ࡱ> _a^ @?bjbj >T؝؝ 5 8>$b,0|:0000000$1RP4'0 '04<000,y. G U}U-/LR000i-$5$5(y.y.$5.(hxJ<'0'0$ X WHOLEMOVEMENT APPROACH TO CREATIVITY IN MATH EDUCATION BRADFORD HANSEN-SMITH Abstract: Creativity and mathematics education will be addressed as parts within the contextual Wholeness of the circle. Creative activity connects us to spiritual potential of greater vision, presently missing in math education. Through questioning and looking closer at the broader implications of what we already know will often generate connections to new information. Folding circles reveals practical and experiential demonstration about the creative process, offering a principled model towards facilitating progressive development in mathematics education. Key words: whole, creativity, education, mathematics, process, circles, principle, play INTRODUCTION Possibly a practical way to support creativity in mathematics would be to think about mathematics as an art. This means to fully engage young students by exploring principles and playing with the interactive functions of mathematics; expanding boundaries allowing for a greater variety of perceptual input. Life is creatively purposeful as a principled direct expression of consistent interaction, relative to conditional movements of change organized to structural pattern. Playing allows our creative, curious, and courageous nature access to deeper connections towards higher ideals. There is much serious effort invested in mathematics that inhibits the enjoyment of just playing with it; particularly on the educational level. We get stuck in old habits. Through creative effort we find beauty reflected in revealed truths that are embodied in the structural association of physical, mental and spiritual unity. The proof is in the ideal, not the idea; it is not in the part, but the Whole. To question why we teach mathematics will help define how to educate teachers to help students learn for themselves. What do we want to pass on to the next generation? What mathematical tools and thinking will uplift the human experience towards what ideals? Where is the greater social meaning that supports increasing individual value in mathematics? What is the measure of mathematical ethics and the moral implications? These are a few questions requiring thoughtful consideration about the meaning and value of mathematics to the individual within the social progression of human development towards higher good. Creativity in mathematics education is not a math problem; it is broad-spectrum, individual human development problem. As social animals we fear what is difference, yet acknowledgement for individual differences is essential. Security holds back progress and the creative necessity to move out into unknown potential. To create is to reside in what is uniquely individual; groups of like mind, making connections and giving expression to what nobody else sees, contributing meaning towards greater value. Mathematics is a rigorous discipline with a defined knowledge base. It is not an open and fluid means of easily expressing new ideas. It is a closed, fragmented, largely an outmoded system of logic that does not reflect our current experience. A new worldview will eventually unify mathematics with all human activities. Learning is a creative activity; it creates new pathways in the brain, new habits of responding. Responsibility is inherent to all creative insight in the appropriate use of knowledge gained, often ignored in the excitement of something new. Without individual and collective alignment to the Whole, creativity has little benefit. It is no longer acceptable to teach only the tools of mathematics without the responsibility of a greater good in using mathematics. If math education is not about progressive values and growth, then it is simply job training. The creative spirit Creativity is a mind and spirit coordinated function. The spirit is limited by conditions of fear, anger, prejudice, hate, jealousy, pride, and other attitudes intertwined with how we teach and use mathematics. Creative imagination requires intellectual co-operation with spiritual influence; the ability for the mind to work with information outside of the normal range of human experience, expectations, and responses. This larger context provides stability to individual insight, purposeful exploration and the enjoyment of interactive playing towards discover. Creation reveals what previously did not exist giving benefit beyond the individual. Whether by slow evolution or spontaneous insight, the generation of new forms of anything is not controlled by human manipulation or design alone; it requires co-operation with the unknown; faith in the presents of what is greater than that which is created. The direction of spirit is beyond human understanding; not always a conscious activity, but it does require thoughtful means of expression. Creativity requires insight and stimulation of intuition that is beyond the limitations of common mind functions; it is an association of the human mind and spirit, giving to unique physical expression. To enlarge and uplift without violating what is; acknowledging revealed and unrevealed parts to the Whole brings up many questions surrounding mathematics education about the higher values of human potential. Direction is often found in discovering a constant that gives balance, reveals symmetry, and reconciles differences between variables. The most stable constant is the unrevealed Whole for which there is no accounting and is un-measurable by any human standards. We must learn to see what we do not recognize, to look into the visible for the invisible, to find the infinite within the finite. Creativeness occurs on all levels of practical and abstract functioning when parameters are open giving room to play, time to observe, to ponder, to think about why mathematics, what of it is essential, to what delight, and to what supreme practical value does it serve? A creative mind is one that can become fully engaged in what is unknown about what we already know; a mind with curiosity, excitement to explore, stimulated by discovery of imagination, and the need to question. These qualities that naturally drive individual capacity to learn have been suppressed by cultural and educational limitations that lack spiritual clarity and moral responsibility. WHOLEMOVEMENT Looking to the inclusive nature of Wholeness requires observation of principled interaction beyond the generalizations of mathematical functions. It is necessary to look for the extraordinary in the ordinary, where greater meaning is revealed in relationships between parts to Whole. The example I give is Wholemovement. There is no congruency between the word geometry and our present world experience. We have been off the planet and are now measuring universes we didnt know existed ten years ago. Still, we hold to geometry meaning earth measure; measuring things of the earth. Geo means earth, which is spherical. The sphere is an undifferentiated Whole. Metry means measure; measuring is movement. Wholemovement is an updated and comprehensive understanding of the word geometry; a self-referencing and self-distributive, differentiating movement of the Whole that provides the largest context possible for understanding the universe. What practical application does Wholemovement have to math education or creativeness? We now have to consider the Whole, everything. Traditionally unity of the sphere is destroyed through a truncation process. Parts are represented by symbols showing generalizations about geometry and mathematical functions. Without cutting, through compression, the sphere is reformed to a circle disc; nothing is lost, unity remains intact, only the form changes. Spherical reformation to circle displays axial orientation and a circular band defining two circle planes. Compressing sphere-to-circle is a right angle, tri-unified, transformational function. Using a paper plate circle and curving the plane, touching any two opposite points on the edge will form a smaller circle opening. Using end points of the diameter, the circle surface defines a cylinder pattern where the lesser open circle is perpendicular to the surface. By rolling the connecting point on the circumference away from the diameter line the cylinder moves to a cone in one of two possible directions; positive and negative if you will. The difference between cylinder and cone is in surface movement in or out of parallel. The cylinder is a special case cone reconfiguration of the self-referencing circle/sphere Whole. Any two circumference points touching, when flattened will crease the circle in half. This diameter hinge allows reflective movement forming a spherical pattern of movement in space. Assuming all spatial information is held within a compressed circle/sphere Whole, by folding the circle we are then decompressing that information without destroying unity. Two touching points and resulting end points of the folded diameter form minimum description of a tetrahedron; four points in space. The circle reveals in one fold spherical origin and structural pattern in polyhedral expression; where it comes from and where it is going. We can observe well over fifty geometry and mathematical functions that can be described within the relationships formed by this first self-referencing movement of the circle. Being unfamiliar with the nature of the circle we do not know this. The seven primary qualities observed in this first fold are principle to all subsequent folding, transformations and information generation. There are only three options to continue a 1:2 proportional folding; 3, 4, and 5 diameters. Counting and set functions are important to identify pattern development of relationships and properties of various circle interactions. It is from this information that we find the directives and options for continuing the folding process. Through sequential folding nine creases are generated that form a solid tetrahedron. All five Platonic Solids can be formed from multiple circles folded to this same pattern. This expands our understanding about regular and semi-regular polyhedra. They are all individualized interrelated expressions of structurally formed circle/sphere unity. Each is the same folded pattern reconfigured differently through a principled process of touching points and creasing. The circle experientially reveals functions that have been conceptualized and abstracted to symbols, as well as revealing forms, relationships, and transformations not part of our traditional knowledge. This first fold barely indicates the amount of information and the variety of reformations possible by continuing this sequential folding and reforming process. One cannot imagine by looking at the drawing of a circle, a symbol for zero, for nothing, that anything is there. Any child that can fold a circle in half can do this and through their own experience, observations, and understanding discover what would otherwise be out of reach,. Wholemovement is one example of looking deeper into what is commonplace and questioning the excessive importance given to symbols over direct observational experience. CONCLUSIONS The concept of Wholemovement brings into question fragmented information and how we think about what we teach, and what is not taught. Learning is an individual and collective creative interaction using body, mind, and spirit through small acts of imagination, working through experiential understanding in an ethically responsible manner. Movement of the Whole, revealing within, suggests a greater purpose for which current mathematics might possibly serve. In math as in art, creativity looks towards making unity of parts often perceived as separated without connection. There is no formula for creativity. The mind makes observations, connecting to broader realms beyond the usually confinement of discipline and expected outcome. Creativeness is seeing and giving outward expression to inner revelations about things that have meaning. Within circle/sphere unity of Wholemovement endless parts are revealed in full context, proportional to all other parts formed and unformed. What the circle allows is connections and recognition of the beauty of elegance, economy of structural form, interrelated and changing symmetries, all reflecting something of the truth about purposeful movement. When mathematics is only defined by logical abstractions of proofs, there is little room for imaginative questions about what is unformed, left out, unrealized. Eventually we must go back to origin with questions about where, why, what, and how we are, and for what purpose. These questions become critical to understandings truth that proportionally drives the need for progressive development in mathematics, and in all areas. Creativity is often equated with self-expression, which is natural to all people, rather than thinking that it might be more about meaningful play with intelligent observations and reflections about the comprehensive nature of what we play with. Creative will is often towards ethical and moral decisions about unique concepts and ideas having some practical value to the Whole. Mathematics becomes creative when imagination stimulates insight to quicken the mind, finding purposeful expression about specifically unique connections between parts and discovering unknowns within the greater context of what is ultimately, inclusively, Wholemovement. ABOUT THE AUTHOR Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com     Wholemovement Approach to Creativity in Math Education Bradford Hansen-Smith PAGE 60 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 1 PAGE 61 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 56 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 1 %+.:;QR[  ƻ҄|uqb|TLh?%HCJaJheKhx6CJ]aJheKhx5;CJ\aJhx heKhxheKhx5heKhxCJaJmH sH hg5hx6CJ]aJheKhx6CJaJheKhx56CJaJheKhxCJaJheKhx5CJ aJ heKhx5CJaJheKhxCJ$OJQJaJ$$heKhxCJ$OJQJaJ$mH sH ':;QR B;$d\$`a$gdx $d\$a$gdxd\$gdx $d\$a$gdx$d\$]^a$gdx $d\$a$gdxd\$gdx =?~ !`$&d)*.F1335:<$ vd\$^a$gdxd\$gdx$d\$^a$gdx$d\$^a$gdx $d\$a$gdx $d\$a$gdx$dd\$^a$gdx <#A#g#o#q#~#33<<<<<<<< = = = =======M=߹vmeaeaeaeaRhg5hxCJOJQJaJhCjhCUhxhma aJ#h %hx5CJOJQJ^JaJ%hx5CJOJQJ^JaJmH sH #heKhx5CJOJQJ^JaJh`khx5CJaJh %hx5CJaJheKhx6CJ]aJheKhx>*CJaJheKhxCJaJheKhx6OJQJ^JhxCJaJ<<<<<<< = = = =======M=O=P=f=g=h=$a$gd1F]$a$gd1F] $xa$gdx$a$$a$gdxgdxgdx $d\$a$gdxM=O=P=e=f=g=h=i=o=p=r=s=t=x=z=============obTEh;hQCJOJQJaJhQ0JCJOJQJaJhd)p6CJOJQJaJ'h;hQ6CJOJQJaJmHsHh;hQ6CJOJQJaJ%h?0JCJOJQJaJmHnHu hAoIhQ0JCJOJQJaJ)jhAoIhQ0JCJOJQJUaJhQ5mH sH hg5hx5CJOJQJaJhg5hxCJOJQJaJhQhQOJQJh=t======>Q>R>>>>>>{$ 9r  n!&#$+Dq]a$gd+y,&#$+D,gd+$ 9r  ]a$gdN$  ]a$gdN$a$gd+$',&#$+D,a$gd+$a$gdd)py,&#$+D,gd*$=Q>R>>>>>>>>>>>>>?????˺˧˺xdx`Whxhma aJhC'h;hQ6CJOJQJaJmHsHh;hQ6CJOJQJaJhd)p6CJOJQJaJ#h;hQ0J6CJOJQJaJ%h?0JCJOJQJaJmHnHu hAoIhQ0JCJOJQJaJ)jhAoIhQ0JCJOJQJUaJhQh-ChQ56CJaJh-ChQ56CJ aJ >????gdx$a$gd+< 00&P 1hP8:p?. 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