Thousands of years ago, mathematicians thought about numbers in terms of geometry. Numbers had shapes, and by thinking about the shapes, mathematicians could do a lot of things with numbers that are now recognized as algebra. This document contains a lesson plan for students to learn the relationship between numbers--all while developing algebraic…

With both "Cabri" and "Autograph" now venturing into 3D, the dimension that previously was only demonstrated in the classroom with a lot of arm waving and crude wire cages can now be explored dynamically on screen. "Cabri 3D" concentrates on constructions, using the principles of Euclidian geometry, whereas "Autograph" creates objects using a…

This article discusses how paper folding can be used in the classroom to introduce the standard results of school geometry, such as the transversal and parallel lines results, along with results concerning angles in convex polygons and centres of triangles, for example. Angle bisectors, midpoints, perpendiculars are all straightforward…

Several examples proved Van Hiele theory to be a useful ingredient in teaching of a summer course for high school students who had failed geometry during the school year are discussed. The theory provided a framework to help organize and reflect upon instruction for some key concepts in geometry.

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…