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 1, 1000, or 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 one of the many Montessori materials which help children learn to associate numerals with the proper quantities. In some materials, the quantities are fixed and the student’s task is to assign the proper numerals to each fixed quantity. In the Spindle Boxes, 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 One, two spindles with the label Two, etc. The first compartment is labeled Zero and this is the child’s first introduction to this symbol.
The Seguin Boards
To learn the “teen” numbers, the child uses a material known as the Seguin Boards. The boards have the numeral 10 printed nine times in a row. On separate cards are printed the numbers 1 through 9. The child forms the number eleven by sliding the number 1 over the zero in the first ten. This shows her concretely that the number eleven is made up of 10 plus 1. Then she forms twelve by sliding the number 2 over the zero in the second ten, etc. The teacher helps her learn the words eleven, twelve, thirteen and so forth. The student may also place a ten-bead bar and the corresponding number of units to create the quantity that matches each of the numbers in the Seguin Boards.
The Hundred Board
The Hundred Board challenges the young child who can count aloud from one to one hundred to lay out the numerals in the same sequence.
The Board is a square divided into ten rows with ten small squares along each row. The children work with a set of one hundred wooden tiles that are labeled from one through one hundred.
Students spread the tiles out on the rug, arrange them in numerical order, and place them, one tile at a time, on 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.
When they are comfortable with this, they attempt the same exercise by filling in the squares on a blank chart drawn to duplicate the surface of the Hundred Board.
The Bead Chains
The Bead Chains give children the opportunity to begin learning their multiplication tables, the concept of squares and of cubes. There is a short and long bead chain for every number from 1 through 10.
The short bead chain for the number four, for example, goes to the square of the number (16) in increments of four. The long chain of four goes to the cube of the number (64).
The task of the child is to place a small label at each interval (4, 8, 12, 16, etc.). This skip counting skill not only prepares children for higher math functioning, but also invites them to learn about the 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 (cube) chain of 9 goes all the way to 729, and the long (cube) chain of 10 to 1000. This work often takes at least two to three hours to complete – a major accomplishment.
The Golden Bead and Fraction Materials
Children are introduced to the decimal system using the Golden Bead Material. A single bead on the right represents a Unit. A bar made up of 10 Units in a row represents a Ten, Ten of the Ten Bars fastened together to form a square represent a Hundred, and a pile of 10 Hundred Squares forms the cube on the left which represents a Thousand. The children already know the terms square and cube from their work with the geometric materials.
This material is used with corresponding numeral cards printed in different colors to indicate the columns of the decimal system. The Units are printed in green, the Tens in blue, the Hundreds in red, and the Thousands in green again (because Thousands are actually Units of Thousands, followed by Tens of Thousands, etc.)
Children use the Golden Bead material to learn about the concept of units, tens, hundreds and thousands. First they build quantities with the bead material. The teacher starts with simple numbers. She says, for example, “Bring me 3 Units (3).” Soon she can combine numbers in different columns: “Bring me 5 Tens and 7 Units (57).” Eventually the children enjoy accumulating large quantities on a tray, such as 8 Thousands, 4 Hundreds, 3 Tens and 7 Units (8,437).
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).