This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

This study, based on observation of 90 fifth-grade mathematics classes in Chinese elementary schools, examined how the task features, high cognitive demand, multiple representations, and multiple solution methods may relate to classroom discourse. Results indicate that high cognitive demand tasks were associated with teachers' use higher…

Achieving fluency with basic subtraction and add-with-8 or -9 combinations is difficult for primary grade children. A 9-month training experiment entailed evaluating the efficacy of software designed to promote such fluency via guided learning of reasoning strategies. Seventy-five eligible first graders were randomly assigned to one of three…

While preschool-aged children display some skills with quantitative relations, later learning of related fraction concepts is difficult for many students. We present two studies that investigate young children's tendency of Spontaneous Focusing On quantitative Relations (SFOR), which may help explain individual differences in the development…

External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one…

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In…

This article describes the initial implementation of an innovative program for elementary-age children involving origami and pop-up paper engineering to promote visuospatial thinking. While spatial ability measures correlate with science, technology, engineering, and math (STEM) success, a focus on spatial thinking is all but missing in elementary…

Extensive research has shown that elementary students struggle to learn the basic principles of length measurement. However, where patterns of errors have been documented, the origins of students' difficulties have not been identified. This study investigated the hypothesis that written elementary mathematics curricula contribute to the…

In two experiments, we examined fourth and fifth graders' comprehension of the source of information in texts presenting controversial issues. In Experiment 1, participants read short texts in which two people presented different arguments regarding an issue. Participants identified who said what and evaluated each source's knowledge of the issue.…

This study explores changes in students' strategies as they solve different types of volume problems. Fifth graders were presented with pictures showing 3D objects and a unit cube; they determined how many cubes made up the object and explained their responses. We examined whether children transferred strategies across problem types, varying in…

Reasoning about mechanisms is one of the hallmarks of disciplined inquiry in science and engineering, but comparatively little is known about its precursors and development. Children at grades 2 and 5 predicted and explained the motion of simple mechanical systems composed entirely of visible linkages (levers). Students' explanations of device…

Children's estimation patterns on a number line estimation task may provide information about the mental representation of the magnitude of numbers. Siegler and his colleagues concluded that children's mental representations shift from a logarithmic-ruler representation to a linear-ruler representation. However, there are important methodological…

This article presents a theoretical model of the process by which students construct and elaborate explanations of scientific phenomena using visual representations. The model describes progress in the underlying conceptual processes in students' explanations as a reorganization of fine-grained knowledge elements based on the Knowledge in Pieces…

This article examines the pedagogical practices of two science inquiry teachers and their students using a Web-based system called Web of Inquiry (WOI). There is a need to build a collective repertoire of pedagogical practices that can assist elementary and middle school teachers as they support students to develop a complex model of inquiry based…

This study extends prior research on both individual self-regulation and socially shared regulation during group learning to examine the range and quality of the cognitive and behavioral social regulatory sub-processes employed by six small collaborative groups of upper-elementary students (n = 24). Qualitative analyses were conducted based on…