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<p>Some basic notions of algebra are covered in typical high school courses like Algebra I and Algebra II, which are taken by more students than Calculus courses.</p>
<p>The initial course to the field of algebra would be something like “Introduction to Algebraic Structures” or the like. This class is usually a bit difficult and abstract, and high school students would mostly struggle with it. IME, it’s a pretty common course for undergraduate math majors.</p>
<p>I don’t think it’d be a big disaster if such a class replaced Calculus for the advanced 12th grade seniors. It’s certainly an interesting area, and it’s a more “beautiful” subject than introductory calculus. The major downside is that it’s a lot less practical. All those future science and engineering majors have to learn calculus, but the overwhelming majority won’t ever touch abstract algebra again.</p>
<p>My family comes from Romania, and I know a little bit about the high school math curriculums there from my parents (at least the ones that were common in the 1970s and 1980s.) Math courses did not focus on one subject for an entire year - they integrated several different areas every year. Calculus was introduced in 11th grade and continued in 12th grade, but 12th grade math also introduced algebraic structures. I think this type of approach could be useful in the US as well - but it’s a pretty radical change. Additionally, Romania followed the European approach of having “focused” high schools. So the high schools with these math curricula were technical high schools geared toward future math, science, and engineering majors.</p>
<p>Another interesting compromise might be linear algebra. Linear algebra has many more practical applications than introductory abstract algebra, and many students do end up having to learn it. Introductory linear algebra courses are more or less matrix algebra, and a lot of the subject currently comes up in Algebra II, which is why I suppose that very few high schools bother offering a separate linear algebra course.</p>