The counting of numbers builds into arithmetic and arithmetic grows into math. Arithmetic is the simple use of numbers and mathematics is the systematic study of all the ways to use those numbers.

The beauty is in the repeating patterns and the ways they harmonize. The fitting together of all the parts to make a whole understanding is a wonderful thing to behold.

The hard thing to understand is one plus one. That concept can not be proven. Once it is accepted that one plus one equals two, the rest is easy.

Think about it. It's a lot easier to build the walls of a house than it is to lay the foundation and keep it plumb and square and anchored securely so it won't fall apart.

The foundational understanding always requires the best teacher, the kind who gives understanding of what's happening, not just procedure. Any monkey can memorize a trained procedure. It takes a wise man to understand what he's doing.

Such things are the basis, the foundation, of mathematics. The further growth of knowledge and understanding is gained from studying these beginning ideas.

It is good to be aware that knowledge is one thing, and understanding is quite another. It takes more to understand a thing than just to recognize it.

7.      after connecting many dots together so that the first connects with the last,
         then we can eventually arrive at a circle.
        This is considered the most complex of the foundational shapes.
        Why?
       Because it is another scale representation of a dot, composed of many dots.
        It has no apparent beginning and no end, and no direction.

The formula used with the foundational shapes are repeated in various complexities at the highest levels of engineering and science of all kinds. Perhaps the most impressive to me was shown by Dr. Bassicus (spelling?)at a Texas A&M physics class. He showed the growth in complexity of the use of the formula for the radius of a circle. From the simple circle through layers of calculus and into orbiting satellite calculations.

The person who thoroughly familiarized himself with those formula at the foundational levels would be able to more easily understand their applications at the higher levels. Those who only memorized external ideas were having to just fake it when it comes to understanding the higher complexities of use. The beauty and the joy is in the understanding. Understand what you're doing.

There are only a few foundational formula to learn from the foundational shapes. Knowing them well and understanding why they are as they are can give you great rewards in later things. If you take the time to notice, unlike many, you can see the later, more complex formula are combinations of what you've already learned, expressed in other forms. Some new things may be added as you go along the way, but recognizing how those new things are built upon the old is a tremendous help. Do not under-estimate this.

How shall the untrained mind make the correct choice when faced with a decision making situation? Mathematics helps greatly to teach and train the way you think. Cause and Effect, Continuity and Dis-continuity are important in all affairs of your life and the way you will live. Math helps. Written by Larry A. Rice.
The basics of this were recorded on paper by me in 1982 for my own children. During their early years I tried continually to show them shapes, patterns, cause and effect. To ask them questions in such a way as to train them to seek cause, effect, continuity, dis-continuity, and patterns and repeating patterns, and the growth of patterns. I believe this is good for all parents to do for their children.

I showed them rolling balls, ie, dots, and levers and made sure they understood how levers worked (did this as toys and games and play things) and triangles and squares. Once even took them out in the yard with a twelve foot piece of plastic pipe, stood holding one end and told them to keep up with me as I turned. They got tired before I did, obviously, and then let me explain why. I think that was one of the last physical play teaching lessons I was ever able to give my children, but I'm sure they will remember it when they're grown and raising their children. Angular momentum and many such concepts can be explained with simple toys while the child is young and quick to grasp huge simple concepts. Yes, the simple concepts are the biggest, such as that of a dot and a circle.

        A complex skyscraper building is nothing more than a compilation of many simple parts. We often get it backwards, thinking the big things are the more important, when actually the big things are arrangements of many small things. Which is easiest to understand, a mountain or a handful of dirt and gravel? The mountain is a million times accumulation of the dirt and gravel, but what is the dirt and gravel? Do you see what I'm saying?

Despise not the day of small things, for the mightiest ocean is held in its place by an accumulation of tiny grains of sand.

Isaiah 41:20... that you might SEE, and KNOW, and CONSIDER, and UNDERSTAND together..... the steps of wisdom. see the engine, know it, it's parts and environment, consider how it all works together, take it apart, analyze it till you can take it apart and put it back together with understanding why and how it works, .... then you understand the engine.

Daniel 9:23

Psalm 111:10 a good understanding have all they that do His commandments
This can be used to say the child will understand why the father told him to do a thing,... after he does it.

Zechariah 4:10 for who has despised the day of small things?
... is used to say, do not despise small things... remember the ocean contained by grains of sand.

Genesis 1:11 seed thoughts, repeating patterns, growth, multiplication, increase, fruitfulness, productivity

Psalm 119:27

Proverb 14:8 It is the wisdom of the prudent to understand his way...

Isaiah 56:11

Isaiah 40:22