The basic Pythagorean theorem for right-angled triangles is well-known in mathematical terms as a[squared]+b[squared]+c[squared] were "a," "b," and "c" are the lengths of the sides of the triangle with "c" as the hypotenuse. When "a," "b," and "c" are all integers and obey this…

Simplicity arguments are to be found in most geometrical works, from those of Proclus in his "Commentaries on the First Book of Euclid's Elements," up to those of contemporary manuals. Our goal is to read these arguments in their historical contexts to analyze agreements, disagreements and the multiplicity of points of view. For a better…

In this note, it is shown that in a symmetric topological space, the pairs of sets separated by the topology determine the topology itself. It is then shown that when the codomain is symmetric, functions which separate only those pairs of sets that are already separated are continuous, generalizing a result found by M. Lynch.

Vieta's famous product using factors that are nested radicals is the oldest infinite product as well as the first non-iterative method for finding [pi]. In this paper a simple geometric construction intimately related to this product is described. The construction provides the same approximations to [pi] as are given by partial products from…

The overreliance on linear methods in students' reasoning and problem solving has been documented and discussed by several scholars in the field. So far, however, there have been no attempts to assemble the evidence and to analyze it is a systematic way. This article provides an overview and a conceptual analysis of students' tendency to use…

Two studies investigated the effects of a storytelling-context for teaching geometry skills to kindergarten girls and boys. In Study 1, the story+geometry intervention consisted of an adventure story teaching geometry through part-whole-relations puzzles. Learning was assessed through transfer of skills, using a pre-/post design comparing…

A globe, in a school or university yard, which simulates the Earth's orientation in space, could be a very useful and helpful device for teaching physics, geometry, astronomy and the history of science. It would be very useful for science education to utilize the forthcoming International Year of the Planet Earth 2008 and the International Year of…

Good teaching is responsive to individual differences, tailoring instruction to meet the needs of individual learners. In gifted education, students need a curriculum that is differentiated (by level, complexity, breadth, and depth), developmentally appropriate, and conducted at a more rapid rate. This collection of essays from experts in the…

Teachers know that Differentiated Instruction (DI) helps all students to learn. Yet DI challenges teachers, and nowhere more than in mathematics. In this new book, written specifically for secondary mathematics teachers, the authors cut through the difficulties with two powerful and universal strategies that teachers can use across all math…

By translating inequalities into equations (e.g., x greater than 0 can be written x- absolute value of x = 0) and forming equations for unions and intersections of solution sets, students can develop equations for polygons. The method can be generalized to yield equations in three dimensions. (SD)

This is part one of a two-part SMSG geometry text for high school students. One of the goals of the text is the development of analytic geometry hand-in-hand with synthetic geometry. The authors emphasize that both are deductive systems and that it is useful to have more than one mode of attack in solving problems. The text begins the development…

This is part two of a two-part SMSG geometry text for high school students. One of the goals of the text is the development of analytic geometry hand-in-hand with synthetic geometry. The authors emphasize that both are deductive systems and that it is useful to have more than one mode of attack in solving problems. The text begins the development…

This is part one of a two-part manual for teachers using SMSG high school text materials. The commentary is organized into four parts. The first part contains an introduction and a short section on estimates of class time needed to cover each chapter. The second or main part consists of a chapter-by-chapter commentary on the text. The third part…

This is part two of a two-part manual for teachers using SMSG high school text materials. The commentary is organized into four parts. The first part contains an introduction and a short section on estimates of class time needed to cover each chapter. The second or main part consists of a chapter-by-chapter commentary on the text. The third part…