Math-Multiplying Play
As they play the day away, preschoolers pick up six major math concepts.
Teachers call it "enumeration" — he calls it fun.
Four-year-old Nita is playing with four dolls from a set of six. Passing by, her mother asks, "Where are the others?" Nita says, "Um ... I'm calling you 'one,'" as she points to each doll. "You're 'two,' you're 'three,' and you're 'four.' Where are your sisters 'five' and 'six'?" She plays with the dolls for another minute. "Oh! You're 'six'? And you are 'five'? Well, let's go find sisters 'three' and 'four.'"
Nita incorporated counting into her play to keep track of her dolls. Given that play is so important to young children's development, it's not surprising that it's also the source of their first "pre-math" experiences.
The Big Six: Math Concepts that Emerge from Play
During free play, preschoolers frequently engage in informal math activities. Researchers have found that six categories of math emerge through play:
1. Classification: grouping, sorting, or categorizing by attributes. Anna takes out all the plastic bugs from the container and sorts them by type of bug and then by color.
2. Magnitude: describing or comparing the size of objects. When Brian brings a newspaper to the art table to cover it, Andy remarks, "This isn't big enough."
3. Enumeration: saying number words, counting, instantly recognizing a number of objects, or reading or writing numbers. Three children draw pictures of their families and discuss how many brothers and sisters they have and how old the siblings are.
4. Dynamics: putting things together, taking them apart, or exploring motions such as flipping. Teddy flattens a ball of clay into a disk, cuts it, and makes "pizza."
5. Pattern and shape: identifying or creating patterns or shapes, or exploring geometry concepts. Jennie makes a bead necklace, creating a yellow-red color pattern.
6. Spatial relations: describing or drawing a location or direction. When Teresa puts a dollhouse couch beside a window, Katie moves it to the center of the living room, saying, "The couch should be in front of the TV."
Math can be a seamless part of your child's play, but it truly blossoms in a supportive environment in which you provide challenges, suggestions, activities, and vocabulary. Think about how math emerges in all these situations:
Movement
Blocks
Materials
Pretend Play
Games
Computers
Movement
At first it might appear that sensorimotor play, such as the game pat-a-cake, is only distantly related to math. However, these activities provide direct experience with mathematical ideas. For example, very young kids love to jump up and down, march, or chant; these rhythmic actions are the basis for the concept of patterns. Preschoolers might chant "up" as they jump high and "down" as they crouch low. As they say "up, down; up, down," they are creating a connected movement-verbal pattern. Older preschoolers can add more complicated, deliberate patterns, such as "clap, clap, slap; clap, clap, slap" to their repertoires. They can talk about these patterns, representing the pattern with words. Music can help deepen these patterns.
Kindergartners enjoy making up new motions to fit the same pattern, so "clap, clap, slap" is transformed to "jump, jump, fall down; jump, jump, fall down." At this age, children can also describe patterns with numbers, such as "two marbles and one block," showing the first clear link between patterns and numbers.
Children who have played in this way will intentionally recreate and discuss patterns that show up in their artwork. One 4 year old loved knowing the rainbow colors and painted rainbows, flowers, and designs that repeated this sequence several times.
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Blocks
When they build with blocks, preschoolers also build their math, science, and general reasoning abilities. Classic wooden blocks and other construction materials, such as connecting blocks, give children entry into a world where objects have predictable similarities and relationships.
Consider how block building develops. Stacking generally begins at about 1 year. At 2, children place each successive block on or next to the one previously placed and appear to recognize that blocks do not fall when so placed. At 3 to 4 years old, children regularly build vertical and horizontal components within a building. When asked to build "a tall tower," they use long blocks vertically because they have added "tall" to the goal of making a stable tower.
Preschoolers use, at least at the intuitive level, more sophistical geometric concepts than most children experience throughout elementary school. For example, 4-year-old José is making the bottom floor of a block building. He lays two long blocks on the floor, going in the same direction. Then he tries to connect them with a short block. It doesn't reach, so he carefully adjusts one end of a long block. He tries the short block again. It reaches, so he quickly places many short blocks, creating the floor of his building.
Like José, many children intuitively use concepts of parallel and perpendicular. José even seemed to understand that parallel lines are always the same distance apart.
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Materials
Water, sand, and play dough offer many rich opportunities for exploring mathematical thinking, reasoning, and concepts. Measurement, for example, underlies water or sand play. Five-year-old Mary Claire fills different containers with the same cup of water, counting how many cupfuls she could fit in each container. Her mom asks, "I wonder which of these holds the most cupfuls of water?" Excitedly, Mary Claire begins experimenting and estimating.
By asking questions and providing interesting materials, such as cookie cutters, you can extend your child's thinking. For example, try making multiple copies of the same shape in play dough with the cookie cutters, or transforming sand or play dough into objects. Jeremy and his mom are playing with play dough. She tells Jeremy that she is going to "hide" his ball of play dough. She covers the ball with a flat piece of play dough and presses down. The boy says the ball is still there, but when she lifts the piece, the ball is "gone." This delights him, and as he copies her actions, they discuss how the ball is "in" the flat piece.
Children's play with flat blocks (or tangrams), combining them to make pictures and designs and complete puzzles, reveals another developmental progression. Children at first are unable to combine shapes, but as they become familiar with the tangrams they gradually learn to see both individual pieces and the whole — they learn that parts can make a whole, yet still be parts.
By about 4, most children can solve puzzles by trial and error and make pictures by arranging shapes next to one another. With experience, they learn to combine shapes to make larger shapes. They become increasingly intentional, building mental images of the shapes and their attributes, such as side length and angles.
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Pretend Play
Mathematics in constructive play is often enhanced when dramatic play is added. Two children making block buildings next to each other, for example, may begin arguing that their own skyscraper is the biggest.
Consider this example from a preschool classroom in which some children are playing store. Gabi is the shopkeeper. Tamika hands her a "five card," which shows the number 5, along with five dots, as her order. Gabi counts out five toy dinosaurs.
The teacher asks Tamika, "How many did you buy?"
"Five," she responds.
"How do you know?"
"I know because Gabi counted," says Tamika.
Tamika is still working on her counting skills, and she trusts Gabi's counting more than her own knowledge of "five." By exploring this concept within the safety of familiar pretend play, she can further develop her knowledge.
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Games
Games with rules can easily be modified to create opportunities to learn math ideas, skills, and reasoning. Try Compare ("War"), Odd Card ("Old Maid"), and Go Fish. Your child can fish for matching numbers, or try to make a sum, such as 5 (a child with a 2 would ask for a 3).
Games such as Memory (or Concentration) encourage your child to use memory strategies and gain experience with arrays (rows and columns). Encourage him to declare whether he found a match and how he knows. Games such as Tic-Tac-Toe also promote thinking about spatial relations and strategies. "Race" or "path" games (like Candy Land) are similarly valuable. They usually involve generating a number with dice or a spinner and moving the number of spaces indicated. This provides a different, complementary way of making sense of numbers, closely connected to measurement. Games such as "I Spy" (something with 4 sides the same length, for example) or "I'm Thinking of a Number" (with clues about whether the right number is "smaller" or "larger") sharpen older children's knowledge of attributes and logical reasoning.
Good games promote more than concepts and skills — they encourage children to invent and test multiple strategies, communicate, negotiate rules and meanings, cooperate, and reason. Talk about the rules with your child and make up new ones together. When you play games, encourage her to discuss and evaluate her strategies, considering new approaches and solutions.
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Computers
Computer games can engage children in different ways than toys and paper do. They help encourage positive social interaction, cooperation, and problem-solving. For example, two preschoolers are playing a game called "Party Time" in which they can set out any number of items, with the computer counting and labeling for them. "I have an idea!" says Emma, dragging placemats to every chair on the screen. "You have to put out cups for everybody. But first you have to tell me how many cups that'll be." Before her friend can start counting, she interrupts, "And everyone needs one cup for milk and one for juice!" The girls work hard to figure it out, finally counting two times on each placemat on the screen. These children play with the mathematics in the situation, with solutions, just as they play with each other. Cooperation in a computer center sometimes provides a context for initiating and sustaining interaction that can be transferred to play in other areas as well — especially for boys.
Computer games can give immediate feedback that is very helpful — for example, making shapes transparent so that children can see the puzzle. Computer activity is often more effective in stimulating discussion than toys and also evokes higher levels of social play. Children tend to talk more and explain more of what they are doing on computers than when using other materials. At higher levels, computers allow children to break apart and put together shapes in ways not possible with physical blocks.
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Douglas H. Clements, Ph.D., is a professor of early childhood education at the State University of New York at Buffalo. He has also taught preschool and kindergarten.
