Here is the first milestone to be reached in the learning of math:
MATH MILESTONE #1: NUMBERS & PLACE VALUES
The purpose of Mathematics is help one learn to think and reason in a systematic manner. This starts with learning to think systematically with numbers. The first part of Mathematics is called Arithmetic. The word ARITHMETIC (arithmos number + techne skill) means, “Skill with numbers.”
Arithmetic helps us determine “how many” or “how much” of something. Therefore, it introduces the ideas of unit, digits, number and place values. The first action of Arithmetic is counting. The next action is to develop a system of writing numbers. Arithmetic builds upon the concept of place values not only to develop a remarkable system of writing numbers, but also to devise a number logic that helps solve problems quite simply. The simple “logic” of place values is expressed in the Rule of Regrouping. This rule may be demonstrated on an abacus as follows.
WHENEVER ALL THE TEN BEADS ARE TO THE RIGHT ON A WIRE, THEY ARE RETURNED TO THE LEFT AND REPLACED BY ONE BEAD TO THE RIGHT ON THE NEXT WIRE.
The place value system makes it possible to write large numbers in shorthand. It also simplifies computation. This was a great advance over the Roman numerals used earlier. It became possible with the discovery of zero. The place values in numbers are as follows.
Note the repeating pattern of “one, ten, hundred” above. The first group of “one, ten, hundred” is the Basic Group. Next, we have the “one, ten, hundred” group of Thousands. Beyond that we have “one, ten, hundred” groups of Millions, Billions, Trillions, Quadrillion, Quintillion, Sextillion, Septillion, Octillion, Nonillion, Decillion, etc.
To develop skill with numbers one may use fingers at first, and then move to the next step of abacus. The use of abacus helps one visually see the system of place values. The next level is mental math where one learns to think systematically with numbers, assisted by paper and pencil, and calculators.
I feel strongly that a student should first learn to do mental math before using calculator as an aid. Sole dependence on calculators and flash cards would prevent the student from developing the ability to think systematically. With the ability to think with numbers hampered, the student would not be able to learn math beyond the elementary level.
Today, we take this system for granted, but the brilliance of the concepts of zero, the digits, the Rule of Regrouping, and the place values is simply astounding when fully understood.
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