# Your Child's Mathematical Mind

How math, and its connections to everything, can help kids learn, understand, and think more deeply about the world.
By Douglas Clements, PhD

Ages

3-7

Everything around us can be better understood with mathematics. Math can help children make sense of and think about the many aspects of their world through its connections to them. When we see — and help our children see — those connections, we enrich their overall learning and development. You give your child a great gift when you help him make connections between the math in everyday situations and the math he learns in school.

The Math-Language Connection
Math requires people of all ages to think about what words meanWhen you "talk math," you have to be precise in your language and thinking. You have to explain your reasoning.Consider a rectangle. It has four straight sides and four right angles, but just because mostshapes we call rectangles have two long sides and two short sides, it doesn't mean that they have to. Math is an ideal context in which to discuss exactly what words mean — and that words can have different meanings. For example, sometimes people say "straight" when they mean vertical or horizontal, as when a picture is hanging straight on the wall. But other times, straight means not curving, as a straight side of a shape.

Learning how to use language and mathematical thinking benefits children in many areas. If a child doesn't understand why a toy car does not go down a ramp, then using mathematical ideas, such as height or angle (how slanted is the ramp?) can help her see the situation in new ways. When two kids are trying to figure out how to share something, such as a tricycle, math can be helpful: Set a timer for each child's turn. Sharing blocks could involve counting or dealing out blocks to each person.

Thinking Inventively
Everything can have a connection to math, and math connects to everything. Jumping, marching, and climbing stairs, for example, are all ways to practice counting. When children recognize, draw, play with, and combine shapes, they are not only learning about geometry, but also might be experimenting with visual art, architecture, and science. When children follow a story, they make mental pictures of the scenes and characters, using such phrases as "eyes as big as saucers," or the troll is "under the bridge."

These are all "spatial" ideas, which literally shape our view of the world-we use spatial concepts in almost all thinking. Later in their lives, children will use spatial ideas to think about communication networks, the structure of molecules, geography, and so forth. But spatial thinking is also basic to children's early cognitive development. In fact, research shows that working with and combining shapes actually improves young children's math achievements two to three years later-in addition to improving their writing and even their IQ scores!

Can it be true that all thinking involves mathematics? Yes. It all comes down to logic-a branch of mathematics that also happens to be a key aspect of the human thought process. Although logic might seem like the most abstract, least likely area of mathematics for young children to learn to use, researchers see implicit use of logic in all children from an early age. An 18-month-old child pulling a blanket to bring a toy within reach, for example, shows the beginnings of "means-end" analysis. A more explicit example of this early problem-solving ability is displayed by 3-year-old Luke: As he watches his father unsuccessfully look under the van for a washer that had fallen, he says, "Why don't you just move the car back so you can find it?" Luke used means-end analysis better than his father!

Young children show an impressive ability to think inventively. Encouraging your child to think mathematically at his own pace, rather than "rushing" him or showing him how to solve a problem, is an excellent way to meet his need for creative intellectual activity. If we pose problems and encourage kids to solve them in their own way, we help kids connect their informal knowledge with the more formal, in-school mathematics they'll learn later. We will ensure that children won't suffer the fate illustrated by Bill Cosby's line: "One and one make two. That's great. What's a two?"

Making Math Connections Every Day
Throughout the day, you can help your child connect her understandings to math by helping her represent her ideas. In other words, her intuitive ideas can become mathematical. Young children represent their ideas by talking, reading, writing, drawing, and playing. For example, think about some common stories and their connections to math. The Three Billy Goats Gruff includes a number right in the title. To understand the story, a child also needs to understand the concepts of ordering (small, medium, large), correspondences (between the goats' sizes and voices), relationships (the larger the goat, the louder their hooves),patterning (repetitive dialogue), and so forth.

In many books, the connections are clear. For example, the title character in The Very Hungry Caterpillar, by Eric Carle, eats from one to five items of food. For other books, the math might not be so obvious, as is the case with Blueberries for Sal, by Robert McCloskey. As Sal drops blueberries into her pail — "kerplink, kerplank, kerplunk" — you can help make a connection by showing your child a "pail and blueberries" (a tin can and marbles work well). Invite herto close her eyes and listen as you drop a number of marbles into the pail.

Here are some other ways to help your child make math connections through your everyday activities:

• Provide blocks and open-ended materials. Standard wooden blocks and Legos encourage children to build structures, learn about and combine shapes, compare sizes, and count. Playing with less structured materials, such as clay, sand, and water help children develop the foundations of measurement concepts. Encourage your child to use blocks and toys to act out and talk about his play scenarios, such as "these three cars are on the road to grandma's house." You could ask your child how many vehicles there are in all if he has three cars on the road to grandma's house and two trucks on the highway to the factory. Children often compare their block buildings, too. Ask, "How do you know your building is taller than mine?" They also naturally create symmetric designs and buildings. They will notice this symmetry and do more of it, more intentionally, if you discuss it with them.
• Use counting motions. If your child is in motion, perhaps going up stairs, help her count the steps or ask her to climb a certain number. Encourage her to hop on one foot seven times and to play games such as hopscotch that provide opportunities to work with numbers and patterning. Count how many times you can bounce a ball or skip rope without stopping. Don't forget the long jump: "What's the farthest you can jump?" "How will you remember how far you jumped?"
• Count everything. How many apples are in the basket? How many trees do you see outside your window? Count out food items during snack and meal times. Invite your child to set out enough snacks or cups for each family member, as it helps him see real meaning in that number. Matching straws to cups and plates to people develops the concept of one-to-one correspondence. Count the number of windows in a building or chairs around your home — just about everything your child is interested in, he can count.
• Play games and solve puzzles. Play games that involve counting, such as those with spinners, dice, or cards. Try "War" with a regular card deck, or play Uno. Puzzles, especially shape puzzles, build problem-solving skills, shape recognition, and spatial concepts.
• Set up a play store. As your child pretends to buy and sell groceries, toys, and so forth, he learns about counting, arithmetic, problem solving, and simple money concepts.
• Pave new paths. When your child plays in the sand, invite her to make roads for small cars. Then encourage her to talk about the paths she has made. These roads can be described geometrically (straight, curved, closed, and so on).
• Encourage computer use. Use computers wisely to "mathematize" situations.
• Collect, classify, and sort things. Encourage your child to sort his collections — rocks, marbles, pennies, shells, gummy bears, or anything. How many different shapes of leaves can he find? Can he sort apples by their color — green, red, or a little of both?
• Find shapes around you. Look for shapes at home and outside (for example, street signs). Look for shapes inside of things, like windows and bricks in buildings, or triangles in bridges. Cut sandwiches into different shapes, and, before they eat them, invite children to arrange them to make as many shapes as they can.
• Ask your child to show numbers with her fingers. Ask: "How old are you? How many pets do you have? How many cookies did you eat?" Ask her to show numbers in different ways — four can be three fingers on one hand and one finger on the other.
• Measure everything. Collect empty jars of various sizes and shapes and let your child explore and compare how much they hold by pouring water from one jar to another. Use a growth chart to mark your child's height — mark your own, too. Count how many steps it takes to get from the bedroom to the bathroom.
• Classify during clean up. When it's time to put away toys, books, or art supplies, encourage your child to classify things. Put all the blocks that are the same shape in the same box. This is also a good time to use spatial vocabulary, such as "next to," "inside of," or "on top of."

Children are impressive problem solvers. They are beginning to learn the rules of the "reasoning game." Through problem-posing and problem-solving your child learns to express his inventiveness. He will build connections among mathematics, language, and creativity — the essence of learning to think.

Measurement
Shapes
Thinking Skills & Learning Styles
Counting
Patterns
Age 7
Age 6
Age 5
Age 4
Age 3
Block Play
Math
Sorting
Early Math
Learning and Cognitive Development
Shapes and Sizes

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