Developing Children's Understanding of the Rational Numbers: A New Model and an Experimental Curriculum

Abstract

A new curriculum to introduce rational numbers was devised, using developmental theory as a guide. The 1st topic in the curriculum was percent in a linear-measurement context, in which halving as a computational strategy was emphasized. Two-place decimals were introduced next, followed by 3-and 1-place decimals. Fractional notation was introduced last, as an alternative form for representing decimals. Sixteen 4th-grade students received the experimental curriculum. Thirteen carefully matched control students received a traditional curriculum. After instruction, students in the treatment group showed a deeper understanding of rational numbers than those in the control group, showed less reliance on whole number strategies when solving novel problems, and made more frequent reference to proportional concepts in justifying their answers. No differences were found in conventional computation between the 2 groups.

Children’s multiplicative transformations of discrete and continuous quantities

“…An articulated model of fractions based on division is strongly related to middle school children's understanding of other aspects of rational number (Smith, Solomon, & Carey, 2005), and many researchers suggest that early intuitions about transformations of continuous physical amounts support later fraction learning (Resnick & Singer, 1993;Confrey, 1994;Moss & Case, 1999). But children's great difficulty in understanding fractions emphasizes the conceptual distance between intuitions about physical quantities and formal reasoning about rational numbers.…”

Conceptual structure and the procedural affordances of rational numbers: Relational reasoning with fractions and decimals.

“…In fact, Moss and Case (1999) implemented a curriculum with 4th graders in Canada that reorganized the usual order of instruction in rational numbers. Children were first taught percentages (in the context of volumes, and on number lines), then decimals, and lastly fractions.…”

Physically Distributed Learning: Adapting and Reinterpreting Physical Environments in the Development of Fraction Concepts

“…Fractions, however, require new interpretations. Researchers have identified several interpretations of rational numbers, including part-whole, ratio, quotient, and operator (e.g., Carpenter, Fennema, & Romberg, 1993;Confrey & Smith, 1995;Kieren, 1995;Moss & Case, 1999;Streefland, 1993;Thompson & Saldanha, 2003). Our purpose here is not to focus on one of these schemas, but to examine whether physical activity can contribute to the development of a viable fraction interpretation.…”