The sequence of exercises through which the child is introduced to group operations with golden beads. “If education is always to be conceived along the same antiquated lines of a mere transmission of knowledge, there is life to be hoped from it in the bettering of man’s future” (Maria Montessori – The discovery of the child) Dr. Montessori recognized that children are born with a particular kind of mind, one that is naturally inclined towards order.

This ‘special’ mind is what gives humans the ability to make judgments and to calculate; it is how we have progressed in fields such as engineering and architecture. Dr. Montessori called this ‘the mathematical mind’ – a term borrowed from the French physicist and philosopher Blaise Pascal. Montessori felt that, if we are to support development, then we must offer mathematics at an early age since this is the kind of support that is appropriate for the kind of mind that we have. She observed: Great creations come from the mathematical mind, so we must always consider all that is mathematical as a means of mental development. It is certain that mathematics organizes the abstract path of the mind, so we must offer it at an early age, in a clear and very accessible manner, as a stimulus to the child whose mind is yet to be organized. ’ (Maria Montessori – The discovery of the child) Dr.

Montessori believed that children pass through sensitive periods when they possess a unique and amazing aptitude for learning; to take advantage of these sensitive periods, we must prepare environment to simulate their particular interests and allow them to exercise their innate ability to learn. I began to understand to that the shelf-works are actually the practical life and sensorial exercises that are as necessary as the materials insolated qualities physically and prepare the children before they begin the mathematics learning.

From the course manual and the various reading materials, I began to discover some important elements that will lead the children to progress mathematics learning from concrete to abstract using Montessori methods in teaching. These elements are:1) The Montessori Sensorial materials and Exercises 2) The Practical life materials and Exercises 3) The unique Montessori mathematics curriculum and materials When we think of Montessori Mathematics, we cannot just consider math materials only because sensorial training is of great importance in teaming the basics of mathematics.

The entire sensorial materials for dimension (knobbed cylinders, pink tower, broad stairs and long rods) are in sets of ten. When the child works with the sensorial materials, he is indirectly learning the units of measurement, the ‘tenness’ in the materials. For example, when the child has mastered the arrangement of the long rods, he has a sensorial basis for counting tens. The long rods are identical top the number rods: the child already has an impression and is familiar with the long rods which make learning concepts of 1-10 easier when the child is being introduced to number rods.

The long rods act as an indirect preparation for a more advanced activity. Dr. Montessori stated that the long rods provide an absolute and a relative concept of numbers. As the children handled and compared would help them to various combinations and contracts. (The Discovery of the Child, Pg. 264) The sensorial exercises are based on a logical learning sequence. It goes from the concrete to the abstract an example of geometry: In the traditional system, when we introduce ‘triangle’, we would show children a plane figure enclosed by 3 straight lines but is quite an intellectual concept and children are being ‘told’ then ‘discover’.

However, Montessori Method introduces the triangle in the form of solid wooden insets which can be taken out and fitted into corresponding wooden sockets. When the child ‘tough’ and ‘see’, he forms a visual image of the object and make it easier to remember. Secondly, the child is given the ‘triangle’ printed on cards wholly filled in. thirdly, the ‘triangle’ is not filled in now, but drawn with thick outlines. Fourthly, the ‘triangle’ is shown by a thin outline only and lastly it arrives at the Euclidean Definition of a triangle, i. e. a plane figure enclosed by 3 straight lines.

In this example, the child has gone through five stages before reaching the abstract concept. (The Course Manual, Pg. 73) The materials also contain a control of error, e. g. in the cylinder blocks, directress will not tell a child if he makes a mistake in placing the cylinder back, the child will make his own judgment when he discovers on his own if the cylinder will not fit in a hole that is too small, this guides him to accurate observation and also builds up the child’s perseverance skills, that teachers him not to give up so easily.

The sensorial materials basically help the child to learn through their senses. According to Dr. Maria Montessori, there is a purpose for sensory education: “It is exactly in the repetition of the exercises that the education of the sense consists the aims of the exercises are not that the child shall know colours, form and the different qualities of objects, but that he refines his sense through an exercise of attention, of comparison of judgment. Maria Montessori The sensorial materials provide the child with plenty of opportunities to improve his attention p as the child is allowed to work with the material as long as the child’s interest may hold. The sensorial didactic materials such as the cylinder blocks, the Colour tablets, the geometric cabinet and many other exercises provide the child the skills to make comparison and judgment that are so essential to prepare the child to the next level of learning.

When the children work with the sensorial materials, they are being introduced to the idealization of things and isolation of qualities. It fosters concentration and thinking skills, which develop the mathematical mind. The Practical life exercises in everyday living skills help the child to improve his fine motor skills, eye-hand co-coordination and concentration. The activities are familiar tasks to their home settings, such as pouring, transferring, sweeping or even cleaning the shoes. For example, the child begins with simple pouring exercise like pouring water into a bottle with a funnel.

These simple exercises prepare the child indirectly for mathematical concepts such as volume and capacity. The practical life materials are mostly practical and creatively made or assembled by the teachers. The various exercises often begin from simple to more complexes in design and usage. Each material has a definite purpose and meaningful to the child. The directress will also observe different children’s capabilities and arrange the shelves with activities helping to progress in their learning paths. These purposeful activities help to assist children in their development: physically, cognitively, socially and emotionally.

They increase the children’s attention p, help them to understand achieving objectives through set sequence and gain a sense of “I can do it” independence. These Practical life exercises ultimately help the children with three basic mathematical skills: exactness, calculations and repetition. Obviously, the Practical Life and Sensorial activities and materials have indirectly preparing them for learning the mathematical concepts and building a strong platform of foundation even before the child begins mathematical learning. Dr.

Montessori also knew that the child aged six and under learns through his senses and through movement, that is, through hands-on, manipulation. She concluded that she needed to provide mathematical concepts in a concrete form, which would be accessible to the children’s senses. A prime example is the material used to introduce the concept of quantity: the Number Rods. These wooden rods are painted in sections of red and blue so that each section represents the addition of a unit. The rod for two is therefore twice as long and twice as heavy as the rod of one; that the rod for ten is ten times larger than that for one is strikingly apparent.

In traditional education, on the other hand, mathematics is taught in a less hands-on manner. The child is given the abstract symbol as a starting point. Beads on a thread may be used to practice counting to ten, but it is more often done aloud or in the form of songs. Recognizing the symbol and counting up to ten does not imply an understanding of what these numbers mean; they are simply symbols and words to be said in sequence. Furthermore, counting individual objects such as beads requires the child to make the additional mental step of grouping objects together in order to come up with the quantity.

Far clearer is the Montessori approach of presenting the idea of the quantities as a whole using the Number Rods. As Dr. Montessori wrote, ‘When, on the other hand, in ordinary schools, to make the calculation easier, they present the child with different objects to count, such as beans, marbles etc. , and when, he takes a group of eight marbles and adds two more marbles to it, the natural impression in his mind is not that he has added 8 to 2, but that he has added 1+1+1+1+1+1+1+1 to 1+1.

The result is not so clear, and the child is required to make the effort of holding in his mind the idea of a group of eight objects as one united whole, corresponding to a single number, 8. This effort often puts the child back, and delays his understanding of number by months or even years (Maria Montessori – The discovery of the child) The Montessori mathematics curriculum is organized into six groups.

It begins with Group one introducing units of quantity from 1 – 10 using various concrete materials like the number rods, number cards, sandpaper numerals, spindle box, cars and counters and number games to learn the names of the numbers and repetition learning in counting from 1 – 10. The materials are designed to prepare the child indirectly for further learning. For example, the sandpaper numerals are used to teach children the name of the numbers from 1 – 10. By tracing the numerals with his fingers; he builds a muscular memory of the numerals and therefore prepare him to write numbers.

The number ‘Zero’ is being introduced through the spindle box when the child learns that ‘zero’ means ‘nothing’ and he does not put any spindles into the spindle box compartment that shows the numeral ‘0’. Each topic is presented to the child individually. The teacher also uses the ‘three period lesson’ to teach the concept of the numbers 1 – 10. For Example by using the sandpaper numerals. First period: The teacher places the sandpaper numeral 1 in front of the child and says, “This is one. ” She also traces 1 using two fingers. She continues with numeral 2 and numeral 3.

Second period : To test if the name of the numerals is associated in the child’s mind. The teacher places both the numerals 1 and 2 in front of the child and asks, “Show me the numeral 2” and asks child to trace 2, followed by “Show me the numeral 1” and trace 1. The teacher moves on to the next stage should the child succeed in the association. Third Period : The teacher points to the numeral 1 and asks, “What numeral is this? ” If the child is able to name the numerals, she replaces it with the other numerals and asks the same question.

Once the child has mastered the thorough knowledge of the units, then he will be introduced to the whole decimal system, learning the names of the power of ten using the bead materials which is Group two. The children are given concrete experiences with the units, tens, hundreds and thousands; and these learning are made possible and easy because of the golden beads and colorful bead stair. Montessori golden beads are golden beads of the same size, which are used to help children understand unit, quantities of tens, hundreds and thousands.

A single bead comprises one unit. Ten beads strung on a wire indicate ten. One hundred is indicated by ten bars side by side, which makes a square, and one thousand is ten one hundreds staked on top of each other, making a cube. It uses sensorial approach as the different bead materials can be differentiated in depth, weight and quantity. (www. montessori. com/goldenbeads) The children will explore the quantities and place values as they count though the units, tens hundreds and thousands using the materials.

As the children can see visually, making it easier for them to associate the name of the numerals with the quantities. With the use of the materials like the large number cards, children also begin to link written symbol with quantities. All these exercises help to prepare the children as they progress towards abstract learning later on. Next, in Group three, the child first learn with the beads and the beads are arranged vertically to help the child to associate his learning later when he is being introduced to use sequin board A to learn 11 – 19 and sequin board B to learn 10,20,30,…. 0. Correspondingly, the child is also using the sequin board B and the golden beads to reinforce concepts of learning numerals 11 -99. The correct number of ten bars and short bead stair provide control of error and give the child the muscular impression of the quantities. In Group four, the children are being introduced to not just addition and subtraction but also to multiplication and division. It is not yet written in the book but as a sensorial level with an interesting set-up of a banker and two players.

The children will fetch the beads, large numeral cards, small numeral cards on trays, making it fun like a trading game, but most importantly, the children are learning. All the directress has to do is to guide the children, observe their learning capacities and enjoy playing in the game as well. When the children have enough learning and understanding with the operation of decimal system, the children progress on to the next level. This is Group five. Lastly in Group six, the children will begin to transit to abstraction as they move on to internalize the function of arithmetic.

At this point, the children would have a good foundation practicing the individual sums with golden beads. Conclusion Indeed, the Montessori mathematics materials and methods have proven its efficiency that is necessary for laying the foundation of counting and arithmetical operations. These key elements of sensorial learning, practical life exercises and mathematics materials are important process for children to learn from concrete to abstract in mathematical learning. The child is free to explore the material for mathematics material at his own pace, without pressure.

The materials are designed with their own ‘control of error’ so the child is always able to assess his own progress. He is introduced to the concepts of addition, subtraction, multiplication and division in group activities where he is given an actual concrete experience of the meaning of these functions. For example, he experiences addition as the putting together of two quantities that results in the production of a larger quantity and multiplication as a special addition in that it is the putting together of quantities that are all the same.

The working in groups appeals to his need to interact socially at this age in sharp contrast to the solitary approach of the traditional ‘worksheet’. The Montessori approach results in the concepts being fully understood at a time when it is easy for the child to understand as long as the ideas are presented to him through the manipulation of concrete materials. By the time the children are six years old they have a solid knowledge of mathematics that will stand them in good stead not only for further study, but also for many other aspects of everyday life

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