I said: " The immense impact and usefulness of Piagets stages of cognitive development are orginally his and not to be found in Montessoris Teachings"

Montessori via your source on mathematics: "Mathematics: Formation of the concepts of quantity and operations (addition, subtraction, multiplication, and division) from the uses of concrete learning materials. (birth to 6)" Quite general.

Piaget has a few sub-skills concerning Quantities:

  * conservation (xxx has the same amount of x'es as x x x)
  * classification (x is a letter and lowercase)
  * seriation (x X y Y could be orderes XY; xy, but also Xx, Yy)
  * transitivity and others

This insights are very useful if you want to make - for an example - a pre-numeric mathematical game concerning quantities.

Montessori does not provide such deep insights into pre-numeric mathematics.