Then i'd refer you the 1914s Text from Kilpatrick (From the same time than Montessori, a progressive Educator himself) https://archive.org/details/montessorisystem00kilprich
I just chuckeled as i re-read his accounts of her arithmetics material:
"On the whole, the arithmetic work seemed good, but not remarkable; probably not equal to the better work done in this country. In particular there is very slight effort to connect arithmetic with the immediate life of the child. Certainly, in the teaching of this subject, there is for us no funda- mental suggestion. "
Mind you, that's from 1914. Pedagogics have improved since then...
Or do you mean a better understanding how pre-numeric mathematic works? Most efforts are built around Piagets Teachings (Not very scientific as he "just" observed his own three children) https://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive...
One sample of such a adaptations for pre-numerics (not Geometry) are the works of Carin de Vries (sadly only in German): http://oops.uni-oldenburg.de/1014/1/vridia10.pdf