A young girl's small hand grasping beautiful objects, sensing the world around her – shapes, dimensions, relationships, amounts - all represented by concrete objects that prepare the mind for a deep understanding of the principles of mathematics. Maria Montessori discovered that children understand the world, through all their senses which she referred as the ‘Sensitive Periods’, and that only by making use of this knowledge can their understanding be developed to its fullest potential. Nowhere is this more evident than in the Montessori mathematics curriculum. As the child progresses from early childhood through the elementary years, the Montessori Math curriculum moves from the concrete, to increasingly abstract concepts of mathematics. As with all Montessori materials, the mathematics curriculum is performed by the individual, using self-teaching and self-correcting materials. These materials are presented to the child as interest is expressed or observed and then the adult moves away, to allow exploration through repetition, until the concept is mastered. In small sequential steps, each learner develops a mathematical mind at his or her own pace.
The Montessori sensorial materials form the foundation, in preparing the child for the abstract world of numbers. Montessori gave the Sensorial equipment the credit for helping the child explore his environment and also for developing the mathematical mind - the mathematical structures necessary for the order, sequencing and precision of mathematics. During the day-to-day manipulation of concrete objects the child builds her concepts of numbers and the ability to concentrate. Montessori emphasized that all materials must be beautiful, and stimulating to the senses, inviting children to take them from the shelves to explore them again and again. Small children naturally love repeating activities and thoroughly absorb the intended concepts. The Sensorial work is a preparation for the study of sequence and progression. It helps the child build up spatial representations of quantities and to form images. When a child starts working with the sensorial material the first numbers he deals with in most of the apparatus are from 1 – 10 like in 10 cylinders, 10 cubes, 10 prisms etc. As the basic numbers of the decimal system are from 1 – 10, the child unknowingly starts to deal with mathematical concepts through sensorial materials. Other than the numbers, geometrical shapes, their names and complex mathematical concepts like one-to-one correspondence, seriation, sorting & grouping are also initiated with the sensorial materials in a concrete way where the child is actually experiencing them visually or using his muscular memory. Along with these, the Long rods in Sensorial being same in the length as the number rods in maths lays the foundation for the child to receive Maths program in Montessori environment.
By the age of three and half to four, the child is ready for the receiving mathematics. A series of preparations have been made using other areas of Montessori
1. The child has established internal order.
2. The child has developed precise movement.
3. The child has established the work habit.
4. The child is able to follow and complete a work cycle.
5. The child has the ability to concentrate.
6. The child has learned to follow a process.
7. The child has used symbols.
8. The child is informally introduced to the primary numbers.
All of this previous development has brought the child to a maturity of mind and a readiness of work.
Montessori math curriculum is designed to support children's natural interest in math and provides a strong foundation in numbers through the use of specially designed, hands-on/minds-on math materials. Through concrete mathematical experiences, the youngest children learn about dimension, size, number, shape and sequence. Then, with the use of increasingly abstract materials, children make the transition from concrete experience to paper-and-pencil exercises. Each math concept is taught with corresponding materials.
The Montessori maths program is divided into parts to facilitate a sequential and gradual progress in the maths concepts starting from simple to complex. During circle time, informal activities or games are introduced to initiate complex maths concepts like seriation, one-to-one correspondence, sorting and more in the simplest way. Without counting or even uttering a number name, the child is actually introduced to maths through preliminary maths activities.
Then he is actually introduced to the basic numbers of decimal system through the presentations in ‘Early maths in separate sessions or depending on his personal cognitive development level. Using the sensitive periods of order, manipulating small objects and locomotion the child is able to grasp the basic number concepts from 1 – 10 easily. In early maths program, a child needs to know numbers to 10, in order to be able to work in the decimal system. The long rods introduce the concept of comparative length. Through laying out the rods one-by-one, the child learns, at the most basic concrete level, how the numbers one to ten relate to one another. The red and blue segmented rods add the concept of one-to-one correspondence and allow for the memorization of the 1 through 10 counting sequence. Each rod represents a given number and the relative difference between each number is clearly seen. Then Sandpaper numbers are traced by the child one at a time to associate number names with the fixed quantities they have experienced with the number rods. When both the number rods and the number cards are merged together in a single presentation, the child then & there associates loose number names with fixed quantities which gives them absolute clarity in the accurate association of number and quantity. ‘0’ being an essential and probably one of the most important fundamental concept of maths are taken in isolation with the presentation of ‘Spindle Box’ to make the child understand the meaning of ‘0’ i.e. ‘0’ on its own means nothing but when it is associated with a number, it certainly indicates a bigger number. Some number games and ‘0’ games are played during early maths program to reinforce the concept of ‘0’ and also to make the child experience the bigness and smallness of basic numbers from 1 – 10. Next the child is introduced to the cards & counters where the level of abstraction requires the child to demonstrate that he recognizes the numbers one through ten, knows the order of one through ten, understands one to one correspondence, and can relate the proper quantity to the numbers. The basic four operations – addition, subtraction, multiplication and division are shown in simple yet in an interesting way using only the number rods highlighting the importance of 10, which make the child familiarized with the four operations we need to use in our daily lives probably in every step of it. The whole early maths program reinforces the importance of 10 as the base of introducing the formal demonstrations of Decimal system.
Once the child has developed a secure knowledge of numbers to ten, the exciting world of the static golden bead materials comes to her. This part is known as the formal introduction of the Decimal system i.e. a number system based on ten in which numbers are expressed by combinations of the digits from 0 -9. In this system, all the derived units are the multiples of 10. With the introduction to the Static Golden Bead Material, the child learns to handle numbers till 1000 without being introduced to the traditional number names. From this stage, he gets absolute clarity in each and every hierarchies of the Decimal System and his thorough understanding of the primary numbers (1 - 10) helps him to understand the interrelationship between the hierarchies. While working with the static material, the child forms quantities with the concrete and then with the abstract to gain a considerable mastery in associating both. Like in early maths, in this stage also when the child is comfortable with the concrete, abstract symbols of the hierarchies with the importance of the number of zeros are introduced to him. A number of activities are shown with the concrete golden beads and the static cards to reinforce the child’s concepts of hierarchies, sequencing of numbers, each hierarchy is 10 times than the previous one and the column concept of maths which leads to a natural transition to the complex tasks with the dynamic material.
The dynamic material facilitates the change at 10 in maths. While handling and working with the dynamic materials repeated times, the child is able to understand clearly that when a number reaches 10, it goes to the next hierarchy. The child practices hierarchies changing at 10 many times with the dynamic material and cards. It should be noticeable that without being introduced to the traditional number names, the child easily handles bigger numbers in thousands and hundreds while doing all types of operations with or without change just because he has a clear concept of the primary numbers from 1 – 10 from the beginning of the maths program. All the operations concepts are reinforced and the child handles bigger numbers but still he is counting in tens.
In the later part of the maths program, traditional number names are introduced again with forming quantities with the concrete material first and then associating with their written symbols through the ‘Seguin Board’. Addition, Multiplication, Division and Subtraction Boards are used to reinforce the valuable concepts of these basic four operations. The child is introduced to multiplication tables with interesting materials like Multi-coloured bead bars.
It is clearly visible with the break ups of the Montessori maths curriculum, that all the maths material are graded from simple to complex and from concrete to abstract. Montessori took the idea that the human has a mathematical mind from the French philosopher Pascal. The mathematical mind tends to estimate, needs to quantify, to see identity, similarity, difference, and patterns, to make order and sequence and to control error. By comparing all the facts that a mathematical mind should have, we can say that the Montessori maths curriculum is beautifully designed to suit the child’s need for growing up with a mathematical mind