Counting was the earliest mathematical activity. Early humans needed counts to keep track of herds and for trade. Primitive counting systems almost certainly used the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. The first advances in arithmetic were the conceptualization of numbers and the invention of the four fundamental operations: addition, subtraction, multiplication, and division. The earliest advances in geometry dealt with simple concepts such as the line and the circle. Much of our civilization is based on mathematics.
Montessori believed that the child’s mind was mathematical and based on the order with an awareness of the senses. The understanding of the mathematical principles is seen as a gradual ascent from concrete to abstract or simple to complex. The child that has mastered the fundamental concepts involved with the practical life and sensorial materials is drawn to the math activities.
According to Montessori, the process of abstraction depends on two factors- absolute clarity in the concrete and a maturity of the mind. Montessori goes on to compare the mind of the child to an airplane entering into the realm of the abstract. And like an airplane the mind of the child must come down from its ascent and refuel and refresh itself with new experiences in the concrete suggesting that there is interdependence between the intellectual and the senses. The Montessori directress always insures that there is always available a sequence of tasks sufficiently challenging and interesting to engage and develop the child’s potentialities fully.
All that the child has grasped from their earlier experiences in practical life and sensorial comes to fruition – classification, discrimination, perception and dexterity. All are important factors that contribute to the confidence and fascination with math.
The child comes in contact with the quantities and their symbols every day through their environment. So, they are automatically pulled toward what they do not yet fully grasp. The child is now able to reinforce the concepts of one-to-one correspondence which is vital when working with all of the advanced Montessori math materials. With hands on experience and concrete materials to make abstract concepts clear, the child can literally see and explore the math process unfurl before their very eyes. The very first introduction to the math materials comes from the red and blue number rods which have the same dimensions as the red rods from sensorial with segmented proportions and a fixed quantity. In this activity the abstract number is not introduced until the child has had numerous experiences with the number rods. Montessori found that at such an early age it was difficult for the child to grasp the concept of numbers just by counting simple objects. This is simply avoided by representing the numbers as a series of segmented rods at increasing lengths. After this activity along with the introduction of the sandpaper numerals, the child then moves onto the spindle boxes which are fixed symbols and unfixed quantity. This exercise entails counting the correct number of spindles to go in each compartment all the way from zero to nine. After the child has shown their readiness to move on to more abstract material, then the cards and counters are introduced. This introduces the concept of odd and even numbers. The sequence of the math materials moves from the concrete concepts of quantity and number symbols. Then on to the abstract with the introduction to the decimal system, the four operations all with the golden bead material. After a considerable amount of time has been spent in the early math material it is time to move on to the more abstract material. For example, the introduction to the decimal system Montessori felt that it was very important for children to grasp the concept of the decimal system and the carrying of place values because it is within the understanding of these concepts that the material becomes most practical. The "crisis of nine" sounds like a natural disaster. And it really could be if it were introduced in an abstract form. By adding one to nine the amount automatically carries over in place value to the tens place. The children will learn later that these early carrying over and exchanging activities will be of great value in the addition and subtraction realm of the Montessori math material. The materials continue to gradually move the child forward toward the realm of the abstract with the reinforcement of the concrete with materials like the teen and ten boards, the hundred board, the hundred and thousand chain (where the concept of skip counting is imprinted in their minds), the birds eye view, and sequencing of numerals 1-9000. The child becomes more and more comfortable with the material and will eventually begin to add the quantities together and with the help of the directress will be informally introduced to the four operations: addition, subtraction, multiplication, and division.
The dynamic process of the Montessori approach and the preparation that has led up to this point gradually begins to reveal itself parallel to the mind of the child that is also methodical. As the concepts throughout the classroom become more abstract so does the mind of the child. The fluidness of the materials can be seen in the ten-ness of the sensorial materials. The mind that will awaken will not be forced into fatigue by rote motor exercises but will awaken well rested and ready for the world.  The awesomeness of the Montessori approach is the willingness to meet and nurture each child’s individual learning styles, rate of progress, wellbeing, and interests. 

"The child should love everything he learns, for his mental and emotional growths are linked."
-Maria Montessori