Using the modular ring Zeta[subscript 4], simple algebra is used to study diophantine equations of the form (x[cubed]-a=y[squared]). Fermat challenged his contemporaries to solve this equation when a = 2. They were unable to do so, although Fermat had devised a rather complicated proof himself. (Contains 2 tables.)

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…

This article shows how Microsoft Excel's ability to vary parameters with sliders allows students to "stand on the shoulders of giants" and discover characteristics of polynomial functions.

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

This article describes the rich thinking that can go into solving a difficult equation. The authors also give some sources for challenging mathematical problems.

Algebra student achievement has become a matter for increased focus in recent years, as educators answer calls for reform, including the standards documents of the National Council of Teachers of Mathematics and the No Child Left Behind mandates. A meta-analysis of 35 independent experimental studies shows that six teaching method categories have…

The central question of this study was whether there was a difference in mathematics achievement between students taught in an ICL classroom and those taught in a traditional classroom. The author was also interested in the experience students had while using the software. Thus, students in the ICL curriculum were surveyed about their attitudes…

Peters (1992) reports that students in the intervention and control groups showed gains on the Orleans-Hanna test during the course of the school year (that is, from pretest to posttest). However, the test score gains of the two groups did not differ significantly. There was no evidence that the Saxon Algebra curriculum (intervention) was more or…

Through an examination of one institution's efforts to strengthen teaching and learning on campus, the author makes a strong case for the use of common examinations as a powerful form of assessment as well as a fruitful context for faculty deliberations. Providing a continuing occasion for faculty inquiry and discussion, insuring grade…

The relationship between teachers and Computer Algebra Systems is generally problematic. The extensive capabilities of CAS provide opportunities for learning but also bring a new complexity that makes it difficult for teachers to take advantage of these opportunities. Barzel's paper contrasts with this observation: in a "Lernwerkstatt"…

In 4 experiments, the authors explored the role of visual layout in rule-based syntactic judgments. Participants judged the validity of a set of algebraic equations that tested their ability to apply the order of operations. In each experiment, a nonmathematical grouping pressure was manipulated to support or interfere with the mathematical…

For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with…

With the help of technology and a basic high school algebra method for finding the vertex of a quadratic polynomial, students can develop and prove the formula for least-squares lines. Students are exposed to the power of a computer algebra system to generalize processes they understand and to see deeper patterns in those processes. (Contains 4…

The concept of a function is fundamental to virtually every aspect of mathematics and every branch of quantitative science. Presently this type of thinking is carolled at the secondary level, and yet it has many benefits for deepening the understanding of early arithmetic. This is particularly so in the way that operations can be considered as…

In part 1 of this work we showed how modern mathematical research could, with a suitably chosen problem, be included in the first year curriculum of undergraduate mathematicians. With the use of Computer Algebra Systems, even the average undergraduate mathematician can aspire to discover interesting yet still unexplained behaviour in many areas of…