Introduction to Mathematics

     A child can learn basic concepts of mathematics in either of two ways. He can learn by using concrete materials during the years when he enjoys manipulating equipment; or he can learn by abstract methods when he is in the elementary grades. Dr. Montessori demonstrated that if a child has access to mathematical  equipment in his early years, he can easily and joyfully assimilate many facts and skills of arithmetic. On the other hand, these same facts and skills may require long hours of drudgery and drill if they are introduced to him later in abstract form.
     After she observed that the child who becomes interested in counting likes to touch or move the items as he enumerates them, Dr. Montessori designed concrete materials to represent all types of quantities. In a Montessori environment, a child not only sees the symbol for 1, 10000, or 1/2, he can also hold each of the corresponding quantities in his hand.
     Later, by combining this equipment, separating it, sharing it, counting it, and comparing it, he can demonstrate to himself the basic operations of arithmetic. This activity gives him the satisfaction of learning by discovery rather than by being told. Eventually he develops an early enthusiasm for the world of numbers.

THE RED AND BLUE RODS

     In the Montessori classroom the child's first introduction to numbers is made with a set of red and blue rods representing the quantities one through ten. The teacher helps the child to count the alternating red and blue sections of each rod as he arranges them in stair-like formation. The child calls the smallest rod One, the next rod Two, and so forth. The Number Two Rod is a unit, yet it is equal to two of the Number One Rods.
     At about the same time, the child learns the corresponding figures by tracing the numerals in sandpaper. The teacher helps him to place each of these numerals beside the rod illustrating that quantity.
     Working with this equipment gives the child an opportunity to discover many mathematical facts. For example, if he places the Number One Rod end to end with the Number Two Rod it will be exactly the same length as the Number Three Rod. The child is also able to see basic multiplication and division; for example, the Number Two Rod will fit on the Number Six Rod exactly three times.
     He can also use the rods to demonstrate the various combinations that equal the Number Ten Rod. He can place the Number One Rod beside the Number Nine Rod, the Number Two Rod beside the Number Eight Rod, the Number Seven Rod beside the Number Three Rod, the Number Six Rod beside the Number Four Rod and the Number Five Rod taken twice.

    THE SPINDLE BOXES

     The Spindle Boxes represent a parallel exercise in associating the numerals with the proper quantities. This tie the numerals are in a fixed order and the quantities are loose. The Spindle Boxes have ten compartments labeled with the figures Zero through Nine. In a separate box there are forty-five spindles. The child puts one spindle in the compartment labeled 1, two spindles with the label 2, etc. The first compartment is labeled 0 and this is the child's first introduction to this symbol. He usually wants to put a spindle in this compartment but has to learn that Zero means none or nothing.

THE NUMERALS AND COUNTERS

     In this exercise both the symbols and the quantities are loose and both must be placed in order by the child doing exercise. First, he arranges the numerals in ascending order. When placing the appropriate number of red discs under each figure, the child puts the discs in row of two. Each odd number has only one disc in the bottom row. This arrangement automatically illustrates the odd ans even numbers.

THE SEGUIN BOARDS

     To learn the "teen" numbers, the child uses equipment known as the Seguin Boards, The boards have the numeral 10 printed nine times in a vertical row. On separate cards are printed the numerals 1 through 9. The child forms the number 11 by sliding the figure 1 over the 0 of the 10. This shows her concretely that the number 11  is made up of 10 plus 1. Then she forms 12 by sliding the figure 2 over the 0 of the second 10. The teacher helps her with the words eleven, twelve, thirteen and so forth.

     Another set of Seguin Boards is available for learning the numerals 21  through 99. To build the corresponding quantities in this exercise, the child uses colored bead bars. Therefore, work with the Seguin Boards usually begins after the child has been introduced to the Golden Bead Material.