In a 1-hour teaching interview, 20 children (aged 7 to 11) discovered how to tell whether objects might be made of the same material by using ratios of measures of weight and size. We examine progress in the children's reasoning about measurement and proportional relations, as well as design features of instruments, materials, and tasks crafted to…

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three…

This article describes a detailed analysis of knowledge building in a problem-based learning group. Knowledge building involves increasing the collective knowledge of a group through social discourse. For knowledge building to occur in the classroom, the teacher needs to create opportunities for constructive discourse in order to support student…

This article describes how children build increasingly abstract knowledge of linear measurement, emphasizing ways they relate space and number. Assessments indicate children struggle to understand measurement, especially concepts related to complex paths as in perimeter tasks. This article draws on developmental accounts of children's knowledge of…

Discusses children's design of mathematical representations on paper. Suggests that the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and that knowledge of mathematical representations is not simply recalled and applied to problem solving, but also emerges out of one's interactions…

Two experiments examined children's early judgments about numerical relations. Found that children as young as three years old are already adept at reasoning about relations between sets, independently of their ability to form numerical representations. Results support the existence of protoquantitative schemas, or ways of thinking about relations…

Assessed the impact of elementary science experiences on the development of sixth-graders' epistemological views. Found that, compared to students in a traditional science classroom, those in a constructivist classroom developed an epistemological stance toward science that focused on the central role of ideas in the knowledge acquisition process.…

Focuses on a single case study of a mentally handicapped child to examine directly the relation between the development of number-after rules and a counting-on from the larger addend (COL) strategy. Examines the development of children's proficiency with n+ and 1+n combinations and the evolution of their counting-on strategies over time. (AA)

Evaluated the PREG conceptual model of human question asking. Found the model was sufficient as it accounted for nearly all of the questions produced by students, and was discriminating in that it could identify the conditions in which particular classes of questions are or are not generated. (Author/SD)

Fourth, sixth, and eighth graders and adults estimated answers to multiplication problems and explained their estimation procedures. Found performance improved with age, and sixth graders understood the simplification principle in estimation. The most striking developmental changes were in conceptual knowledge used for estimating. Proposes a…