It is fascinating to read Ms. Montessori’s observation and summation of what happened when students worked with rods and beads in learning arithmetic.
From pp. 207-208 of The Advanced Montessori Method, Volume II by Maria Montessori (Clio Press, Oxford England, ISBN 1-85109-233-1, (c) 1965 Montessori-Pierson Estates):
The “bead work” became at once an established element in our method, scientifically determined as a conquest brought to maturity by the child in the very act of making it. Our success in amplifying and making more complex the early exercises with the rods has made the child’s mental calculation more rapid, more certain, and more comprehensive. Mental calculation develops spontaneously, as if by a law of conservation tending to realize the “minimum of effort.” Indeed, little by little the child ceases counting the beads and recognizes the numbers by their color: the dark blue he knows is 9, the yellow 4, etc. Almost without realizing it he comes now to count by colors instead of by quantities of beads, and thus performs actual operations in mental arithmetic. As soon as the child becomes conscious of this power, he joyfully announces his transition to the higher plane, exclaiming, “I can count in my head and I can do it more quickly!” This declaration indicates that he has conquered the first bead material.
Our rational form of consciousness allowed the students to associate a concept (e.g., “four”) with a color, as it allows us to associate a concept (again, “four”) with a sound or visual shape: “four.” This association made it possible for the students’ minds to work more efficiently — to “realize the ‘minimum of effort’ “. And this association and efficiency allowed the students to go on to counting and adding.
Our rational form of consciousness allows us to leverage our perceptual, subconscious ability to form mental associations and to identify similarities and differences.
Forming concepts and forming principles are methods of cognitive integration. Integration takes the joint work of conscious awareness and subconscious processing.