The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.

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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 (

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