Maria Montessori observed a child's natural tendency toward numerical order, precision, classification, and measurement. She especially noted that learning is an active process that necessitates the child's direct manipulation of a prepared environment.
The Montessori environment encourages and nurtures the child's natural desire to engage in activities that develop skills and attain concepts. Skills are those specific abilities that can be taught by modeling or demonstrating and which the child can improve with practice. Concepts, on the other hand, are attained--they cannot be taught directly and are building on prior experience by adding bits of understanding to the child's conceptual foundation. The more a child can repeat an activity or experience, the more they internalized the knowledge.
Let's compare a child's conceptual attainment to a weak flame, that wavers and often goes out. The flame needs rekindling again and again until finally, it glows brightly. The mathematical concepts the child builds in the 3 to 6 classroom environment, form the groundwork for future mathematical achievements.
Mathematical concepts and relationships are constructions of the mind, and they cannot be transmitted verbally or taught directly to the child. The concepts and relationships require the coordination of mental and physical activity and action with objects. The development of logical reasoning unfolds from concrete reality to abstraction, which the child derives from the interaction with materials.
The child's progress in counting, numbers, and mathematical concepts vary according to their ability and developmental level. Once the child masters previous activities, new lessons are introduced.
There are five primary areas of math:
One-to-One Correspondence: As children learn to count, they must, at the same time, learn not only the concept associated with each number but the permanency of the number as well. Learning to count is essential, but simply counting will not help the child learn number concepts. They must also understand what the number they said stands for. For example, they learn that four forks are the same quantity as four spoons. The words "four" always mean the same quantity, and it's permanent in its meaning no matter the object. The young child learns this notion of numbers through lessons on one-to-one correspondence, which are divided by levels of difficulty.
Numeration: This includes teaching quantities, practicing counting, an association of a name with the quantity, and the notion that numbers consist of other numbers.
Decimal system: Using concrete golden glass beads, the child learns to identify the categories of the decimal system--thousand, hundred, ten, unit, and the quantities and composition. Visually and sensorily, the child perceives the concept of the size of the beads, which represent specific quantities. The child also sees the relationship of each quantity, as it is made up of "ten" of the previous quantity. Also, the child gets the weight perception (baric quality) by actually feeling the difference between one, ten, hundred, and thousand. Ultimately the child is introduced to all of the operations using golden beads.
Linear Counting: Teachers introduce a variety of materials to children to help them practice in linear counting. Ultimately students count not only one-by-one but skip counting by twos, 3's, 4's, etc.
Facts: Using concrete materials, children learn and later memorize vocabulary, including addend, sum--add with vertical and horizontal equations. They also discover various components of a number and experience commutative law.