Perhaps, of all the Montessori apparatus, the Math Materials are the most glamorous. They are beautiful, showy, and in their simplicity supremely intelligent. They give the children a sensorial experience of the abstraction that is mathematics, allowing them to store concepts so that when the time comes to deal exclusively in abstract terms, the understanding is already there.
Before the children begin to work with the Math Materials, they are well prepared. The Exercises of Practical Life have given them the opportunity to develop logical and sequential thought patterns. The logical order of the Practical Life activities has been complemented by the mathematical order inherent in the Sensorial Materials. These materials allow the children to work with the quantities 1 to 10 in several dimensions and with the Math Materials, they are given their numerical value.
The Montessori Materials offer a clear example of indirect preparation, a principle that is rooted in the child's natural manner of learning.
With the mathematical apparatus, every piece of material isolates one concept, and these isolated concepts integrate to form the basis for a further step in the development of the child's mathematical understanding.
Among other areas of indirect preparation, in this series of exercises, the children are presented with the four operations in concrete form. When doing addition, multiplication, subtraction and division, they literally carry and borrow and change the quantities involved. As they put them together and take them apart, they perceive unconsciously the interplay of the numbers, which prepares them, at a later stage, to explore and memorize addition, subtraction, multiplication and division tables with another series of exercises which again isolate the particularities of each operation.
They have now reached a new plane of development, and they will consciously pursue the exploration of mathematics in a manner appropriate to the characteristics of their age. The transitional materials are a means that will help them function in this dimension, representing, as it were, the end of the runway where the child's luminous, adventurous mind can take off to explore the untold reaches of the mathematical sciences.