A few years ago I designed a G.A.T.E. (Gifted and Academically Talented) pilot program in Tucson, Arizona. It was limited to artistic exercises without any math lessons. This program developed into the FOUNDATIONS section, with some overflow into the DESIGNS section. It was designed to assess not only student reaction but its ease of teaching (teachability) as well. The teacher chosen to implement the program later commented:
Most of my students took the G.A.T.E. Test in January. We were very surprised at the results. A very high percentage of my students scored in the 7th, 8th, or 9th stanine in spatial reasoning. Approximately one-third of them scored in the top two stanines.
The only apparent difference in that group of students was that they had had the geometry program before they took the test. It would be interesting to see whether the program makes a difference with other groups. Sincerely, J. C. 6th Grade Teacher, Tucson, Az
The method that governs Native American geometry must be the MOTHER of all connect-the-dot/manipulative traditions. It belongs to a tradition of spatial inquiry that's been around for a long time: a tradition that will make you learn math while you learn to play and explore. For some reason it has been largely missed or ignored as a teaching tool by generations of textbook writers and officials charged with the education of our children. Yet Native Americans have practiced it for millennia, as have oth er cultures around the world.
This site is especially designed for 4th-9th graders, their teachers, and their parents. The ages between eight and fourteen have proven a critical age range where students who become turned off to math and science are unlikely to rekindle that interest for the duration of their educational careers. The materials presented here will not only maintain or rekindle that interest, it will rekindle that interest in teachers as well.
The site is also designed to counter the threat currently posed to art programs throughout the country. Everywhere it seems that art programs are under fire, through budget cuts or complete cessation by the educational "powers-that-be." It is deeply hope d that the materials presented here may serve to breathe life into the art programs of America through the wisdom of its original inhabitants. When you work with the materials furnished here, you will find evidence that we, as a culture, have been surely misguided in our belief, our worldview, that science and art are necessarily opposed.
While arguments of the values of subjectivity versus objectivity often run rampant at budget meetings, this trend can be reversed when we realize the importance of spatial reasoning as a fundamental intelligence. Spatial reasoning is an intelligence that effortlessly blends, weaves, decorates and measures geometric constructs once students learn the basic connect-the-dots knowledge that governs this system. Put in other words:
When you do the art, math happens!
Consider this site a free on-line workbook or workshop. Try it out on yourself, your children, your students. Sample tests are provided but the material is provided in the hopes that it will inspire the creativity to draw up your own. When you do, I would like to see what you come up with. Further, the geometry/math materials should be taught simultaneously with the exercises in the Designs section. Take seriously the hands-on approach, because you will find that the common denominator between the math and the art is precision. Once students know that the shapes they are making are perfect in theory, they will strive to make them perfect on paper. This is not a tall order to ask of them; within an hour, 2nd graders were making 60 degree triangles and hexagons within a degree or two of accuracy.
Once familiar with the fundamentals, there's nothing to prevent taking the knowledge outdoors to landscape your yard and school grounds. Not only will you beautify the area, you will have built a mathematical resource. If you have two sticks and a rope you can duplicate everything you do with a compass and straightedge. Students can design their ideas in class, testing their accuracy with protractors. Outside, when the constructions are built on a much larger scale, you will find yourself using square root proportions to ensure the precision of the design.
If you can make a circle, draw lines and connect dots, you can learn and teach a great deal about square roots, proportional constants, and irrational numbers. Instead of introducing these fearsome concepts as numerical abstractions, square roots are built into the shapes that you are constructing on paper during art class. To accomplish many of the designs, students will be working with square roots and other proportional constants without fear. Because the shapes are introduced as artistic exercises, without any kind of mathematical prerequisites, all mathematical content is optional and dispensed as you see fit.
The exercises that follow have been field tested on students from the 2nd Grade through 9th Grade, ranging from G.A.T.E. students to those with A.D.D. requiring daily medication. Middle school L.D. students who could not read a word, picked it up in minutes, proving to themselves they could indeed learn. The friend who introduced me to this tradition in graduate school is successfully applying the tradition to the architectural analyses of an ancient civilization, and learning the traditions first hand from the living surviviors of that civilization. Soon, he will complete his doctoral dissertation. He suffers from dyslexia.
One of the hopes that inspired this work was the chance that maybe some LD children can be saved from a life of delinquency if they find something in the classroom they can do well. Self esteem must be experienced in the classroom. If not, it will be sought in the streets or in some other negative environment. The art work gained through field trials attest to the creativity and intelligence that has been otherwise locked away, bypassed, or neglected by traditional channels of education. If you are a teacher of LD children, or if you are LD yourself, I hope you will at least attempt the exercises. If you are dyslexic, find comfort in the fact that an upside-down, backwards square is still a square.
Multi-cultural programs should also benefit from this approach to ancient geometric traditions and wisdom. Multi-cultural education programs were originally designed to promote a dignified intercultural appreciation and respect for cultural and ethnic differences that exist in America's schools. In California alone, there are over 80 different languages represented among students enrolled in English as a Second Language (ESL) programs. Over a dozen languages can be present among the student body of a single urban classroom. The brief program that is provided here is guaranteed to unify the minds in these classrooms because the logic transcends linguistic and other cultural obstacles: all lessons are passed on by visual demonstration and sustained by a self-evident logic that unfolds through the creativity of simple hands-on activities.
Over the years, particularistic multi-cultural education programs have come under fire for accentuating the differences between cultures and ethnic groups, and thus fueling the felt antagonism and separation between those groups. Though there are calls for more pluralistic multi-cultural programs, few if any have come to light. By learning the rudiments of the geometry and taking it upon yourself to examine the references provided, pluralistic approaches to multi-cultural programs will be at your fingertips. In a single lesson, you will find symbols and iconographic systems that can unite Native American, Asian, African, Mexican, Semitic and European students. As a advocate of the metaphor that American society represents a salad bowl (rather than a melting pot), the geometry program offered here could well be considered salad dressing.
The non-random geometry of Native America is both a tradition and a science. Its uses and forms are consistent with the nature of the geometry outlined in Kappraff's, Connections: The Geometric Bridge Between Art And Science (1991). This type of geometry falls under the subject discipline of Design Science.
E-mail: Chris Hardaker
Copyright 1994-2000, Chris Hardaker