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“The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)” Oxford Dictionary of English 3rd Edition, OUP, 2010
Mathematics makes it possible to observe, consider and appreciate parts and aspects of entities with precision, to make comparisons and establish relationships between entities. Mathematics is a science of structure, order and relations which has evolved from counting, measuring and describing number and shape, it encompasses logical reasoning and quantitative calculation.
It evolved from practical life movements, co-ordination and comparison and connects with all other aspects of human culture – the sciences, music, handicraft, cookery, construction, sports, dance and throughout the universe where macro and micro patterns of relationship exist in harmony, Mathematics has an aesthetic and spiritual dimension. Like the Sensorial curricula Mathematics is a study of universal phenomena.
The Mathematical Mind develops in contact with the environment, if nurtured the child will be able to express her reality and have strong foundations from which to reason and apply Mathematics.
Humans search for the nature of reality and it’s expression though symbols which abstract meaningful facts with precision, allowing us to communicate this meaning and preserve it. Precision, clarity and accuracy allow us to represent reality and reduce it to it’s essence. Finding simple, natural laws about reality has been a pinnacle of human development; the mind is able to generalise an idea, distill it into an explanation and apply it universally, drawing humanity together. This ability is described by Pascal as the ‘Mathematical Mind’ and the term is used by Montessori to describe a tendency of all humans.
Montessori’s approach to Mathematics
Abstracting the nature of reality is a human attribute, which all people should have the opportunity to understand and apply to their daily reality. The child is not born with the capacity to use Mathematics but has a Mathematical Mind, which given appropriate experiences in precise observation, comparison and opportunities to explore and compare different entities, the child sees the relationship between things.
When the child compares quantitatively she uses arithmetic, leading to abstract intellectual activity. When we compare the shape and dimensions of objects we use geometry, contact with the environment necessary for the later ability to abstract. The child is born with tendencies towards order, calculation, abstraction and generalisation, in the first plane she lays the foundation for logic, judgement and discrimination.
Exactness, precision, and manipulation are finely tuned as the child uses the purposeful, active Sensorial and Practical Life materials. Purposeful, precise actions hold the child’s interest and challenge her to repeat work, making further progress.
Indirect Preparation for Mathematics
To arouse a deep enthusiasm for Mathematics we introduce the Science of Number and give precise language to express her understanding of the full intellectual idea; we create opportunities for the child to develop her understanding of the relationship between the concrete and abstract.
The Exercises of Practical Life show the child how to move exactly and precisely, developing some degree of concentration aided by the Sensitive Period for Order. The sequencing of the activities lay the foundation for logical thinking, following procedure and applying judgement, especially in relation to the Points of Interest, for instance when the child sees objects laid out in sets, they can be separated or put together – the outer order she sees becomes internalised and an inner order develops.
The Sensorial Materials are made with scientific precision, with gradations which convey Scientific and Mathematical information. The materials of dimension introduce the decimal system. The activities develop the child’s capacity to discriminate differences and similarities, to sort and make fine qualitative and quantitative comparisons, self-creating the thinking processes necessary to comprehend Mathematics. The child becomes familiar with the precise unit of difference between two members of a graded series, bringing an understanding of the importance of order and the ability to see patterns, relationships and sequences.
The child studies the relationship between objects which are necessary for her to understand shape (geometry cabinet), form (materials of dimension) colour (tablets) and to be aware of their interplay (knobless cylinders), carrying out calculations, extracting abstract ideas from the qualities of matter (Baric tablets). The Sensorial Materials are designed to express these Mathematical facts and introduce algebra (the cube of the binomial) geometric relationships and finding area (constructive triangles) and fractions (small insets), which will be further explored in the Second Plane. Finally naming these Sensorial-Mathematical experiences helps to form clearer concepts and enable the child to communicate her experiences precisely. Reasoning within sharply defined limits provides the basis for intellectual, joyful abstraction and imaginative creations.
In order to compare quantities precisely addition is necessary, so the child is introduced to Numeration which is conventionally difficult for the child to grasp and as a result of which many adults dislike Mathematics. Montessori believed that we have a Tendency towards Mathematics and that if we can be taught it in a way which manages the difficulties it will be both enjoyable and simple for the child to handle. The first difficulties she identifies are that, (1) the succession of numbers and the placement of digits (units, tens, hundreds, thousands, decimals and the use of zero as a placeholder), (2) the sequence being not understood but simply memorises the sequence, masking the difficulties but which causes even more problems to the child’s understanding and confidence when it later when it emerges that the child has not grasped the basic concepts. (3) the nature of the unit itself – any entity we choose can be the ‘standard unit of measurement’ and it often has other associations, so that seven coins can be ‘one coin, seven times’, ‘coins’ or the numbers ‘one’ to ‘seven’ with each cat being given a different number. This lack of understanding can be very frustrating for the child, teacher and parent and the more examples exacerbates the problem.
Montessori solves these problems by presenting the quality of the unit as ‘an unbroken multiple’ by giving the child work with the ‘Addition of Variables’, in which the whole object can be seen to be one object made up of many identical parts, a real life example would be that a sari measures 6 yards, while being one item, so a sari is an unbroken unit of the multiple of 1 yard. This unbroken whole is offered first with the Number Rods, developing from the child’s familiarity with the Red Rods.
Later qualities are offered in loose identical unites, here they can be broken, this is known as the ‘Arithmetic of Groups’. The child is helped to establish contact with counting thorough the ‘Arithmetic of Variables” before the ‘Addition of Groups’, allowing the child to form the concept of numeration.
Concept of Numeration
Before learning to evaluate quantities and count with understanding the child needs to understand;
- The true meaning of a unit (standard of measurement)
- The role the unit plays in counting (How many units make a quantity)
- The relationship between the unit and the quantities above it
- The reciprocal relations between different quantities above the unit
After sufficient preparation direct help is sensorially offered for Mathematics. Objects with other associations (beads, marbles etc.) are not used, as they distract from the quantitative. The activators offer concrete opportunities to work, manipulate repeat and see the result of acting on numbers (operations). After the concrete has been understood a symbolic representation of these experiences is offered (the names of numbers and the numerals, then the numbers and symbols together). Abstraction and creative imagination result from the child’s keen sense of reality and her ability to hold this understanding in her mind. The whole of Mathematics is offered first and later it’s parts, with one difficulty isolated and presented, moving step by step towards individual work and abstract understanding.
The six groups of Arithmetic
Group 1 – Numbers from one to ten and zero
Group 2 – Decimal system of numeration and operations
Group 3 – Linear counting
Group 4 – Memorising base combinations
Group 5 – Passage to abstraction
Group 6 – Fractions
Group 1 – Numbers from one to ten and zero
For the child to abstract into units and quantities we give the numbers one to ten and zero, by the ‘Arithmetic of Variables’. The Number Rods, which are identical to the Red Rods in the sensorial materials, are rendered countable by being divided into ten centimetre sections with blue and red paint. Each Number Rod is ‘an unbroken multiple’ of the first unit, e.g. the ‘Rod of Seven’ is a single rod with seven sections, each section materialising the unit. The child sensorially realises that the first (smallest) rod is one unit, in terms of which the other rods can be measured and therefore she understands the succession of the numbers one to ten.
The Number Rods represent the physical qualities which are complimented by the Sand Paper Cyphers (numerals), symbolic representations of the qualities she has counted with the Number Rods. When the child can recognise each symbol she is given the Number Cards and uses the quantity and symbol together. Later the child develops this understanding when she puts combinations of Rods together to form the lengths of other Rods, (e.g. 7 + 3 = 10), this is the basis for the sensorial understanding of addition and subtraction.
Next loose quantities of identical objects are given, these are the ‘Addition of Groups’ they are sorted into groups with the Spindle Boxes, here the concept of Zero (not anything) is introduced. Group activities reinforce this concept. The Cards and Counters gives the Key to the World of Mathematics to the child leaving her free to explore and understand the Decimal System.
Group 2 – Decimal system of numeration and operations
The number ten is the lynchpin of this system and our understanding of many processes depends on our ability to apply the decimal system confidently. The child’s understanding of how to quantify individual numbers and groups of numbers and name them reaches a new point here, where the child handles large numbers. The child already understands how to make the numbers 1 to 10 and that 0 means ‘not anything’, she comes to understand here that these numbers can be compiled to represent all numbers, of any size. The immensity of the decimal system and it’s manipulation is opened up to the child.
Aided by the ‘golden bead material’ order and classification are offered to the child in sensorial experiences which show a simple yet profound idea.
1 Golden Bead is a unit (point)
10 Golden Beads make a ‘bar of ten’
10 ‘bars of ten’ make a ‘hundred square’
10 ‘hundred squares’ make a ‘thousand cube’
Why is this given early?
The young child has a right to appreciate the beauty of the Mathematical System, knowing only the first ten digits helps to show it’s simplicity. The sensorial handling of quantities at a concrete level (beads) comes first, followed by their abstracted numerical symbols and these two are associated in the ‘Formation of Large Numbers’, after a solid understanding of what each quantity represents has been reached, the children repeat the presentation until the idea is very clear.
A group of children who have reached this level of the Mathematical Mind and have some social abilities are helped to make the operations of putting together, splitting, putting the same number together and splitting a number into equal portions. These exercises are shown collectively to small groups of children at around five years, laying down the understanding of the exact processes. At this stage an appreciation of the process is what is sought; rather than the ‘correct answer’, the children should attain a ‘feel for’ what happens. Once a group of children have been introduced to the operations they become aware that no more than the number 9 can exist in any one category, and once the next figure is reached the nine must be exchanged from one of the next category in an active process which holds the child’s interest.
The child is led to independent work with the operations using symbols which are further abstracted from the quantities in the ‘Stamp Game’ and the “Dot Game”. Now she is familiar with making dynamic shifts and she is able to record her mental movements on paper. The decimal system, it’s laws and simple ordering effects are presented sensorially, at the right moment in the child’s development.
Group 3 – Linear counting
Certain groups of numbers have names which ‘jar’ with the simple effect of sounding the decimal names – e.g.’ eleven’ rather than ‘one ten and one unit’, these names are more convenient linguistically but do not express Mathematics directly. The child can use either wording to describe the graphical representation in order to perform the operations, however, the Director introduces the conventional name when she reflects the answers from the ‘Formation of Large Numbers’ onwards, as she coaches the child indirectly to express her understanding. The child is guided by herself to choose either to pursue the names or continue with the operations (Group 2), the director and materials are available to help her learn these lessons when she is ready.
Group 3 offers the names of the teen numbers and groups of tens e.g. ‘fourteen’ and ‘forty’, this is more complex in some languages than English, which takes less time. Once comfortable with the names chains are given to allow the child to count with linear progression or using skip counting.
Group 4 – Memorising base combinations
Basic tables help with later mental arithmetic work at the Elementary Level. Patterns with operations are shown through games in which the child discovers the basic and essential combinations, the commutative law, as it applies to addition and multiplication, and the relationships in subtraction and division.
The materials again move from concrete to the more abstract. As the child repeats the same patterns in different ways she internalises the information. The tables are memorised with enjoyment and ease, through engaging activities, rather than learning by rote.
Group 5 – Passage to abstraction
After sufficient experience, the child has gained enough skills and has formed sufficient abstract impressions to apply the knowledge to the material to complete all the operations independently. The material is a bridge for the child’s realisation that she does not need the materials to perform the operations on paper. She realises that her understanding is complete.
Group 6 – Fractions
The child works with a quantity below the unit.
Nurturing the inner processes
There is no benefit in the child reciting numbers by rote without understanding their significance. The child is able to distinguish one from many sensorially with as much ease a she can distinguish small from large. If she can do so with precision the basis for all of Mathematics is within her grasp. Confidence, a quiet and secure understanding, is needed to enjoy Mathematics, which if nurtured by the Director imparting knowledge clearly and with wonder, motivates the universal Mathematical Mind, available to each child. Mathematics is presented holistically, spanning Geometry, Numeration and the Algebraic, with difficulties resolved one at a time.
All levels of Mathematics are given in the three step process, firstly the idea is introduced, secondly experience is provided and thirdly the child is brought to awareness of what she knows. It is necessary to prevent mental barriers obstructing the natural acquisition of knowledge, so Mathematics is presented as a world to enjoy and explore not as a series of agonising problems demanding to be solved. The materials often have an inbuilt Control of Error freeing the child from external evaluation as she has her attention drawn to her own error and is able to ‘fix’ it. The naturally builds her confidence and she develops a positive self image of herself and her ability to apply Mathematical reasoning, logic and her ability to calculate. This not only resonates in her later abilities in Mathematics and The Sciences but also on her capacity to judge, evaluate and think clearly.