Last night I woke in the middle of the night dreaming that I had a classroom full of unruly students. It's been a while since I've had one of those dreams. I use to have them all the time. Part of it is probably because of feelings of inadequacy at work combined with running into one of my former students.

When I woke up I was thinking about how to teach Algebra. I wanted to relate the algebraic thinking to the world of a student. I know that none of them need "algebra" as math at that age, so I asked myself what algebraic thinking is outside of math. We say that it relates to everything and yet there are thousands of people who say that they don't know "algebra" and can't do "algebra". How are these people getting by without it if it is so important?

Simply put, they aren't. Sure, they don't have to manipulate equations with x's and y's. Instead they have to make substitutions where they set up scenarios in such a way that anyone could fill the role. e.g. Who could play a certain positon if so and so were injured? What would a dream team look like. What would it be like if so and so was on the date with so and so instead of so and so? They make mappings in their head of relations. e.g. Bret -> Quarter Back, Joe -> Fullback, etc. Susan -> Jack, April -> Manuel, Eric -> Jason...

One way to introduce the concept of "relations" (mappings) is to have the students find things that are related and display those things in some sort of form. You are almost guarenteed to get tables and maps. Then you can introduce a grid and represent the relation as a point in the grid. (can you say cartisian coordinates?) Finally, start mapping things in large numbers. You can use a computer and excel. You can relate every student in the school to a "zipcode" or a phone number, or the number of phone numbers in the phone, or any other collection of information about them. Then expand that to the city.

Once you start dealing with things that are too big to fit into a graph or a table or even a database, you would introduce sets of ordered pairs (domain and range) that have an infinite domain or an infinite range and show that only a piece of it is representable on a graph or a table or a map at a time. Then introduce the concept of a symbolic formula that represents the entire relation. Number of molecules in a volume is a constant times the volume.

Stick with writing out the descriptions for a while, then introduce the concept of using letters to stand for the quantites. Eventually, you will have to manipulate some of the formule and you will be able to introduce the letter x and y to stand for any quantities (abstract) and talk about generic manipulation of symbolic equations.

That...is how algebraic thinking leads to the need to learn "algebra".