Hi Alan, You digressed into 'new math' and I disagree
You wrote:
Separate issues are: what parts of the real stuff should be taught to children, how should the teaching be done, etc. This is very important in its own right - recall the very bad choices made by real mathematicians when they chose set theory, numerals as short-hand for polynomials, etc. during the "new math" debacle.
When I was 16 I moved schools, and joined a cohort who had been educated in the Schools Mathematics Project, a English incarnation of 'new math'. I had kind of a traditional classical math education up to that point, and I felt like a fish out of water for a few weeks. My first impression was that my new classmates thought much more like real mathematicians, and at first that seemed like a pointless stuffy homage to academia.
Later I learned to enjoy the math for its own sake, but I had another surprise a couple of years later. The SMP kids seemed much better equipped for the world of applied math at university and technical college. Set theory and number theory are vital for computer scientists (as I understand), matrix algebra and numerical methods for engineers. So when I got to college (to study engineering), I was glad to have had a chance to try my hand at real nineteenth century math in high school.
By the way, I never learned, even today, any kind of general algebra or shorthand for polynomials, so I cannot comment on that.
It didn't hurt that in those days, most math teachers in England were math major graduates (so perhaps an example of the benefits of the Hawthorne effect we discussed.)
By the way, the SMP still exists in a cut down form: http://www.smpmaths.org.uk/ It didn't go down in a public fireball like 'new math' in the US, but instead seems to have been quietly squashed by the all powerful National Curriculum steamroller.
Best, David