A test item exhibits differential item functioning (DIF) if students with the same ability find it differentially difficult. When the item is administered in French and English, differences in language difficulty and meaning are the most likely explanations. However, curriculum differences may also contribute to DIF. The responses of Ontario…

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.

A simple problem relating to birds chasing each other gives rise to a homogeneous differential equation. The solution draws on student skills in differential equations and basic co-ordinate geometry.

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…

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

This note could find use as enrichment material in a course on the classical geometries; its preliminary results could also be used in an advanced calculus course. It is proved that if a , b and c are positive real numbers such that a[squared] + b[squared] = c[squared] , then cosh ( a ) cosh ( b ) greater than cosh ( c ). The proof of this result…

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…

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

Pascal's Triangle is, without question, the most well-known triangular array of numbers in all of mathematics. A well-known algorithm for constructing Pascal's Triangle is based on the following two observations. The outer edges of the triangle consist of all 1's. Each number not lying on the outer edges is the sum of the two numbers above it in…

The geometry textbooks of the new system of secondary education in Iran differed dramatically from the old ones considering the aims, the visions, the content, the approach, and the educational purposes. Four hundred eighty mathematics teachers participated in a nationwide professional development program conducted by the author to facilitate the…

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…

The interpretive framework for the study of learning introduced in this article and called "commognitive" is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather…

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,…

Mathematics is both a beautiful language and the simplest systematic discipline men ever created. The simplicity of mathematical concepts almost guarantees that the facts it establishes about those concepts will also be elemental. Despite this simplicity, most people complain about the difficulty of mastering the subject and shun the study of…

High school students in the United States have been taking more challenging courses in recent years, but academic achievement has been stagnant. At the heart of the matter is the quality of curriculum, instruction, and assessment. Some courses tend to be more challenging in name than in practice. High schools also have a history of autonomy that…