Mathematics in Everyday Life is Yet Another Item of Human Culture

Mathematics is yet another aspect of human society presented in a Montessori environment. However, it is a field created entirely by mankind, having evolved in response to human needs. Precision and order, fundamental aspects of human nature, have informed the development of arithmetic. The abstract science of Mathematics enables us to analyze, appreciate, and consider elements and aspects of various entities with precision. Therefore, we encounter Mathematics in all aspects of our daily lives. Mankind is dissatisfied with an imprecise understanding of quantities; Mathematics helps to satisfy this need by providing precise quantities. A child is born with innate tendencies towards precision and order.

This is also a period characterized by an enhanced sense of order. We find that a child with a well-structured routine feels more secure and peaceful than one without such a routine. From an early age, the child is accustomed to experiencing things sequentially, a basic principle of Mathematics. In this way, we see that even a child just over a year old can understand the difference between a few and many, less and more, etc.

As a baby, the child begins to perceive the quantity of food they consume. They also hear numerical instructions, like "You get one banana". A child who has started to speak might say "I am two years old" but show all five fingers. All of these developments take place in a state of unfamiliarity. Often, adults hearing children use such terms become excited to teach them numbers. However, adults need to understand that, at this stage, for the child, it is like reciting names without any understanding of precise quantities. The preparation to work with Mathematics begins from birth.

With assistance from his first, second and finally, the Montessori environment he has been part of, the child undergoes a series of preparations before being formally introduced to Mathematics. He has established an internal order and honed specific movements by working with some of the Exercises of Practical Life material. He has developed the ability to choose and complete a task cycle. He has found balance and developed the ability to concentrate. He has also been introduced to some of the Sensorial material (for example, the Red Rods, Geometric solids, etc.), which illustrate the mathematical gradation visually. However, all of this is indirect preparation for the introduction to Mathematics. Around the age of 3.5 years, what Dr. Montessori calls "the awakening of the mathematical mind" occurs. The child is no longer satisfied with vague concepts such as some more, a little more, etc.; he now wants to know exact quantities.

At this phase, arithmetic comes as a response to his demand for exact, accurate statements. It is offered to fulfill what he requires right now, in the fashion he needs for the purpose of his self-formation. Considering the instruments of discovery he has at his disposal, which are his hands and his senses, it has to be offered as what Dr. Montessori refers to as psycho-arithmetic. So how is this aid supplied in the House of Children? Similar to the material for Sensorial Activities, the tangible materials for Math are visible abstractions. In the realm of Math, the sequence of presentations of varied materials always follows the rule of first introducing the quantity alone by associating it with a name. Then, the symbol is introduced separately and designated the same name. Ultimately, the child is provided with an opportunity to relate the quantity with the symbol.

1) To comprehend the concept of a unit as the foundation of counting, the child must be introduced to a system of numeration. The preparatory stage comprises helping him comprehend the initial ten natural numbers. With the support of a unit, we enumerate the quantities and utilize the names of numbers.

In a Montessori environment, however, materials like the Number Rods, Sandpaper numbers, Pin Boxes, and Cards and Counters are utilized to introduce the fundamentals of counting, unlike traditional schools. In this manner, the child not only learns to count but is also genuinely able to understand precisely how much, as he interacts with physical materials using his hands and senses.

2) To grasp how the decimal system operates, it should be understood that the decimal system is the foundation on which we base ourselves to establish order among numerical quantities. It is straightforward, cogent, and recurring and falls within the child's comprehension at this developmental stage. Here, we employ the decimal system's tangible grain and card materials to represent the quantities and symbols for figures above nine. Again, the child is introduced to the notion of place value through materials first, followed by the symbols, and lastly the association. He comprehends that in the decimal system, there is an absolute value that does not alter, but the relative value of a number changes based on the position it's in, such as hundreds, tens, ones, etc. At this juncture, the child will be capable of transporting quantities and forming numbers up to 9000.

3) To understand common names, which are names that we frequently use in society, such as eleven (one ten and one), twelve (one ten and two), etc., the child must recognize them in order to comprehend and communicate how names of numbers are used in society. Here, the child is introduced to the quantities and symbols through materials like the Seguin Boards, Hundred Board, and Short and Long bead chains.

This also makes the material countable and more acceptable for the child, as opposed to merely stating the names and numbers. When the child is comfortable with these common names, they are provided with the opportunity to count with either direct progression or by skip counting.

4) To understand the nature, as well as the application of the four arithmetical operations, the youngster needs to be familiar with the four operations in arithmetic: addition, subtraction, multiplication, and division. This comprehension serves as the foundation for carrying out essential calculations. The procedures are introduced in a group activity, where large amounts are combined or split by various children, and the results are counted and represented with symbols. At this stage, it is the understanding of the process that is valued over the correct answer.

The opportunity for the child to 'feel' or experience what actually happens provides a solid foundation in understanding the four operations. The child sees and experiences the amounts being combined or divided using the Decimal system's dynamic component material. They are further introduced to the stamp game, which reinforces the four basic operations. The stamp game reflects the gold grain material, initially introducing the process concept. Although it's more abstract, it demonstrates the same process to the child.

The representation of the quantities in the form of stamps enhances the child's understanding, enabling him to operate at a higher level of abstraction. Concurrently, the child is introduced to the basic combinations in all four operations through the Stripboards and the different charts. The child learns to understand the combinations, the commutative law, its relation to addition and multiplication, and the relationship among the four operations. Participating in meaningful activities helps the child memorize the combinations instead of merely relying on rote learning.

5) Moving into abstraction: After mastering the previous steps through repetition, the child develops necessary skills and forms abstract concepts to perform the operations independently. Working with materials like bead frames and the dot game verifies that the child, who has gone through the entire process, has achieved this level of independence. By this point, the child has been introduced to different levels of hierarchy up to one million and is prepared to work with large amounts.

These materials still have a concrete representation but are symbolically depicted to a greater extent. The child has steadily transitioned from working with beads and other physical materials, which served as learning aids. At the point of readiness, the child lets go of the physical aids to work in abstraction. It is at this juncture that the child's understanding becomes comprehensive.

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