Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…

This research aims to determine second-year university students' understanding in interpreting and representing fractions. A set of fraction tests was given to students through two direct learning interventions. An unstructured interview was used as an instrument to obtain explanations and confirmations from the purposive participants. A total of…

The present study tests two predictions stemming from the hypothesis that a source of difficulty with rational numbers is interference from whole number magnitude knowledge. First, inhibitory control should be an independent predictor of fraction understanding, even after controlling for working memory. Second, if the source of interference is…

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as…

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of…

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non-symbolic ratio processing and…

Existing frameworks of teachers' knowledge required to teach mathematics do not adequately capture the dynamic aspects of knowledge manifested in teaching practice. In this paper, we examine the knowledge demands that arise in situ, in the course of a teacher listening and responding to students' thinking, while teaching the topic of decimal…

A formative pilot study of a fractions intervention was conducted to address the intervention's potential for improving fifth-grade struggling students' knowledge of fractions, and to identify any logistical or instructional design issues, before scaling up for a randomized control trial. In the first section, we provide details of the formative…

The focus of the present synthesis reflects the research on programs, practices, and policies intended to improve mathematics outcomes funded through the Institute of Education Sciences' (IES's) National Center for Education Research (NCER) and National Center for Special Education Research (NCSER). The authors were asked to review those published…

In this article, we present some mathematical tasks designed to support pre-service teachers (PSTs) in obtaining a broader, more connected understanding of decimal numbers and quantities. We describe classroom scenarios from a content course in which PSTs experienced these cognitively demanding decimal tasks, enacted in ways that maintain high…

Investigates validity of scores derived from the measurement procedure involving number lines by assessing its unique contributions to performance differences in criterion measures of rational number knowledge and skills, including fraction computation, application, and explanation. States that 205 fifth-grade and 208 sixth-grade students…

Many elementary and junior high school students do not become proficient with common and decimal fractions because they have established few connections between the form they learn in the classroom and understandings they already have. (Author/GC)

Reports results of interviews in which students were asked to illustrate three interpretations of fractions: a unit subdivided into equal-sized parts; a set subdivided into equal-sized parts; and a ratio. Two areas of difficulty in giving reasonable representations are discussed. Proposes teaching suggestions to overcome these difficulties. (MDH)

The purpose of this study was to characterize and compare novice and experienced elementary teachers' pedagogical knowledge and pedagogical content knowledge of three major topics in mathematics: whole number operation, fractions, and geometry. The study participants were 26 preservice elementary teachers and 28 experienced kindergarten through…

Does conceptually oriented instruction jeopardize students' computational competence? If it does, then why are so many reform efforts continuing to emphasize the importance of teaching for conceptual understanding? If it does not, then why are the majority of teachers at all grade levels continuing to teach for computational competence without…