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…

While many studies have considered the association between teachers' and students' conceptions of teaching and learning and classroom practices, few studies have researched the influence of teachers' conceptions on students' conceptions. Our objective was to analyze the influence of music teachers' conceptions on student…

This study investigates whether Virtual Manipulatives (VM) within a Physical Manipulatives (PM)-oriented curriculum affect conceptual understanding of electric circuits and related experimentation processes. A pre-post comparison study randomly assigned 194 undergraduates in an introductory physics course to one of five conditions: three…

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs,…

This article aims to account for students' assessments of the plausibility and applicability of analogical explanations, and individual differences in these assessments, by analyzing properties of students' underlying knowledge systems. We developed a model of explanation and change in explanation focusing on knowledge elements that provide a…

This article identifies aspects of argumentation in scientific practice that are key for scientific sense-making and articulates how engagement in these aspects happens both inter-mentally (between people) and intra-mentally (an individual's reasoning). Institutionally, peer review exerts critique on new knowledge claims in science and is…

The current study examines the personal epistemology of fourth-grade elementary school teachers from Germany (n = 10) and the United States (n = 10) to gain a more nuanced understanding of teachers' beliefs about knowledge and knowing through a cross-cultural lens. Analyses of semi-structured interviews reveal similarities and differences in the…

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…

There are infinitely many ways of talking about infinity. The assortment of discourses on learning infinity is infinite as well. When the author says "way of talking" or "discourse," she is concerned with much more than the question of how words are chosen and combined. Ways of talking are not just innocent "auxiliaries" to thinking--they shape…

This article reports on a case study from a design-based research project that investigated how students make sense of the disciplinary tools they are taught to use, and specifically, what personal, interpersonal, and material resources support this process. The probability topic of binomial distribution was selected due to robust documentation of…

Findings about mathematics teachers' beliefs typically involve broad characterizations of those beliefs that are compared with general descriptions of practices. Teacher development research suggests that changes happen effectively from attention to specific practices. Few investigations of beliefs and practices are done at this level of detail.…

There has been a body of emerging research describing students' understanding of complex systems. This research has primarily studied students understanding of complex phenomena in science. However, complex phenomena are also pervasive in everyday life. Children observe and participate in them daily. How do they reason about such ordinary complex…

In a 3-year longitudinal, mixed-method study, 67 children in two schools were observed during literacy activities in Grades 1-3. Children and their teachers were interviewed each year about the children's motivation to read and write. Taking a grounded theory approach, content analysis of the child interview protocols identified the motivations…

Clinical interviewing is viewed here as a social interactional pattern in order to examine the nature and limits of the technique as a means of scientific data acquisition. I defend the technique against criticisms that it is ecologically suspect and prone to systematic biases, mainly due to influence of the interviewer on the interviewee or to…

This paper reports the mathematical generalizations of two groups of algebra students, one which focused primarily on quantitative relationships, and one which focused primarily on number patterns disconnected from quantities. Results indicate that instruction encouraging a focus on number patterns supported generalizations about patterns,…