What does K-12 math class even teach until (optionally?) 8th grade then? Surely it's not all just basic arithmetic until kids are 12 years old?

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Algebra concepts are taught starting around 6th grade (for most students). The first proper algebra class (actually called that, and not pre-algebra or just math) is 8th or 9th grade.

> Algebra concepts are taught starting around 6th grade (for most students)

That's way too late, BTW. In most of the developed world, loosely "algebraic" thinking is introduced starting from the earliest grades, generally phrased as "here's how you should reason to solve these complex, 'multiple step' word problems". "Single-step" word problems (as we'd call them in the U.S.) are effectively unknown, since they're pointless (except as a curiosity); the whole point of word problems is to introduce complex reasoning about mathematical operations, which then seamlessly motivates formal algebraic reasoning.

(A good review article on this approach: Persson, Ulf and Toom, André: Word Problems in Russian Mathematical Education, available at: https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/my-a... )