Describes how students use crazy cars to confirm that the sum of measures of the angles of any triangle is 180 degrees. (Author)

Diagrams that illustrate characteristics that are always, sometimes, or never present in a concept can be categorized as examples or nonexamples to broaden students' understanding of basic geometric concepts. (MKR)

Takes a critical look at the assessment scheme by discussing underlying philosophies of teaching and what effects they may have on students. (ASK)

Describes the most enduring link between Napoleon and mathematics as the geometric result known as Napoleon's Theorem, which states that if equilateral triangles are drawn on the three sides of any triangle, the line segments joining the centers of these equilateral triangles will themselves form an equilateral triangle. (ASK)

Analyzes data concerning a comparison task where two points--the intersection point of two lines--and the intersection of six lines were compared. Suggests that students tend to grasp the two points as unequal; moreover, the intersection point of six lines was frequently viewed as bigger and heavier than the intersection point of two lines.…

Explains the two ontological convictions that played the role of epistemological obstacle for Bolzano, a contemporary of the founders of non-Euclidean geometry. Forms part of the body of work aimed at identifying obstacles in the history of mathematics in order to confront them with obstacles to learning and to establish their epistemological…

Discusses the mathematics of Theo van Doesburg's painting, Arithmetic Composition 1. Investigates ratios and symmetries as well as relationships among geometrical forms in the painting. (MM)

Between the 17th and 19th centuries, the Japanese government closed its borders to the outside world in an attempt to become more powerful. Foreign books were banned, people could not travel, and foreigners were not allowed to enter the country. One result of this isolation was the flourishing of sangaku--wooden tablets inscribed with intricately…

As time has progressed, the role of applied mathematics has become increasingly important. Indeed there are now more students enrolled in applied mathematics courses in senior high schools and colleges than in pure mathematics. Such courses become more relevant both to the student and to future employers, if the same constants and equations that…

The author of this article, while recently working through some problem sets on determining volumes by triple integrals in cylindrical and spherical coordinate systems, realized that, although the textbook he was using included many interesting problems involving spheres, cylinders and cones and the increasingly complex solids that arose from the…

Direct sum decomposition of Abelian groups appears in almost all textbooks on algebra for undergraduate students. This concept plays an important role in group theory. One simple example of this decomposition is obtained by using the kernel and range of a projection map on an Abelian group. The aim in this pedagogical note is to establish a direct…

A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.

An exploration of nonconvex, crossed, and degenerate polygons (NCCDPs) are described with the help of examples with pedagogical tips and recommendations that are found useful when teaching the mathematical process of extending geometric patterns to NCCDPs. The study concludes that investigating such extensions with interactive geometry software…

Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". This paper discusses a simple theorem of Descartes, which enables students to easily determine the number of vertices of almost every polyhedron. (Contains 1 table and 2 figures.)

Prediction is a great skill to have in any walk of life: it can, in fact, save lives at times. While the two investigations posed in this column may not be that dramatic, they might just increase one's appreciation of some important connections between grids and rectangles and the divisors of numbers that appear in the dimensions of those…