Paul Lockhart hits the bull's eye in his essay titled A Mathematician’s Lament. If a musician doesn't know every tiny detail about music theory, is she really a musician in the true sense? If you can't build beautiful music sheets, can you really create exceptional music? Of course, these questions are ridiculous but to wake up in a world where this is the norm would be a nightmare.
Unfortunately, we do live in a world close to this. Pedagogy needs an overhaul. Hoping that elementary schoolers will realize the importance of concepts later in life is the worst way to sustain subject interest. Mathematics is taught in a similar fashion - all the disparate pieces from geometry, number theory, algebra and the likes are taught in a way that doesn't put them in the context of larger picture. But this isn't some new extraordinary insight. The sentiment against modern schooling is (not?) surprisingly shared by parents and students alike. Everyone wants to solve it from teachers to politicians, or at least they acknowledge that the problem exists.
Quoting some blunt statements from the essay.
There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it.
Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it.
I feel strongly about this now as I pursue research as a profession. I have had to unlearn everything in the past two decades of schooling. Research has opened up my mind in ways that I couldn't have predicted. I've learned to accept that not everything has an explanation or possibly ever will. The pursuit of those explanations is what makes this effort worthwhile nonetheless.
A lot of us have been led to believe that young students need structure. Free-form classes don't work. Well, let me burst the bubble. Structured classes in schools don't work either.