Montessori understood that in order to teach an abstract concept such as math, the child must first work with concrete, hands-on material. It begins at the toddler level by teaching Pre-Math concepts. These concepts include temporal relations (sequencing), spatial relationships, seriation (ability to order in a series), class inclusion, one-to-one correspondence, and equivalence. These concepts are built into the materials and as the children explore, they begin to gain an innate understanding. As the child gets older we begin to introduce the concept of number. Not just at the “Sesame Street” level of counting, but the understanding of the association between quantity and numeral.
Once this understanding is attained, the child can now explore the different operations and manipulations of number, again using hands on materials. As they work more and more with the material, the children will soon be able to abstract the process, working without the need of manipulatives. At this point students can go beyond the realm of just equations on a page and see how the different facets of math are used in the real world and apply their mathematical knowledge to real world problems.