In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a "network of mathematical resources" by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics…

An earlier paper discussed the case of a flexible but inextensible membrane filled to capacity with incompressible fluid. It was found that the resulting shape satisfies a set of three simultaneous partial differential equations. This article gives a more general derivation of these equations and shows their form in an interesting special case.

Research on toddlers' mathematical knowledge is sparse. Studies on children's mathematical competencies before school age have mostly focused on older children. Few of the previous studies have included large groups of toddlers, few have been conducted in natural settings, and few have been directed at a broad field of mathematical knowledge. The…

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students' perceptions of proofs in the geometry. The study explores…

Video recordings of instruction have been a mainstay for supporting conversations about teaching by representing particularities of instruction and by presenting the viewer with a multitude of details for interpretation. In this essay, we contrast non-fictional videotapes of actual classroom interaction with fictional animations of classroom…

We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…

Kinesthetic intelligence is one of the seven kinds of intelligence identified by Gardner's multiple intelligence theory (1983). The kinesthetic approach to teaching has numerous pedagogical advantages and can be adapted to the teaching of mathematics. This article describes a series of kinesthetic activities designed to explore the properties of…

This study presents findings from three cohorts of students--the classes of 2003, 2004, and 2005, in the School District of Philadelphia--that were admitted to the district's career and technical education (CTE) schools through a randomized lottery process. This study takes advantage of this so-called "'natural experiment' to compare high…

When hoping to initiate or sustain students' interest in mathematics teachers should always consider relevance, relevance to students' lives and in the middle and later years of instruction in high school and university, accessibility. A topic such as the mathematics behind networks science, more specifically scale-free graphs, is up-to-date,…

For students who have difficulty envisioning how the mathematics that they are learning is used outside the classroom, an adventure as field mathematicians can be enlightening. Measuring stream discharge is a field experience that allows students to engage in a hands-on (and boots-on) real-world, problem-solving activity and that integrates…

Although teachers' professional knowledge and instructional beliefs are known to have a variety of subcomponents, research on the structure and interrelations of these subcomponents is relatively scarce. This article presents some findings of a preliminary, quantitative empirical study examining the relationships between mathematics teachers'…

Generations have been inspired by Edwin A. Abbott's profound tour of the dimensions in his novella "Flatland: A Romance of Many Dimensions" (1884). This well-known satire is the story of a flat land inhabited by geometric shapes trying to navigate the subtleties of their geometric, social, and political positions. In this article, the authors…

Exploring mathematical symmetry is one way of increasing students' understanding of art. By asking students to search designs and become pattern detectives, teachers can potentially increase their appreciation of art while reinforcing their perception of the use of math in their day-to-day lives. This article shows teachers how they can interest…

Determining how to provide good tutoring functions is an important research direction of intelligent tutoring systems. In this study, the authors develop an intelligent tutoring system with good tutoring functions, called "FUDAOWANG." The research domain that FUDAOWANG treats is junior middle school mathematics, which belongs to the objective…

The purpose of this study is to identify the effect of using metacognitive strategies for problem solving in "geometry" on fifth grade students' achievement, metacognitive skills and attitude. Experimental method was used with a pretest/posttest control group design. Firstly, both groups were subject to a pretest that was comprised of…