Dr. Montessori knew that the child aged six and under learns through their 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.
The Montessori approach using unique materials 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.
Before the child even touches a piece of mathematics material they have spent lots of time preparing themself indirectly to work in a mathematical way. When, aged three years they spend time pouring water from jug to jug they observe and judge relative quantities. When they scrub a table or polish a mirror they learn how to set about a task in a logical way and to concentrate on a problem until it is solved. When they work with the Sensorial materials they are constantly required to sort, to look for similarities and differences and compare and contrast different series – all of these critical for their later work with mathematics.
The child is free to explore the mathematics material at their own pace, without pressure. The materials are designed with their own ‘control of error’ so the child is always able to assess their own progress. They are introduced to the concepts of addition, subtraction, multiplication and division in group activities where they are given an actual concrete experience of the meaning of these functions. For example, they experience 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.
Montessori materials consolidate mathematical concepts in a systematic way, which leads the child from the concrete towards abstraction, originally allowing them to physically manipulate the material until a time when they can deal with the problem exclusively in abstract terms. Every piece of material isolates one concept, which integrates to form the basis for a further step in the child’s understanding of mathematics. The children are thus able to take on challenging concepts in geometry (volume, cube, roots), algebra (including BODMAS) and explore the mathematics that features in the world all around us.