Due to the learning paradox, students cannot have real difficulty in understanding a mathematical concept that they have not yet understood. There is a gap between real difficulties, directly experienced by students, and illusionary ones, only observed by researchers. This paper aims to offer a critical reflection on our understanding of the term…

Grade 8 students were given a task that had pairs of shapes that had been transformed (i.e., preimages and images). Students were to identify which of the three rigid transformations--reflections, rotations, or translations--were used to map the preimage to the image. Students were also asked to give a justification for their choice of…

This article presents theoretical knowledge, sample activities, and reflections from the implementation process on how teachers can benefit from the Variation Theory to help students enhance their mathematical comprehension. In this context, "congruence of triangles" was conceptualized as an object of learning. Critical aspects were…

A "math congress" is a pedagogical approach in which students present their solutions from their mathematical work completed individually, in pairs, or in small groups, and share and defend their mathematical thinking. Mathematical artifacts presented during math congress remain on display as community records of practice. Math congress…

A core component of informal statistical inference is the recognition that judgments based on sample data are inherently uncertain. This implies that instruction aimed at developing informal inference needs to foster basic probabilistic reasoning. In this article, we analyze and critique the now-common practice of introducing students to both…

This article is concerned with "commonsense science knowledge", the informally gained knowledge of the natural world that students possess prior to formal instruction in a scientific discipline. Although commonsense science has been the focus of substantial study for more than two decades, there are still profound disagreements about its nature…

Regional Australian students were surveyed to explore their understanding and knowledge of the greenhouse effect, ozone depletion and climate change. Results were compared with a parallel study undertaken in 1991 in a regional UK city. The comparison was conducted to investigate whether more awareness and understanding of these issues is…

The purpose of this study is to examine the difference in effectiveness between two on-line scientific learning programs--one with an argumentation component and one without an argumentation component--on students' scientific argumentation ability and conceptual change. A quasi-experimental design was used in this study. Two classes of 8th grade…

Dividing by zero can be confusing for students at all levels. If teachers are to provide clear and understandable explanations of this topic, they must possess a strong conceptual understanding of it themselves. In this article, the authors present qualitative data on fourth- through eighth-grade teachers' perceptions of division by zero. They…

This article examines a classroom discussion of multiple interpretations of the scales on two distance versus time graphs. The analysis describes how two students and a teacher used multiple meanings for phrases of the form "I went by" and coordinated these meanings with different views of the scales. Students' ambiguous and shifting meanings did…

Using Freudenthal's method of historical phenomenology, the history of statistics was investigated as a source of inspiration for instructional design. Based on systematically selected historical examples, hypotheses were formulated about how students could be supported in learning to reason with particular statistical concepts and graphs. Such a…

The 1994 U.S. History Framework encourages students to learn about ideas and people and to understand how the present world is the result of choices and convictions of people, and the focus of the assessment is to see what diverse U.S. students in grades 4, 8, and 12 understand about U.S. history. The framework proposes that students studying U.S.…

This document provides the framework to guide the development of assessment instruments for the 1994 National Assessment of Educational Progress (NAEP) in geography. This framework is designed to assess the outcomes of students' education in geography in grades 4, 8, and 12 as part of the NAEP. The document draws from the five themes of geography,…

This report explains an evaluative study of the conceptual and affective development of students associated with the Floating Lab Program, an experiential field project sponsored by the University of New Hampshire and the Maine Sea Grant Program. The field program involved an opportunity for students to have hands-on experiences aboard a 65-foot…

The aims of this study were considered under three headings. The first was to elicit misconceptions that students had about the terms "solute", "solvent" and "solution." The second was to understand how students' prior learning affected their misconceptions. The third was to determine if students were able to make a…