THE 100,000 CRANES
What is X? Why is Y? is a transition book, straddling Class 7 primary and Class 8 high school. As such, it seems reasonable to review the number path traveled by children to this point. The following is a brief exposition on Geometry – “Earth Measure”.
A 7-year-old girl in Hiroshima was suffering the agonies of terminal cancer, the result of the greatest single act of evil in human history, the dropping of the atomic bomb on August 8, 1945.
Little Skaura (or whatever her name) heard that child sufferers, if they fashioned 100,000 paper cranes – symbol of immortality – would be healed. Even though making thousands of the beautiful birds, sadly she died short of the amount. Today millions of paper cranes, from tiny to tall, festoon the parks and gardens of the City of Peace. Sakura (“Cherry Blossom”) was not only beseeching her gods, she was learning Geometry, though of a limited kind.
Paper Cutting and folding is one of the first geometric experiences of Steiner-educated children – in their 7thyear indeed. Here they create more than just cranes; the whole horizon of Origami is scanned for imaginative and stimulating ideas. Even practical paper cutting and folding designs, like aesthetically pleasing serviette folding, arouses the creative juices of both teacher and taught. Then there are fans, flexagons and frost crystals. This practicality is in accord with Steiner’s ideas on maths teaching: “We should develop the children’s thinking by means of external things which they can see; keeping them as far away as possible from abstract ideas.”
It is also in accord with the meaning of geometry, “earth measure”. Geo is an anagram of ego, or self-consciousness. There are few better know-thyself activities than geometry, which Steiner Education exploits fully in a peerless path of ego-awakening; comprehensive, cohesive, consecutive and coherent.
In Form-Dynamic Drawing. 7-year-olds are, in their formal learning needs, like opening flowers; or to use a less embroidered term, they have a radially symmetrical consciousness. This omni-directionality is informed by the children experiencing the dynamics of expansion and contraction through their drawings, whether of dandelions, starfish, fire wheels, or invented forms.
This transforms a year later to one of bi-lateral symmetry, such as we see in the forms of a head of wheat, or a centipede. It must be emphasized that these children’s drawings, large enough to embrace whole-arm movement, are not realistic; capture being but one of serval inspiration sources for what are essentially geometric, though not strictly formalized, patterns. As they are done in colored chalks or crayons, there is also a strong artistic component. Some might even be thought of as glorified doodling!
On linear exercise helps the 8-year-old incarnate an elegant writing style, by drawing strings of single letters; joined repetition of the work ‘eel’ (eeleeleeleel) perhaps? There is some dispute about whether much of form-dynamic drawing is Language or Maths. Happily the boundaries are elusively blurred! Traditional designs are also an ideas treasury, such as those lovely Celtic waves and volutes decorating jewelry and gravestone.
The next year, Class 3, the child’s soul advances to a-symmetry, where drawings such as mirror images are perennially popular. Much Aboriginal art is highly patterned and stylized but a symmetric. Another favorite is those designs of, say, a stairway, where, if you look at it with one focus it is going up, with another descending – these drawings were elevated to art form by M.C. Escher.
The equivalent unit in Class 4 is Freehand Geometric Drawing. This more formalized skill meets the child’s developing cognitive needs, which demand that the number principles in star, circle or square designs are self-evident. The beautiful, new (now colored pencil) renditions are also far more disciplined than in earlier years. One fun exercise is to design a set of interlocking tiles or tables, based on repeating geometric elements.
11-year-olds are further focusing, with a need for precision only geometric instruments can provide. How the children eagerly receive the presentation of their shiny new ruler, compass, set squares and protractor! With these, the Class 5s manifest quite wonderful creations; one example being the seemingly limitless configurations of stars.
The first could be based on the ever-popular 6-pointed Star of David. This is easy to construct, based on the enclosing circle being six times the radius. Harder though is the pentagram. One can make illuminating discoveries in this star creation by drawing a (nice large) circle; mark off 12 points, as in a clockface, then proceed with the “miss number” method.
What kind of star do you have if, starting at 12 you draw a line and miss, say, two numbers (1 and 2) to land on 3, then 6, 9, and 12? A 4-pointed star, or diamond! Miss 4, and we have a dazzling 12-pointed star! What about a circle with seven points? How many kinds of 7-element forms (gons and grams) can you find with the “miss number” method here? Discovery is the essence of this star-studded voyage! Again, the artistic rendering of these number-perfect drawings – or creating them as string patterns etc. – can be very beautiful.
A great favorite – if artistically presented – is the Theorem of Pythagoras. This leads to the construction of brightly-colored logarithmic spirals, based on the progression of square numbers, square roots, golden section rectangles, or right-angle triangles.
With 12-year-olds, we enter the cryptic world of crystals – well, solid geometry actually. Crystals just happen to be ideal expressions of the Five Platonic Solids and other myriad configurations. (Class 6 also enjoy their – not so-incidentally! – first formal Geology main lesson this year). Why cryptic?
Consider just one miracle of numerical symmetry, the S – E + P = 2 equation. When S is surface, E edges and P points, the answer for tetrahedron, icosahedron, hexahedron and even the spiritually symbolic dodecahedron, is always 2. An example is the hexahedron (cube): S – E + P = 2 … 6 – 12 + 8 = 2.
Again a popular element of this Class 6 unit is the artistic, where children actually construct the solids. This can be in all kinds of materials and structures, from simple cardboard templates to large geodesic domes of timber slats and stretched fabric. These glorious creations can be either just decorative, or practical as well. The 20-surfaced icosahedron is the favored form of disco mirror balls, a small dodecahedron can be a handy 12-surface desk calendar.
Here’s another example of Steiner’s practical over abstract imperative: Why do they sell cool drinks in those cardboard tetrahedrons? Because they have the largest surface-to-volume ratio, permitting heat loss, as well as permitting cooling surface evaporation. What is the best container in which to retain the heat of liquids? The sphere, with its lowest S:V ration of all solids. This helps explain why spherical planets maintain their vulcanism for eons.
Now from Maths to Language: Who is the most published author in the history of literature? If you answered, “Beauty Quest”, you’re right. Actually the Greek translation is Euclid. This genius’s definitive work on geometric theorems headed the best-selling lists for over two millennia – both in the East and West. Snippets of this kind can impress 13-year-olds; as can their own “quest for beauty” of the world of perfection that is the geometric theorem.
In adult life, the likelihood of using almost any of this geometric knowledge is remote. So even though this 7-year form-dynamic odyssey has been highly enjoyable, some might see it as a waste of time. In Steiner education, the issue is not usefulness – who can predict what any individual will eventually draw from their education? Rather than goal is the awakening of each child’s geo-ego in all areas of human endeavor. This is the essence of the Universal Curriculum.
You can learn how to make paper window stars using this book here
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