In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…

This paper introduces a class of matrices, the rows and columns of which add up to identical constants (generally speaking, different for rows and columns). Some properties of these matrices, which will be called "magic rectangles" are discussed.

Ensuring that students have a strong foundation in mathematics by the end of 7th grade is essential if they are to move successfully into an advanced math curriculum that begins with algebra. But student achievement at the end of 7th grade varies widely in California--and this has consequences. Some 7th graders are taking a full Algebra I course;…

The purposes of this study were to (1) investigate non-accelerated middle schools students' understanding of pre-algebraic concepts, in particular, addition and subtraction of integers and the notion of equality; (2) describe and analyze how non-accelerated middle school students interact with The Geometer's Sketchpad (GSP) applets depicting…

As a follow up to the National Council on Teacher Quality (NCTQ) national studies of how well elementary teachers are prepared to teach reading and mathematics, NCTQ looked at preparation in both subjects in all undergraduate teacher preparation programs in Utah. This study evaluated Utah's nine undergraduate elementary teacher preparation…

The implementation of a new mathematics syllabus in the elementary context is problematic, especially if it contains a new content area. A professional development model, Transformative Teaching in the Early Years Mathematics (TTEYM) was specifically developed to support the implementation of the new Patterns and Algebra strand. The model was…

In early 2008 researchers from the University of Melbourne's "New Technologies for Teaching Mathematics" project created a lesson for the Year 10 students at their Victorian research schools. Two important goals of secondary school mathematics education are to build students' conceptual knowledge and to teach students to think…

Computer algebra systems (CASs) have been used by thousands of teachers and students for teaching and learning algebra. They have the ability to perform efficiently almost all of the algebraic expansions and simplifications. Nevertheless, the traditional approach of using paper and pencil in acquiring procedural knowledge is still widely…

Teachers all over the world are aware that students struggle with fractional concepts and with elementary algebra. Support for this assertion can be found in a variety of research reports. The National Assessment of Educational Progress (NAEP), a United States report, indicates that students have recurrently demonstrated a lack of proficiency in…

For a given orbital period and eccentricity, we determine the maximum time lapse between the winter solstice and the spring equinox on a planet. In addition, given an axial precession path, we determine the effects on the seasons. This material can be used at various levels to illustrate ideas such as periodicity, eccentricity, polar coordinates,…

In this paper, the author investigates the algebraic curve defined by the relation y[superscript 3] - 3y + 2x = 0. Treating this relation as a reduced cubic in the variable y, he uses a procedure first discovered by the mathematician Scipione del Ferro (Nahin, 1998, pp. 8-10) to obtain an expression for y in terms of x, namely y = (-x + [square…

Previous research, which typically overestimated competence, indicates that preschoolers have an unreliable or a localized understanding of the addition-subtraction inverse principle (e.g., 2 + 1 - 1 = 2). Forty-eight Taiwanese 4- to 6- year-old participants were tested with a relatively conservative measure to gauge when a reliable and general…

This paper, the first of a two-part article, follows the trail in history of the development of a graphical representation of the complex roots of a function. Root calculation and root representation are traced through millennia, including the development of the notion of complex numbers and subsequent graphical representation thereof. The…

Hill ciphers are linear codes that use as input a "plaintext" vector [p-right arrow above] of size n, which is encrypted with an invertible n x n matrix E to produce a "ciphertext" vector [c-right arrow above] = E [middle dot] [p-right arrow above]. Informally, a near-field is a triple [left angle bracket]N; +, *[right angle bracket] that…

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers' belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I…