Algebra is: an art, a detached language, a way of thinking, a foundation of the sciences. With very little effort you can find other “definitions” of algebra. According to Keith Devlin “The important thing to realize is that doing algebra is a **way of thinking** and that it is a way of thinking that is **different** from arithmetical thinking.”

— http://profkeithdevlin.org/2011/11/20/what-is-algebra/

I and many other mathematicians and teachers of mathematics have long contended that “getting the answer” is not the primary goal in algebra classes. A very good exposition of this point of view is an excellent 17 min. video by Phil Daro. http://vimeo.com/79916037. Everyone with an opinion about what should be happening in our algebra classrooms should watch this video.

In the past I have written that “The purpose of early college level algebra courses is therefore to introduce the student to the use of abstraction, generalization, and deductive reasoning while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions.”

— http://www.college-algebra.com/essays/what_are_we_studying.htm

I apologize for not regularly including considerations of structure in previous writings. An important goal when teaching algebra (indeed all math) is to help students to see the elegant structure of mathematics and to recognize how such a structural view is important in many situations. Consider the operator of a pump station on a large cross-country pipeline. The operator must view his facility as a part of a much larger structure and must understand how his facility interacts with the rest of the pipeline structure. With a comprehensive structural view the operator is an asset to the company, without that view he may in fact be a hazard.

Should the student learn a bunch of isolated formulas and receive training for solving a hundred or so silly problems or would it be better to become a critical thinker capable of using abstraction, generalization, and deductive reasoning with the ability to examine an entire structure when needed in problem solving situations?

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