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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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A pattern comes about because of some repeating characteristic, such as color, shape, size, etc. Different types of repetitions generate different patterns regardless of what is being repeated.

A relation is an existing connection; or a significant association between or among things. Because of a relation, if something is present then we may expect another thing to be present as well. A pattern is made up of relations. To see a pattern is to have a sense of relations among things.

A number line is a pattern that presents relations among numbers. With the help of this pattern we may locate a missing or hidden number.

A symbol may be used to represent a missing or hidden number in a pattern. The position of that symbol in that relationship may then help discover that number. Here are some exercises in this subject for the kindergarten level.

LEVEL K6: PATTERNS & RELATIONAL SENSE

“Patterns and Relational Sense” forms the foundation of the subject of ALGEBRA. It is a study of patterns underlying numbers, and quantitative relationships.

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To measure a distance would be to determine how close or far a location is, such as, the distance of school from home. To measure a size would be to determine how small or big something is, such as, the size of a house. To measure capacity would be to determine how much something can hold, such as, the capacity of a pool. In short, to measure is to determine the extent, size, capacity, etc. of something.

To measure anything we need a unit. For example, to measure the length of a table, we may use span as a unit. To measure the size of the floor we may use square tile as a unit. To measure the capacity of a tub, we may use bucket as a unit. In short, the unit used in measurement is a small but definite amount of what is being measured.

Suppose we want to know how long a football field is. We may walk the length of the field and count how many steps it took. This will give us the number of steps that fit in that length. We can then say that the measure of the field is so many steps. Here a small length called ‘a step’ is the unit used to measure the length of the field. 

If we want to compare two lengths then we must measure them using the same unit, and the unit must be of the same size each time it is applied. To be able to compare measurements around the world, we must use the same unit the same way anywhere in the world. Thus, we have standard units of meter for length, kilogram for weight, hour for time, etc. Here are some exercises in measurement for the kindergarten level.

LEVEL K4: COUNTING & MEASUREMENT  

The measurements make the numbers meaningful. It is important to get the sense of numbers in terms of units by measuring as many things as you can, using as many different types of units as you can.

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