Your Child's Geometric World

Douglas H. Clements

Geometry at three? You bet! From an early age, children make sense of the shapes they see in the world around them.

Your toddler, after some experimentation, puts a square block into a square hole. It fits! She tries again. Success is a great feeling.

Watching this thoughtful exploration, you might ask yourself: What does my child know about shapes? What is she learning? What more will she learn in preschool? What might she learn if given the opportunity?

Your children "do" mathematics spontaneously in their lives and in their play. We've all seen preschoolers exploring shares and patterns, drawing and creating geometric designs, taking joy in recognizing and naming specific shapes they see. This is geometry -- an area of mathematics that is one of the most natural and fun for young children.

During the preschool years, children's intuitive knowledge of geometry frequently exceeds their numerical skills. By building on strengths and interests that are already present, parents and teachers can foster enthusiasm for mathematics and provide a logical context to develop number ideas.

In a preschool classroom you might see a scene that looks something like this: A teacher challenges Michelle and Debbie to use their bodies to make a shape together. The children sit down facing each other and stretch their legs apart. With feet touching, they create a diamond. Another child takes a look, sees the diamond shape, and says: "If we put someone inside, we can make two triangles." Immediately, they ask Ray, the smallest child, to scrunch in and lie across the middle. It works! A diamond can be divided into two triangles. Michelle notes that there's a shape that has six sides, and she wants to try making one of those. Another child may even know that the shape is called a hexagon. After a brief discussion, Michelle gets five other children together. Then, under her direction, they all lie down on the floor and create a six-sided shape.

Building your child's geometric imagination
Preschoolers are learning to make mental images -- pictures they can carry in their minds. Young children ten to form static images -- "still" mental pictures they can refer to. Older children are learning to form dynamic images that they can move or change. For examples: Five-year-old Brian eyes the gerbil cage, trying to figure out if it will fit in the back of his mother's car. He thinks so, and later that day he finds out he's right. Building children's geometric imaginations is an important part of exploring spatial relations and experiencing mathematics.

By age six, children often have stable yet limiting ideas about shapes. Four-year-old Tina tells her teacher, "That's not a square. It's too big. A square looks like this." Her classmate Charlie adds, "Triangles have to be this way. That's not a triangle. It's too upside down." You can broaden your child's understanding by pointing out to her a variety of examples -- squares that are many sizes and triangles that are "long," "skinny," "fat," and turned in many directions.

You can also encourage deeper thinking about shapes not just through hands-on activities and discussion, but through picture books. In The Greedy Triangle by Marilyn Burns (Scholastic Inc.), a triangle gets so bored being the same old shape that it asks a shapeshifter for one more side and one more angle. Then it becomes a quadrilateral. After demanding more sides and angles and becoming a pentagon, hexagon, and so on, the shape learns that being a triangle is the most interesting after all. [For more books about math, see "Good math reads" on page 51.]

Building imagery
Your child's teacher will be engaging the class in various activities that encourage mathematical thinking. For instance, she will be helping children continue to build static and dynamic images through manipulative work with shapes such as unit blocks, puzzles, and tangrams.

After children have plenty of time to explore and work with these materials, she'll introduce "quick-image" activities. This is something that you and your child can easily do at home. Just draw a simple set of shapes on a piece of paper. Then give your child a brief glimpse and cover the image back up. You might draw a square divided into two small squares and a rectangle. After your child has had a quick look, ask her to share what she has seen. She may respond be describing the shapes or talking about the images she is reminded of. Remember: There really aren't any incorrect responses. You simply want to involve her in thinking about shapes and what she sees.

Building with blocks
When your child plays with blocks, he is creating forms and structures that are based on mathematical relationships. For example: When children are working to make a roof for their block building, they are struggling with the concept of "length relationships." When they substitute two shorter blocks for one long one, they are working with the concept of "equivalence." As they continue building -- higher and higher, wider and wider -- they are considering height, area, and volume.

Encourage your child to discuss his thinking and help him put words to his actions. For example if your child and a friend are arguing over whose block tower is biggest, ask if they mean whose building is tallest or widest or whose tower has the most blocks. They may be surprised to find that the tallest tower doesn't have the most blocks! This gives them a new way to consider what they're doing. You can also engage your child in thinking about the similarities and differences among the blocks they use and the structures they make. Suggest that he try a variety of challenges -- putting blocks in order by length or using shorter blocks to make a wall that's a s long as the longest block.

Remember that these activities are not meant to determine your child's aptitude for math or to see how many right responses he comes up with. They are, instead, opportunities to explore the properties of mathematics and encourage children to participate in logical, creative, and critical thinking.

Mapping the world
As all parents and teachers know, young children learn to get around early. But what can they understand about place and movement mathematically? Is it reasonable to think that your young child could actually make and use maps? Consider this:

  • Many three-year olds can build simple but meaningful maps with landscape toys such as houses, cars, and trees. Some threes can even replicate a room in miniature using blocks, toys, and other props.
  • Many older preschoolers know about relative distances between landmarks. For example, without having been specifically told, four-year-old Andrea many know that there is a greater distance between her house and Grandma's than there is between her house and the stop sign.
  • Many children can also place locations based on a route. Let's say you take a walk to the store with your child. You leave your house, turn right, go down the street one block, turn right again, and walk one more block to the store. Back at home, stand in front of the house with your child and ask him where the store is. Without ever having walked the route "as the crow flies," your child points diagonally in the direction of the store.
  • When challenged to learn a route through a playhouse that had six rooms, four-year-olds who examined a map beforehand learned a route more quickly than those who did not.

Geometry can be the most fun and naturally engaging aspect of mathematics to explore with your child. As children learn about the structure of shapes and space, they are building on what they already know. But we must all keep in mind that children learn these ideas most effectively through active engagement with toys, blocks, puzzles, manipulatives, drawings, computers -- and, of course, you!


Math around the house

Your child can learn math through everyday activities and by using ordinary household items!

Tableware Setting the table helps your child grasp the concept of one-to-one correspondence. How many people are going to eat? That's how many forks, spoons, plates, napkins, and so on he'll need so that there can be one at each place setting. Through this task your child will also practice patterning: At each place, he'll put the napkin and fork on the left, the plate in the middle, and the knife and spoon on the right. (Every now and then you can defy convention and have your child create a new place-setting pattern.)

Socks Empty out a drawer of socks (or have your child pick out all the socks from the clean laundry). Your preschooler will observe that two of something makes a pair -- an important math word. Use the socks to teach him other math terms. For instance, is the number of red socks more than, less than, or equal to the number of blue socks?

Cereal boxes With cereal and other food boxes, your child can build towers, create a for, stack together doll furniture, and so on -- all of which will help him visualize three-dimensional geometric relationships and learn the concepts of "little," "big," and "bigger."

Buttons You can put to good use all those miscellaneous fasteners you've saved over the years. Empty your button box and encourage your child to see how many ways they "go together." He can sort them by size, by number of holes, or by color. This will help him see that items can be sorted and classified inn many ways. (This activity is for children over age three who do not put small objects in their mouth.)


Good math reads

Here are some great books about shapes and maps:

As the Crow Flies by Gail Hartman, illustrate by Harvey Stevenson (Bradbury Press)
Grandfather Tang's Story by Amy Tompert, illustrated by Robert Andrew Parker (Crown Publishers, Inc.)
The Shape of Things by D.A. Dodds (Candlewick Press)


Getting around!

Young children are not necessarily natural map readers, but they do possess impressive abilities that can be fun and exciting to explore. Here are some activities to do just that:

Where are we? Offer your child cut-out felt shapes of items in her backyard, such as trees, a picnic table, a swing set, the sandbox, and so on. Encourage her to lay out the shapes on a felt board to make a simple map. (The same can be done for her bedroom.) Point out that changing any item would change the map.

Which way is up? Your child may be ready to learn environmental directions such as "above," "over," and "behind," and develop navigation ideas such as "font," "back," "going forward," and "turning." To foster this learning, help her represent these concepts on simple maps. For instance: Ask her to use masking tape and help you mark a path from the door to the wastebasket. Together decide where to glue the door and the wastebasket on a large sheet of paper. Then look for and list the items on the map. As you child gets the hang of map making, she may want to work on some more complicated maps.

What's your perspective? Provide opportunities for your child to explore perspective. Take photographs of your home from different perspectives and spread them out on a table to discuss and think about. Another idea is to photograph your child's favorite playground equipment from different perspectives. Mix up the photographs and let her sort them according to structure.


Did you know?

  • Given a mixed grouping of shapes, 82 to 92 percent of preschoolers can pick out all of one type: rectangles, triangles, squares, or circles.
  • Young children are accurate at naming squares even when the squares are tilted.
  • Many young children recognize rectangles even when they are tilted.

Surprise! Statistics show that as children get older, their knowledge about shapes doesn't necessarily increase. That tells us we're not doing an effective job of building on children's great beginnings.


Promoting math thinking at home

Here are some guidelines to keep in mind:

  • Acknowledge your child's work by describing it rather than just saying it is "wonderful" or "good." This helps her learn to value feedback and to really listen to what you are saying.
  • Ask questions instead of making corrections.
  • Accept unusual or unexpected solutions.
  • Comment on your child's work with observations like these:
  • "I see that you have lined up three triangles and two circles. Tell me about your pattern."
  • "Would you tell me why you decided to do it that way?"
  • "That's a creative way of thinking about it. Let me know what you decide."
  • "That's an interesting plan. Is there anything you want me to do?"
  • "I noticed that you matched a yellow sock with a blue one. Can you tell me why you decided to do that?"
  • "Thank you for setting the table. I see you placed a napkin on each plate."

Good questions to ask:

  • "What do you suppose would happen if…?"
  • "What will you do next?"
  • "I wonder…"
  • "How can we check to see how close your guess is?"
  • "Why do you think that?"
  • "How did you figure that out?"
  • "Do you have any ideas about how we might begin?"
  • "How can we do this differently?"
  • "Hmm. I hadn't thought of that. Can you tell me about it?"
  • "Tell me about your pattern."
  • "What other things can we find shaped like a square/circle/triangle?"

Adapted from Family Math for Young Children by Grace Davila Coates and Jean Kerr Stenmark (Lawrence Hall of Science, university of California at Berkeley, 1997).


Author's note: Time to prepare this materials was partially provided by National Science Foundation Research Grant NSF MDS-8954664, "An Investigation of the Development of Elementary Children's Geometry Thinking in Computer and Noncomputer Environments" and National Science Foundation, grant ESI-9730804, "Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.

Also note that the photos by John Fortunato, that were part of the original article, are not included in this online version.

Author Info:

Douglas H. Clements, Ph.D., is a professor of early childhood education at the State University of New York at Buffalo and an active research and writer in mathematics and computer education. He has also taught preschool and kindergarten.

Other Articles by Douglas H. Clements:

7 Ways to Add Math to Everyday Play

Analyzing Children's Length Strategies with Two-Dimensional Tasks: What Counts For Length?

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A Case for a Logo-Based Elementary School Geometry Curriculum

Challenges for Teachers Attempting to Integrate a Mathematics Innovation

Computers in Early Childhood Mathematics PDF

Computers and Mathematical Assessment

Computers Support Algebraic Thinking

"Concrete" Manipulatives, Concrete Ideas

Constructing Geometric Concepts in Logo

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The Effective Use of Computers with Young Children

The Future of Educational Computing Research: The Case of Computer Programming PDF

The Geometric World of Young Children PDF

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Mathematics for Young Children

Mathematics in the Preschool PDF

Metacognition, Learning, and Educational Computer Environments

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Preschool and Kindergarten Mathematics: A National Conference PDF

Research on Logo: A Decade of Progress

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Students' Development of Length Concepts in a Logo-Based Unit on Geometric Paths PDF

Subitizing: What Is It? Why Teach It? PDF

Teaching Length Measurement: Research Challenges PDF

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