In this review, we attempt to integrate two crucial aspects of numerical development: learning the magnitudes of individual numbers and learning arithmetic. Numerical magnitude development involves gaining increasingly precise knowledge of increasing ranges and types of numbers: from non-symbolic to small symbolic numbers, from smaller to larger…

Understanding of numerical development is growing rapidly, but the volume and diversity of findings can make it difficult to perceive any coherence in the process. The integrative theory of numerical development posits that a coherent theme is present, however--progressive broadening of the set of numbers whose magnitudes can be accurately…

The processes and strategies used by Finnish second graders in solving verbal multiplication and division tasks were investigated, and the relationship of these processes and strategies to Piagetian abilities, memory capacity skills, and rational number concepts was charted. Three categories of strategies were classified: operations based on…

A teaching experiment with six third graders to analyze their ways of operating while solving fraction tasks. Children's quantitative reasoning with fractions was based on their quantitative reasoning with natural numbers. Presents the constructive itinerary of one of the most advanced children in the group. (Contains 44 references.) (Author/MDH)

A nine-year-old's conception of fractions was compared with his knowledge of units. He had effective schemes for solving some partition problems but did not consistently use units of different sizes in interpreting fractions. His solutions to equivalence problems showed no coherent method of verification. (MNS)

Discusses convergence between instructional practices suggested by research on achievement motivation and practices promoted in mathematics-instruction reform literature by focusing on fourth- through sixth-grade students (N=624) and their teachers (N=24). Concludes that the instructional practices suggested in the literature of both research…

This study compared 6- to 7-year-olds' knowledge of numerical fractions prior to school instruction in Croatia, Korea, and United States. Results suggested that the Korean vocabulary of fractions may influence the meaning children ascribe to numerical fractions and that this results in children being able to associated numerical fractions with…

This report describes an investigation of how young children respond to two types of tasks: (1) finding one-half of a continuous and a discrete material; and (2) attempting to share continuous and discrete material equally between two dolls. Continuous material, such as string, paper, or liquid, is quantified by adults using measurement units. A…

Basic fraction concepts are viewed as the seedbed for many important mathematical ideas including the notions of equivalence, inverse, decimals, probability, ratio, and proportion. It is viewed that a critical component of children's mathematical knowledge necessary for success in certain fractional number contexts appears to be intellectual…

Tested 3- to 7-year-olds' ability to calculate with whole numbers, fractions, and mixed-numbers, in a task in which an amount was displayed, then hidden. Subjects were to determine the hidden amount resulting when numbers were added or substracted. Found that, although fraction problems were more difficult than whole-number problems, competence on…

Two experiments tested claim that transparency of Korean fraction names promotes fraction concepts. Findings indicated that U.S. and Korean first- and second-graders erred similarly on a fraction-identification task, by treating fractions as whole numbers. Korean children performed at chance when whole-number representation was included but…

Discussed is why students have the tendency to apply an "add the numerators and add the denominators" approach to adding fractions. Suggested is providing examples exemplifying this intuitive approach from ratio, concentration, and distance problems to demonstrate under what conditions it is applicable in contrast to the addition algorithm. (MDH)

Presents teaching methods to rectify the tendency of students and even teachers to divide the smaller number into the larger in problem situations requiring division, while recognizing the impossibility of the answer in the situation. (MDH)

Reports on a study designed to investigate aspects of young children's knowledge of fractions and partitions of quantities. Findings suggest that although children did not have a deep understanding of the concept of one half, a significant number demonstrated a powerful systematic procedure leading to the creation of accurate equal fractional…

Concrete experience should be a first step in the development of new abstract concepts and their symbolization. Presents concrete activities based on Hyde and Nelson's work with egg cartons and Steiner's work with money to develop students' understanding of partitive division when using fractions. (MDH)