Issues in Place-Value and Multidigit Addition and Subtraction Learning and Teaching

Abstract

Research on multidigit addition and subtraction is now sufficient to question some present textbook practices and suggest alternatives. These practices revolve around the organization and placement of topics within the curriculum and around teaching/learning methods. These questions are being raised because the evidence indicates that U.S. children do not learn place-value concepts or multidigit addition and subtraction adequately and even many children who calculate correctly show little understanding of the procedures they are using (e.g., Cauley, 1988; Kamii & Joseph, 1988; Kouba et al., 1988; Labinowicz, 1985; Lindquist, 1989; Resnick, 1983; Resnick & Omanson, 1987; Ross, 1989; Stigler, Lee, & Stevenson, in press; Tougher, 1981). Initially, children's conceptual structures for number words are only unitary conceptual structures in which the meanings or referents of the number words are single objects (as in counting objects) or a collection of single objects (as in the cardinal reference to a collection of objects). These unitary conceptual structures become more integrated, abbreviated, and abstract and progress through a developmental sequence that enables children to carry out increasingly abstract and efficient addition and subtraction solution procedures (Carpenter & Moser, 1984; Fuson & Hall, 1983; Fuson, 1988, in press b; Kamii, 1985; Steffe & Cobb, 1988; Steffe, von Glasersfeld, Richards, & Cobb, 1983). However, even the most sophisticated of the procedures based on these unitary conceptual structures is time-consuming and error-prone for adding and subtracting two-digit numbers greater than 20 or so. For such two-digit and larger multidigit numbers, children need to construct multiunit conceptual structures in which the meanings or referents of the number words are a collection of entities (such as counting one two three hundred, in which the referent for each is a collection of 100 entities of some kind) or a collection of collections of objects (as in the cardinal reference of seven hundred to a collection of collections of 100 entities). Both English number words and written number marks for multidigit numbers are built up of increasingly larger multiunits related to ten (ten, thousand, etc.). Understanding multidigit numbers requires being able to think about these various sizes of multiunits, and understanding operations on multidigit numbers requires understanding how to compose and decompose multidigit numbers into these multiunits in order to carry out the various operations. With present school instruction many children in the United States do not build any multiunit conceptual structures for multidigit numbers but instead treat them