The prealgebra curriculum presents many opportunities for encouraging students to justify their inferences. Requiring students to communicate clearly the reasoning behind their solutions, with appropriate mathematical language and notation, helps lay the groundwork for future, proof-based mathematics courses. Prealgebra students can even be…

It is common to start a course on computer programming logic by teaching the algorithm concept from the point of view of natural languages, but in a schematic way. In this sense we note that the students have difficulties in understanding and implementation of the problems proposed by the teacher. The main idea of this paper is to show that the…

This paper presents a tool for drawing dynamic geometric figures by understanding the texts of geometry problems. With the tool, teachers and students can construct dynamic geometric figures on a web page by inputting a geometry problem in natural language. First we need to build the knowledge base for understanding geometry problems. With the…

In this article we consider data arising from student-teacher-researcher interactions taking place in the context of an experimental teaching program making use of multiple modes of communication and representation to explore three-dimensional (3D) shape. As teachers/researchers attempted to support student use of a logo-like formal language for…

This paper is a descriptive and preliminary report on recent efforts to address two questions: 1) Can school mathematics be used to enhance our students' ability to understand their changing world? and 2) What role might computer technology play in this regard? After recounting some of the mathematical tools that led to a better understanding of…

This study examines the origins of geometry in and out of the intuitively given everyday lifeworlds of children in a second-grade mathematics class. These lifeworlds, though pre-geometric, are not without model objects that denote and come to anchor geometric idealities that they will understand at later points in their lives. Roth's analyses…

In this paper, we summarize the findings of a research synthesis specifically related to the geometry and measurement content knowledge of elementary/middle school preservice teachers. Findings from this synthesis show that preservice teachers have weak conceptions in geometry and measurement content knowledge. However, instructional strategies…

Ellipses vary in shape from circular to nearly parabolic. An ellipse's eccentricity indicates the location of its foci, but its aspect ratio is a direct measure of its shape. This article takes a careful look at the shape of an ellipse and offers practical suggestions and specific activities to deepen students' understanding of the geometry of an…

Tiny prisms in reflective road signs and safety vests have interesting geometrical properties that can be discussed at any level of high school mathematics. At the beginning of the school year, the author teaches a unit on these reflective materials in her precalculus class so that students can review and strengthen their geometry and trigonometry…

This study investigated and compared the geometry knowledge and levels of pre-service elementary teachers from the United States and Taiwan. Forty pre-service teachers in Taiwan and 48 pre-service teachers in the United States at the beginning of their teacher education programs completed the Entering Geometry Test (EGT) and the van Hiele Geometry…

Teacher professional development activities in the USA take many forms from half-day workshops that focus on particular topics or classroom techniques to long term course work that offers university level credit. With few exceptions, the primary goal of such activities is to enhance the teachers' classroom effectiveness and improve student…

Phenomena that occur in microdroplets are described to the undergraduate chemistry community. Droplets having a diameter in the micrometer range can have unique and interesting properties, which arise because of their small size and, especially, their high surface area-to-volume ratio. Students are generally unfamiliar with the characteristics of…

The 2005 National Assessment of Educational Progress (NAEP) High School Transcript Study (HSTS) found that high school graduates in 2005 earned more mathematics credits, took higher level mathematics courses, and obtained higher grades in mathematics courses than in 1990. The report also noted that these improvements in students' academic records…

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given tau greater than 0, the free plate eigenvalues omega and eigenfunctions upsilon are determined by the equation Delta Delta upsilon - tau Delta upsilon = omega upsilon together with…

A shape grammar is a formalism that has been widely applied, in many different fields, to analyzing designs. Computer implementation of a shape grammar interpreter is vital to both research and application. However, implementing a shape grammar interpreter is hard, especially for parametric shapes defined by open terms. This dissertation…