The natural numbers are about order. The Common Core Standards don’t get it. So they don’t get natural numbers.
Keith Devlin’s repeated writings on multiplication ain’t repeated addition display a learned approach to not getting it either. Devlin doesn’t get that everything about natural numbers is about keeping your place in the count, i.e. is about order.
Order needs to be taught from grade K onwards. This includes initial segments, or head segments from 0 to n, and tail segments from n onwards. Head segments are closed under predecessor and tails are closed under successor. Both predecessor and successor are functions.
Order must be emphasized starting from grade K. Students need to be taught repeatedly by demonstration that natural number concepts and algorithms are all techniques of keeping your place in the count. This is the basis of all ye know and all ye need to know in arithmetic.
When they are not taught this, they are going to hit a brick wall when they get to algebra one. Algebra one under Common Core is just a set of behaviors. Algebra one is a stimulus and response subject when order is not taught in numbers as the foundation of numbers. The Peano Axioms codify this. Peano Axioms are about order not addition. Addition is a technique of keeping one’s place in the count. So is multiplication.
Not teaching this means arithmetic is not understood. Not understanding arithmetic means algebra can’t be understood. The result is drill factories to teach stimulus response. The high stakes testing just tests the drill factory and is the whip on the teachers.