The best place to look at what is being taught as "Algebra" at the high school level is to go to the (new) source: the Common Core State Standards Initiative[1]. 45 states (and DC) are using the standards. The best places to start are Math Appendix A[2], which explains the Initiative's recommendations for a High School curriculum, and the Introduction to the Algebra Strand[3], which describes (in brief) the goals for actual algebra instruction.

>If High Schools truly wanted to introduce Algebra to students in a "pure" form then Math teachers would need to know/understand Abstract Algebra. And then be able to teach introductory ideas to Abstract Algebra which, in all honesty, anyone can come to understand.

They don't. Not on the general level. You're worrying overmuch about semantics. The name "Algebra" isn't ever going away; no high school guidance department wants to explain to every single college that their program teaches the same thing as a normal "Algebra 1" class but just calling it "Practical Mathematics". (Incidentally, "Practical Mathematics" would imply much more basic mathematics, your traditional "Home Economics" class with taxes, investments, credit cards, balancing checkbooks, etc.)

Beyond that, abstraction is much, much, much, harder for the average student than you realize. Students have trouble seeing the relationship between the Distance Formula and the Pythagorean Theorem. Some have trouble even manipulating basic formulas, such as solving the Ideal Gas Law for a given variable. They will insist on plugging in the numbers into PV=nRT every time, and then solving the equation over and over again. Dividing by 5 is tangible and easily visualized to these students, dividing by R is not.

Math teachers take the job because we love math and want to share that with our students, but practical concerns come first. Symbolic manipulation is much more widely used in the average high school student's future education than pure math. I, and many teachers, include facets of pure mathematics in our courses. When my Honors Geometry students begin working with infinity, I have them read Strogatz's excellent piece on Hilbert[4], which is one of the most popular assignments each year. (And yes, there is next to no Geometry in there, but you have to keep minds sharp somehow.) I also throw in some basic Real Analysis when discussing the concept of rigor in proofs. When my Algebra 2 classes have to trudge through a brief review of Algebra 1, we spend the time talking about why closure matters, and what number systems are and are not closed over what operations. My PreCalc classes do a decent amount of Number Theory.

Finally, most schools offer Discrete Mathematics (i.e., an introduction to Pure Math) as a senior-level math elective. However, it's competing for the brightest minds with AP Calculus and AP Statistics. If you'd like to see more students study Pure Mathematics, the best way to do that would probably be to gather a group of like-minded educators and petition the College Board to create an AP course covering said material

[1]http://www.corestandards.org

[2]http://www.corestandards.org/assets/CCSSI_Mathematics_Append...

[3]http://www.corestandards.org/Math/Content/HSA/introduction

[4]http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-...