in the early years of education it needs to be shown what math is for, using real life examples......then teach all the trickier stuff

I majored in mathematics in college and took twenty-one semester credit hours my first year. That included summer school. The most math I took in one semester as an undergraduate was nine hours. Fifty-four credit hours total after three years out of a total of 120 hours.

It was history phobia for me -- the last two undergraduate courses I took were history courses. Took them after starting graduate school.

on teaching math in preschool. basically, it is about teaching numbers in real world ways.
a lot of her research is included in the common core standards.

dr jennifer mccray of the erickson institute.

If you blow the Revolutionary War, you can make it up on the Civil War test. If you goof up photosynthesis, you can recover with mammalian anatomy. If you don't fully get the first chapter in any math book, you can forget about understanding any of the rest of it. My husband, who in retirement studies n-dimensional geometry for fun, flunked calculus because on the first day he couldn't figure out with dy/dx why the "d's" didn't cancel out and never managed to catch up.

Math needs to be taught individually, with no moving to chapter 2 until you get chapter 1.

high school, math quickly and unceasingly becomes and remains a nightmare.

Math (as well as foreign languages and some sciences) must be taught sequentially.

I've long advocated for a completely different set-up at the junior high and high school level than the current system of two semesters with almost every course a full year sequence. Change to 6 week segments. Each segment taught as an independent unit. In things like math and foreign languages, you must complete them sequentially. Other things, like history, literature, and social sciences, most of the time the segments can be taken out of order. Some history will be sequential.

The real problem comes if the schools only offer the first algebra segment in the first six weeks of the school year. It must be offered at least twice, so a student who fails the first time doesn't have to postpone completing algebra for a full year.

Once you hit the AP level, which are college level classes anyway, you go back to the traditional full year system.

One is that math is mechanical, you learn the mechanics from methods of construction plus axioms 1,2,3,.. to theorems 1,2,3,..., and you combine one totally mechanical system with others to get more complex 100% mechanical, in fact mindless systems that robots can follow. So we've got our computers.
The other is that things like numbers like Pi, facts like the incommensurability of diagonals, and math associated with geometry (and physics) in general seem to be hardcoded into the universe.