The nature of the development of arithmetic performance has long been intensively studied, and available scientific evidence can be evaluated and synthesized in light of Nelson and Narens' model of metacognition. According to the Nelson-Narens model, human cognition can be split into two or more interrelated levels. Obviously, in the case of more…

The purpose of this article is to show how the philosophical theory of inferentialism can be used to understand students' conceptual development in the field of mathematics. Based on the works of philosophers such as Robert Brandom, an epistemological theory in mathematics education is presented that offers the opportunity to trace students'…

The development of numerical concepts is described from infancy to preschool age. Infants a few days old exhibit an early sensitivity for numerosities. In the course of development, nonverbal mental models allow for the exact representation of small quantities as well as changes in these quantities. Subitising, as the accurate recognition of small…

Children's thinking follows natural developmental paths in learning math. When teachers understand those paths and offer activities based on children's progress along them, they build developmentally appropriate math environments. The authors explain math learning trajectories and why teaching math using the trajectories approach is effective. A…

This book includes two main sections: a discussion of problem solving and a section on computation with whole numbers. A primary theme of the text is that problem solving sets the stage for meaning and conceptual development with respect to numbers. The section on problem solving includes numerous problem-solving activities that have a dual…

To build cognitive foundation for number, twenty-six low-performing, low-SES first graders did mathematical physical-knowledge activities, such as "bowling," during the first half of the year. As their arithmetic readiness developed, they tried more word problems and games. At the end of the year, these children did better in mental arithmetic and…

In this paper four programs are described in which children learn multidigit number concepts and operations with understanding: (1) the Supporting Ten-Structured Thinking projects, (2) the Conceptually Based Instruction project, (3) Cognitively Guided Instruction projects, and (4) the Problem Centered Mathematics Project. The diversity in these…

Describes the characteristics of progressive schematization with regard to column multiplication and column division. Contrasts this with column arithmetic based on progressive complexity. Presents a summary of research data concerning column arithmetic. (TW)

In the first chapter Brownell critically examines the psychological bases of the three most common theories of arithmetic instruction: drill, incidental learning, and meaning. In chapter 2 the results of a nation-wide survey of actual teaching practices are reported. Chapter 3 presents a contrast between "informational arithmetic" and…

This brief report from the Indigenous Mathematics Project focuses on the way in which numerical reasoning is changing in the Oksapmin community of Papua New Guinea as a function of participation in new social institutions: economic exchange with currency and enrollment in school. Each of these new institutions means that arithmetic problems are…