By: Lori Woellhaf
The educational charity Every Child a Chance Trust has commissioned a report prepared by KPMG, which says that 30,000 children leave primary school each year unable to do simple calculations. Furthermore, the research said that there was a significant link between poor numeracy and antisocial behavior.
As for older children, it was determined that teenagers leaving school without basic maths cost the taxpayer £1.9 billion a year due to unemployment. With this apparent failing in mind, we ask how Montessori approaches mathematics.
Dr. Montessori recognised 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 organises 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 organised.’
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 practise counting to ten, but it is more often done aloud or in the form of songs. Recognising 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.
The Montessori approach offers another concrete experience in the form of the Golden Bead material used to introduce the Decimal system. A child of four can see without being told the differences between one, ten, one hundred and one thousand: one unit is represented with one golden bead whereas one thousand is a cube made up of one thousand golden beads. As the child handles the material in a series of different activities the contrasts are enforced by the comparative weights and volumes of the items. The fact that the child has been given a vision of the whole scope of the Decimal system inspires wonder and a desire to explore further. In traditional schools the larger quantities are not introduced until the child is much older; this child is proud to say ‘I can count to 100’ whereas the Montessori child, having truly grasped the idea of the Decimal system can count on indefinitely.
Mario Montessori Jr made the following remarks on his grandmother’s method:
‘It is to her credit that she devised a means of introducing highly abstract concepts in a concrete way so that children could explore them at this early stage. A child manipulates them, performing actions and in the meantime, through this sensomotoric experience, gets acquainted with the principle or concept involved.
Numerals are generally introduced in a traditional school by giving dotted outlines of the numerals which the children trace over with a pencil. This offers the dual challenge of controlling the pencil whilst also following the outline provided; in Montessori this skill is learnt as a separate activity. The Montessori approach to introducing numerals focuses only on the symbols themselves using the Sandpaper Numerals: cut-out sandpaper numerals from zero to nine mounted on painted wooden boards. The child learns the shapes of the symbols using his senses as he feels the rough sandpaper and simultaneously absorbs its appearance and its name as he listens to the teacher saying each numeral aloud.
Although initially the child sits with the teacher to learn these numerals, he will also take part in games with them and will practise writing them on a chalkboard and on paper. Repetition is an integral part of the Montessori approach; Dr. Montessori observed that the child possesses a mind capable of effortlessly taking in unlimited information and so the more frequent his experiences, the deeper they are imprinted in his subconscious. In addition, children actually enjoy repeating these activities and will choose to do so unprompted.
But before the child even touches a piece of mathematics material he has spent lots of time preparing himself indirectly to work in a mathematical way. When, aged three years he spends time pouring water from jug to jug he observes and judges relative quantities. When he scrubs a table or polishes a mirror he learns how to set about a task in a logical way and to concentrate on a problem until it is solved. When he works with the Sensorial materials he is constantly required to sort, to look for similarities and differences and compare and contrast different series – all of these critical for his later work with mathematics.
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.