I think what originally got interested in math was the infamous "new math" program  that existed in the 60s.
And the thing with New Math is that was abolished through a backlash of parent and teachers because ... the parents and teachers couldn't understand the new concepts.
And this is the thing. Math is both an abstract enterprise and an enterprise of bookkeepers adding numbers on ledgers, parents adding up grocery bills and so-forth. While New Math might have had some flaws in execution, any effort to produce curriculum suited to students learning abstract mathematics is going to run-up against the "readin', writin' and arithmetic" crowd in a similar fashion to New Math.
One simple example (which I'm probably mixing up) is "making tens"; when doing addition or subtraction, rather than blindly memorizing a bunch of facts like "3+8=11", they learn to break the numbers up, so you take 3 and 8, "make a ten", and have one left over.
This is actually a great thing to help with understanding of what's going on with math, and after my kids showed me a few examples, I totally got it. But for those who are less open to new ways of teaching, it may just seem like change for the sake of change. And it's not just math; they're also teaching other subjects (such as writing) in newer ways, which I honestly think are fantastic - having kids spend time writing every day is great.
There are also quite a few bad questions in the workbooks; this has probably always been true, but the combination of bad question plus unfamiliar (to parents) concepts causes a strong reaction. Plus the whole "the gubmint is trying to brainwash mah kids!" contingent overreacting about everything....
My, I'd be happy if we had military dictatorship which imposed a new order including the metric system, radians instead of degrees and algebra from 1st grade onward. But I don't think that program would have a sufficient constituency.
I know that it has a terrible reputation, but was it a plain bad idea, or a good idea implemented poorly, or a good idea implemented well but unfairly maligned?
The controversy was whether these concepts are useful or helpful to non-mathematicians, and mathematicians may be the least qualified people to answer that.
My sense is that a lot of the reaction to the New Math was the same as the reaction to dictionaries that came out at the time and included words like "groovy" or said that it is not the crime of the century to write "which" when not preceded by a comma. That is, it was more of a political effect than pedagogical.
That said, there were a lot of people who thought on a lower or more local level, that math teaching that asked students to notice the Associative Law, or that asked students to think about sets, was not directed to immediate gain. That made it wrong in these folks's view, or at least puzzling. They had not learned those things and one thing about Math is that it is unchanging so ... .
I am excited by the Common Core, myself, but I hear a lot of rejection that seems to me to be just general opposition to doing it differently than when the speaker was in school. There was a lot of that in the 60's, for sure.
My evaluation of Common Core so far is that it's terrible. Students are forced to learn multiple ways of doing basic arithmetic, which is highly counterproductive and confusing if the child mastered the topic (and concept) with the first approach. It also lacks any aspect of what this essay stresses - helping children see the beauty and creativity of math. The emphasis is on learning processes by rote.
We'll see how things progress in the next grade, but I'm not optimistic. Private school is looking very attractive.
Maybe I have a mistaken impression of how significant "new math" was, but I'd expect people to have thought about it. I mean they've all heard the Tom Lehrer song, right?
Extremes of either seem to be destructive, but hitting the sweet spot where the balance is just right is incredibly hard, and also incredibly demanding of time and resources.
The world leaders during WW2 recognized this - basically, when critical work needed to be done, they knew whom they needed and roped in a large number of the leading "dreamers/thinkers" of the time to work on the Manhattan Project.
The thing that saddens me is that it is often conflict and animosity that awaken this recognition - other examples include the Cold War, artillery calculations in WW1, etc.
FWIW, the math major graduates at the college where I teach, to my knowledge, have never had serious trouble getting employment.
An example of a good effort to combat these tendencies is a course by Prof. Bhargava at Princeton, where he introduces good mathematics through card tricks and games:
... although I don't think we'd heard from a high-school classmate of the author before.
Some of the nest mathematicians are the worst teachers. They focus too much on small details, as if they are carrying out their own usual role; and don't intuitively explain the basics concepts that are now too ingrained for them to easily recall the original difficulties they had with learning them.
Music and mathematics are not similar. I also imagine music is more subjective / less standardized. You might be better off with the majority of your time spend with a music teacher than Elton John, for example.