In building models of students' fractions knowledge, two prominent frameworks have arisen: Kieren's rational number subconstructs, and Steffe's fractions schemes. The purpose of this paper is to clarify and reconcile aspects of those frameworks through a quantitative analysis. In particular, we focus on the measurement subconstruct and the…

This study concerns 182 first year mathematics undergraduates' perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils' perspectives. The aim was the investigation of tangency images that settle after undergraduates' distancing from the notion for a few months and after their participation in…

While the general notion of "metaphor" may offer a thoughtful analysis of the nature of mathematical thinking, this paper suggests that it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has "met-before." The notion of "met-before"…

This paper reports on a research study based on data from experimental teaching. Undergraduate dance majors were invited, through real-world problem tasks that raised central conceptual issues, to invent major ideas of calculus. This study focuses on work and thinking by these students, as they sought to build key ideas, representations and…

This study investigates teachers' argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible…

This case study discusses an activity that makes up one of five lessons in an ongoing classroom teaching experiment. The goal of the teaching experiment is (a) to gain insight into kindergartners' spatial structuring abilities, and (b) to design an educational setting that can support kindergartners in becoming aware of spatial structures and in…

An operational definition offered in this paper posits learning as a multi-dimensional and multi-phase phenomenon occurring when individuals attempt to solve what they view as a problem. To model someone's learning accordingly to the definition, it suffices to characterize a particular sequence of that person's disequilibrium-equilibrium phases in…

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the…

College students' epistemological belief in their academic performance of mathematics has been documented and is receiving increased attention. However, to what extent and in what ways problem solvers' beliefs about the nature of mathematical knowledge and thinking impact their performances and behavior is not clear and deserves further…

Though written accounts of collegiate mathematics teaching exist (e.g., mathematicians' reflections and analyses of learning and teaching in innovative courses), research on collegiate teachers' actual classroom teaching practice is virtually non-existent. We advance this claim based on a thorough review of peer-reviewed journals where scholarship…

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate…

In our work in teacher education and professional development, we aim to help teachers to learn to participate in, and create, classroom ecologies that support students' learning. In this article we focus on the challenges of developing a classroom ecology that provides mathematical sustenance for students. We pay particular attention to the ways…

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an…

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem posing activities in geometry, using the "What If Not?" (WIN) strategy, as part of their work on computerized inquiry-based activities. Data received from the PTs'…

In discussion-oriented classrooms, students create mathematical ideas through conversations that reflect growing collective knowledge. Linguistic forms known as indexicals assist in the analysis of this collective, negotiated understanding. Indexical words and phrases create meaning through reference to the physical, verbal and ideational context.…