This system Montessori Mathematics 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.”
Introduction to Montessori Mathematics
Mathematics deals with shape, space, members, their relationships and attributes by the use of numbers and symbols. It is a study of the science pattern and includes patterns of all kinds, such as numerical patterns, abstract patterns, patterns of shape and motion. In the Montessori classroom, almost all type math activities are presented to the child including counting, numeration, the decimal system, computation, the arithmetic tables, whole members, fractions geometry , etc.
We begin to offer mathematics to the child in the final two years of preschool i.e. age four to age five and six.
Mind is mathematical in Nature
The ability to count, to compute and to use numerical relationships are among the most significant gifts to the human mind . Little children are naturally attracted to the science of numbers. Mathematics, lille language, is the product of the human intellect. It is therefore part of the nature of a human being. It gives humans the capacity to understand the world in which they live . All humans exhibit this mathematical inclination, even little children.
It can therefore be said that human beings have a mathematical mind. Montessori took this idea, that humans have a mathematical mind, from the French philosopher Pascal. Maria Montessori said that a mathematical mind was “a sort of mind built up with exactly.” The mathematical mind tends to estimate, to quantify, to see identity . similarity, difference, and patterns, to make order and sequence and control error .
In Casa dei Bambini ( the first Montessori house of children established in 1907 ) children responded to the academic scenario. The children at Casa dei Bambini were too young to go to public schools, yet they expressed a great desire and fascination with numbers. To respond to their interest , Dr. Montessori developed a series of concrete mathematical learning materials that surprise many mathematicians and educators to this day. In a short span to time, the children were able to add and subtract four-digit numbers up to 9999. Soon they jumped further to multiplication, division, skip counting and mastered even more advanced and abstract mathematical concepts.
Indirect Preparation of Mathematical Mind
Understanding of numbers develops through experience with concrete objects and progressive they become abstract ideas. various areas of Montessori Method facilitate the development of mathematical mind and lead children to success in math activities. For example, Exercise of Practical Life (EPL) contributes effectively in the development of mathematical mind. EPL are the everyday household activities and include the courtesies of day to day living .
The child is naturally attracted to these activities as these experiences are essential for full development and self dependence. As he progresses with his work with Practical Life Exercises, they engage his intellect, satisfy inner urges to maintain order and practice required for mathematical activities. Participation in these activities help the child become a responsible and helping member of everyday living .
Along with it, at the unconscious level, practicing these activities forms the essential patterns in the nervous system that leads to the conceptual development of “Order” “Concentration” and “Independence”, which are all crucial elements of mathematical mindset. These skills are learned by the child through EPL , without even knowing him extensively in mathematical progress. For example, order is one of the basic elements for math , as we cannot do any mathematics without the capacity of sequencing and ordering .
Similarly, the ability to concentrate on a task is also essential for math as it helps develop logical thinking and problem solving. EPL also gives children good eye-hand coordination and controlled movements, which the math material presents in a Montessori classroom.
All Sensorial Material is also mathematical in nature. It is exact in dimensions and levels of progressions . It is presented with exactness and will be used by the child with exactness which prepares for the study of sequence, precision and progression. The progressive patterns in the materials, the relationships between each part of the other, their relative position in space, ordering and sequencing help prepare the mathematical mind . For example, red rods, pink, tower, broad stairs, geometric solid, binomial cube, and trinomial are considered as tools of preparation for mathematics.
Working with EPL and sensorial activities provide the child with a series of preparation for starting direct math activities, e.g.
- The child has established internal order.
- He has developed precise movement.
- He has established work habits .
- He is able to follow and complete a work cycle.
- He has the ability to concentrate.
- He has learned to follow a process.
- He has used symbols.
- He has learnt to follow a process.
- He has used symbols
All of this previous development brings the child to a maturity level of mind and a readiness of work with math materials.
Montessori Mathematical Experiences
The Montessori approach follows developmentally appropriate ways for the child to explore arithmetic. The following rules are observed.
. The match journey begins with concrete experiences and then eads the child towards abstractions
. There is also a progression of difficulty… simple to complex. Almost all exercises are preparations for the next steps.
. All exercises follow the following sequential progression.
. Firstly the child is introduced to certain quantities in isolation, and is given the name for it.
. Next, symbols (numerals ) for relevant quantities are introduced in isolation .
. Finally, the association between the quantities and symbols is made.
Group of Montessori Mathematical Exercises
The exercises in Montessori math are grouped. The progression from one group to another is mostly sequential, however sometimes parallel exercises from different groups may also be carried out. The preschool (age 3 to 6 years) curriculum normally includes the first three to four groups. While the rest is introduced at later stages.
Group 1 (Numbers through Ten)
The exercises in this group introduce the child to counting and numerals from 0 to 10 through concrete experiences. The exercises include Number Rods, Sandpaper Numbers, Printed Numerals, Spindle Box, Cards and Counters, Golden & Colored Bead Bars and the Memory Games.
By manipulation of these material, children build the basic concept from to ten not only to memorize the natural order of numbers, but also to recognize the relationships
between quantity and numerals.
Group 2 ( decimal System )
This group is introduced when the child has full understanding of numbers till 10. The child is introduced to the hierarchies system and how the system functions. The four operations of arithmetic i.e addition, multiplication, subtraction and division are also introduced at this level.
Group 3 (counting Bey; oun Ten)
The exercise of the third group can be introduced parallel to the exercises of the decimal system 9group), but once those exercises are well underway . These exercises include the tens, and liner and skip counting .
Group 4(Memorization of Arithmetic Tables)
This group can be introduced parallel to the later work of the decimal system and while group 3 exercises are continued.
Group 5 (Passage to Abstraction)
This group of exercises allowed the child to stop the use of the material as he is ready to start work with symbols on paper . These exercises may again overlap with the exercises of group 4. For example , the child who knows the process and tables for addition can begin to do the addition in this group also, while he may be working with other operations in group 4.
Group 6 (Fractions)
Fractions can be introduced parallel to the exercises of group 5.Some beginning exercises like sensorial work with fractions may even begin sooner than that. The advanced exercises in this group such as writing of fractions and the operations with fractions can be introduced later as the child gains more understanding of abstractions.
Role of the Directress
As for other areas of Montessori curriculum, the directress is responsible for setting up the prepared environment . She understand the role of presenting other areas like EPL and sensorial to the child for the indirect preparation of the mathematical mind before math exercises are introduced . Math materials are introduced only when the child is ready. She should present the exercises carefully and clearly with great zeal
. She should not unnecessarily repeat the presentations or correct the child . She should observe the children carefully and note down progress. She should allow the children to work with the materials for as long as they want and as many times as they want . She should encourage repetition until mastery.
Montessori Vs Traditional Math Teaching
An inappropriate approach to teaching math can easily lead children to a state of antagonism towards mathematics. In most of our schools, children feel uncomfortable doing mathematic exercises because mathematics is abstractly presented as a cold and boring subject. Hardly any concrete experiences are offered before children are put to work with abstract symbols. The traditional method, used normally in schools, includes the teacher taking an equation from the textbook and expecting children to internalize steps as he solves it on the blackboard . Then, he demands children to solve the remaining equations in the textbook according to the given steps in their notebooks.
He fails to understand that math is all about problem solving. Providing steps to solve a problem is like solving half of the problem, and making children dependent on the provision of steps to solve problems. The children should be encouraged to concentrate and find their own solutions. Secondly, young children need concrete experiences for the full development of the mathematical mind which is hindered if abstract symbols are presented right in the beginning .
The child should be provided with opportunities to feel , see and internalize concrete mathematical experiences without hurry or pressure, how numbers change and grow and relate to each other. In The Absorbent Mind , Dr. Montessori wrote that
“Children at an early age are urged by the laws of their nature to find active experiences in the world about them. For this use their hands, and not only for practical purposes, but also for acquiring knowledge.”
Technically, this idea is employed universally in Montessori classrooms. Mathematical concepts are presented in a fun and interesting way by using concrete materials. This helps children develop from concrete to abstract concepts. According to Piaget “Knowledge is not a copy of reality”. To know an object, to know an event, is not simply to look at it and make a mental copy, or it . To know an object is to act on it, to modify, to transform it and to understand the process of this transformation, and as a consequence to understand the way the object is constructed.
Three Period Lesson
Three Period Lessons are used for teaching new words to the children. The related vocabulary of a material is given after the child has mastered using it , we teach both the names of materials and the names of the qualities (positive, comparative and superlatives) related to these materials. The three period Lesson is divided into three steps, which makes language absorption easier for the child. The technique has been explained below with the example of vocabulary introduced with Tactile Numbers. Three Period Lesson for Naming
Period 1: Naming Period (This is……………)
The directress presents the child with three contrasting objects, in this case the numerals, and places them on a mat/table leaving some distance between them. She, then, traces the first number in the right way of writing it with two finger of her dominant hand.While, tracing she introduces the name of the numeral, e.g. “This is one”. She also asks the child to trace the number. In the same way she gives the names of the other two numbers. She repeats the names of each of the numbers, sometimes also changing their position on the table/mat.
Period 2 : Recognition and Association Period (Show me———-)
In nature’s course o;f development, the child learns to point out things before he begins to say the names of things. Thus, pointing towards things is a relatively easier skill than actually naming them. For example, respond to questions like “Where the fan?” etc. They either look towards those ;objects or point towards them even before speaking . Therefore, we also move to “pointing/showing” steps after naming, rather than pronouncing.
After naming the object a few times, the directress challenges the child to recognize the objects by their names …. saying . “Show me ————-”. For example, “show me two”. If the child makes an error, the directress should go back to period 1 without correcting the child.
The second period should be and the directress should spend more time in this period as compared to any other period in the lesson. She should change positions of the numbers and ask the child again to show a particular number.
She can also ask the child to do something with the number she names. For example, “please place three in my hands,” Or another example could be, “Please place three over there”.
She may move one step ahead and associate the language with the environment, asking if the child can point to another number 2 written in the classroom. The second period may be extended to several days.
Period 3 : Pronouncing Period (What is this?)
Once the directress is sure that the child can name the objects, she challenges the child objects himself. For example the director might point to one of the objects and ask : “What is this?” The child should then respond, for example, this is two.”
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