Something not often mentioned is that one of the divisions between people is that some have brains wired to comprehend “abstracts”, symbolic representations of other things, be they mathematical, musical, or many other things. Other people have brains that have different wiring, so abstracts are difficult at best, and they will never develop great skill with them, because they can’t.
Importantly, basic mathematics and geometry, as well as a little basic musical notation can be taught to most everyone, but only those wired to understand it will be able to take it further.
For this reason, skills such as algebra, trigonometry, calculus, and advanced geometry should only be taught to those with an aptitude for it. In turn careers dependent on knowing these abstracts are limited to those who can master them.
Otherwise you have the “Pearls before Swine” rule, that you may offer expensive pearls to pigs, but it just wastes your time and annoys the pigs.
And please divorce the idea that such abstracts are indicative of intelligence. Or talent. Or creativity. Or success in many endeavors.
Many musicians “play by ear”, and are able to replicate the notes played and the spirit in which they are played, or better, but are incapable of reading sheet music. And most can use the basic parts of the notation as guides, but have no connection between them and music they play other than in timing.
So the bottom line is that while algebra is used as a discriminator of scholarship, it is a poor one, so it should be taught instead to those who grasp it, allowing them to go further in its study without being delayed by those who cannot.
Yes, and tangential to your point, many sixth-graders' brains will not have developed the capability for abstract thinking that algebra requires. So, very-early introduction of the subject may not be wise.
“So the bottom line is that while algebra is used as a discriminator of scholarship, it is a poor one, so it should be taught instead to those who grasp it, allowing them to go further in its study without being delayed by those who cannot.”
We bring different skills to school/college/life and those good skills should be advanced. Students should not be hammered by those pushing Algebra or the opposite!
One of our younger relatives thrived on the complex math and physics and sucked at using written English. He was told that he had to master the written English. He said that he would hire a good secretary.
He has two engineering disciplines and is a vice president in a field that uses heavy duty math for real building issues 24/7.
He hired a good secretary and kept her as his executive assistant as he went through the ranks to a VP role.
His brother made D’s in Algebra and never understood it. His ability is using real math to solve real problems in making
premimum products while meeting strict standards. He, then translates those abilities into written protocols. He does that with a computer and has no need for a secretary.
Two men with the same basic DNA, in the same family and raised in the same house by the same two parents. They have a lot in common, but their job skill sets are totally different.
One was told by outsiders that he would fail due to his inability to write with English. The other was told he would fail because he could not do Algebra.
Yet both didn’t listen to the know it alls. They maximized their abilities and skills and have been successful at it.