The concept of number is the product of a gradual, social evolution. It began with a basic human need to understand and express ideas about the world in which they lived. Humans needed a language to express quantitative experiences like measurement, the passage of time, keeping track of possessions and formulas for inventions. The number system we use today is the outcome of thousands of years of creation. It began with the simple realization of the quantity one, and then, more than one.
In our civilization, the basic feature of the system of numbers used is ten. This decimal system is not an accident. It is easy to see that our fingers are a handy tool for counting, a built in abacus. Ten, in fact, has been the basis of most counting systems in history.
“In its natural state the human mind is already mathematical: it tends toward exactness, measure and comparison.”
Maria Montessori discovered that children between the ages of two and four were in their sensitive period for order. Order is the fundamental basis of the development of the mathematical mind. She wrote, “That the mathematical mind is active from the first, becomes apparent not only from the attraction that exactitude exerts on every action the child performs, but we see it also in the fact that the little child’s need for order is one of the most powerful incentives to dominate his early life.”
Montessori also discovered that children were in their sensitive period for sensorial discoveries. To meet the needs of these sensitive periods, she designed didactic materials that involve the use of the senses, the hand, and the mind. Sensory motor activities supply the mind with information, furthering a child’s intellectual development. As they absorb this sensory information, they begin to discriminate and create an internal order. Montessori believed children who absorbed information using their senses during this sensitive period would later on use this information for higher intellectual faculties, which she called, the mathematical mind.
“There is abundant research showing that movement and cognition are closely intertwined. People represent spaces and objects more accurately, remember information better, and show superior social cognition when their movements are aligned with what they are thinking about or learning.”
~Paula Polk Lillard
The Montessori math curriculum is intricately intertwined with the sensorial and the practical life curriculum for this exact reason. Both of these curriculum areas allow the child to explore objects and their relationships in a mathematical way. Practical life work feeds the child’s need for order and precision. Working with these materials also allows the child time to break down a task and create order in their work. This repetition sets patterns and creates work habits. Simple exercises in practical life reinforce the math concepts of one to one correspondence and one to many.
The sensorial curriculum teaches children about relationships between objects, shapes, dimensions and amounts. The concrete materials in sensorial are exact in construction. Many materials, like the tower of cubes, refine the visual sense to discriminate between sizes and help the child experience the concept of 10, the basis of our decimal system.
Montessori math begins with the concept of 1 – 10. The child is first introduced to these quantities in isolation and is given a name for them. The number rods are an example of a material that shows a visual difference between the quantities 1 – 10. Besides showing the one to ten relationships, it shows one to one correspondence. Next, symbol is introduced in isolation and is given a name. Sand paper numbers are a great material and used for this exercise. The child is given the opportunity to associate the quantity and symbol with materials like the number rods and cards or the spindle box only after they have been given the above isolated lessons. The sequence of numbers is automatically incorporated in both works.
After the child has mastered the concept of 1 – 10, the decimal system can be introduced. Concrete materials are used to give a visual sense of unit, ten, hundred and thousand. As the lessons progress, children are shown how numbers can be put together to make a larger number through addition and multiplication. Children are also shown how numbers can be sub-divided into smaller quantities with subtraction and division.
All these exercises include concrete materials which allow the child to practice their skills, after an initial lesson, with as much repetition as needed. The math materials build on each other, providing a foundation of concrete references for the abstract concepts that emerge with greater understanding and experiences, as a child moves from unconscious thought.
“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 can thus carryout an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations.”