Mathematics

Counters

Equipment for this exercise is a small wooden box with two compartments. In one side are red wooden numbers from 1 to 10. In the other side are red plastic disks called counters. The child is shown how to space the numbers out in a row across the top of his workspace. The counters are laid out under the numbers in two vertical columns. Therefore, the 1 has one counter under it, slightly to the left. The 2 has two counters under it, one in each of the two columns. The 3 has three counters, two in the left column and one in the right column. The purpose of this lesson is to introduce the child to the concept of odd and even numbers, to having 1 left over, and to the concept of counting by some number other than 1, in this case by 2’s. These concepts will be used again and again as binary systems, set theory, and division and multiplication are introduced. At this age, the child is enjoying laying the numbers and counters in orderly rows. Eye/hand coordination and counting are reinforced.

At home, enjoy letting your child lay things out in orderly fashion, such as setting the table, arranging books on a table, or lining up his toys on the shelf.

Golden Bead Material

The golden bead material is the introduction to our decimal system. A single bead represents the unit. A strand of ten beads on a wire is a ten bar. Ten of the ten bars fastened together to form a square make a hundred square. The hundred squares form a cube of 1000. These terms ‘square’ and ‘cube’ are familiar already from the sensorial material. The first lessons with this material are simple. The children build quantities with the bead material. The teacher illustrates how much easier it is to work with a ten bar than with ten individual beads. Soon the child can combine different numbers of sets, such as 2 thousands, 7 hundreds, 2 tens, and 3 units. Eventually, the children enjoy accumulating large quantities on a tray. They learn to represent these quantities with corresponding cards. When the cards are stacked, the number is shown as 2723.

The bank game is an exercise in exchanging the beads. The large quantity of material which the children use as a source for the game is called the bank. The children use the bank whenever they want to change units to tens, tens to hundreds, hundreds to thousands, or vice versa. Adding and subtracting four place numbers can all be done with the golden beads.
If two children wish to add, each one puts a quantity of bead material on a tray and selects the corresponding cards to represent the quantity. They then select large cards to represent their total. For subtraction, the teacher places a large quantity of bead material with the corresponding large cards on a large tray. She then gives the child a smaller tray with a number written on a small card. The child ‘takes away’ this quantity of bead material from the large tray and puts it with the small cards. The quantity remaining on the large tray is the answer. The child then finds the number cards to represent the answer. With this game, the children gain a real impression that subtraction is the breaking up on one large number into smaller ones.
At home, you may see your child delight in very large numbers. Share with him how many times MacDonald’s hamburgers go around the earth or how heavy a pickup truck is. Our world is full of mathematical concepts that the child is eager to explore.

Bead Cabinet

When the child has mastered the bead stair, he is introduced to the bead cabinet. The beads are in the same colors (the 1 bead is red, the 2 beads are green, the 3 bar is pink, etc.) but they present the concepts of squared, cubed, set theory, and skip counting. The beads in the cabinet are arranged in squares of each number, cubes of each number, and bars chained together to equal the number in the cube. The child is shown how to take small plastic pointers and count by 2’s or 3’s or 6’s. He may also learn the squares and cubes of the numbers 1 through 10. The thousand chain is a particularly fun exercise. It reaches 27 feet and generally takes up a large part of floor space. The child is impressed with the size of 1000. There is a great deal of self-esteem involved in completing this long exercise. Counting by 10’s to 1000 and writing the numbers in a book is a big job for a 5 or 6 year old. It is not a task an adult would set for such a young child. But he chooses to do it himself with great joy.

At home, validate your child’s interest in numbers by listening and responding to his stories. Share mathematical concepts by pointing out cubes, squares, and sets. If you enjoy fun games with numbers, by all means share them now with your child. As your child begins to approach the time when abstract concepts appeal to him (around 6), introduce ideas about eternity, light years, and space.

Hundred Board

When the child has developed a good concept of teens and how tens progress, he is introduced to the hundred board. This is a blue board marked off into10 rows of ten. Small white markers in the shape and size of the marked off squares on the board have numbers from 1 to 100. The child is shown how to place the numbers correctly on the board. He is delighted with the orderliness of counting by tens down the right side of the board and of how all the 2’s and 6’s and 9’s line up under each other. He is shown how to write the numbers to take home for himself. As his confidence with numbers increases, he has no fear of moving into addition and subtraction, even of large numbers, for after all, hasn’t he written all the numbers up to 100?

The language development is the ability to count to l00 without saying ‘twenty-ten’, ‘twenty-eleven’, ‘twenty-twelve’. At home, have patience with telling your child for the fourth time what comes after 59. Delight with him in the wonderful expansion of his knowledge and thrill with him at the largeness of 53 or the great old age of 87. Help him understand the relativity between his brother’s 8 birthday candles and his father’s 33 candles or between 3 miles to school and 67 miles to the beach.

Strip Boards

The mathematic operations are taught using beads and strip boards. The strip boards look like large game boards. Red and blue rulers of graduated lengths represent quantities from 1 through 9. These are laid along the base board and the child quickly sees what the result of addition or subtraction is (depending on the board used). He may then transfer the concrete form to his number book, so that he can begin to understand numbers at the abstract level.

The most successful use of this material is to get the child to the point where he no longer needs it. It is a significant maturity level for the child to become independent of the equipment. Sometimes parents become excited that their child is adding and subtracting four-digit numbers and carrying tens. At the beginning, however, the child is still dependent on the equipment and cannot duplicate his efforts at home. Take care to be gentle with his new-found ability. Be aware that you do not put him in a position where he must fail. The best way to know how to help your child is to follow his lead. Let him show you what his interests are.

Sequin Boards

There are two sets of Seguin boards, named after Carl Seguin, a contemporary of Maria Montessori and a noted educator. There is the ‘teen’ board, which is a board with nine slots painted with 1 and 0 in the background. There are nine plaques painted with the numbers 1 through 9 to slide into the slots and create the numbers from 11 to 19. The child is shown how to lay a golden bead ten bar on the left side of the board to correspond to the ten place and to lay unit beads beside the board to correspond to the number to be slid into the slot. For example, the child forms 11 by sliding a 1 into the top slot over the 0. This is a concrete illustration that 11 consists of 1 ten and 1 unit. With the ten board, the child can create two digit numbers from 21 through 99.

There is a great deal of language introduced with these boards. The child learns to concretely demonstrate numbers up to 20 and to count properly to 99. (There is no ‘twenty-ten’, for example.) He learns to associate such large numbers with concrete materials. The logic and beauty of numbers begins to form in the child’s mind. He may also begin to delight in counting verbally to 99 or in singing nonsensical songs that count to 99. Have patience. You probably will not go insane before this stage passes.