How Children Utilize Their Mathematical Mind as Part of Their Natural Progression
How Children Utilize Their Mathematical Mind as Part of Their Natural Progression
“Dr Maria Montessori took this idea that the human has a mathematical mind from a French philosopher Pascal and developed a revolutionary math learning material for children as young as 3 years old. Her mathematical materials allow the children to begin their mathematical journey from a concrete concept to abstract idea”. With reference to the above statement please discuss how these children utilize their mathematical mind as part of their natural progression, to reason, to calculate and estimate with these Montessori mathematical materials in conjunction with their aims and presentations? What is a mathematical mind? The Mathematical Mind’ refers to the unique tendencies of the human mind. The French philosopher Blaise Pascal said that ‘every human being is born with a mathematical mind’. Dr. Montessori borrowing this concept, further explained that the mathematical mind is the “sort of mind which is built up with ‘exactitude’”. “In our work therefore, we have given a name to this part of the mind which is built up with exactitude, and we call it the ‘mathematical mind’.
I take the term from Pascal, the French Philosopher, Physicist and Mathematician, who said that the man’s mind was mathematical by nature, and that knowledge and progress came from accurate observation.” – Maria Montessori, The Absorbent Mind, Chapter 17, Pg. 169 She said the qualities of a mathematical mind was such that always tends to estimate; needs to quantify, to see identity, similarity, difference, and patterns to make order and sequence. The concepts within the mathematical mind do not simply refer to common associations with math, such as basic operations. Instead, Montessori believed that the human tendencies lead one to be mathematical in thought. That is, basic human tendencies such as order, orientation, exactness, repetition, activity, and manipulation of objects, all lead to the development of a mathematical process of thought. “The child perceives, without conscious reasoning, patterns of relationships: things to things, things to people, people to people…
The mathematical mind [therefore] is a power to organize, classify and quantify within the context of our life experience” Mathematics is not only about additions or subtractions a child learns at the school, it is all around the child from the day he is born (or may be well before that). It is a well known fact that an embryo can hear its mother. So the mother says “the baby kicked me twelve times today” or “my delivery is within another two weeks” when he was in her stomach. And then after he was born he may hear ‘you were born on the second’ or ‘at eight you go to the bed’ or ‘one button is missing in your pajama shirt’ or in the society he may be questioned ‘how many sisters or brothers do you have?’ etc., A child’s day to day life is all connected with mathematics and all the basic conversations he has is very much involved with mathematics. In that case the child is born to a world that is full of math, created by human for their benefits and the child needs to adapt to it.
Children need math to sort, categorize and group things within his environment. They need to count, they need to learn the time and then gradually they need to work with arithmetic’s, geometry and algebra in the school when they grow up. “We must convey to the child the belief that we have made mathematics ourselves, and that we re-make it every time we move, think, work or play. We should help the child understand that it is simply part of our being human to have a mathematical mind”. – Gettman D, BASIC MONTESSORI, Chapter 1, Page 159. Teaching mathematics to a young Montessori child is not a difficult task as he is very much exposed to numbers during his day to day life. By the time they enter into the Montessori school most of them are able to count one to ten (we call this “rote counting”, they just count without knowing the real meaning of the counting). Even in the prepared environment, though the child does not directly work with the materials within the math shelf as he enters, he however indirectly learns math concepts such as repetition, calculation, exactness, fraction, estimation and classification and most importantly order through the practical life activities.
A significant discovery that Dr. Montessori made was the importance of offering indirect preparation for the math materials while children were in the sensitive periods for movement and the refinement of the senses. It is through children’s work with the Exercises of Practical Life and Sensorial materials that they first encounter and experience the concepts of measurement, sequence, exactness, and calculation Sensorial education is the basis of mathematics. Dr. Montessori said that children are sensorial learners. They learn and experience the world through their five senses. So sensorial education helps the child to create a mental order of the concepts he grasps using his five senses. “The skill of man’s hand is bound up with the development of his mind, and in the light of history we see it connected with the development of civilization.” – Maria Montessori, THE ABSORBENT MIND, Chap 14. pg. 138 Montessori firmly believed that the ‘hands’ are the mother of skills.
By providing Montessori sensorial materials to the child she was convinced that correct manipulation with quality and quantity would certainly create a lasting impression in the child’s mind with the understanding of mathematics. We place materials quite intentionally on trays, we color code activities, materials are displayed in a logical sequence, and we break down movements during presentations into series of sequential steps. The sensorial materials simply present three mathematics concepts of completeness, geometry and early algebra.
Dr. Montessori was convinced that there are two things to be introduced before working with mathematics. “Before beginning mathematics work, the child must therefore do two things: explore and accept the notion of idealized things with isolated qualities, and gain practice in the requisite intellectual.” – MMI Mathematics Course Manual pg. 6 The child’s intellectual skills are developed through both practical life and sensorial activities. In practical life activities, children practice calculation skills when determining how much water to pour when carrying out exercises like pouring water from jug to bottle with an indicator line, or spooning beans from bowl to bowl with an indicator line, or from jug to jug; up to the more complex activities of sweeping which have the qualities of repetition, calculation and exactness.
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 of their magnitudes such as with the Pink Tower, knobbed cylinder etc. These sensorial materials also provides the child with the skills of calculation with the pink tower and red rods; as the child judges the size and length of the cubes and rods respectively, as well as repetition with baric tablets etc., All of the materials in the Montessori classroom have been specifically designed to attract the interest of the student, while at the same time teaching an important concept. The purpose of each material is to isolate a certain concept the child is bound to discover. 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. Dr. Montessori also said, ‘what the hand does the mind remembers’. The very first math material to be presented to the child is the number rods. Number rods are very concrete and help the child to feel and understand meaningful counting. It is also not very new to the child as he has already worked with the red rods before. The only difference is number rods are colour coded with red and blue, which helps the child to visually discriminate the difference in length and then to count the rod. The teacher presents the material by a three period lesson, and by repeating the same activity again and again, the child understands that two means two things and three means three things and so on and so forth.
The aim of the number rod is to help the child Learn the names of numbers 1-10 and visually associate the numbers with the quantity as well as to show that each number is represented by a single object, as a whole, separate from others. The number rods help the child memorize the sequence of numbers from 1 to 10. When the child counts one rod as a single unit, he immediately notices an increment in the number rod “2” even though it is still a single unit thereby helping him to associate the numbers to the quantity. “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.”……….Maria Montessori, The Discovery of the Child, Chapter 18, pg. 265.
“The satisfaction of discovery leads to an enthusiastic interest in numbers when the child is able to demonstrate the fundamental mathematical operations, rather than simply 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 grasping and reinforcing the ideas in a way that is concrete, rather than abstract.”……..Teaching Montessori at Home. Now the child is working with the concrete materials to understand the quantities of numerals one to ten and then he knows the written symbols too.
The next step is to teach him how to combine the quantities with the written symbols. This is done through a set of fun games. The Teacher invites the child to bring the number cards and the rods to the mat and then gets the child to identify the concrete value (the rod) first and then find and match the number card with the rod. Next the teacher requests the child to identify the number cards randomly and match them with the rods. This activity helps the teacher to observe how thoroughly the child is familiar with the numbers. The next two games help the child to understand the sequence of numbers. When the numbers and the rods are randomly scattered on the mat, the teacher requests the child to identify the number rods in sequence and then match the numbers with it and build the stair then in the next activity the child identifies the number cards in sequence and then matches with the respective rods and builds the stair.
The aims of these exercises is to establish the child in the recognition of numerical symbols 1-10., as well as help him learn association of quantity to symbol and also help the child understand quantity and sequence of numbers using manipulatives. Once the child is very clear with numerals one to ten, the next step is to teach the decimal system. Decimals are introduced to the child with the concrete manipulation using the golden beads. Through a three period lesson, the child is introduced to one, ten, hundred and thousand. The child feels and sees what one means by a small unit and then sees that ten is a long bar and then hundred is a flat square of ten ten-bars bound together and finally the thousand is a cube made up of ten 100 squares. The child can visually discriminate the difference in the sizes of different value and then feels it too. ‘Counting through’ helps them to further internalize the concept of decimal system. The teacher counts up to nine units and then says ‘if we have one more unit we will have a ten bar’. So this helps the child to understand that to make ten we need ten units.
Then to make hundred we need ten ten-bars and then finally the thousand cube is made out of ten hundred-squares. The great deal begins with the decimal system operations. Here the child is introduced to additions, subtractions, multiplications and divisions. The child learns the exact abstract way of additions or subtractions using the golden beads and large and small number banks. All these activities are teacher directed and working with these activities, helps the child understand that addition means combining two amounts together and then have a big amount at last; that subtractions means giving some amount away from what he had and then what remains is a small amount; that multiplication means having the same amount in to different numbers of times and gets a large amount as the answer; and finally, that divisions are giving the amount away equally or unequally among two or three people. These operations are very concrete to the child since he sees and manipulates the material. After manipulating with the concrete materials, the child moves to the abstract counting.
Using the large number cards, the teacher introduces the written symbols of power of ten (the decimal system). Then moves to the ‘counting through’ with the written symbols. Once the child is through with quantities and the written symbols the teacher shows the child to link concrete with abstract making the ‘Bird’s eye view’. Through the bird’s eye view the child can clearly see the process of the quantity increases with the written symbols. It gives the child the sensorial impression that when the symbol increases from one to ten, ten to hundred and hundred to thousand value of the quantity also goes higher. The aim of introducing the decimal system, is to help the child understand the concept of ten, learn the composition of numbers as well as the place value system and their equivalencies. After the decimal system operations, the child progresses to informal recording. By this time, the child knows the numbers very well and he is familiar working with sums too.
The informal recording introduces the child to small number rods. In the first presentation, he is concretely introduced to composition keeping ten as a guide and showing him how to make ten using rods up to six. Decomposition is also equally concrete, first he makes ten and then takes one away the child sees he is left with nine. During this presentation, the symbols of plus, minus and equal to, are also introduced and in the second presentation he is introduced to recording. The teen board is introduced to the child when he is through with the decimal system. It is also called ‘linear counting’. The short bead stairs varying in colour and quantity (one is red, two is green, three is pink, four is yellow, five is light blue, six is purple, seven is white, eight is brown and nine is deep blue) The coloured bead bars show clearly the separate entities from 1 to 9 and the ten-bars are the main concrete materials involved with the linear counting.
First of all, the child learns to build the short bead stair and then combines the short bead stairs with ten bars to teach the names of quantities eleven to nineteen. When the child understands the names of values, the written symbols are introduced through the ‘sequin board A’. Similarly the names of quantities from ten to ninety are also introduced and then the ‘sequin board B’ is used to teach the abstract concept of written symbols. The hundred and thousand bead chains reinforce the child’s counting from one to a thousand and also helps the teacher to evaluate child’s standards with understanding counting. The coloured bead bars show clearly the separate entities from 1 to 9, in combination with the tens they show the child that numbers 11 to 19 are made of ten and a number 1 to 9 The purpose of introducing the child to the linear counting exercises is to develop the child’s ability to recognize and count to any number.
As well as learn skip counting. The child’s own sound knowledge of the numbers 1 to 10 and their numerical order acts as a guide “This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child’s special aptitude for mathematics. When they leave the material, 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.” – Montessori M., The Discovery Of The Child, Chapter 19, pg. 279
Maria Montessori, The Absorbent Mind, Montessori Pierson Publishing Company, the Netherlands, Reprinted 2007 Maria Montessori, The Discovery of the Child, Montessori Pierson Publishing Company, the Netherlands, Reprinted 2007 Modern Montessori Institute, DMT 107 Mathematics Students’ Manual David Gettman, Basic Montessori, Saint Martin’s Press, 1987 Elizabeth Hainstock, Teaching Montessori in the Home, Random House Publishing Group, 2013