Well, he only talks about school algebra. I'm not familiar with the US high school curriculum but I doubt it goes much beyond polynomials…
FTA (3rd paragraph):
> (I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)
That's almost clear from the question. If this was about higher math, the question would likely have been "What is an algebra?", as "What is algebra?" has the simple but almost content-free answer "The study of algebras." (http://mathworld.wolfram.com/Algebra.html)
