A crowd of children cluster around the math game table in their preschool classroom. Two children roll a die to determine how many blocks to stack on the back of a Duplo® elephant. The tower grows higher and higher. Tiffany rolls a three and carefully counts three blocks to balance on the elephant’s back. Willy rolls a four. Instead of counting, he points to each dot on the die and takes one block each time he touches a dot. As Willy stretches to place his last block on the top of the stack, the block tower begins to sway. Suddenly, it crashes to the table. The crowd of children roar with delight and quickly begin picking up the blocks for the next game.

New and exciting approaches to math education are emerging in early childhood classrooms. They have been developed by teachers responding to research that documents how children construct mathematical knowledge. Educators now know that children pass through mathematical reasoning stages, and the old materials, such as workbooks and many of the commercial manipulatives, don’t facilitate this process. However, math games developed by teachers who understand how to make math materials developmentally appropriate not only encourage mathematical reasoning, but are fun and exciting for children. Teachers and parents can easily create these games once they understand the mathematical stages of young children.

Stages of Quantification

Constance Kamii’s (1982) research indicates that children progress through three stages of quantification: global, one-to-one correspondence, and counting. As adults, we often think of counting as the only way to quantify. Children, on the other hand, begin to solve mathematical problems long before they are able to understand and employ counting.

• Global—Young children are guided initially by their perceptions when quantifying. This is reflected in the global stage of quantification. At this stage, children make a visual approximation of the quantity they are attempting to match. When asked to take as many objects as are in another group, they may create a pile or row that looks about the same as the model set. At the snack table, for example, a child might look at the teddy bear crackers on his or her neighbor’s napkin and take either a large or small handful to approximate the same amount for him- or herself.

• One-to-One Correspondence—At this stage, children attempt to make an equivalent set by taking one object for each item in the original set. They may employ different strategies to establish one-to-one correspondence relationships. For example, children may point to an item in the original set each time they select a new object for their own set. They may also line up the materials in rows so that each object they take is opposite one item in the original set. In the water table, a child might place one toy duck next to each of his or her friend’s ducks so that he or she is sure that they have the same number.

• Counting—Children in the counting stage of quantification understand that the last item they count represents the total. In order to use counting successfully to quantify, children must also realize that there is a particular order to saying the counting words, and each object can be counted one, and only one, time. In the block area, a child at this stage might count all of the cars and then count out an equivalent number of people so that each car has a driver.

Children cannot be taught how to quantify. They need to develop this reasoning on their own. However, the materials that parents and teachers provide can encourage young children to stretch their thinking and move forward in mathematical reasoning. New research on brain development indicates that this is happening at the youngest ages (Shore, 1997).

Math Games

Math manipulative games and board games channel children’s thinking toward mathematical concepts through active play. They allow children to make sets with movable objects and observe the results. The materials are also self-leveling, meaning that children in all three stages of quantification can play the same games. The materials often incorporate dice or spinners to encourage children to quantify and construct equivalent sets with the game pieces. Without realizing it, children solve math problems over and over again, all within the context of an exciting game. The games below were developed and field tested by teachers at the University of Cincinnati’s Arlitt Child and Family Research and Education Center. They are described in detail in the book More Than Counting (Moomaw & Hieronymus, 1995).

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Strawberry Game

In the strawberry game, children roll a die to determine how many cloth strawberries they can put in their individual vegetable baskets. They use ice tongs to pick up the strawberries. At the end of the game, children can compare quantities of strawberries if they choose. The teacher-made die consists of quarter-inch-round file stickers applied to a one-inch cube in sets from one to three. The larger dots are easier for young children to point to and quantify. The die only goes up to three since very young children are often overwhelmed by quantities larger than three. However, a standard one to six die can by substituted for children who are ready.

Mail a Letter Game

In this game, each player has a game board with a short path leading from a child sticker to a mailbox. The path spaces are valentine envelope and letter stickers. Children spin a spinner and move a small spool person along the path. The movers are made by gluing a macramé bead to a spool and adding a face and hair. Children quantify and create equivalent sets as they attempt to move their person the same number of spaces as indicated on the spinner. This game correlates well with the popular children’s book, A Letter to Amy (Keats, 1968).

Baby Game

The baby game is an example of a grid game. In addition to rolling a die and taking counters, children must place the counters on bingo-like playing boards. This gives children the opportunity to align counters with pictures on the boards in a one-to-one correspondence relationship and visually compare sets. The game boards consist of 15 pictures of multicultural babies, cut from wrapping paper and mounted onto poster board. The counters are tiny pacifiers, bottles, and rattles often used as shower favors. Young children can roll a one to three die and select one item for each baby picture. Older preschool or kindergarten children can roll two dice, add them together, and attempt to collect all three baby items for each baby on their boards.

Balloon Game

The balloon game is an example of a long path math game. All players use the same game board. Children move small people along a path made from balloon stickers and try to reach the party at the end of the game. If they land on a space that says "POP," they can decide the consequences—lose a turn, move back a space, go back to the beginning, whatever! Older or more advanced children may play this game by using two dice and adding them together. Most children add by counting all the dots initially. Eventually children begin to remember the addition combinations without ever having to memorize them.

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Why Use Different Types of Games?

In addition to providing variety, the numerous types of games vary somewhat in difficulty. The easiest games are manipulative and grid games, such as the strawberry and baby games described here. They are more concrete. Long path games, such as the balloon game, are the most difficult. Moving along a path is more abstract for young children than taking actual objects to form sets. Short path games, such as the mail-a-letter game, provide an intermediary step between manipulative or grid games and long path games.

What Is the Adult’s Role?

When playing math games with children, teachers or parents can carefully observe a child’s level of quantification and model the next stage. If a child is using a global method of quantifying, for example, the adult could model one-to-one correspondence by pointing to each dot on the die and taking one counter for each dot. If the child is already at the one-to-one correspondence stage, the parent or teacher might model counting. Adults can also ask leading questions to encourage thinking and problem solving. The following are examples of leading questions:

• Which basket has the most strawberries?
• How did you decide how many strawberries to take?
• If you roll a two, will you have a bottle for each baby?
• How many more spaces do you need to go to reach the mailbox?

What If a Child Makes a Mistake?

Children do make errors when playing math games. All of us pass through periods of making mistakes when we learn something new. The important thing to remember is that the error is a reflection of the child’s current level of thinking. Teachers and parents should avoid correcting children’s errors. Such corrections undermine children’s self-confidence and do not help them understand the underlying concepts. With experience, children correct their own errors as they move forward in their thinking. Children who are not corrected, but who are constantly encouraged to think and evaluate, become enthusiastic, confident problem solvers.

Conclusion

Math is an exciting part of the early childhood curriculum. Research has moved early childhood professionals forward in their understanding of how children construct mathematical concepts. Teacher-made math materials contribute to children’s development by encouraging them to think about real math problems they encounter through playing math games. Teachers facilitate this development by asking leading questions that draw attention to mathematical relationships imbedded in the materials. Children learn that math is fun and interesting—not something to avoid.

References

Kamii, C. (1989). Numbers in preschool and kindergarten. Washington, DC: NAEYC.

Keats, E.J. (1968). A letter to Amy. New York: Harper.

Moomaw, S. & Hieronymus, B. (1995). More than counting: Whole math activities for preschool and kindergarten. St. Paul, MN: Redleaf Press.

Shore, R. (1997). Rethinking the brain. New York: Families and Work Institute.