Alright, my head is about to explode. For the next two days I'll be sitting here in a room all alone trying to figure out how the heck we are supposed to bridge this gap. I have all the tools ready to go(computers, legal pads, pens, state framework,etc.). And all of a sudden it hits me. The rules of math have come about because they were necessary. For example, Natural numbers work until you try to subtract. Then you have to have integers. Integers are fine until you divide, which leads to Rational numbers. The Rational numbers break down when you try to find the side length of a square with an area of 15. Take the kids on this tour and we get to say: "Okay class, we have just discovered the Real Number system."
We have exponents and scientific notation because it get really tedious to multiply (5,000,000,000)(8,000,000) by hand. We introduce the symbols and variables because we don't want to have to work out every single case for every single situation. We generalize because mathematicians are inherently lazy. We truly find the shortest distance between two points. Kids are inherently lazy; they know the shortest distance between homework and their XBox. Hey, we have someting in common. How do we expolit that commonality in order to have kids "discover" algebra for themselves? How do we scaffold our entire curriculum, so that kids move from the Natural Numbers to Projectile Motion in such a way that they actually see how there is a need for it?
I realize that I am probably not saying anything you all haven't already discovered for yourselves. But, before I go and reinvent this wheel, I would like to know what you have all been doing to get this Algebra bus rollin'.