The National council of Teachers of Mathematics shares these important standards in its latest document on optimum environments for math learning.

Between infancy and kindergarten the foundation for children's mathematical development is established. Mathematics learning should build on children's curiosity and enthusiasm and should grow naturally from their experiences. If done well, early mathematics is more than "getting ready" for school. Mathematical experiences for children, if appropriately connected to children's worlds, challenge them to explore ideas in more sophisticated and rich ways than previously believed possible. These ideas include number as well as quantitative relationships, shape, space, symmetry, and patterns.

Why Focus on Mathematics Before Kindergarten?

Many mathematics concepts, at least in their intuitive beginnings, develop before school. Infants spontaneously use the ability to recognize and discriminate among small numbers of objects. Before they enter school, many children develop knowledge about numbers and geometry. Young children use mathematical ideas in everyday life and develop informal mathematical knowledge that can be quite complex and sophisticated. The long-term success of children's learning and development requires quality experiences during their "early years of promise" (Carnegie Corporation, 1998). Quality experiences occur in environments that are rich in language, encourage children's thinking, value children's uniqueness, and nurture children's explorations.

What Types of Mathematical Experiences Are Appropriate?

Because young children develop a disposition for mathematics based upon their early experiences, opportunities for learning need to be positive and supportive so that children learn to trust their own abilities to make sense of mathematics. Parents, teachers, and other caregivers can help children become aware of mathematics in their activities through sensitive observation, conversation, and guidance. An adult places crackers in a toddler's hands while saying, "Here are two crackers-one, two." A three-year-old chooses how she wants her sandwich cut-into triangles, rectangles, or small squares. As a child arranges stuffed animals by size, an adult might ask, "Which animal is the smallest? Which is the largest?" Children often compile a wealth of knowledge about topics that interest them. Mathematics should be developed from those interests and activities that build on them.

Mathematical understanding develops as children create streets and buildings in the sand or make playhouses with empty boxes. Knowledge grows as children count steps across the room, sort collections of rocks and other "treasures," or put away toys or groceries. Through their everyday activities children learn mathematical concepts: sorting as they put toys or groceries away, spatial relations and comparisons of solids when building with blocks, representation through drawing and painting their ideas, patterns as they go about their daily routines, directional terms by singing motion songs, and spatial visualization when working with puzzles. Children show what they know by talking as well as through models, role playing, and pictures. In the preschool years, quality learning is often incidental and informal. This does not mean unplanned or unsystematic. The most powerful mathematics learning for young children is seldom acquired sitting down in a group lesson.

It is especially important to provide introductions to the language and conventions of mathematics, always starting with and maintaining a connection to children's informal knowledge and language.

Young children are grappling with number words and counting for the first time. They need to learn words used to compare and to indicate position and direction. For example; a friend may be taller than they are, but the same friend may be shorter than another child. Activities that encourage mathematical learning, such as books and stories with numbers and patterns, music with actions and directions, or games that involve rules and taking turns, help children understand a range of mathematical ideas. Things to count, sort, compare, match, put together, and take apart should be accessible to children. It is also important to remember that children gain both competence and ownership as they return to favorite activities repeatedly

If children have rich mathematical experiences during the preschool years, the typically wide variation in children's knowledge upon entering school can be narrowed. "Not knowing" is a reflection of not having had opportunity to learn rather than inability to learn. At-risk children who need additional support so they do not start school at a disadvantage should be identified with appropriate assessments. Children showing early signs of developmental or learning problems require immediate attention. Pediatricians and other healthcare providers can recognize indicators of early learning difficulties and can suggest resources to accommodate these challenges.

Assessments used for screening must be adapted to the needs and characteristics of young children. They should include a variety of mathematical emphases beyond numbers. Such early assessments should be used to gain information for teaching and for potential early interventions rather than for sorting children. Interviews and observations are more appropriate assessment strategies than conventional group tests, which are not likely to yield reliable, valid, or complete data.

A Climate for Young Learners

Young children are active, resourceful individuals who construct, modify, and integrate ideas by interacting with the physical world and with other children and adults. They learn by talking about what they are thinking and doing and by collaborating and sharing ideas. Their ability to communicate through language, pictures, and other symbolic means develops rapidly during these years. Young children also make connections that clarify and extend their knowledge; they add new meaning to past experiences by writing and interpreting others' written records. Adults facilitate children's development of mathematics by asking questions that lead to clarifications, extensions, and the development of new understandings. Language is very important to learning mathematics.

Young children make sense of the world by reasoning and problem solving, and teachers should recognize that young children think in ways that can be very sophisticated. At the same time, it should be recognized that a child's ways of knowing and communicating are different from those of an adult. Children will find their own ways of representing and communicating their ideas. By the end of the second grade, children should begin to use many conventional representations with understanding.

Mathematical concepts develop throughout the pre-K-2 years and at different times and rates for each child. All children need adequate amounts of time and opportunities to develop, construct, test, and reflect upon their understanding of mathematics. Since a wide range of cognitive development is related to previous experiences and opportunities to learn, primary education must build on the idea that all children can learn significant mathematics, and the school must take the responsibility for supporting that learning. That is, strong mathematics programs provide the support and resources so that all children leave grade two confident and competent in mathematics.

Starting Out Strong

During their early years (pre-k through second grade) children are building beliefs about what mathematics is, what it means to know and do mathematics, and about themselves as mathematics learners. These beliefs influence their thinking, performance, attitudes, and decisions related to studying mathematics in later years. High-quality early childhood mathematics programs:

• Build upon and extend children's intuitive and informal mathematical knowledge.
• Are grounded in knowledge of child development.
• Provide environments that encourage children to lie active learners who are eager for new challenges.
• Develop a strong conceptual framework that provides anchoring for skills acquisition.
• Nurture and develop children's natural inclination to solve problems.

A complete copy of the national mathematics standards can be obtained from the National Council of Teachers of Mathematics, http:/standards-e.nctm.org.