The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers' understanding of other functions of proof than just the traditional function of 'verification'. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of…

The global psychophysical theory of summation and magnitude production of R. D. Luce (2002) had joint presentations of pairs of intensities (measured above threshold) being matched asymmetrically, with 1 component being 0 intensity and the other the matching intensity. For loudness, an intensity pair to the 2 ears is matched by an intensity in…

The topic of centers of similarity can be treated synthetically or analytically. While the synthetic method is more practiced, the analytic approach is more appropriate when the problem is given in an analytic geometry setting. In this article, two non-congruent squares ABCD and A'B'C'D' are given, where A(0,0), B(3,0), C(3,3), D(0,3) and A'(5,4),…

This note uses some elementary concepts in linear algebra to prove some well-known results in two- and three-dimensional geometry.

This article was written as a result of the author reading "MT177," a special issue dedicated to the teaching of "proof" in mathematics. He used the ideas in this special issue for planning his session "mathematical reasoning and proof," which was part of a weekend course for primary trainees. It consisted of three activities: (1) How many…

Students are asked to analyze a photo of a beer stein, computing volumes and areas based on the photo.

This article details an exploration of Baravelle spirals as visual representations of infinite geometric series, focusing on a variety of strategies used by preservice teachers in discovering patterns and investigating relationships of variables.

The article discusses the use of L-systems, which provide students with a unique method to construct line fractals, including the Koch snowflake, the Sierpinski triangle, and the Harter-Heighway dragon. Applets that use L-system theory offer a graphics tool that promotes geometric reasoning, sparks enthusiasm, and connects to historical themes in…

This article describes how the concept of "key idea" can be used in high school geometry to connect students' informal explorations with rigorous mathematical proof. (Contains 6 figures.)

We propose to use didactical theory for the design of educational software. Here we present a set of didactical conditions, and explain how they shape the software design of Cabri-Euclide, a microworld used to learn "mathematical proof" in a geometry setting. The aim is to design software that does not include a predefined knowledge of problem…

The author recounts experiences introducing a first grade class to the concept of symmetry. The lesson includes activities for developing recognition of objects that have symmetry as well as visualization of both horizontal and vertical lines of symmetry.

In this paper, we will focus on an innovative way to teach some of the engineering fundamentals at the freshman level in an introductory engineering class. Unfortunately, today many students graduate without a good grasp of these fundamental concepts--concepts that every engineer, regardless of his or her area of specialization, should know. In…

This report focuses on the performance of fourth-and eighth-grade students in Puerto Rico in various mathematics content areas on the 2005 National Assessment of Educational Progress (NAEP) in mathematics. The NAEP mathematics assessment was administered to public school students in Puerto Rico for the first time in 2003. Although NAEP had…

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary…