# Arithmetic described

Basic arithmetic is critical in most aspects of life, and algebra has nearly the same usefulness.

## What school was supposed to teach

As early as kindergarten education, we start math with what we observe:

1. Descriptions and demarcations for the things we observe, which become the beginnings of [geometry](math-geotrig.md).
2. Groupings of similar things we observe, which become the basics of arithmetic through identifying quantities (e.g., 3 apples).
3. Incrementing more of those quantities as an interval of 1 (e.g., 1, 2, 3, 4, 5, 6...).
4. Using [language](language.md) to define the concepts as expressions (e.g., 5 + 1 to declare 5 things and 1 thing).
5. Addition by combining those groups together (e.g., 2 + 3).
6. Subtraction by removing groups from other groups (e.g., 7 - 2).
7. Composing numbers in different ways to come to the same answer (e.g., [fact families](https://www.splashlearn.com/math-vocabulary/number-sense/fact-family)).
8. Exploring applied mathematics by giving [story-based](stories.md) problems.
9. Using a base-10 number system to familiarize everyone with it as a mathematical convention.

From there, most of the effort is applied toward repetition and scaling upward:

- Counting increasingly higher values (i.e., 100, then 1,000).
- Adding and subtracting more proficiently and with higher numbers.
- Giving more story problems to sufficiently expand on how it can be useful.
- Further applied mathematics with things like time and money.

A bit later, we expand on the ideas:

1. Defining the concept of equality using equal groups of various types (e.g., 3 boxes with 2 things each in them).
2. Learning about commutative properties within mathematics (e.g., 4 + 3 = 3 + 4).
3. Introducing multiplication as compounding addition (e.g., 5 + 5 + 5 = 5 x 3 = 15).
4. Reversing the grouping of multiplication to create division (e.g., 15 / 3 is 3 groups, which are 5 each).
5. Demonstrating fractions as partial integers that represent unresolved division (e.g., 1/3).
6. Partial base-12 and base-60 calculations with time.
7. Factoring values into smaller or larger components, which form the beginnings of algebra.
8. The standard addition and subtraction algorithm:
       8 5 4 3
     - 1 3 2 2
       _______
       7 2 2 1
9. Indicating place value relationship (e.g., 0.001 is 10 times more than 0.0001).
10. Comparing, ordering, and rounding numbers (e.g., 513 rounded to the nearest 100 is 500).
11. Multiplicative comparison of values, which is the beginnings of mathematical analysis (e.g., A is twice as large as B, 1,000 is 100 times more than 10).
12. Multiplying and dividing multi-digit numbers by factoring the numbers.

Around 4th and 5th grade, there's a bit of a divergence into partial numbers:

1. Indicating how fractions can be equivalent (e.g., 1/3 = 2/4).
2. Showing how fractions can decompose (e.g., 4/3 can become 1 1/3)
3. Adding and subtracting fractions with the same denominator.
4. Multiplying fractions together (e.g., 4 x 2/3 = 4 sets of 2/3 combined = 8/3 or 2 2/3).
5. Expressing decimals to the tenth and hundredth (e.g., 0.1, 0.001).
6. Indicating how fractions are both quotients and whole numbers.
7. Multiplying fractions together.
8. Adding, subtracting, multiplying, and dividing decimals.
9. Adding and subtracting fractions with different denominators by factoring to a common denominator (e.g., 1/2 + 1/3 = 3/6 + 2/6 = 5/6).
10. Understanding the concept of scale with fractions relative to each other.

At this point, many people hit a mental wall, usually because they have a hard time imagining the point of representing 8/4 and 2 at the same time, or why 1/2 and 0.5 are both valid numbers. Most mathematicians are *not* aware that math represents a [perspective](image.md) upon reality, and their obsession with certainty and abstraction blinds them from the fact that math itself is expressing different points of view. Since this all heavily compounds, algebra becomes impossible to explain later.

However, whether we understand it or not, the cumulative education continues:

1. Introducing exponents and square roots.
2. Introducing the concept of ratios (e.g., 1:2).
3. Adding to fractions with percentages.
4. Dividing fractions and decimals, which effectively multiplies the numbers.