A response to a comment made at this course site reflects what I know. Once you've had experience with Montessori materials you "feel them in your body."
The numbersleuth.org/universe video makes me want to see something similar done with power comparisons using materials made more than one hundred years ago and still used, successfully, today in classrooms to transfer a sense of quantity, size and dimension from body (hand) to mind.
I dream of a playground/park where children can build with their classroom materials and then find the same in stationary constructs so that they can climb and stand at varying heights in proportion to what they can hold in their hands. @MariaDroujkova