20150914 THE X-BAG:

Tracking the invisible quantity.

Please allow me one more quote from Wikipedia. It's not supposed to be an authoritative source, but you know what Wikipedia really is.

The American Institutes for Research (AIR) reported in 2005 that the NCTM proposals "risk exposing students to unrealistically advanced mathematics content in the early grades."[14] This is in reference to NCTM's recommendation that algebraic concepts, such as understanding patterns and properties like commutativity (2+3=3+2), should be taught as early as first grade.

This is sad, but again, I am not in charge of "most students". TLG knows and uses commutativity since she learned addition, and she learned long addition before she turned five. Putting a bigger number first, always makes computation easier. Associativity? Adding something like 1+7+9, she applies 1 to 9. There is no positional arithmetic without algebra. And let me remind you what a regular minority kid could do with patterns.

TLG can use formulas. Absolutely. The notion that young kids can't learn such things is either ignorance or deliberate manipulation. Big kids, who underwent many years of brain stiffening treatment at school, do have troubles learning to use names and roles for numbers. Small kids, who learn naming things, don't.

When TLG's faint-hearted big sister was learning positional addition on shnumbers, I explained to her elementary algebra using letters for colors. R is the red quantity, B is the blue. It was perfectly fine. Several months later, my suggestion to use letters for numbers made her upset. I reminded her she did it with pegs. She recalled this, but now she knew that letters were pre-algebra, and pre-algebra was to be afraid of. I believe, that's how I've heard about pre-algebra for the first time.

In the beginning of the forth grade, she mentioned that the next year she would start pre-algebra and said she was scared of it. I assured her that she already knew this silly subject, and taught her recurrence relations.

While I was teaching TLG's big sister, the other kids, or, maybe, even the teachers (the art teacher, quite probably), were telling her the horror stories. That's how school controls the population.

Soon after TLG turned four, I took a plain paper bag and crossed it corner to corner with red marker. This device came to be known as the X-bag. Another one was the Y-bag. At first, I was stuffing the bags with counting bears, and TLG was keeping tracks of my additions. Then I started asking questions like: if I put 3 more bears in the bag, there will be 8 of them. What the mystery number X is equal to?

TLG quickly understood equations and started solving them. Guess, for the time being her method is good enough because she is still learning how addition and subtraction work together. By the way, she learned about negative numbers and zero from her mom before she turned 4, and her understanding is very solid. I tried to catch her many times.

Recently, I returned the X and Y bags on stage. TLG remembered them, and was happy to see them again. We moved some pegs back and forth carefully watching and tracking additions and subtractions as they occurred. Next we moved paper cards with big numbers.