This article describes explorations using The Geometer's Sketchpad for algebra and calculus content.

This article begins with a comparison of two groups of teachers, working on the same tasks in Analytic Geometry. One group has only basic experience in CAS-assisted problem solving, and the other group has extensive experience. The comparison is discussed in terms of the interplay between reflection, operative knowledge and execution. The findings…

In this article, the author describes a 200-year-old ladder problem that can carry learners to high levels of mathematical thinking and activity. This problem requires learners to go from a word problem to an equation to a graph and from there to a solution. As this problem of specifics is turned into a problem using variables, technology,…

This paper describes aspects of 13 year-olds activity in mathematics as emerged during the implementation of proportional geometric tasks in the classroom. Pupils were working in pairs using a piece of software specially designed for multiple representation (symbolic and graphical) of the variation in parametric procedures with dynamic…

Some of the graphing capabilities of The Geometer's Sketchpad (GSP) in the "Technology Tips" are introduced. The new graphing features of GSP allow teachers to implement the software not only in geometry classrooms but also into their algebra, precalculus and calculus classes.

Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given…

Given two circles C 1 and C 2 in a plane such that neither one of the two circles is contained in the other, there are either four common tangents when the circles do not intersect at all or the circles have three common tangents when they touch each other externally or only two common tangents when the circles intersect exactly at two points. The…

In this study, I present an analysis of high school geometry curricula regarding mathematical proof opportunities. I examined eight U.S. high school level geometry textbooks, which were categorized into three main groups: technology-intensive, standards-based, and traditional curricula. I conceptualized "ideal" proving activity combining…

This paper was prepared in response to the Conference Board of Mathematical Sciences recommendations for the preparation of secondary teachers. It shows how using trigonometry as a conceptual tool in spreadsheet-based applications enables one to develop mathematical understanding in the context of constructing geometric representations of unit…

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…

In the early stages of working with dynamical geometry environments, students make many more mistakes than if they thought out and implemented the same constructions on paper. Most Czech teachers have very little experience of doing geometry using computers. A methodology which could help them to teach students to avoid mistakes dependent on the…

Although the nature of the research varies, as do concepts of angle, research in general supports the supposition that angle is a complex idea, best understood from a variety of perspectives. In fact, the concept of angle tends to be threefold, consisting of: (1) the traditional, static notion of two rays meeting at a common vertex; (2) the idea…

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…

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…

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…