Tuesday, June 15, 2010

Mathematics ~ From Finger Counting to Fibonacci Sequence

From finger counting to Fibonacci sequence one lives the delight of awareness, exploration and discovery of mathematics, the art and science of space, form, structure and time.

When asked by a student why do I have to know math, the response is drawn from nature's ever present quest of balance, order, and harmony. For the younger child, fingers, toes, eyes, ears, hands, feet, legs, arms, knees, elbows, knuckles, fingernails, toenails, groups of two, groups of five, groups of ten become the basis of understanding the 'words' and 'symbols' that 'count' and 'measure' time, space, form and number. From this body awareness comes the exploration of these numbers in the child's immediate environment. Moving from body, to immediate environment, to the sense of self in time and space, provide experiences upon which abstract concepts, symbols, are eventually and appropriately introduced and integrated.

The art of calculation lies in understanding pattern, sequence, and constancy. It is not surprising that with what appears to be little instruction and or 'drill' a child readily learns counting by twos, fives, and tens. The identification of pattern, the meaning of an 'odd' number, the meaning of an 'even' number, how to 'double' are prerequisite to mastery of addition, subtraction, multiplication and division facts. It has been my experience that those children who have difficulty in mastering the basic addition, subtraction, multiplication, division facts, need to be given the opportunity of discovering particular patterns through materials that can be handled, arranged, changed. Using what is 'real' in the counting world such as actual coins have brought understanding and mastery of all basic math facts to scores of children I have worked with since the founding of APOGEE Learning Enhancement Training Systems™ in 1983.

Children are natural explorers and delight in discovery. Allowing such exploration and discovery through observing 'patterns,' 'order,' and 'balance' in nature, provide the most natural introduction and understanding of 'number' and mathematics. The exploration of the Fibonacci sequence of numbers and its relationship to patterns, allows the growing child to observe and experience first hand, the role of pattern in living forms, structure and music. APOGEE sessions provide opportunities to observe the certainty of existence. Such exploration and discovery provide motivation and a confident reference for further learning.

Leonardo Fibonacci of Pisa was a mathematician in 13th century Italy. The first written record of the Fibonacci Sequence was in the book Liber Abaci, written by Leonardo of Pisa and published in 1202. The Fibonacci Sequence is a series of numbers where each consecutive number is equal to the total of the previous two. It begins with the numbers 0, 1. When these numbers are added the answer is again 1, so the sequence becomes 0, 1, 1. This process is repeated and the sequence becomes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on.

Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition Fibonacci numbers or golden sections in nature are found relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees. Connections have been found between the Fibonacci numbers and aspects of engineering, architecture, painting and music. They occur regularly within nature from the petals on a flower to the spirals of a pine cone. Trace back a honeybee’s family tree and each generation will represent a Fibonacci number. You can even see what is known as a Fibonacci spiral in the formation of galaxies.

The florets within the sunflower's cluster are arranged in a spiral pattern. Typically each floret is oriented toward the next by approximately the golden angle, 137.5°, producing a pattern of interconnecting spirals where the number of left spirals and the number of right spirals are successive Fibonacci numbers. Typically, there are 34 spirals in one direction and 55 in the other; on a very large sunflower there could be 89 in one direction and 144 in the other. This pattern produces the most efficient packing of seeds within the flower head.

The Fibonacci Sequence allows students to experience the seemingly magical properties of numbers. By applying simple equations to this number sequence students uncover unexpected connections. Opportunities to predict answers and to locate patterns within and between sequences of numbers, to observe these patterns and sequences in plant life, sea life, in galaxies, in one's own body, is to bring meaning and life to the art and science of mathematics.

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  1. Excellent post! I envy the student that gets to study in your environment!

    Joel Morrison

  2. Joel, The "Interface-Philosophy" you present in 'Spinbitz" is very much a part of each APOGEE session. Each discussion in your publication is valued, applauded and applied. Thank you for being here.