It's important to teach abstract thinking throughout the day, in every learning center, by talking about and helping children reflect on their experiences. Here are some math activity ideas that support abstract thinking.

Use number words everywhere. Very young children do not always understand that number is an important attribute of collections. Even after they do, they may not have an exact sense of numeration. Using number words helps them build both of these ideas and teaches them that numbers can be used to categorize things.

Help children abstract the counting rules. Activities and discussions can help children develop the ability to apply the counting rules. For example, have a pup- pet, "Mr. Mixup," count incorrectly, and invite the children to correct him. Ask them to describe what he did wrong.

Play with routes and maps. Very young children can talk about landmarks they see when they take walks, within or out- side of the school building. They can begin to create models using toys that represent these landmarks. Older children can try to build a model of the classroom and eventually start to draw simple maps. Emphasize that models and maps are "shrunken" versions of a space.

Use manipulatives of all types. Good abstract thinking is connected to concrete experiences. Manipulatives, such as pattern blocks, shape sets, and unit blocks, as well as real-world objects, such as buttons, help children build representations of mathematical ideas.

Choose manlpulatives carefully. Manipulatives should not limit children's solutions. To build their knowledge, children should be in control of their solutions. "Open" manipulatives, such as connecting cubes, may lead to greater abstraction than do more structured materials, such as base-ten blocks.

Use manipulatives wisely. Encourage children to use manipulatives in many different ways. Blocks can be used for counting, arithmetic, patterning, and building geometric forms.

Talk about manipulative use. Encourage children to use manipulatives to solve a variety of problems, and then to reflect on and justify their solutions. This is an essential step in abstracting the ideas that the manipulatives should help develop.

Classify-for a reason. Sort and classify everything, in as many ways as the children can imagine. Whenever possible, classify for a reason, such as by putting away blocks of the same shape together.

Talk, talk, talk. Children, especially those with fewer resources at home, need to separate themselves from the here and now and use ever more abstract words, symbols, and images to describe what is not immediately present. Ask children to reflect on their day and plan what they will do tomorrow. Ask them to consider new ways of approaching problems. Encourage children to represent their ideas in a different way, such as by singing.

Ask "Why?" "Why not?" and "What if?" Asking such questions prompts children to abstract, attend to, and describe mathematical objects, such as shapes.

Help children learn to ask good questions. Young children rarely ask for more information when they do not understand something, but, given explicit encouragement, they learn to do so.

Talk about talking; think about thinking. Children have to learn about the sounds that make up words, the words that make up sentences, and the sentences that make up stories. Talk about all of these.

Get physical. All that talk should be in the context of concrete and physical actions. Act out concepts, such as patterning, and then talk about them. Remember the paradox that abstract thinking yields the greatest riches when you are doing something physical at the same time.

(Adapted from Clements, Sarama, and DiBiase, 2004)

See main article, Building Abstract Thinking Through Math