Euclidean geometry is still taught in American high schools, but I am strongly against it. I think it should be replaced with linear algebra.

Arguments against Euclidean geometry:

• Most of what you prove in a high school Euclidean geometry class seems pretty obvious until you learn about non-Euclidean geometry. It makes students think that proofs are pedantry for its own sake.

• Euclidean geometry is basically useless. There was undoubtedly a time when people used ruler and compass constructions in architecture or design, but that time is long gone.

• Euclidean geometry is obsolete. Even those students who go into mathematics will probably never use it again.

Arguments for linear algebra:

• $\mathbb{R}^2$ with the standard inner product is a model for the Euclidean axioms, so in particular you can still prove the same theorems if you really want to.

• Linear algebra generalizes easily to dimensions larger than 3 where most students' geometric intuition breaks down, so it is easier for them to appreciate the need for axioms and theorems.

• Linear algebra - particularly eigenvalues and eigenvectors - is ubiquitous in modern science and engineering. I would argue that the average person is much more likely to encounter an eigenvalue problem than a calculus problem.

• Linear algebra is, of course, still the basic language in which most of mathematics is expressed and thus a linear algebra class is a more honest taste of what math is all about.

• Providing students with an early foundation in linear algebra would make later education run more smoothly. Even many non-scientists use software that is based on solving linear systems or computing matrix decompositions, and it might help for such people to have a little more context. And those who go on to take further science classes - particularly physics - would more obviously benefit. If nothing else, we might finally be able to teach our students the correct second derivative test in multivariable calculus classes...