A considerable number of children invent counting-up procedures for situations in which an unknown quantity is added to a known quantity.24 Many of these children later count up in taking-away subtraction situations (13–8=? becomes 8+?=13). When counting up is not introduced, many children may not invent it until the second or third grade, if at all. Intervention studies with U.S. first graders that helped them see subtraction situations as taking away the first x objects enabled them to learn and understand counting-up-to procedures for subtraction. Their subtraction accuracy became as high as that for addition.25

Experiences that focus on part-part-whole relations have also been shown to help students develop more efficient thinking strategies, especially for subtraction.26 Students examine a join or separate situation and identify which number represents the whole quantity and which numbers represent the parts. These experiences help students see how addition and subtraction are related and help them recognize when to add and when to subtract. For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic.27

For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplish-ments in arithmetic.

Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations.28

Single-Digit Multiplication

Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction. U.S. children progress through a sequence of multiplication procedures that are somewhat similar to those for addition.29 They make equal groups and count them all. They learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20,…to multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know.

As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4×3 by counting “3, 6, 9, 12”). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables (see Box 6–4 for some examples).30 Moreover, finding and describing



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