Describes a number line activity based on students' individual timelines to help students understand the concepts of integers and rational numbers. Middle school students and their parents construct a number line using positive and negative rational numbers to represent dates of events before and after the student's birth. (KHR)

Synthesizes research on content knowledge of rational numbers, the learning of rational number order and equivalence concepts, and teachers' pedagogical reasoning about these concepts. Uses teachers' descriptions of how a hypothetical student would understand rational numbers to sketch a formative model of pedagogical reasoning. (16 references)…

This paper reports on a study of seventh grade students (N=4) who attend a rural K-12 school. Students participated in a 5-week teaching experiment designed to build on their existing knowledge of the unit concept and extend it to the rational number operations of multiplication and division. Data collected comes in the form of videotapes,…

Children and adolescents tested to determine the development of the operator construct of rational numbers employed different problem-solving strategies depending on test presentation. Three major phases in the rational number construct appear to be a primitive fractional construct, a unit operator phase, and a general operator phase. (SB)

Based on a theory of rational numbers which sees the rational numbers construct of a person as potentially built on a number of subconstructs, this study explored the view of rational numbers as operators. Significant changes in performance level and quality appeared between ages 11 and 12 and 12 and 13. (JC)

This paper provides some guidance as to what to listen for to help students make sense of expressions in ways that connect to their ideas and honestly address the mathematics of rational numbers. It offers a reasonable initial answer to the question, "Where do students' ideas about fractions and ratios come from, and how can we work productively…

Describes the constructs of referents, relationships, and modes and illustrates how these constructs may be reflected in students' written responses to a decimal task that requests an explanation. Examines sets of responses from two classrooms using the proposed framework to illustrate the type of information that teachers may acquire through the…

This chapter presents a yearlong teaching experiment involving a 4th grade class in a New Jersey school that focused on fractions. The 25 children worked in pairs or in small groups, then came together as a whole class for sharing in larger discussions. The development of children's thinking is explored through analyzing videotapes of class…

The principles of cognitive psychology as they relate to rational number learning and instruction are discussed. Issues of knowledge acquisition, organization, retrieval and transfer are reviewed as they affect this area of the elementary school curriculum. Specific problems in learning about rational numbers and systematic student errors are…

This paper describes the Rational Number Project (RNP), teaching experiments concerned with the teaching and learning of fractions among 4th and 5th grade students. Interviews with 4th grade students who used the RNP curriculum and with students who used a traditional curriculum were conducted by RNP staff as well as classroom teachers. This paper…

The development of proportional reasoning is a major focus of the middle grades curriculum. The challenge for educators is to find contexts that engage students and that facilitate the study of proportional reasoning. This chapter explores proportional thinking with students in grades 3-8 by using a number of books in which the underlying stories…

This yearbook contains articles that give insight into students' thinking about factions, ratios, and proportions. Suggestions are offered on how to develop the concepts and skills associated with these topics. The book is divided into elementary, middle school, and professional development sections. Chapters include: (1) "The Development of…

Analyzes three textbook series to find what meaningful learning experiences with fractions the books offer relative to modes of representation, pictorial models, qualitative reasoning, and students' informal knowledge of partitioning. Discusses implications for teacher use, including supplementation, teacher-support materials, alternative…

What makes a fraction meaningful and a definition of the quantitative notion of fractions are discussed. Then observations from teaching experiments are presented. (MNS)

This study investigates the development of 5th grade children's understanding of quotient and the classroom norms and practices that constrain or enable that understanding. It reports not only how the children's understandings develop, but also why and under what conditions they develop. The results of this study indicate that children progressed…