Dr. Maria Montessori said — after discussing how to introduce and develop the concept of number (with materials like beads and rods) and the concepts of addition and subtraction (again, with Montessori materials like beads and an abacus (a “counting frame”)) — that once a child gets to complex multiplication and has done some basic multiplication (yup…beads and abacus):
When there is a number to be multiplied by more than one figure, the child not only [already] knows the multiplication table, but he easily distinguishes the units from the tens, hundreds, etc., and he is familiar with their reciprocal relations. He knows all the numbers up to a million and also their positions in relation to their value. He knows from habitual practice that a unit of higher order can be exchanged for ten of a lower value.
To have the child attack this new difficulty successfully one need only tell him that each figure of the multiplier must multiply in turn each figure of the multiplicand and that the separate products are placed in columns and then added. The analytical processes hold the child’s attention for a long period of time; and for this reason they have too great a formative value not to be made use of in the highest degree. They are the processes which lead to that inner maturation which gives a deeper realization of cognitions and which results in bursts of spontaneous synthesis and abstraction.
(p. 228 of The Advanced Montessori Method, Volume II by Maria Montessori (Clio Press, Oxford England, ISBN 1-85109-233-1, (c) 1965 Montessori-Pierson Estates))
Math develops a person’s ability to think conceptually and abstractly, and develops a person’s ability to think through complex problems. And helps develop the child’s power and patience to engage in sustained mental effort.