How do people learn novel mathematical information that contradicts prior knowledge? The focus of this thesis is the role of structure in the acquisition of knowledge about hyperbolic geometry, a non-Euclidean geometry. In a series of three experiments, I contrast a more holistic structure--training based on closed figures--with a mathematically…

Hyperbolic geometry occurs on hyperbolic planes--the most commonly cited one being a saddle shape. In this article, the author explores negative hyperbolic curvature, and provides a detailed description of how she constructed two hyperbolic paraboloids. Hyperbolic geometry occurs on surfaces that have negative curvature. (Contains 11 figures and 4…

This is the first of two articles which describe how to use "JavaSketchPad" to explore the centres of a triangle. This introductory exercise is suggested in the GSP "Workshop Guide". Students can use "JavaSketchPad Interactive Geometry" (JSP) at home at no cost. They are likely to impress their parents with their enthusiasm for geometry and all…

This dissertation was conducted in a gendered academy in a low-income community in northern Indiana. The study examined how involvement with inquiry-based mathematical tasks embedded in a curricular unit impacted the geometric reasoning of elementary aged African-American males. The curriculum included activities focused on angles, lines and…

Studying geometry is an integral component of learning mathematics because it allows students to analyse and interpret the world they live in as well as equip them with tools they can apply in other areas of mathematics. Therefore, students need to develop an understanding of geometric concepts as well as gaining adequate geometry related skills.…

The purpose of this study was to enhance pre-service secondary mathematics teachers' skills of using the Geometer's Sketchpad (GSP) through Lesson Study. Twenty-four Lesson Study groups each comprising four or five pre-service secondary mathematics teachers who attended a mathematics teaching methods course in a local public university were set up…

Students' lack of understanding about the relationships between geometry in two and three dimensions led the author to a surprising source of inspiration--the ancient philosopher and geometer Plato. From a theoretical perspective, the author's approach embodies the four instructional strategies that Eggen and Kauchak (2001) suggest for engaging…

Given the importance of algebra, middle school mathematics teachers have a responsibility to help students transition from understanding arithmetic to understanding the algebra that will be necessary for success in high school. One method of transition involves introducing algebraic concepts in concrete ways using meaningful contexts. The series…

Although there are increasing numbers of professional development (PD) materials intended to foster teachers' mathematical knowledge for teaching within the topics of number and algebra, little attention has been given to geometry. In this article we describe the Learning and Teaching Geometry project's approach to the development of PD materials…

This article presents a web-based tool designed to compute certified equations and graphs of geometric loci specified using standard Dynamic Geometry Systems (DGS). Complementing the graphing abilities of the considered DGS, the equations of the loci produced by the application are remotely computed using symbolic algebraic techniques from the…

The use of difference and differential equations in the modelling is a topic usually studied by advanced students in mathematics. However difference and differential equations appear in the school curriculum in many direct or hidden ways. Difference equations first enter in the curriculum when studying arithmetic sequences. Moreover Newtonian…

At a high school level, sometimes students can perceive math as simply another step towards graduation and learn just what is needed to get by. If realistic examples were used more often in classroom settings, then teachable moments would happen--and that is where inspiration can be fostered. This article provides thoughts and ideas behind the…

A case study examines a 5th-grade teacher's orchestration of discourse and interaction to create opportunities for English language learners to participate in the repair of mathematical errors during a unit on finding the area of geometric shapes. The findings detail how the teacher used gestures in grounding her questioning, revoicing students'…

Using an apparently simple problem, ''Design a cylindrical can that will hold a liter of milk,'' this paper demonstrates how engineering design may facilitate the teaching of the following ideas to secondary students: linear and non-linear relationships; basic geometry of circles, rectangles, and cylinders; unit measures of area and volume;…

In this article, we present an experiment carried out in an 11th grade class in the curricular subject of DG (descriptive geometry). In a context where failure is more in the national exam, as well as in the discipline's attendance, the main objectives that motivated the implementation of this educational experience, were, on the one hand, to…