Montessori way of teaching mathematic is truly a wonderful and interesting way to learn the concepts of mathematics. Mathematic is very important for our daily life. Mathematic relates to numbers and numbers are around us all the time, so it is very important to introduce numbers to the child at very early age, but before you introduce numbers it is important to lay a strong foundation of practical life activities and sensorial training to the child.
Practical life activities allows the child to gain precision in movements and strength of hand as well as the ability to focus on the work, it also helps in development of sense of order and logical and sequential thought pattern that indirectly prepare the child for mathematic .
It also provides an opportunity for the child to develop the social skills to work well with others. On the other hand sensorial training helps in child’s development of the senses that allows the child to better investigate the world around him.
Sensorial training also helps the child to learn mathematics when exploring dimensions with the red rod, pink tower, he experiences exactness & perception through cylinder block, sense of weight through baric tablet and so on. It also sets the ground work for geometry and algebra with the geometric cabinet and triangle boxes and binomial and trinomial cubes. The exercises in this area assist the child in building up their Mathematical mind.
Montessori looked at the Mathematical Mind as that of a mind that works with precision taking precise observations and working the new information into their existing framework of knowledge.
Work with the Sensorial materials builds up this precise mind by sharpening the observational skills of all the senses. “And if we look now at the sensorial apparatus which is able to evoke such a deep concentration, there is no doubt that this apparatus may be regarded not only as a help to exploring the environment, but to development of the mathematical mind. (Absorbent mind, Ch 17, 186)Activities in both the areas help the child to move with precision and to work exactness of movement and thoughts. Although number are around a child when he does not understand math at all, the child come around questions such how old are you? how many eyes do you have ??
But Maria Montessori believed that the sensitive period to mathematic in a child is in between 3 to 5 ? year of age, at that age it is easier to him to be introduced to the concepts of numerals, decimal system, place value, linear counting and so on…hence it is very important that the child gets correct nvironment and appropriate material necessary to enhance his mathematical skill and in-order to cater to this need of the child Montessori organized the Montessori mathematics into 6 groups, Introduction of numbers, decimal systems, teen board, operations, arithmetic tables and recording and abstractions. Montessori strongly believed that mathematic is a reality based where the concrete material is used to present abstract ideas for child’s better understanding, hence she designed the material in this area on the same approach. The activities designed proceeded from concrete to abstract.
Again with in each group the exercise moves from concrete to abstract. Maria Montessori initially introduced activities like number rods, sandpaper numeral, spindle box, cards and counter to the child, where he grasp the order from one to ten, he understands the value of each number, understand odd & even and is able to recognize the relationship of numbers and quantities, she then introduced decimal system to the child, child prior experience with the sensorial training makes these activities more interesting and acts as a bridge in his better understanding.
The child is introduced decimal system using golden bead material to teach the names of the quantities one, ten, hundred, & thousand through three period naming lesson. “Upon a thorough knowledge of units is achieved, the material is limited to ten. Once the units have been mastered, the whole decimal system is introduced. ” (course manual) The golden bead material consists of individual bead to represent unit, ten beads stuck together to make ten making line, a hundred square and three dimensional cube for thousand.
She designed this kind of material because she believed that the child cannot normally conceive the size of hundred or thousand. This give the child the idea thousand is equal to ten hundred or one hundred ten bars. Also the material helps the child to know that unit, ten, hundred and a cube completes one whole cycle of decimal hierarchy. As quoted by Montessori in her book, “Thus when the ten is repeated ten times, it formed a square of ten rows containing a hundred beads.
Finally, ten square placed over each other and tied together formed a cube. The cube of ten is thus one thousand” (The Discover of the child, Ch 19, pg 276). Once the child is familiar with the names of the quantities and since the child by now he already knows number from one to ten and has the ability to count, he is introduced with counting though activity where by he is given lesson to illustrate that in order to go beyond nine in any one category, it is necessary to go to the next higher category.
Upon the child’s familiarity with the tangible version, he is introduced to more abstract version in form of numbers cards. The number cards consist of four sets of cards representing the hierarchy of numbers in the base ten system, one to nine in green, ten to ninety in blue, hundred to nine hundred in red, and thousand to nine thousand in green. These written symbols help the child to relate how the quantities are written.
The same activities are repeated with theses number cards to now teach the child the names of power of ten of the written symbols and to reinforce the cycle of decimal hierarchy through counting through activities. The child is then graduated to association of quantities to their written symbols. Once the child has a understood the full concept, the child is then able to compose numbers by combing number cards and able to relate to its corresponding quantity or vise verse where the child is given the quantity of one or more category and he locates the appropriate number card.
Examples if the child is given the quantity four thousand, three, hundred, twenty two, the child is expected to bring the number cards of corresponding quantity, hence this activity not only helps in child’s power to familiarize the formation of complex numbers and match to its relevant quantity but also form an impression of the hierarchy of numbers, place value & role of zero in the place value. Once the child is clear with the concepts of decimal system, he is then introduced the next category of linear counting, Linear counting is counting objects in line or sequence.
This is introduced to the child with the activity forming a stair pattern with colored short beads to reinforce the number 1-10, followed by the next activity where by the child is introduced to the concept that when tens are added to units, they make teens. This is done through the activity where the child is given ten bars and shorts beads to connect and form teens. This concept is further reinforced by sequin board, which is also called the ‘ten board’ along with wooden number cards. The board has number ten running down.
The number cards can be slipped into the slots to form teen. The main purpose to do this activity is to show how the teens are formed and how they are written. The sequence of activities in this area is same as the decimal system, first the child to taught the name of the quantities and then shown the child how they are written, once the child gains proficiency in the number from 11-19, the child is then introduced to names of ten’s and sequin board b though naming lesson and are given activities to further practice the numbers in sequence from 11-99 using sequin board with numbers.
Colored bead chain gives a further good practice in linear counting to the child, in this activity the child is expected to mark a label after counting every 10 beads, He applies all his understanding and is able counts the beads up to 1000, This activity is also an indirectly prepares the child for addition, subtraction, multiplication, though children needs concrete material to understand and grasp the concepts of mathematics. Gaining his confidence over simple counting, sequencing of numbers and understanding a place value of the numbers, the child is then moved on to decimal system operations.
Maria Montessori divided operations into four categories, addition, multiplication, subtraction and division. She designed a variety of activities in each area with increasing complexity. Montessori’s hand on material makes abstract concepts clear and concrete to the child. The first activity in this area is addition operation. The activity requires 1 set of large number cards and 2 set of small numbers cards, and mat that that defines column for thousand, hundred, tens and unit and golden bead material bank. The child is given a two complex four digit number made with small numbers as the addends to be combined for the addition operation.
The child is required to get the corresponding quantity for both the addends from the golden bead bank and place them on the mat in their respective column, the two addends combine in the process of addition, beginning with the units, the child counts the combined quantity and determine the result by adding the two together, then the child brings the large number cards that represents result, the child visually see the concrete form, he also understand from this operation that two small quantities combine together to form a large quantity.
Once the child completely has understood the concept of addition, he moves to little complex activity. In the next presentation the procedure remains the only difference is if the addition has resulted in the quantity of ten or more, the child stops at count of ten and exchanges ten unit beads for one ten bar, this teaches the child the concept of carry forward or changing. It is very important that the child understands the concept of carry forward clearly as all the operations having carry forward are based on the same formula.
The child then moves to multiplication, multiplication is an extension of addition but here the child must understand that the term multiplication means the merging of several smaller quantities to make a large quantity. The child is provided with the same dividend two times or three times and following the same procedure as for the addition. First the child does the operation without change and then with child, then the child learns subtraction with and without change using the same material, but here the concept is not combining the quantities but taking away small quantity from the big quantity.
In this area the child learns to borrow the quantity from the succeeding hierarchy and finally to division where the child understands division means sharing the quantity, he also leans that if a quantity cannot be shared, there is a reminder. This way of teaching decimal operation to the child develop a strong understanding in mathematic, whereby his abstract form of mathematic becomes concrete to the child. “The satisfaction of discovery leads to an enthusiastic interest in numbers when the child is able to demonstrate the fundamental mathematical operations, rather than simple being told seemingly dull and meaningless facts.
He physically holds the quantities that he sees represented by written symbols. he combines the materials, counts, separates and compares them while visually graspng and reinforcing the ideas in a way that is concrete, rather than abstract. “. ( http://www. montessori-book. com/Math. html) After the child has developed the concept of the four basic operation addition multiplications, subtraction, and division the child is slowly transitioned toward abstraction. In the next activity, the child still works with concrete material using the small number rods but gradually makes a passage to paper and pencil.
This activity further reinforces addition and subtraction. The first activity the child does in this area is addition using small number rods to compose ten, the aim of the activity is to make the child understand how different numbers combine to get the same result, the child learns to form simple equation like 6+4=10 using small number rods on the mat and records the same on the paper. This is then followed by the next activity where his understanding is revived by asking the child ‘how many more he needs to compose 10’, in the process the child is also introduced to plus, minus and equal sign through three period naming lesson.
The composition and decomposition activities also help the child understand the meaning of these symbols. Snake game enhances this concept visually. The complexity further increases with the next activity as the child now start to do addition using series cards, the child by now understand the formation of a sum, he reads the sum on the series card, combines the quantity and records the answers. He learns to do subtraction and multiplication with the same principle. These activities further helps child’s gives practice in addition, subtraction and multiplication but with more abstraction.
The child then moves a step ahead toward abstraction where the child is introduced to addition strip board and red and blue rulers, and here the child is given table sums which lead the child through all possible combination in addition. None exist beyond combination nine plus nine, It also helps the child learn addition tables. Using the same technique, he then proceed to learn subtraction tables and multiplication tables with the introduction of subtraction strip board and multiplication board, In the strip board activities, the child has no quantity to relate to, he works with the written symbols and does sums and records the answers.
Group 6 is a transition to abstraction, helping the child internalize the function of arithmetic and gradually discrete the physical manipulation of material (Basic Montessori, Ch 5, pg 161) Once the child has full understanding of addition, multiplication, subtraction and division, he is ready to do complex four digit sums with all four categories, by now the child completely understands the decimal hierarchy and place value of the number.
The child is introduced with individual sums. Individual sums are four digit sums having all four hierarchy of decimal system namely thousand, hundred, tens and units. Each hierarchy is color coded with the same colors as on the felt melt, the child is given the color pencil to record the answers with the same color. The procedure to do individual sums is same as decimal operation but in here the child records the answers on the individual sum cards.
The repetition of these exercises not only solidified his foundation over understanding on decimal system operation but most leads to toward the most abstract form of mathematic for that age of a child, Upon his mastery over decimal system operation he is then introduced to stamp game which is the most abstract form, the child no more requires the aid of physical demonstration of its working. In the stamp game the child does the addition multiplication, subtraction and division in the same manner as in decimal system operations, only the “stamps” are used to represent the golden bead material.
The material here is more symbolic, the stamps has written symbols of 1, 10, 100 & 1000 on them, The child now understand very well the function of each symbol and is able to conduct the addition, multiplication, subtraction and division operation using just the symbols and record his answers on the sum card with the appropriate color pencil with no intervinience of an adult. Teacher only correct the answers if required using the control card Laying a strong foundation of decimal system upon a child plays an important factor in his development for mathematic skills and metal development.
This method of learning fosters deep understanding rather that memorizing. Child’s sensorial training and practical life activities acts as a bridge to enhance his aptitude towards his learning and the hand on material makes the abstract concepts clear and concrete, where the child can visually see the logical strategies and implement them in his understand mathematics in concrete Beginning with the sensorial foundations of mathematics, the exploration of order, sequence, and precision is viewed as a manifestation of the human tendencies.
Problem solving, formula derivation, discovery approaches, the role of the materials, and the connections between arithmetic, geometry, and algebra will be viewed developmentally in response to the mathematical mind and its evolving needs from early childhood to adolescence. ity. “This system in which a child is constantly moving objects with his hands and actively exercising his senses, also take into account a child’s special aptitude for mathematics.
When they leave the materials, the children very easily reach the point where they wish to write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations. ” Maria Montessori, The Discovery of the Child, Chap 19. Group 6: Abstraction The beads frame is the last set of materials before the child goes into the passage of abstraction. The exercises in Group 6 help the child internalise the functions of arithmetic, and he The final group of exercises in the arly childhood Montessori math curriculum provide further experience with arithmetic and move the child closer to abstraction and symbolic representation. Three activities – the stamp game, the dot board, and the small bead frame – make up the area of work that leads the child to abstraction. While each of these materials is initially introduced using addition, they can also be used to perform subtraction and multiplication, and in the case of the stamp game, division.
The stamp game is used by an individual child to do addition in the same manner as in the decimal system operations only “stamps” are used to represent the golden bead material in the following manner: (a) green stamps with 1 written on them represent units, (b) blue stamps with 10 printed on them stand for a ten bar, (c) red stamps with 100 written on them represent a hundred square, and (d) green stamps with 1,000 written on them stand for a thousand cube. The dot game is also representational and introduces the child to column addition and the decimal category of 10,000 using paper specifically prepared for this exercise.
Rather than combining the quantities of two addends using the golden bead material or stamps, in this activity the child combines two amounts by placing different colored dots representing the decimal categories in the appropriate columns. Dynamic addition is carried out by crossing out any rows of ten found in a column and placing a one for each row in the next higher category. “Rarely, however, can he count with certainty the fingers of one hand, and when he does succeed, in doing this, there is always the difficulty of knowing why,…
The extreme exactness and correctness of a child’s mind need clear and precise help. When numerical rods are given to children, we see them even the smallest take a lively interest in counting. “………. The Discovery of the Child, pg 264. “The satisfaction of discovery leads to an enthusiastic interest in numbers when the child is able to demonstrate the fundamental mathematical operations, rather than simple being told seemingly dull and meaningless facts. He physically holds the quantities that he sees represented by written symbols. e combines the materials, counts, separates and compares them while visually graspng and reinforcing the ideas in a way that is concrete, rather than abstract. “…….. Teaching Montessori at home. Abstract concepts become concrete as the student demonstrates his understanding of fractions. Maria Montessori often used color coding to help students solidify new concepts introduced. The color coding of the BarCulator halves, thirds, fourths, sixths, eighths, twelfths and twenty-fourths is thus very consistent with the Montessori method.
By working with this manipulative the student is first introduced to comparing fractions. “That the mathematical mind is active from the first, becomes apparent not only from the attraction that exactitude exerts on every action the child performs, but we see it also in the fact that the little child’s need for order is one of the most powerful incentives to dominate his early life” (Montessori, 1967, 189,190). The short bead stair: The Short Bead Stair represents the quantities one through ten in easily recognized colors.
A single red bead represents 1. A bar of two green beads represents 2. A bar of three pink beads represents 3, and so forth, with the bar of ten golden beads representing 10. If a child wishes to do the sum, 3+5, she places the bar of three pink beads beside the bar of five blue beads, counts the total number of beads and records her result: 3+5=8. Materials for futher practice: Parallel activities for addition and/or subtraction are always ready for the child who is eager to use different material.
These include the Strip Boards, the Snake Game, the Dot Game and the Stamp Game. Various boards are also available for multiplication and factoring whenever the child is ready and interested. The child can do simple addition with this Strip Board which features red and blue rulers of graduated lengths representing the quantities 1 through 9. On the large square board, a child can place the numbers 1 through 100 in sequential order.