Mathematics is an area then which precise relationships between entities can be established. The process would involve evaluation, comparison, calculation and precise conclusions. As a corollary the verification is itself a process.

Numbers are introduced in three distinct stages. At this very beginning the child is given activities and experiences to appreciate the quantitative aspect of an entity. Simultaneously, he is also given the appropriate nomenclature. This is opposed to the earlier practice of the other areas wherein nomenclature is introduced after sufficient experience is not complete without the characteristic activity of counting. In order to count the child needs the number names.

### Arithmetic is divided into 6 Groups

Concept of Numeration

In this groups the child is introduced to the various aspects of the basic numbers 1 to 9. These include the quantitative value, the number names, the numerical symbols, the sequence & the concept of zero. Therefore the child is offered the basic unit. He is also introduced to the inter relationships between the basic numbers & also as they are related to the basic unit.

Decimal system of numeration

The child is introduced to the decimal system which is very simple and yet powerful. It makes reckoning very easy. It is also the basis for the most popular and universal system of measurement namely and the metric system. It has a regular structure. If the concept of numeration (first group) had enabled the child to evaluate the entities from the point of view of their absolute value, it is the decimal system which would help them identify the place value. When these are put together the child is able to work with large numbers with same ease with which he had worked with the basic numbers. Around this period the child experiences are urge and skill to handle large numbers. Taking this into cognizance, the child has to be introduced to various arithmetical operations namely addition, subtraction, multiplication, division. This is the 1st and formal introduction to arithmetic of operation.

The Teens & Tens

As the child is moving towards the end of the II group this is offered. The conventional names of the numbers have no mathematical significance. With English, conventional names are limited to a combination of one ten (11 to 19) and the different groups of tens (20 to 90). Introduction of the conventional names has also given the child the opportunity to appreciate numbers in a linear fashion. In this manner child is able to master linear counting. Though to begin with this group does not have mathematical significance, the later part of the group offers the child scope for memorizing the basic multiples of the basic numbers 1 to 9.

Memorization (basic combinations)

Having been introduced to the nature of the 4 arithmetical operations the child is able to apply them to the various hierarchies. The various combinations of each of these operations as related to the basic numbers are given.

Passage to Abstraction

Having worked with the second group and absorbed the concept of decimal system of numeration and the fourth group where in the child has scope to memorize the combinations for the various mathematical operations. The child is now offered scope to bring these 2 groups together. The child will now perform operations on larger numbers with ease

f) Fractions

As part of the sensorial material, the child is introduced to fractions. He also comes across fractions while working with napkins, constructive triangles, binomial cube, trinomial cube and decanomial, square. However a formal arithmetic introduction is given. Having been introduced to the basic unit growing in one direction till infinity the child is also given the sub divisions of the basic unit in the other direction, also going up to infinity. Having been given the basic understanding of the various factors involved in a fraction the child is given scope to utilize them. This is done by introducing the 4 arithmetical operations.

This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child's special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations.
Dr.Maria Montessori