Rational Numbers: An Integration of Research

By Thomas A. Romberg; Elizabeth Fennema et al. | Go to book overview

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RATIONAL NUMBERS: TOWARD A SEMANTIC ANALYSIS -- EMPHASIS ON THE OPERATOR CONSTRUCT
Merlyn J. Behr
Louisiana State University
Thomas Post
University of Minnesota
Guershon Harel
Purdue University
Richard Lesh
Educational Testing Service

Two notational systems are used to present a semantic/mathematical analysis of the operator construct of rational number. The one system constitutes a generic manipulative aid, the other is an abstract representation for a mathematics of quantity model. Different conceptions of the numerator and denominator lead to new interpretations of the operator construct. One of two interpretations elaborated upon is duplicator/partition-reducer, the other stretcher/shrinker. Both present a notion of rational number as an exchange function -- that is, they exchange the operand quantity of a rational number operator to a conceptually new quantity that has a ratio to the original quantity equal to the numerator-to-denominator ratio. Differences between problem situations that require one or the other interpretation for an accurate mathematical model are illustrated through example problems and solutions. Computational algorithms implied by the analysis are illustrated and discussed.

When one considers the question of what experiences a child needs in order to have a complete understanding of rational number, the notion that a rational number is an element of an infinite quotient field is overly simplistic. When the concept of rational number is used in real-world situations it takes on personalities that are not captured by that mathematical characterization. In order to be in a position to develop experiences from which children can gain a complete understanding of the concept of rational number, researchers need to explore children's ability to acquire knowledge of these personalities and determine what their informal knowledge of these personalities is. Moreover, teachers and curriculum developers need to be aware of these personalities of rational number. Questions of how to develop learning situations so that elementary and middle-grade

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