Children possess and build mathematical competencies from their first year and keep on learning mathematical ideas throughout their preschool years and beyond. This is not surprising. Mathematics helps children make more sense of their physical and social worlds. "That doesn't fit me - 4 grew too big!" "No fair! She has more than I do!"

Young children invent mathematical ideas and strategies. For example, take 5-year-old Alex:

Alex's brother, Paul, is 3. Alex bounds into the classroom and announces, "When Paul is 6, I'll be 8; when Paul is 9, I'll be 11; when Paul is 12, I'll be 14."

Teacher: My word! How on earth did you figure all that out?

Alex: It's easy. You just go "three - FOUR - five" [saying the "four" very loudly and clapping hands at the same time], you go "six - SEVEN [clap] - eight" you go "nine - TEN [clap!] eleven"....

This small but remarkable dialogue reflects the potential all young children have to learn - and even to create - mathematics.

Before they enter school, many children develop early abilities in number and geometry, from accurate counting of objects to finding their way through their environment to making shapes. They use mathematical ideas in everyday life and develop informal mathematical knowledge that is surprisingly complex and sophisticated. With your guidance, children can become more acutely aware of this knowledge-an awareness that is crucial for mathematical understanding and learning.

Zachary's grandmother saw this awareness when she was walking him out of preschool. He stopped, pointed, and exclaimed, "Look, grandma! Hexagons! Hexagons all over the walk. You can put them together with no spaces!"

As such examples show, young children like doing mathematics. They all do. Boys and girls alike, in all socio-economic situations all exhibit spontaneous interest in mathematical ideas.

Young children can and should engage in mathematical thinking. All young children possess an informal knowledge of mathematics. Instruction should build upon and extend children's daily activities, interests, and questions, bringing the mathematics in such activities to the fore. This approach ensures that mathematical content will be meaningful for very young children.

You Hold the Key!

You can structure the classroom environment so that the potential for mathematics surrounds children. Show them the math in their everyday activities and plan special activities that focus on mathematics. Support their curiosity and offer appropriate challenges. You can:

• provide lots of unit blocks, along with time to use them.
• ask a child to get just enough scissors for every child who is in the group.
• challenge children to guess and check how many steps it is to the playground.
• sit down with children in large and small groups to pose, solve, and discuss mathematical problems.

It's also important to make sure mathematically oriented materials such as blocks are readily available. Notice that moment when building mathematical language and concepts requires intervention. For example, when two children each claim that his building is the largest, you might discuss how one is "taller" but the other is "wider" (or "contains more blocks"). You may decide to add materials after observing children. For example, when you see children comparing the length of two rugs, make sure that connecting cubes, string, and other objects that might be used for measuring are close by.

Math Around the Room

You can help children connect their informal knowledge to their budding explicit knowledge of mathematics. For example, children might be able to manipulate blocks to find that adding one block to a group of three blocks results in a group of four blocks. Later, they can be asked to do similar problems even when the three blocks are hidden. Eventually, they will be able to "count on." Asked what two more than three is, they might say, "Threeeeee ... four ... five. Five!"

Children should also be encouraged to connect mathematics topics to each other For example, children connect number to geometry by counting the sides of shapes, using rows and columns to understand number combinations, or measuring the length of a rug. This helps strengthen concepts in these areas as well as beliefs about mathematics as a coherent system.

Our world can be better understood with mathematics. Early childhood is a good time for children to become interested in counting, sorting, building shapes, patterning, measuring, and estimating. Quality preschool mathematics is not elementary arithmetic pushed down onto younger children. Instead, it invites children to experience mathematics as they play in, describe, and think about their world.

Linking mathematics to literacy and other areas strengthens both. Most good mathematics activities also develop language and vocabulary. For example, when children are lining up, teachers can build in many opportunities to develop an understanding of mathematics. Children wearing something red can be asked to get in line first, those wearing blue to get in line second, and so on. Or, children wearing both something red and sneakers can be asked to head up the line.

Understanding stories involves mathematical understandings, such as conditionals (if/then), classification, patterning, order, and number. Think of the numbers, size relationships, sequences, and repetitious patterns in "Goldilocks and the Three Bears" and other favorite stories. It's no wonder that research shows that early mathematics experiences, especially geometric ones, result in later improvements in language and literacy, as well as general intelligence.

Connecting With Families

Here are some ways you can involve families in children's math learning:

• Feature math nights. During these events, you might want to:
• Talk with families about your mathematics curriculum, including the wide range of mathematical concepts (see Chart: Development of Mathematical Concepts) and mathematical thinking that children will be involved in.
• Engage families in making some of the mathematics materials you'll be using (for example, cutting out colorful paper pattern blocks).
• Give families the opportunity to solve mathematical problems themselves, such as shape puzzle problems, so that they can experience the learning firsthand.
• Have a "mathematics show" in which children share some of the mathematics projects they've been involved with.