The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.

The purpose of this paper is to present an analysis of the difficulties faced by students when working with different representations of vectors, planes and their intersections in R[superscript 3]. Duval's theoretical framework on semiotic representations is used to design a set of evaluating activities, and later to analyze student work. The…

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the "Elements" of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in…

Students are faced with many transitions in their middle school mathematics classes. To build knowledge, skills, and confidence in the key areas of algebra and geometry, students often need to practice using numbers and polygons in a variety of contexts. Teachers also want students to explore ideas from probability and statistics. Teachers know…

In some secondary mathematics curricula, there is a topic called Stereometry that deals with investigating the position and finding the intersection, angle, and distance of lines and planes defined within a prism or pyramid. Coordinate system is not used. The metric tasks are solved using Pythagoras' theorem, trigonometric functions, and sine and…

Determining how to provide good tutoring functions is an important research direction of intelligent tutoring systems. In this study, the authors develop an intelligent tutoring system with good tutoring functions, called "FUDAOWANG." The research domain that FUDAOWANG treats is junior middle school mathematics, which belongs to the objective…

The Shepherd's Principle (to count the number of sheep in a field, count the number of legs and divide by four) is applied to the problem of finding the number of different rectangles on an N by N checkerboard. (MM)

This book, photographically reproduced from its original 1942 edition, is an extended essay on one of the three problems of the ancients. The first chapter reduces the problem of trisecting an angle to the solution of a cubic equation, shows that straightedge and compasses constructions can only give lengths of a certain form, and then proves that…