The usual line up.

Reading about the way children learn to count sends me to sleep, to be honest. Maybe it was the book - Gelman and Gallistel are not the most accessible of authors. I think I have deciphered it and what I have taken from the reading is that there is a set order of principles

which children need to grasp. So to move onto the stable order principle, they need to be applying the one-on-one principle.

I would think (drawing on experience from my own children), the majority of children have stable order up to 10 by the time they start school. Will they also have the cardinal principle (i.e. understanding the significance of the the final number they count)? I'm not sure. I think you can definitely notice it when they are singing counting songs as they tend to stress the last number.

A teacher of early maths can encourage abstract principle by getting them to count the marbles going into the jar, or simple clapping games.

The implications of this, I think would be that if a child is yet to develop stable order, then there is very little point in trying to get them to understand the cardinal principle as it would mean nothing to them! You would take a step back and engage them with work which is attainment-appropriate for them.