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 materials in his early years, he can easily and joyfully assimilate many facts and skills of arithmetic.
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 one (1), one thousand (1000), or one half (½), 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.
The Spindle Boxes are a foundational Montessori material which teaches children to associate numerals with quantities. In some materials, the quantities are fixed and the child’s task is to assign the proper numeral to each fixed quantity. However, in the Spindle Boxes, the numerals are in a fixed order and the quantities are loose. The boxes have ten compartments labeled with numerals zero to nine. In a separate basket, there are forty-five individual spindles. The first compartment is labeled with the numeral 0; this is the child’s first introduction to this symbol. Next, the child places one spindle in the compartment labeled with the numeral 1, two spindles with the numeral 2, and so forth.
To learn about teen quantities and numerals, the child uses a material known as the Seguin Boards. The boards have the numeral 10 printed nine times in a single column. On separate cards are printed the numerals 1 through 9. The child forms “eleven” by sliding the numeral 1 over the zero in the first 10. This shows concretely that the numeral eleven is made up of 10 and 1. Next, the child forms “twelve” by sliding the numeral 2 over the zero in the second 10, etc. The child is introduced to the words for teen quantities (eleven, twelve, thirteen, etc. up to twenty). The student may also place a ten-bead bar and the corresponding number of unit beads to create the matching quantity.
The Hundred Board challenges the young child who can count aloud from one to one hundred to recognize and lay out the numerals in the same sequence. The square board is divided into ten rows with ten small squares along each row, totaling one hundred small squares. The children work with a corresponding set of one hundred wooden tiles that are labeled from 1 through 100. The student spreads the tiles out on a rug, arranges them in numerical order, and places them (one tile at a time) onto the Hundred Board, working from the upper left-hand corner along each row to the right, down to next row, and so on until complete.
The Bead Chains give children the opportunity to begin learning multiplication tables (presented as repeated addition) and the concept of squares and cubes. There is a short (square) and long bead chain (cube) for each of the quantities 1 through 10. For example, the short bead chain of four represents the square of the number (16) in four 4-bars of beads. The long chain of four represents the cube of the number (64) in sixteen 4-bars, or four 4-squares. The task of the child is to place a label at each interval (4, 8, 12, 16, etc.). Skip-counting invites children to learn about relationships and patterns between numbers. As they build up to doing the longer bead chains, they are also learning organization, sequencing, and focusing skills. The long chain of 9 represents the cube and goes all the way to 729. The long chain of 10 is a work that often takes at least two to three hours to complete – a major accomplishment.
Children are introduced to the decimal system using the Golden Bead materials, learning about the concept of units, tens, hundreds, and thousands. (A single bead represents a “unit,” a bar of 10 units in a row represents a “ten,” 10 ten-bars form a square representing “one hundred,” and ten 100-squares forms the cube representing a “thousand.”) This material also includes corresponding numeral cards printed in different colors to indicate the place values of the decimal system. The units (or 1s) are printed in green, 10s in blue, 100s in red, and 1000s in green (again, to represent units of thousands, followed by tens of thousands, etc.). Initially, the teacher asks the child to bring simple quantities, such as “Bring me 3 units.” Soon, the child can combine quantities from different columns: “Bring me 5 tens and 7 units.” Eventually, the children enjoy accumulating large quantities on a tray (such as 8 thousands, 4 hundreds, 3 tens, and 7 units to equal 8,43). Once a child is adept at this, it is a simple matter to move on to manipulating these four-digit numbers with the various operations (addition, multiplication, subtraction, and division).