# Mathematics Autobiography

Mathematics is yet another item of human culture introduced in a Montessori environment. However, this is a field which is created entirely by man and has evolved totally based on the needs of the man. It is the innate tendency of precision and order that has helped man create arithmetic. The abstract science of Mathematics helps man to evaluate, appreciate and consider parts and aspects of various entities with precision. Thus we find mathematics in all aspects of our everyday life. Human nature is not satisfied with a vague appreciation of quantities. Mathematics comes as a rescue to address this need by helping a man get precise answers from the perspective of quantity. The human child is born with all the human tendencies of precision and order. This is also the period where he has a heightened sense of order. We find that a child who has a proper, well-set routine feels secure and is at peace compared to a child who does not. Right from this age, the child is used to things happening in a sequence which is one of the basic principles of Mathematics. Soon, we notice a child who is just over a year old can understand the difference between a few & many, less & more etc. As a baby, he gets to see the quantity of food he eats. He also receives instructions from those around him involving numbers such as “You get one banana”. A young child who has started to speak may be heard saying “I am 2 years old” but will show all the five fingers. All of these happen in a state of unconsciousness. Often when adults hear children use such terms, they get excited to teach them numbers. However, the adults need to understand that at this stage, for the child, it is like the chanting of names with no understanding of precise quantities. The preparation for working with Mathematics takes places from birth. With all the help from his first, second and finally the Montessori environment he has spent time in, the child goes through a series of preparations before being formally introduced to Mathematics. He has established an internal order and gained mastery in precise movements by working with some of the Exercises of Practical Life material. He has established the ability to choose and complete a work cycle.

He has been normalized and has developed the ability to concentrate. He has even been introduced to some of the Sensorial material, which represents the mathematical gradation which he can see visually. However, all of this is an indirect preparation for the Introduction of Mathematics. When the child is around 3.5 years old, there takes place what Dr. Montessori calls “the awakening of the mathematical mind”. He now enters into a phase where he is no happier with statements such as some more, little more, etc. He now wants to know exactly how much. At this stage, arithmetic comes as an answer to his need for precise, accurate statements. It is offered to meet what he needs at this time, in the manner he needs for the purpose of his self-formation, taking into consideration the instrument of learning he has at his disposal which is his hands and his senses. Hence it has to be offered as what Dr. Montessori calls as psychoarithmetic. How is the help offered in the House of Children? Similar to the material for Sensorial Activities, the concrete material for Arithmetic are materialized abstractions. In the area of Mathematics, the pattern of presentations of the various materials always follows the rule of first introducing the quantity in isolation by giving it a name. Then the symbol is introduced in isolation and given the same name. Finally, the child is given the opportunity to associate the quantity and symbol. 1) To understand the concept of a unit as the basis of counting The child has to be introduced to a system of numeration. The preparatory period consists of helping him know the first 10 natural numbers. With the help of a unit, we count the quantities and we use the names of numbers. However, in a Montessori environment, materials like the Number Rods, Sandpaper numbers, Spindle Boxes, and Cards and Counters are used to introduce basic of counting, unlike traditional schools. This way the child not only learns to count but is also really able to appreciate exactly how much, as he is working with physical material using his hands and senses. 2) To understand how the decimal system works The decimal system is the foundation on which we base ourselves to establish order among numerical quantities.

It is simple, clear and repetitive and is within the child’s understanding at this stage of development. Here we use the decimal system static bead and card material to introduce the quantities and symbols for numbers over nine. Here again, the child is introduced to the place value concept through the material first, followed by the symbols and finally the association. He understands that in the decimal systems there is an absolute value which does not change but the relative value of the number changes depending on which place it is at i.e hundreds, tens, ones etc. At this stage, the child will be able to bring quantities and the formation of numbers up to 9000. 3) To know the traditional names Traditional names are names that we commonly use in society such as eleven (one ten and one), twelve( one ten and two) etc. The child should know them to be able to understand and be able to communicate how names of numbers are used in society. Here the child is introduced to the quantities and symbols via materials like the Seguin Boards, Hundred Board, Short, and Long bead chains.

This again makes the material countable and easier to accept for the child rather than just mentioning the names and numbers. Once the child is comfortable with the traditional names, an opportunity is given to the child to count with linear progression or using skip counting. 4) To know the nature and application of the four arithmetical operations The child needs to be aware of the four operations in arithmetic namely addition, subtraction, multiplication and division and that even for this counting is the basis to carry out the necessary calculations. Even the operations are introduced as a group activity where large quantities are merged and split by different children and the results are counted and expressed in the form of symbols. At this stage, it is the appreciation of process that it sought after and not the correct answer. It is simply an opportunity for the child to ‘feel’ or experience what actually happens. This provides the child solid foundation in understanding the four operations as they are visually seeing and experiencing the quantities being merged or split using the Decimal system dynamic part material. They are further introduced to the stamp game which helps in further learning and reinforcement of the four basic operations. The stamp game directly corresponds to the golden bead material which first introduced the concept of the operations. It is however more abstract, yet demonstrates the same process to the child. The representation of the quantities in the form of stamps consolidates the child’s learning of the concept and allows him to work at a higher level of abstraction.

Meanwhile, the child is also exposed to the basic combinations in all the four operations through the Stripboards and the various charts. This helps the child understand the combinations, the commutative law, how it applies to addition and multiplication and the relationship between the four operations. The combinations are memorized through engaging in meaningful activities and not just by rote. 5) Moving into abstraction After mastering all of the above by repetition, the child has developed the necessary skills and has formed the necessary abstract impressions to carry out the operations independently. Working with materials like the bead frames and the dot game prove that the child who has undergone the entire process has arrived at this level of independence. By this point, the child has been introduced to different levels of hierarchies up to one million and is equipped to work with large quantities. These materials still have a concrete representation but are also represented more symbolically. The child has slowly but surely moved on from handling the beads and the other such physical material he started his learning with. The material acted as a tool for the child to learn experientially and at a point when he is ready, to let go of that aid to work in abstraction. It is at this point that the child’s understanding is complete.

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