Thomas P. Carpenter, University of Wisconsin-Madison Megan L. Franke, University of California at Los Angeles Victoria R. Jacobs, California State University-San Marcos Elizabeth Fennema, University of Wisconsin-Madison Susan B. Empson, University of Texas at Austin

This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.

An understanding of most fundamental mathematics concepts and skills develops over an extended period of time. Although cross-sectional studies can provide snapshots of the development of these concepts at particular points in time, longitudinal studies provide a more complete perspective; however, relatively few studies have traced the development of fundamental mathematics concepts in children over more than a single year. In this paper we report the results of a 3-year longitudinal study of the growth of children's -understanding of addition and subtraction involving multidigit numbers. We focus particularly on children's construction of invented strategies for adding and subtracting multidigit numbers. The overarching goal of the study was to investigate the role that invented strategies may play in developing an understanding of multidigit addition and subtraction concepts and procedures. We trace the development and use of invented addition and subtraction strategies and examine the relation of these strategies to the development of fundamental knowledge of base-ten number concepts and the use of standard addition and subtraction algorithms. Finally, we consider what the use of invented strategies affords by way of avoiding systematic errors and extending knowledge of basic multidigit operations to new problem situations. BACKGROUND There is mounting evidence that children both in and out of school can construct methods for adding and subtracting multidigit numbers without explicit instruction in specific procedures (Carpenter & Fennema, 1992; Carraher, Carraher, & Schliemann, 1987; Cobb & Wheatley, 1988; Fuson & Burghardt, 1993; Hiebert & Wearne, 1996; Kamii, 1989; Labinowicz, 1985; Nunes, 1992; Olivier, Murray, & Human, 1990; Saxe, 1988). It is hypothesized that these invented strategies can play a central role in making problem solving a focus of learning arithmetic procedures and in helping students develop number sense and understanding of multidigit operations (Carpenter et al., The research reported in this paper was supported in part by the National Science Foundation under Grant Number MDR-8955346. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. An earlier version of this paper was presented at the annual meeting of the American Educational Research Association, New York, 1996. MATH 5392 - Page 37 *

1994). Children employ a number of strategies for solving multidigit problems at varying levels of sophistication. Many of these strategies are constructed by children individually or collectively, without direct instruction by the teacher (for a more...