Evidence contra your second claim, that Piaget's "stages of cognitive development orginally his and not to be found in Montessoris Teachings": Montessori's theory of the sensitive stages [1] and the planes of development [2].
I'm not going to continue to argue with someone who willfully spreads lies (malicious "hyperbole") and poppycock.
[1] http://rmschool.org/content/sensitive-periods
[2] https://ami-global.org/montessori/quotes/four-planes-develop...
Would you mind reading the site guidelines and following them scrupulously when commenting here? We're trying for a better outcome than scorched earth followed by heat death, which seems to be the default for internet forums.
My father worked with Piaget, so I lack tolerance for someone who deliberately spreads lies about Piaget's work.
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
Montessori does not provide such deep insights into pre-numeric mathematics.
You pretend to "infer", from a one-sentence description of how the Montessori method uses concrete materials to teach the basic arithmetic operations, that somehow other mathematical principles are absent from the Montessori pedagogy.
Of course these principles are demonstrated and taught in Montessori schools, also with concrete learning materials.
You may not know much about Montessori, or you may have an axe to grind, or ...? Either way, you should stop making demonstrably false statements about her pedagogy (and about Piaget).