than additive relations. Mack’s39 approach is to engage the students in part–whole activities as a starting point, and to ground these concepts in realistic situations in which students are pressed to consider the multiplicative relations. Finally, Lamon40 has devised programs that address each of the subconstructs separately. All of these programs and others developed by the Rational Number Project have demonstrated a significant impact on the participating students.

Below I present a different approach to teaching rational number that I developed with my colleague Robbie Case. Our approach, shown through controlled experimental trials to be effective in helping students in the fourth, fifth, and sixth grades41 gain a strong initial grounding in the number system, also highlights multiplicative understanding, with an additional focus on the interrelations among fractions, decimals, and percents.42 While there is no one best method or best set of learning activities for rational number,43 our approach provides an opportunity to describe how instruction in rational number can be built around the principles of How People Learn that are the theme of this volume.

First, as will be elaborated, our curriculum is based on our analyses of students’ prior understandings (Principle 1). Our instructional strategy is to help students to further develop these informal understandings and then integrate them into a developmentally sequenced set of activities designed to help them develop a network of concepts and relations for rational numbers (Principle 2). Finally, as will be illustrated throughout our accounts of the lessons, a central feature of this program is the fostering of a metacognitive approach to rational number (Principle 3). By providing students with an understanding of the interconnections among decimals, fractions, and percents, our curriculum helps them develop the ability to make informed decisions on how best to operate with rational numbers.

Pipes, Tubes, and Beakers: A New Approach to Rational-Number Learning
Percents as a Starting Point

In our curriculum, rather than teaching fractions and decimals first, we introduce percents—which we believe to be a “privileged” proportion in that it only involves fractions of the base 100.44 We do this through students’ everyday understandings. We situate the initial learning of percent in linear measurement contexts, in which students are challenged to consider the relative lengths of different quantities. As will be shown below, our initial activities direct students’ attention to ideas of relative amount and proportion from the very beginning of their learning of rational number. For example, we use beakers of water: “If I fill this beaker 50 percent full, approxi-



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