I realize that mathematics evolved and develop a life of its own independent of nature; however, I was only pointing out that nature provided the inspiration for mathematics.
There are simple examples of geometric shapes existing in nature. The triangle is a natural derivative of nature. The pentagon and how it relates to the golden section. Then you have the studies of Fibonacci. The notion of a right angle is a natural phenomenon.
As man started to write down these shapes, he noticed certain patterns. Along with the triangle and the notion of a right angle, the Pythagorean theorem was discovered. It was discovered because it is there in nature. Without the Pythagorean theorem, there would be no calculus. I believe that the concepts of Ratio, Proportion and equality are natural concepts that are observed.
So I believe that children can be involved in more meaningful exercises than arithmetic. You have kids that spent 12 years of schooling and don't understand the most basic concepts of mathematics like a proportion. I don't know if you can separate Algebra from Arithmetic but I certainly wouldn't encourage advanced Algebra till they are ready.
I am always surprised that Rhoda Kellogg studies with children's art was not taken seriously. It shows that children have an innate ability to sense completeness and wholeness.
I understand that the mental processes are not there for children but they can go through the physical processes through exercises and exploration. Then as their minds develop, they will have a more natural grounding of those concepts.
I am certainly no expert in advanced Mathematics nor in education. I am only indulging in the more philosophical aspects of basic Mathematics; and in particular, why so many people have no clue as to what a proportion is. I know people who don't know how to measure. In order to measure something, you have to understand fractions. These people spent 12 years of school. It is just amazing to me.