The rudimentary basics are important for everyone, and mostly those who will not pursue math-related careers. However, it is the content that is taught after the basics (middle school and high school curriculum) that could be spent on more advanced areas with less calculation work.
I feel embarrassed expressing all of these ideas with little background to support them. I'm sure you are a lot more knowledgeable on this subject than I am, so thanks for taking the time to acknowledge to my arguments and discuss them with me. ;)
We don't need to teach more, we need to teach better. We should not chase nominal accomplishments such as whether students have "covered" differential equations and discriminants by high school's end. There is already too much of that. The same is even more true for the foundational material taught in middle school. If you skate across the basics in an effort to cover more, earlier, you risk serious damage to the students' development in mathematics and science.
The problem all comes down to 10% curriculum and 90% teachers. Curriculum only seriously concerns me when it overconstrains good teachers and prevents them from doing their job.
I agree with this in part but the problem is that relying on good teachers is not something that can be scaled across countries. I'm not sure if the introduction of computers in the national education might help solve this problem, but I'm hopeful.
Perhaps like art and music, we should only be teaching beyond the very basics to those who actively seek it out. We don't require all highshool students to learn how to play the piano, read music, or draw nudes, so why should they all learn how to invert a matrix or find its determinant by hand?
My stance is that mathematics is first of all useful in the small for normal people. Everyday arithmetic, basic accounting, that sort of thing. But beyond that we need to teach mathematics for the same reason we teach art, music and literature. For achieving this goal, even if what we aim for is an appreciation of general principles, I was arguing that properly directed hand computation and concrete problem solving plays an important part. Concrete tinkering engages your brain in a complementary way to reflecting on nakedly abstract principles.