Based on the history of mathematics, Ancient Babylon and Egypt are the two places that were at the center of the development of algebra. Both of these civilizations used algebra in different ways and for different reasons, but it's generally accepted that it was the Babylonians who first made basic use of algebra and pioneered its beginnings in the field of mathematics. All the theory and the method came about for its own particular reasons that were always done to solve a problem and make a solution easier to find. For example, the Babylonians used algebra to work out the area of items and the interest on loans, among other things. It had real use and purpose and this why it was developed. I think that is the reason why they have the general mathematical principle in a time before the development of algebra and algebraic notation invented.
In mathematics, we often use the generalization method to introduce students to the mathematical concept starting from something simple moving toward the more general underlying foundations. While abstraction in mathematics is from concrete real-world problems to abstract concepts, it is the process of extracting the underlying structures, patterns, or properties of a mathematical concept, removing any dependence on real-world objects with which it might originally have been connected. Both generalization and abstraction are important and need to be encouraged in mathematical learning. Calculus, Real Analysis, Linear Algebra, Topology are some examples of abstraction in mathematics. Other than generalization and abstraction, a big part of mathematics is the applications.
I think it is very difficult in number theory, geometry, or any of the areas in mathematics stating general and abstract relationships without using algebra. If I have to do it without using algebra, I will use substitution and trial and error methods.
Good work, Megan. You've put a lot of thought into the relationship among practical applications, generalization and abstraction. There is something very interesting about 'removing' the concrete objects -- even in something as basic as counting! Fascinating ideas to explore.
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