• The conceptual bases for operations with numbers and how those operations relate to real situations should be a major focus of the curricu lum. Addition, subtraction, multiplication, and division should be presented initially with real situations. Students should encounter a wide range of situations in which those operations are used.

• Different ways of representing numbers, when to use a specific rep resentation, and how to translate from one representation to another should be included in the curriculum. Students should be given opportunities to use these different representations to carry out operations and to understand and explain these operations. Instructional materials should include visual and linguistic supports to help students develop this representational ability.

Learning to operate with single-digit numbers has long been characterized in the United States as “learning basic facts,” and the emphasis has been on rote memorization of those facts, also known as basic number combinations. For adults the simplicity of calculating with single-digit numbers often masks the complexity of learning those combinations and the many different methods children can use in carrying out such calculations. Research has shown that children move through a fairly well-defined sequence of solution methods in learning to perform operations with single-digit numbers, particularly for addition and subtraction, where rapid general procedures exist. Children progress from using physical objects for representing problem situations to using more sophisticated counting and reasoning strategies, such as deriving one number combination from another (e.g., finding 7+8 by knowing that it is 1 more than 7+7 or, similarly, finding 7×6 as 7 more than 7×5). They know that addition and multiplication are commutative and that there is a relation between addition and subtraction and between multiplication and division. They use patterns in the multiplication table as the basis for learning the products of single-digit numbers. Instruction that takes such research into account is needed if students are to become proficient:

• Children should learn single-digit number combinations with un derstanding.

• Instructional materials and classroom teaching should help students learn increasingly abbreviated procedures for producing number combinations rapidly and accurately without always having to refer to tables or other aids.