activity project. Discussion of desired improvements will follow.

* Students review charts, tables, and graphs which describe the political entity's financial resources and 192 expenditures; they examine the feasibility of funding additional recreational facilities. * Pupils devise strategies for presenting their findings and suggestions to the school board, community leaders, and local press.

Comments * Activity Three addresses a number of the changes in content and emphasis recommended by the National Council of Teachers of Mathematics in its Standards (1989, 70) : Developing and using tables, graphs, and rules to describe situations. Using statistical methods to describe, analyze, evaluate, and make decisions. Connecting mathematics to other subjects and to the world outside the classroom. * Besides enabling pupils to practice data- gathering and analysis techniques, the activity promotes awareness of the interrelationships of knowledge. Use of works of art broadens the students' educational background and promotes intercultural understanding. The activity enhances pupils' ability to function effectively in a democratic society: Mathematics and group cooperation are utilized in order to improve community facilities. Thus students perceive some of the bonds among mathematics, the social sciences, politics, and economics. * By linking mathematics to areas of student concern— recreational facilities, the activity emphasizes 193 the relevance of mathematics to "real world" problems. (Issues of relevance are frequently encountered by educators; see Saunders 1980.) * The activity incorporates conditions providing for cooperative efforts. This feature is particularly useful in view of the emphasis placed by contemporary educators on group endeavors (NCTM 1989). Unfortunately, cooperative projects in mathematics are infrequent: over fifty percent of students surveyed by the National Assessment for Educational Progress in 1986 reported they never had the opportunity to work in small groups (Dossey et al. 1988).

Activitv Four Topic: Symmetry in Band Ornaments Grade Levels: Six and Seven

Objectives Students will be able to: 1. investigate, experiment, and explore symmetry by using physical materials; 2. identify the four rigid motions of plane figures which preserve distance between points, collinearity of points, betweenness of points, and angle measure; 194 3. construct, draw, and create each of the seven types of patterns called band ornaments; 4. describe and recognize the seven types of band ornaments as they appear in various art forms (for example, ceramics and textiles); 5. work cooperatively to produce a report for evaluation purposes.

Materials Felt board and a supply of the letter "p" cut out of felt fabric. Each letter will be at least five inches in length for purposes of visibility; color variety provides contrast. Slides (Suggestions follow. Slides can be made by the Visual-Aid Center using books listed in Bibliography.) Samples of wallpaper borders

(Obtain discontinued samples from home-decorating centers.) Textiles (Locate, for example, a small rug, article of clothing, wall hanging, or quilt with border.) Pottery, picture frame, basket, or dinner plate featuring a repeated border design 195 Construction paper, poster board, paper, pencils, glue

Background Teacher will have read the Chapter Four discussion of symmetry and linear band patterns. Students will be familiar with the basic concepts of symmetry from prior mathematics learning, as suggested by the Standards (NCTM 1989).

Procedures 1. Explain to the class that they are going to investigate symmetry in preparation for a field trip. 2. Divide class into groups of three or four. 3. Place the letter "p" on the felt board and ask students to copy it on a sheet of paper. Invite pupils to visualize the types of motions which could be used to alter the letter's position without changing its size or shape. (These are rigid motions.) Encourage students to draw the result (image) of each motion. 4. Instruct student groups to discuss the motions and list possible solutions. 5. Ask group leaders to demonstrate outcomes of their deliberations at the felt board. Students will take two letters from the supply and affix one to the board. With the remaining letter, students will demonstrate the 196 symmetry motion, placing the letter in an appropriate image position. 6. Each of the following symmetries should be included:

translation: P P • reflection: q P glide reflection: P b rotation: d P (This is just one illustration of rotation: that of 180°. ) 7. Show students the linear band patterns in each of the objects brought to class for illustrative purposes. Explain that each pattern was created by utilizing the four symmetries discussed in item (6) above with rotation limited to that of 180°.

8. Challenge students to consider how many different types of linear repeated patterns could be obtained by using an asymmetric motif (like the letter "p" or the number "7" or the symbol "?" ) and combinations of the four symmetries described. Allow enough time for students to experiment by drawing; by cutting a number of identical asymmetric motifs from construction paper and using them as manipulatives; by comparing and contrasting hypotheses with others in their group. 197 9. After students have had sufficient time to experiment and make conjectures, explain that there are exactly seven discrete types of linear band ornaments. (A mathematical proof of this fact is beyond the scope of an elementary-middle school curriculum. See Chapter Four.) However, the exercise in item (8) above serves to provide an opportunity for students to make conjectures, test hypotheses, and discuss mathematics. The exercise may also stimulate student interest in the power of mathematical proof. 10. Show slides to illustrate each type of linear band ornament. Accompany each type with an appropriate schematic representation of the symmetries involved using the letters "p", "q", "b", and "d." (Suggested by Stevens, 1980.) a) Stevens 1980, 97 (See Figure 17): This simplest of the linear band ornaments is generated by successive translations of an asymmetric motif. Here we observe an Egyptian design from the New Empire period. It features a dolphin-like creature which appears to be balancing a ball on its head. The scheme p p p p . . . can be used to represent this type of repeated border. b) Stevens 1980, 109 (See Figure 18.): This design from ancient Greece looks like a stylized vine or the tracks of a duck walking on wet sand. It is created 198 by successive glide reflections and can be represented by the pattern: p b p b p b . . . c) Washburn and Crowe 1988, 105: A man's shoulder or waist cloth from Sumba features rows of fantastic- looking animals. Each row repeats a motif consisting of two wild-eyed beasts facing each other in reflection symmetry. We can represent this type of linear band ornament as: pqpqpqpq. . . Here, an asymmetric motif is reflected in a mirror; the resulting motif (beast and its reflection image) is itself reflected in a second mirror perpendicular to the direction of translation. (For a different version of this linear band pattern, see Figure 19.) d) Zaslavsky 1973, 184: Here we look at the bands on a hat worn by a bronze Benin flute player. The motif is chevron-like in appearance. It appears that a translation pattern (/ / / . . .) has been reflected in a mirror parallel to the line of translation. It looks like this:

<<<<<<.

Using the feltboard letter "p", we could describe this band or strip pattern as: 199

P P P P • • . b b b b . (For an Inca motif utilizing thesame format, see Figure

20.) e) Mitchell and others 1977, plate 27 d: This is a photograph of a carpet page from the sixth century Book of Durrow, fol. 192v. (A carpet page is one of pure decoration, no script.) The animal interlace design takes the shape of serpent-like creatures, each biting its tail, and forms a border strip above and below the central circular eternity symbol. The serpent-like border consists of successive translations of a motif with 180°

rotational symmetry. It can be represented schematically as:

q b q b q b . As with any linear band ornament of this type, the design

looks the same right-side-up as upside-down. (For a Western-Hemisphere Native pattern of the same classification, see Figure 21.) f) Washburn and Crowe 1980, 120: The San Idelfonso Pueblo ceramic design shown here incorporates all four symmetry elements; its shape is a deceivingly-simple zig­ zag format. Schematically, we can represent the border pattern as:

pdbqpdbq. 200 (For a prehistoric Indian design from Delaware utilizing the same format, see Figure 22.) g) Stevens 1980, 163: In this slide we see four linear band ornaments from different eras and different parts of the world. One is a Victorian ornament, another an Arabian design, and the last two are patterns from mosaics: one Pompeian (Figure 23) and the other Byzantine. The symmetries that these designs share can be represented schematically by the pattern: b d b d . p q p q . . . 11. Instruct each group to create a basic asymmetric motif. This motif is then used by group members to devise an illustration for each strip pattern. The group projects are mounted on poster board and displayed in the classroom.

Evaluation Students will take a field trip to local resource centers. For example, in the Washington, D.C. area, students could visit the Museum of African Art, the Islamic mosque, or the Textile Museum. Students will each have a schematic representation of the seven linear band ornaments, pencil, and paper. Working in teams, pupils will attempt to locate at least three items displaying a band pattern of the types studied. They will: name and 201 date each item; list country of origin; describe medium (fiber, metal, pottery, glass, wood, etc.); sketch the design and classify it. After the group projects have been evaluated, they will be mounted on colored posterboard, protected with transparent covering, and displayed in the halls. For students living in areas where field trips to museums are impractical, the activity can utilize local resources. For example, items displaying border patterns of the type studied can frequently be found in the home (rugs, clothing, blankets, pottery, and quilts). Trips could be taken to shops selling china, wallpaper trim, carpets, and fabric. Often, state or county municipal buildings incorporate tiled border patterns in floor, wall, and ceiling aspects.

Comments * Activity Four includes many of the recommendations made by the National Council of Teachers of Mathematics (1989). The Council suggested that a mathematics curriculum for grades five through eight provide students with opportunities to: a) develop spatial sense by drawing, visualizing, comparing, transforming and classifying geometric figures (p. 112); 202 b) describe extend, analyze, and create a wide variety of patterns (p. 98); c) model situations using oral, written, concrete, and pictorial methods (p. 78); d) communicate mathematical thought processes (p.79). (The small group format provides a forum for students to discuss, compare, criticize, and evaluate each others• ideas.)

As proposed by our model, the activity provides pupils with an interdisciplinary curriculum. Learning experiences combine knowledge from the fields of art, history, and mathematics, encouraging students to view intellectual endeavor from a unified, rather then fragmented, perspective. By utilizing works of art from many different societies, the activity promotes the multi-cultural approach to learning advocated by a growing number of public-school systems in the United States (Kirby 1990, 3[B]). A multi-cultural curriculum may enhance student self-esteem by highlighting the positive contributions made by all groups to the body of human knowledge (Washington State Mathematics Council 1984, preface; Zaslavsky 1990). 203 Activity Five Topic: The Golden Ratio Grade Levels: Six through Eight

Objectives Students will be able to: 1. construct a right angle using a compass and straight edge (Student can construct a perpendicular to a line through a given point on the line) (Figure 29); 2. construct a Golden Rectangle using ruler and compass (Figure 2); 3. define the Golden Ratio in terms of the Golden Rectangle; 4. name and discuss at least two works of art which illustrate use of the Golden Ratio; 5. construct a regular pentagon using ruler and compass (Figure 30);

6. describe the relationship between the Golden Ratio and the regular pentagon;

7. build a model of the Euclidean solid based on a regular pentagon (the dodecahedron) from two-dimensional drawings: both flat plan (two-dimensional representation) and perspective view. 204

#

Figure 29. Construct a perpendicular to a line through a given point on the line. 205

Given circle with center C. Draw two diameters such that IJ is perpendicular to KM. Bisect JC. Call the point of bisection L. With LK as radius, draw an arc intersecting IJ in X. Line segment KX will be equal in length to each side of the pentagon. Start at K and mark off arcs of length KX along the circumference. Connect the points as shown.

K

J

M

Figure 30. Construction of a regular pentagon 206 Materials Slides (Suggestions follow. Slides can be made by the Visual-Aid Center using books listed in the Bibliography.) Construction paper; pencils; rulers; hand-held calculators; tape measures; compasses; glue; lightweight cardboard; rigid vinyl sheet plastic, .030" thick; X-acto knives; heavy chipboard (for backing purposes during cutting the plastic).

Background Teacher will have read the Ratio and Proportion section of Chapter Four to review basic concepts. Students will have been introduced previously to the concept of ratio and proportion, as recommended by the Standards (NCTM 1989). Students will have had some prior experience with use of compass and straightedge. (See Fairfax County Virginia, Public Schools 1989.)

Procedures

1. Review the concepts of ratio and proportion with students. 2. Inform students that the next activity will focus on a special ratio favored throughout the ages in many different cultures. (Note; Typically, students do 207 not encounter the concept of irrational number until later in their mathematical studies. Use 1.618 as an approximation. The Standards (NCTM 1989) suggest that students in grades five through eight extend their understanding of whole number operations to include fractions and decimals. Thus, the approximation is pedagogically appropriate at this point in the educational cycle.) 3. Show slides depicting use of the Golden Ratio in works of art from different lands. Refer to discussion of The Golden Ratio in Chapter Four, where references to particular works of art and associated bibliography are given. In particular, we recommend utilizing; * Stonehenge; the station-stone rectangle and Sarsen archways

* The Parthenon, Athens * The Pyramid of the Sun, Mexico * Pottery created by the Zuni and Pueblo peoples * Native American blankets

* Wood engravings by Durer in the Apocalypse series * Mondrian's Broadway Boogie Woogie 4. Introduce an historical aspect by explaining that the ratio has the name $; tell where the name came from. 208 5. Distribute a compass, straightedge, and paper to each student. Stress safety considerations. Explain how the simple devices, compass and straightedge, were used throughout the years as essential aids in architecture and design. Demonstrate use of these tools. This is an appropriate point at which to show students slides of two specific works of art mentioned in Chapter Four; Raphael's School of Athens and Blake's God as Geometer. 6. Demonstrate; a) construction of a square (If necessary, see Boles and Newman [1987, 62].) b) bisection of a line segment (If necessary, see Boles and Newman [1987], 19.) c) construction of a Golden Rectangle. (See Chapter Four, Figure 2.) 7. Invite students to construct a Golden Rectangle. Suggest they measure the length and width, then calculate the ratio of length to width. How close does it come to 1.6? 8. Make a brief comment concerning measurement and accuracy. As the Standards state; During students' early experiences with counting and operations using whole numbers, they work with precise situations that yield exact counts. Measuring the length of an object is quite different, and it is essential that students understand this difference. The approximate nature of measuring is a concept that 209 takes time and many experiences for students to develop and understand. (NCTM 1989, 117) 9. Divide the students into groups of two. Challenge students to measure the dimensions of rectangular forms found in everyday life. Some suggestions are: credit cards; notebook paper; paper currency, both U.S. and foreign; windows; picture frames; business cards; tiles; bricks; bath towels; neck scarfs; doorways; table tops; classroom floors; magazine covers. 10. Instruct students to organize the data gathered in tabular form. (Supply students with photocopied sheets to simplify presentation of experiment results; uniform organization aids students in comparing their findings.) Columns should be headed: Item--- Length Width Ratio of Length to Width. 11. Allow students some time in class to make measurements. The sheet should be completed as part of the mathematics assignment. During the subsequent mathematics period, students will work in groups of four to six, comparing and consolidating their findings. Students will compute the ratio of items displaying the

Golden Ratio to total number of items examined. Results will be shared with the class. Students will discuss whether or not their findings indicate signs of a modern- day preference for the Golden Rectangle. 210 12. Review the terms polygon and regular polygon. Tell students that they are going to continue exploring the Golden Ratio by means of yet another geometric construction. In part (6) above, they constructed a square in order to produce a Golden Rectangle. The square is a regular polygon of four sides. Now, using only a compass and straightedge, they will construct a regular polygon of five sides: the pentagon. (Challenge students to recall and name the regular polygon with three sides.) Demonstrate construction of a regular pentagon (Figure 31).

13. Inform students that the pentagram, a special five-pointed star, is associated with the regular pentagon. The pentagram exhibits many occurrences of the Golden Ratio. (See Boles and Newman 1987, 50.) Display a transparency depicting a regular pentagon and the pentagram derived from it. Point out one or more of the occurrences of the Golden Ratio using segments of the pentagram. Note that the ratio of any diagonal of the

pentagon (a part of the pentagram) to a side of the pentagon equals $. 14. Relate the Golden Ratio, pentagon, and associated pentagram to art and geography by mentioning that the five-pointed star, as symbol of health and auspiciousness, is frequently found in national flags. 211 (Refer to Chapter Four.) Jasper Johns, an American artist, created a series of paintings featuring his country's flag. One work is Three Flags. (A reproduction is found in Janson 1970, 555.) 15. Challenge students to locate* national flags incorporating the pentagram in their design format. Students, working in teams of two to three, will go to the school (or public) library for a brief session with the research librarian. They will then proceed with the assignment, sketching the flags and noting country of origin. Students might have replicas of some of these flags at home. During the next mathematics session, students will share their findings with others. (Currently, there are over twenty five nations whose flags display the five-pointed star; see Chapter Four.) 16. Introduce construction of the regular dodecahedron (consisting of twelve congruent regular pentagons) by mentioning: * The Etruscans used pentagonal dodecahedra as dice in the first millennium B. C. (Holden 1971, 1). * Euclid discussed it (along with the remaining four regular solids: the tetrahedron, cube, octahedron, and icosahedron) in Book XIII of his Elements (Huntley 1970, 31). 212 * Plato mentioned it in his Timaeus (Jowett Vol II 1937, 35) when delineating construction of the universe. (The five regular solids are thus called the Platonic solids.) Now demonstrate construction of a pop-up dodecahedron using two sheets of cardboard in contrasting colors. (For procedure, see Olson 1989, 43.) Distribute materials and invite each student to make a similar model. Display student products by hanging them in groupings of varying heights in the lunchroom. 17. Demonstrate the "tab and glue" method for construction of a dodecahedron. (For procedure, see Olson 1989, 44, 57; also refer to Pearce and Pearce 1978, 53.)

Evaluation Evaluation for this activity will be both formative and summative in nature. As Tyler explained, "Evaluation is essentially the process of determining to what extent the educational objectives are actually being realized by the program of curriculum and instruction" (Tyler 1964, 106). He stated that student behavior must be appraised many times, both at early points in the educational process and at later points. Thus, the teacher should make evaluations as students participate in Procedure items seven, nine through eleven, fifteen, and sixteen. This approach provides pupils with on-going. 213 constant feedback as they take part in geometric constructions, data-gathering and analysis, model-making, and library research. At the end of the unit associated with Activity Five, the instructor should assess development of the pupils' mathematical progress by using a mixture of means. Following the National Research Council (1989) recommendations, we shall employ quizzes, essays, assignments, and projects. While short quizzes can be used to evaluate attainment of objectives one through three, five and six, the essay form lends itself to objective four. A project involving construction of a regular dodecahedron from cardboard— by the "tab and glue" method— or from colored vinyl— is appropriate for objective seven. The mixture of assessment procedures allows for evaluation of both lower-level skills and important higher-order objectives of the mathematics curriculum.

Extension Extrapolations from Activity Five are numerous and can be tailored to meet needs of all students, from those with weak mathematical skills to the highly gifted. Having made a model of the regular dodecahedron, the student could be encouraged to repeat the process using 214 different materials and altering the size of the basic pentagon. For decorative purposes, students can construct mobiles, sculptural pendant configurations, holiday ornaments, children's toys, or earrings by utilizing such diverse media as silver, brass, or copper wire, stained glass, felt, and Origami paper. As extra credit, students could construct a set of the five Platonic solids. These models could be displayed prominently and shared with other members of the school community.

Enterprising pupils could explore the duality of Platonic solids by first constructing a wire cube. Joining the center point of each face of the cube with additional wire, students discover they have created an octahedron. Repeating the procedure by first constructing a regular octahedron, students see that the dual is a cube. Similar construction techniques indicate that a tetrahedron is self-dual. Very ambitious pupils could investigate the duality of dodecahedron and icosahedron. As an extension of Golden Ratio studies, students could experiment with Penrose tiles. In creating patterns with these tiles, students make an excursion into contemporary mathematical thought. (See Gardner [1989] for an introduction to Penrose tilings.) 215 Comments Activity Five incorporates a multiplicity of approaches, procedures, and goals recommended by the National Council of Teachers of Mathematics in the Standards (1989):

* By uniting the study of mathematics, art, and geography, the activity provides pupils with "... many opportunities to observe the interaction of mathematics with other school subjects and with everyday society" (p. 84). * The activity recognizes that "Perspective, proportion, and the Golden Ratio are ways of learning mathematics in the context of art and design" (p. 85). * Sections of the activity apply ratios in a wide variety of situations. "To provide students with a lasting sense of number and number relationships, learning should be grounded in experience related to aspects of everyday life or to the use of concrete materials designed to reflect underlying mathematical ideas" (p. 87). * Discussion of the Golden Ratio and its applications throughout the ages strengthens pupils' appreciation of ". . . the need for numbers beyond the whole numbers" (p. 91).

* The activity assists students in developing spatial sense by ". . . constructing, visualizing, drawing 216 and measuring two- and three-dimensional figures, relating properties to figures, and contrasting and classifying figures according to their properties" (p. 113). It is well known that mathematical solids are encountered in the fields of engineering, architecture, chemistry, sculpture, design, and virology, to name just a few. (See, for example, Williams 1981; Critchlow 1969; Williams 1979.) Holden (1971) maintains that the most effective way of learning about these solids is by constructing and handling them, and Activity Five provides students with ample opportunities to build models. Holden continues: Space provides no three-dimensional blackboard. We learn about space only by living in it. A child climbing in his jungle gym may learn more about it [space] than he will ever learn again, for his books will be made of two-dimensional sheets of paper. (Holden 1971, i)

The construction of models of mathematical solids, Holden continues, assists development of spatial visualization skills. The ability to visualize an inside by looking at an outside is a generally useful skill and almost necessary for draftsmen, architects, and surgeons. Pictures can give a little help in cultivating the skill. . . . The privilege of seeing what the faces of a solid look like from the inside is commonly reserved for those who make it. (Holden 1971, 143) It is apparent that Activity Five can aid students' understanding of mathematics in many ways, while also furthering knowledge in related fields. CHAPTER 6 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

Summary and Conclusions Opening sections of this work examined the country's need for competent mathematicians, statisticians, engineers, and scientists. 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. To this end, we explored an alternative, innovative 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. As Whittrock commented in his plea for fresh, new approaches: Repetition and reinforcement of behavior sometimes lead to disinterest. The brain responds, at least momentarily, to novelty, to the unexpected event, to discrepant information. For learning in schools, techniques other than reinforced practice, drills, and review seem likely to stimulate interest. (Whittrock 1981, 13)

217 218 Interdisciplinary, cultural, and motivational considerations were reviewed in developing the curriculum model. Recent research was cited from the fields of education, psychology, sociology, physics, and medicine. These studies lend support to the premise that the teaching of mathematics through art is not only feasible but educationally sound. In Chapter Four, we investigated some of the many connections between art and mathematics, then focused on the concepts of ratio, proportion, and symmetry. The chapter can be used by teachers to extend their understanding of these mathematical topics and related extensions in 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 delineated in Chapter Five incorporate the premises of the Standards (NCTM 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. Evaluation is both formative and summative in nature and utilizes a variety of methods for gathering information to assess student progress. Evaluative 219 measures suggested in the activities serve many purposes. As recommended by the National Research Council (1989), evaluation procedures allow students to recognize personal success, enable teachers to judge student achievement, and provide administrators means of measuring effectiveness of instruction.

Recommendations

Although a considerable body of research indicates that the mathematics-art curriculum model is both feasible and educationally sound, ultimate questions concerning its efficacy can only be answered definitively by field testing. Research studies of a quantitative nature must be conducted. In this endeavor, both attitude towards mathematics learning and mastery of content must be assessed. Moreover, analysis of test data should include investigation of differences in mathematics achievement and/or interest by race, gender, and ethnic group. Are there comparable by-products in art?

Matched groups must be utilized in the testing to minimize confounding effects. The sample of subjects must be large enough to provide appropriate precision in hypothesis testing. Obstacles to valid inference-making must be identified and rectified. Two goals of research are to draw valid conclusions concerning the effects of an independent variable and to make valid generalizations to populations and 220 settings of interest. . . . [P]our categories of threats to these goals are: 1) Statistical conclusion validity is concerned with threats to valid inference-making that result from random error and the ill-advised selection of statistical procedures. 2) Internal validity is concerned with correctly concluding that an independent variable is, in fact, responsible for variation in the dependent variable. 3) Construct validity of causes or effects is concerned with the possibility that operations that are meant to represent the manipulation of a particular independent variable or the measurement of a particular dependent variable can be construed in terms of other variables. 4) External validity is concerned with the generalizability of research findings to and across populations of subjects and settings. (Kirk 1982, 20-21) Broad implementation of the mathematics-art interdisciplinary program will require teacher training, administrative cooperation, and the development of additional curriculum materials. Teacher-training workshops should provide content enrichment in both mathematics and art; they should also provide professional staff assistance in creating and producing interdisciplinary thematic units. (The Association for

Supervision and Curriculum Development, for example, offers an institute for integrating the elementary school curriculum. See ASCD fall/winter 1990.) Administrative cooperation at the local and regional levels is required to provide art resource personnel and classroom teachers time for joint planning of activities and thematic units. University faculty from 221 the fields of art, mathematics, and education could bring their expertise to the development of unified curriculum materials. More research and planning are necessary, and we recognize that there is no single curriculum that will prove effective with all children. But, in light of the dismal performance of American youth on both national and international mathematics tests, the perceived need for a more culturally-informed populace (Bloom 1987; Hirsch 1988), and the many requests for a fresh, innovative approach to teaching mathematics, the curriculum discussed in this work merits additional consideration. SELECTED BIBLIOGRAPHY

Arnheim, Rudolph. Art and Visual Perception. A Psychology of the Creative Eve. Los Angeles: University of California Press, 1974. Association for Supervision and Curriculum Development. Update 32, no. 4 (May 1990). ." Integrating the Elementary School Curriculum." ASCD Professional Development Opportunities: 1990 Fall/Winter Catalog: 18. Ayers, Alex. "Childhood at Risk." Educational Leadership 46, no. 8 (May 1989): 70-72.

Baenninger, Maryann, and Nora Newcombe. "The Role of Experience in Spatial Test Performance: A Meta- Analysis." Sex Roles 20, nos. 5/6 (1989): 327-344. Bain, George. Celtic Art: The Methods of Construction. New York: Dover Publications, Inc., 1973. Ball, Deborah L. "Prospective Elementary and Secondary Teachers' Understanding of Division." The Journal for Research in Mathematics Education 21, no. 2 (1990): 132-144.

Battista, M. T., H. W. Grayson, and G. Talsma. "The Importance of Spatial Visualization and Cognitive Development for Geometry Learning in Preservice Elementary Teachers." The Journal for Research in Mathematics Education 13 (1982): 332-340. Beckwith, Jon. "Gender and Math Performance: Does Biology Have Implications for Educational Policy?" Journal of Education 165 (1983): 158-174. Benbow, C.P. and J. C. Stanley. "Sex Differences in Mathematical Ability: Fact or Artifact?" Science 210 (December 1980): 1262-1264.

222 223 Bishop, Alan J. Mathematical Enculturation. Dordrecht: Kluwer Academic Publishers, 1988. ______. "Review of Research on Visualization in Mathematics Education." Focus on Learning Problems in Mathematics 11, no. l (Winter 1989): 7-16. Bloom, Allan. The Closing of the American Mind. New York: Simon and Schuster, Inc. 1987, Boles, Martha, and Rochelle Newman. The Golden Relationship: Art. Math. Nature. Bradford, Massachusetts: Pythagorean Press, 1987. Bouleau, Charles. The Painter's Secret Geometry. New York: Harcourt. Brace, & World, Inc., 1963. Bourgoin, J. Arabic Geometrical Pattern and Design. New York: Dover Publications, Inc., 1973. Bronowski, J. Science and Human Values. New York: Perennial Library, Harper and Row, 1965. Browne, Dauna. Learning Styles and Native Americans. Paper, 1986. Dialog, ERIC ED 297 906. Bruner, Jerome S. "On Learning Mathematics." The Mathematics Teacher 53, no.8 (December 1960): 610-619. ______. Toward a Theorv of Instruction. Cambridge, Massachusetts: The Belknap Press of Harvard University Press, 1966. . "Models of the Learner." Educational Researcher 14, no. 6 (June/July 1985) Burton, Grace M. "Regardless of Sex." The Mathematics Teacher 72, no. 4 (April 1979): 261-270. Campbell, Joseph. The Power of Myth. New York: Doubleday, 1989. Canaday, John. Mainstreams of Modern Art. New York: Holt, Rhinehart and Winston, 1959. Caplan, Paula J., Gael M. MacPherson, and Patricia Tobin. "Do Sex-Related Differences in Spatial Abilities Exist?" American Psychologist 40, no. 7 (July 1985): 786-799. 224 Carnegie Foundation. The Condition of Teaching. A State bv State Analysis. 1988. Princeton, New Jersey: The Carnegie Foundation for the Advancement of Teaching, 1988. ______. Turning Points: Preparing American Youth for the 21st Centurv. Washington, D.C.: The Carnegie Council on Adolescent Development, 1989. Carss, Marjorie, ed. Proceeding of the Fifth International Congress on Mathematics Education. Boston: Birkhauser, 1986. Charbonneau, Manon P., and Vera John-Steiner. "Experience and the Language of Mathematics." In Linguistic and Cultural Influences on Learning Mathematics, ed. Rodney R. Cocking and Jose P. Mestre, 91-100. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1988. Cheek, Helen Neely. "Increasing the Participation of Native Americans in Mathematics." The Journal for Research in Mathematics Education 15, no. 2 (March 1984): 107-113. Cocking, Rodney R., and Jose P. Mestre, ed. Linguistic and Cultural Influences on Learning Mathematics. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1988. Cocking, Rodney R., and Susan Chipman. "Conceptual Issues Related to Mathematics Achievement of Language Minority Children." In Linguistic and Cultural Influences on Learning Mathematics, ed. Rodney R. Cocking and Jose P. Mestre, 17-46. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1988. Coe, Michael D. The Mava. Great Britain: Thames and Hudson, Ltd.,1984. Cole, Herbert M., and Doran H. Ross. The Arts of Ghana. Los Angeles: The Museum of Cultural History, University of California, 1977.

Collins, Debra. 1990. Mathematics Coordinator, Albermarle County Public Schools, Virginia. Personal communication, 17 March. Comiti, Claude, Joseph Payne, Wally Green, and Alistair McIntosh. "Elementary School (Ages 7-12)." In Proceeding of the Fifth International Congress on Mathematics Education, ed. Marjorie Carss, 57-72. Boston: Birkhauser, 1986. 225 Contreras, Maxlmiliano. Hemispheric Learning and the Hispanic Student. Paper, The Center for Quality Education, California State University, Dominguez Hills, California, June 1985. Dialog, ERIC, ED277 505. Cook, Theodore A. The Curves of Life. New York: Dover Publications Inc., 1979. Coop, Richard H., and Irving E. Sigel. "Cognitive style: Implications for Learning and Instruction." Psychology in the Schools 8 (1971): 152-161. Cresswell, John L., Claire Gifford, and Debbie Huffman. "Implications of Right/Left Brain Research for Math Educators." School Science and Mathematics 88, no. 2 (1988): 118-131. Critchlow, Keith. Order in Space: A Design Source Book. New York: The Viking Press, 1973. Crouch-Shinn, Jenella, and M. Shaughnessy. Brain Research: Implications for Education. Paper, the Psychology Department, Eastern New Mexico University, Portales, New Mexico, 1984. Dialog, ERIC, ED 271 707 D'Ambrosio, Ubiratan. "Socio-Cultural Bases for Mathematical Education." In Proceeding of the Fifth International Congress on Mathematics Education, ed. Marjorie Carss, 1-6. Boston: Birkhauser, 1986. Davidson, Basil. African Kingdoms. Alexandria, Virginia: Time/Life Books, 1971. Dewey, John. Experience and Education. New York: Collier Books, 1967.

Divorky, Diane. "The Model Minority Goes to School." Phi Delta Kappan 70, no.3 (November 1988): 219-222. Doczi, Gyorgy. The Power of Limits. Boulder: Shambhala, 1981. Dorf1er, Willibald, Claude Gaulin, Hilary Shuard, and Graham Jones. "Pre-Service Teacher Education." In Proceeding of the Fifth International Congress on Mathematics Education, ed. Marjorie Carss, 111-123. Boston: Birkhauser, 1986. Dormer, Peter, and Ralph Turner. The New Jewelrv. Trends and Traditions. New York: Thames and Hudson, 1986. 226 Dossey, John A., Ina V. S. Mullis, Mary M. Lindquist, and Donald L. Chambers. The Mathematics Report Card. Are We Measuring Up ? Trends and Achievement Based on the 1986 National Assessment. Princeton, New Jersey: The Educational Testing Service, 1988. Duckett, W. "An Interview with Harold Hodgkinson: Using Demographic Data for Long Range Planning." Phi Delta Kappan 70, no.2 (October 1988): 166-170. Dunn, Kenneth, and Rita Dunn. "Dispelling Outmoded Beliefs About Student Learning." Educational Leadership 44, no. 6 (March 1987): 55-62. Dunn, Rita. "Learning Styles:State of the Science." Theorv Into Practice 23, no.l (Winter 1984): 10-19. "Commentary: Teaching Students Through Their Perceptual Strengths or Preferences." Journal of Reading 31, no. 4 (January 1988): 304-309. Dunn,Rita J., J. Beaudry, and A. Klavas. "Survey of Research on Learning Styles." Educational Leadership 46, no.6 (March 1989): 50-58. Dye, Daniel Sheets. Chinese Lattice Designs. Cambridge: Harvard University Press, 1937; reprint. New York: Dover Publications, Inc., 1974. Edwards, Edward B. Pattern and Design With Dynamic Symmetry. New York: Dover Publications Inc., 1967. Eisner, Elliot W. Cognition and Curriculum: A Basis for Deciding What to Teach. New York: Longman Inc., 1982. Elliot, Portia C. "Going 'Back to Basics' in Mathematics Won't Prove Who's 'Right', but Who's 'Left' (Brain Duality and Mathematics Learning)." The International Journal of Mathematical Education in Science and Technology 11, no.2 (1980): 213-219. El-Said, Issam, and Ayse Parman. Geometric Concepts in Islamic Art. Palo Alto, California: Dale Seymour Publications, 1976. Escher, M. C. Escher on Escher. Exploring the Infinite. New York: Harry N. Abrams, Inc., 1989. 227 Eves, Howard. Great Moments in Mathematics Before 1650. Washington, D.C.: Mathematical Association of America, Incorporated, 1983. Fairfax County Public Schools. 1989. Mathematics Curriculum for Grades Five through Eight. Fehervari, Geza, and Yasin H. Safadi. 1400 Years of Islamic Art. London: Khalili Gallery, 1984. Feldman, Edmund Burke. Varieties of Visual Experience. Revised and Enlarged Edition. New York: Harry N. Abrams Inc., 1971. ______. Varieties of Visual Experience. 3d ed. New York: Harry N. Abrams, Inc., 1987. Fennema, Elizabeth. "Teachers and Sex Bias in Mathematics." The Mathematics Teacher 73, no. 3 (March 1980):169 -173.

Fennema, Elizabeth. "Sex-Related Differences in Mathematics Achievement: Where and Why." In Mathematics— People. Problems, Results. (Vol.3), eds. D. M. Campbell and J. C. Higgins, 209-221. Belmont, California: Wadsworth International, 1983. Fennema, Elizabeth, and Julia Sherman. "Sex-Related Differences in Mathematics Achievement, Spatial Visualization, and Affective Factors." American Educational Research Journal 14, no. 1 (Winter 1977): 51-71.

Fleming, John, and Hugh Honour. The Pencmin Dictionary of Decorative Arts. England: Alan Lane, 1977; Penguin Books Ltd., 1979.

Gardner, Helen. Art Through the Ages. 4th ed. New York: Harcourt, Brace and World, Inc., 1959. Gardner, Martin. Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman and Company, 1989. Garfunkel, Solomon A. "Building Arks." UME Trends. News and Reports on Undergraduate Mathematics Education 2, no. 1 (March 1990): 1,5. Garrard, Timothy R. Gold of Africa. Germany: Prestel-Verlag, 1989. 228 Ghyka, Matila. The Geometry of Art and Life. New York, New York; Dover Publications Inc., 1977. Gibney, Thomas, Jon Ginther, and Fred Pigge. "Are Elementary Teachers Better Prepared in the Content of Elementary Mathematics in the 1980's?" School Science and Mathematics 88, no. 7 (November 1988): 595-603. Giedion, Sigfried. Space. Time and Architecture. Cambridge: The Harvard University Press, 1949.

Gitelson, Idy B., Anne C. Petersen, and Maryse H. Tobin- Richards. "Adolescents' Expectancies of Success, Self- Evaluations, and Attributions about Performance on Spatial and Verbal Tasks." Sex Roles 8, no.4 (1982): 411-419. Glennon, Vincent J. Neuropsvcholoov and the Instructional Psvcholoqv of Mathematics. Kent, Ohio: The Research Council for Diagnostic and Prescriptive Mathematics, 1981. Good, Thomas L., E. Grouws, and D. Mason. "Teachers' Beliefs about Sma11-Group Instruction in Elementary-School Mathematics." The Journal for Research in Mathematics Education 21, no. 1 (1990): 2-15. Gough, Sister Mary Fides. "Mathophobia: Causes and Treatments." Clearing House 28 (January 1954): 290-294 Grabar, Oleg. "Architecture and Art." In The Genius of Arab Civilization: Source of Renaissance. 2d ed. John R. Hayes, ed., 77 - 120. London: Eurabia (Publishing) Ltd., 1983. Graeber, Anna 0., D. Tirosh, and R. Glover. "Preservice Teachers' Misconceptions in Solving Verbal Problems in Multiplication and Division." The Journal for Research in Mathematics Education 20, no. 1 (1989): 95-102. Grillo, Paul Jacques. Form. Functon. and Design. New York: Dover Publications, Inc., 1960. Grow, Meda F., and Norbert Johnson. "Math Learning: The Two Hemispheres." The Journal of Humanistic Education and Development 22 (September 1983): 30-39. Guay, R. B., and E. D. McDaniel. "The Relationship Between Mathematics Achievement and Spatial Abilities Among 229 Elementary School Children." The Journal for Research in Mathematics Education 8, no. 3 (1977): 211-215. Hambidge, Jay. The Parthenon and Other Greek Temples; Their Dynamic Symmetry. New Haven, Connecticut: Yale University Press, 1924. _. Practical Applications of Dynamic Symmetry. New Haven, Connecticut: Yale University Press, 1932. Harris, Bruce S., ed. The Collected Drawings of Aubrev Beardsley. New York: Bounty Books, 1967. Hart, Leslie A. Brain-Compatible Education. Paper presented at the National Conference sponsored by the Learning Styles Network, Reston, Virginia, 1982. Hartt, Frederick. History of Italian Renaissance Art. New York: Harry N. Abrams, Inc., 1969. Hawkins, Gerald S. Stonehenge Decoded. New York: Doubleday & Company, Inc., 1965. Hayes, John R., ed. The Genius of Arab Civilization: Source of Renaissance. 2d ed. London: Eurabia (Publishing) Ltd., 1983. Hembree, Ray. "The Nature, Effects and Relief of Mathematics Anxiety." The Journal for Research in Mathematics Education 21, no. 1 (1990): 33-46. Henderson, Linda Dalrymple. The Fourth Dimension and Non- Euclidean Geometry in Modern Art. Princeton, New Jersey: Princeton University Press, 1983. Herstein, I. N. Abstract Algebra. New York: Macmillan Publishing Company, 1986.

Hill, Shirley A. "New Perspectives on the Education of Teachers." In The Teaching of Mathematics: Issues for Today and Tomorrow. The Mathematical Sciences Education Board, 5-14. Washington, D.C.: The National Academy Press, 1987. Hirsch, E. D., Jr. Cultural Literacy; What Everv American Needs to Know. New York: Vintage Books, 1988. Hoffman, Mark S., ed. The World Almanac and Book of Facts. 1990. New York: Scripps Howard, 1990. 230 Hofstadter, Douglas R. Godel. Escher. Bach; An Eternal Golden Braid. New York: Vintage Books, 1979. Holden, Alan. Shanes. Space, and Symmetry. New York: Columbia University Press, 1971 Holt, Michael. Mathematics in Art. London: Studio Vista, 1971. Horvath, Patricia J. "A Look at the Second International Mathematics Results in the U.S.A. and Japan." The Mathematics Teacher. 80, no. 5 (May 1987): 359-368. Hoyle, Fred. On Stonehenge. San Francisco: W.H. Freeman and Company, 1977. Huntley, H. E. The Divine Proportion. New York, New York: Dover Publications Inc., 1970. Jackson, Allyn. "Making Mathematics Work for Minorities." Notices of the American Mathematical Society 37, no. 6 (July/August 1990): 666-668. Janson, H. W. History of Art. Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1970. Jenifer, Franklyn G. "Afrocentricity is No Cause for Alarm." Washington Post. 19 November 1990, 15 (A). Johnson, Susan A. Hemispheric Specialization and Learning Style Theorv: Some Considerations for the Elementary Teacher. Paper presented as Exit Project, Indiana University at South Bend, July 1983. Dialog, ERIC, ED 233 807. Johnson, Virginia R. "Myeline and Maturation." Science Teacher 49, no. 3 (1982): 41-44,49.

Kalsbeek, David H. Linking Learning Stvle Theorv with Retention Research:The TRAILS Project. Paper presented at the Annual Forum of the Association for Institutional Research, Orlando, Florida, 22-25 June 1986. Keefe, James W. "School Applications of the Learning Style Concept." In Student Learning Styles: Diagnosing and Prescribing Programs. 123-132. Reston, Virginia: National Association of Secondary School Principals, 1979. 231 . "Assessment of Learning Style Variables: The NASSP Task Force Model. Theorv Into Practice. Vol.24 No.2.(Spring 1985): 138-144. Learning Stvle Theorv and Practice. Report prepared for the National Association of Secondary School Principals, Reston, Virginia, 1987. Kelly, W. P., and W.K. Tomhave "A Study of Math Anxiety/Math Avoidance in Preservice Elementary Teachers." The Arithmetic Teacher 32, no. 5 (1985): 51-53. Kirby, Joseph. "P.G. Reshaping Teaching Along Cultural Lines." Washington Post. 14 August 1990, 3(B). Kirk, Roger E. Experimental Design: Procedures for the Behavioral Sciences. 2d ed. Belmont, California: Brooks/Cole Publishing Company, 1982. Kline, Morris. "Projective Geometry." In The World of Mathematics. ed. James R. Newman, 622-641. New York: Simon and Schuster, 1956. . Mathematics in Western Culture. New York: Oxford University Press, 1976. Kolb, D. Experiential Learning. New Jersey: Prentice Hall, 1984. Kosslyn, Stephen M. "Aspects of a Cognitive Neuroscience of Mental Imagery," Science. 240, no. 4859 (June 17, 1988): 1621-1626.

Kuntzsch, Ingrid. A Historv of Jewels and Jewellerv. New York: St. Martin's Press, 1981. Laver, James. The Concise Historv of Costume and Fashion. New York: Harry N. Abrams, Inc., 1969.

Leap, William. "Assumption Strategies Guiding Mathematical Problem Solving by Ute Indians." In Linguistic and Cultural Influences on Learning Mathematics, eds. Rodney R. Cocking and Jose P. Mestre, 161-186. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1988.

Learning Styles Network. "Are Different Strategies Needed for Left/Right Brained Students?" In Learning Stvles Network Newsletter 3, no. 3 (Autumn 1982): 1. 232 Leder, Gilah. "Sex-Related Differences in Mathematics, An Overview." Educational Studies in Mathematics 16 (1985): 303-320. Leonard, Jonathan Norton. Ancient America. Alexandria, Virginia: Time/Life Books, 1978. Lester, Frank K. Jr., and Joe Garofolo. The Influence of Affects. Beliefs, and Metacocmition on Problem Solving Behavior. Paper presented at the Annnual Meeting of the American Educational Research Association, Washington, D. C., 20-24 April 1987. Levy, Jerre. Children Think with Whole Brains: Mvth and Reality. Paper presented at the National Conference Sponsored by the Learning Styles Network, Reston, Virginia, 1982.

"Research Synthesis on Right and Left Hemispheres: We Think With Both Sides of the Brain." Educational Leadership 40, no. 4 (January 1983) : 66-71. Linn, Charles F. The Golden Mean. Mathematics and the Fine Arts. Garden City, New York: Doubleday and Company, Inc., 1974. Linn, Marcia, and Anne c. Petersen. "Emergence and Characterization of Sex Differences in Spatial Ability: A Meta-Analysis." Child Development 56, (1985): 1479- 1498.

Maccoby, E. E., and C. N. Jacklin. Psvchologv of Sex Differences. Stanford: Stanford University Press, 1974. MacFarlane Smith, I. Spatial Abilitv. Its Educational and Social Significance. London: University of London Press, 1964.

Marsh, Herbert W. "Sex Differences in the Development of Verbal and Mathematics Constructs: The High School and Beyond Study." American Educational Research Journal 26, no. 2 (Summer 1989): 191-225. Martin, Gary W., and G. Harel. "Proof Frames of Preservice and Elementary Teachers." The Journal for Research in Mathematics Education 20, no. 1 (1989): 41-51. Martin, John Rupert. Barogue. New York: Harper and Row, 1977. 233 Mathematical Sciences Education Board, National Research Council. Reshaping School Mathematics. A Philosophy and Framework for Curriculum. Washington D.C.; National Academy Press, 1990. McCall's Needlework and Crafts Publications, ed. The McCall's Book of Quilts. New York: Simon and Schuster, 1985. McLeod, Douglas B. Affect and Problem Solving: Two Theoretical Perspectives. Paper presented at the Annual Meeting of the American Educational Research Association, Washington, D. C., 20-24 April 1987. Metropolitan Museum of Art. The Mathematics of Islamic Art. A packet for teachers of mathematics, social studies, and art. New York: The Metropolitan Museum of Art, 1979. Miller, Cynthia A. Can Hemispheric Lateralization Be Used as a Predictor of Success for Black Women in College Mathematics Courses? Paper presented at the Annual Conference of the Georgia Council of Teachers of Mathematics, Rock Eagle, Georgia, October 1986. Mitchell, G. Frank, Peter Harbison, Lian de Paor, Maire de Paor, and Roger A. Stalley. Treasures of Irish Art. 1500 B. C.-1500 A. D. New York: The Metropolitan Museum of Art and Alfred A. Knopf, 1977. Moore, Elsie G., and A. Wade Smith. "Sex and Ethnic Group Differences in Mathematics Achievement." The Journal for Research in Mathematics Education 18, no. 1 (January 1987), 25-36. Morris, Janet. "Math Anxiety; Teaching To Avoid It." The Mathematics Teacher 74, no.6 (September 1981): 413-417.

Munsell. Paul E., M. Rauen, and M. Kinjo. "Language Learning and the Brain: A Comprehensive Survey of Recent Conclusions." Language Learning 38, no. 2 (June 1988), 261-278.

National Center for Education Statistics. Elementary and Secondary Education Volume 1. Washington, D.C.: U.S. Department of Education, Office of Educational Research and Improvement, 1988. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, 234 Virginia: The National Council of Teachers of Mathematics, Inc., 1989. National Research Council. Everybody Counts. A Report to the Nation on the Future Mathematics Education. Washington D.C.: National Academy Press, 1989. Naylor, Maria, ed. Authentic Indian Pesions. New York: Dover Publications Inc., 1975. Newcombe, Nora, Mary M. Bandura, and Dawn G. Taylor. "Sex Differences in spatial Ability and Spatial Activities." Sex Roles 9, no. 3 (1983): 377-385. O'Brien, Maire, and Conor Cruise O'Brien. A Concise History of Ireland. New York: Beekman House, 1972. Olson, Alton T. Mathematics Through Paper Folding. Reston, Virginia: Nathional Council of Teachers of Mathematics, 1975. Padwick, Richard, and Trevor Walker. Pattern: Its Structure and Geometry. Sunderland, Great Britain: The Ceolfrith Press, 1981. Pearce, Peter, and Susan Pearce. Polvhedra Primer. Palo Alto, California: Dale Seymour Publications, 1978. Pedersen, Katherine. Parent Careers and Attitude Toward Mathematics as a Male Domain. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, California, 16-20 April 1986. Pedoe, Dan. Geometry and the Visual Arts. New York, New York: Dover Publications Inc., 1976.

Plato, The Dialogues of Plato. Translated by B. Jowett. Vol. Two, Timaeus. New York: Random House, 1920. Posner, Michael, S. Petersen, P. Fox, and M. Raichle. "Localization of Cognitive Operations in the Human Brain," Science 240, no. 4859 (June 17, 1988), 1627 -1631. Ragghianti, Carlo Ludovico, ed. National Gallerv: Washington. New York: Newsweek, Inc. and Arnoldo Mondadori, 1968. 235 Raphael, Dennis, and Merlin Wahlstrom. "The Influence of Instructional Aids on Mathematics Achievement." The Journal for Research in Mathematics Education 20, no. 2 (March 1989): 173-190. Reed, Stanley. All Color Book of Oriental Carpets and Ruas. New York: Crescent Books, 1972. Revault, Jacques. Designs and Patterns from North African Carpets and Textiles. New York: Dover Publications, Inc., 1973. Reyes, Laurie-Hart, and George M. Stanic. "Race, Sex, Socioeconomic Status, and Mathematics." The Journal for Research in Mathematics Education 19, no. 1 (January 1988): 26-43. Roberts, Fred S. "The Introductory Mathematics Curriculum." The College Mathematics Journal: an official publication of the Mathematical Association of America 15, no. 5 (November 1984): 3384-3399. Roebuck, Rhonda. 1990. Art Consulting Teacher, Albermarle County Public Schools, Virginia. Personal communication, 17 March. Rosen, Joe. Svmmetrv Discovered: Concepts and Applications in Nature and Science. London: Cambridge University Press, 1975. Rosenthal, Robert, and Donald R. Rubin. "Further Meta- Analytic Procedures for Assessing Cognitive Gender Differences." Journal of Educational Psvchologv 74, no. 5 (1982): 708-712. Rubenzer, Ronald L. Educating the other Half: Implications of Left/Right Brain Research. Information analysis sponsored by the National Institute of Education, Washington, D. C., 1982. Dialog, ERIC, ED 224 268. Rubin, William, ed. Pablo Picasso: A Retrospective. New York: The Museum of Modern Art,1980. Sadker, Myra, David Sadker, and Sharon Steindam. "Gender Equity and Educational Reform." Educational Leadership 46, no. 6 (March 1989), 44-47. Saunders, Hal. "When Are We Ever Gonna Have To Use This?" The Mathematics Teacher (January 1980): 7-16. 236 Schafer, Alice T., and Mary W. Gray. "Sex and Mathematics." Science 211, no. 4479 (1981): 231. Schillinger, Joseph. The Mathematical Basis of the Arts. New York: Da Capo Press, Inc., 1948; reprint. New York: The Philosophical Library, 1976. Senechal, Marjorie. "Shape." In On the Shoulders of Giants: New Approaches to Numeracy, ed. Lynn Arthur Steen, 139- 182. Washington, D. C.: National Academy Press, 1990. Senechal, Marjorie, and George Fleck, eds. Patterns of Svmmetrv. Amherst, Massachusetts: The University of Massachusetts Press, 1977. Seymour, Dale, and Jill Britton. Introduction to Tessellations. Palo Alto, California: Dale Seymour Publications, 1989. Sherman, Julia A. "Field Articulation, Sex, Spatial Visualization, Dependency, Practice, Laterality of the Brain, and Birth Order." Perceptual and Motor Skills 38 (1974): 1223-1235. Sill, Gertrude Grace. A Handbook of Symbols in Christian Art. New York: Collier Books, 1975. Skemp, Richard R. The Psychology of Learning Mathematics. Bungay, Suffolk: Richard Clay (The Chaucer Press) Ltd., 1986.

The Psvchologv of Learning Mathematics. Hillsdale, New Jersey: Lawrence Erlbaum Associates Inc, 1987. Smith, Elaine C. "Culture in Elementary Mathematics Education for African-American Learners: Enhancing Achievement Through Curricular Design." Ph.D. diss.. The American University, 1988. Smith, G. E. Kidder. A Pictorial History of Architecture in America. New York: American Heritage Publishing Co., Inc., 1976; reprint. New York: Bonanza Books, 1981. Smith, Linda, and J. Renzulli. "Learning Style Preferences: A Practical Approach for Classroom Teachers. " Theory into Practice 23, no.l (Winter 1984), 44-50. Sowell, Evelyn J. "Effects of Manipulative Materials in Mathematics Instruction." The Journal for Research in Mathematics Education 20, no. 5 (1989): 498-505. 237 Spinden, Herbert J. A Study of Maya Art; Its Subject Matter & Historical Development. New York: Dover Publications Inc., 1975. Spring, Christopher. African Textiles. New York: Crescent Books, 1989. Springer, Sally P., and Georg Deutsch. Left Brain. Right Brain. 3d ed. New York: W. H. Freeman and Company, 1989. Steen, Lynn Arthur. "Teaching Mathematics for Tomorrow's World." Educational Leadership 47, no. 1 (1989): 18-22. Stevens, Peter S. Handbook of Regular Patterns: An Introduction to Svmmetrv in Two Dimensions. Cambridge, Massachusetts: M. I. T. Press, 1980. Struik, Dirk J. A Concise History of Mathematics. 4th ed. New York: Dover Publication, Inc., 1987. Suina, Joseph. "Epilogue: And Then I Went to School." In Linguistic and Cultural Influences on Learning Mathematics, ed. Rodney R. Cocking and Jose P. Mestre, 295-299. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1988. Syron, Lisa. Discarded Minds: How Gender. Race and Class Biases Prevent Young Women from Obtaining an Adequate Math and Science Education in New York Citv Public Schools. Report prepared for the Full Access and Rights to Education Coalition, New York, New York, 1987. Thomas, Brian. Geometry in Pictorial Composition. England: Newcastle upon Tyne, 1969. Tobias, Sheila. Overcoming Math Anxiety. Boston: Houghton Mifflin Company, 1978. Tufte, Edward R. The Visual Display of Quantitative Information. Cheshire, Connecticut: Graphics Press, 1983. Tyler, Ralph W. Basic Principles of Curriculum and Instruction. Chicago: Random House, 1964.

VanDevender, Evelyn M. "Problems in Teaching Mathematics in the Elementary Classroom." School Science and Mathematics 88, no. 1 (January 1988): 65-71. 238 Washburn, Dorothy K., and Donald W. Crowe. Symmetries of Culture;Theory and Practice of Plane Pattern Analysis. Seattle, Washington: The University of Washington Press, 1988. Washington State Mathematics Council. Multicultural Mathematics Posters and Activities. Reston, Virginia: National Council of Teachers of Mathematics, 1984. Weyl, Hermann. Svmmetrv. Princeton, New Jersey: Princeton University Press, 1953. Wheeler, Ruric E. Modern Mathematics. Seventh Edition. Pacific Grove, California: Brooks/Cole Publishing Company, 1988. Whittrock, M. C. "Educational Implications of Recent Brain Research." Educational Leadership 39, no. 1 (October 1981): 12-15. Williams, Christopher. Origins of Form. New York: The Architectural Book Publishing Company, 1981. Williams, Robert. The Geometrical Foundation of Natural Structure. New York: Dover Publications, Inc., 1979. Wood, Eric F. "Math Anxiety and Elementary Teachers: What Does Research Tell Us?" For the Learning of Mathematics: an International Journal of Mathematics Education 8 (February 1988): 8-13. Yale, Paul B. Geometry and Svmmetrv. New York: Dover Publications, Inc., 1988. Yee, Doris K. Sex Eouitv in the Home: Parents' Influence on Their Children's Attitudes about Mathematics. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, California, 16-20 April 1986. Zaslavsky, Claudia. Africa Counts. Number and Pattern in African Culture. Westport, Connecticut: Lawrence Hill and Company, 1973.

______. "World Cultures in the Mathematics Class." History and Pedagogy of Mathematics Newsletter 20 (July 1990): 5-7. 239 Zenhausern, Robert. Education and the Left Hemisphere. Paper presented at the National Conference Sponsored by the Learning Styles Network, Reston, Virginia, 1982.