I taught elementary and jr hi for over 30 years. I taught math for about half that time. The problem, as I see it with math teaching as it is done today, is that math teachers, who usually are good at math and enjoy it, asked each other how they should have been taught. They thought (and probably correctly for themselves) that students should be allowed to discover the beauty and patterns of math on their own. So, in the sixties we had “modern math”. I was told to teach 4th graders base-six mathematics so that the little ones would discover how base 10 works on their own. The forced memorization of the times table as I had learned them was forbidden. The children of the seventies as a result could do much less with math than the generations that preceded them. It is even worse today.
The problem with this approach is the simple fact that the large majority of humans don’t particularly like math and are unlikely to ever enjoy the beauty of it. They will have to know how to use it their lives, of course, but for most of us, math is just a tool to get things done, not a “language” for understanding the universe. I believe science teachers tend to make the same error. Today basic science facts are not taught, but are hopefully discovered by the student. Most don’t discover anything and know very few science facts.
To me the biggest problem with teaching math today is that few teachers know math themselves. Past the basics/early elementary school, it appears that most people who know math have opted for something other than teaching as a career; by high school qualified math teachers are few and far between. People who can teach high school math can usually command a much higher salary doing something else in the private sector, and as long as union work rules require the math teachers be paid on the same scale as gym teachers makes it unattractive indeed.
My mom taught piano.
You did not "discover the beauty or patterns of music" on your own because left on your own, you won't do it.
But if you do your drills and practice then one day you will "see" the beauty of the music. She used to call it "the click" and few things made her happier then when she could say that one of her students had heard the click.
Math is very much related to music. While those teachers like to think they would have discovered the joy of math on their own almost none of them would have. They had to do the drills, learn the patterns and then one day they heard the click.