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…

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and…

Algebraic reasoning skills can be enhanced by building connections among symbolic, tabular, and graphical representations. But how do classroom teachers change traditional teaching practices to foster the use of different representational forms? This article explores how I learned to incorporate representational fluency in teaching and assessment.…

We describe a physics investigation project inspired by one of the adventures of Odysseus in Homer's "Odyssey." The investigation uses the laws of mechanics, vector algebra and a simple way to construct a fan-and-sail-cart for experimental verification.

"Cognitive Tutor[R] Algebra I," a full year course, delivers instruction in single variable data, simplifying linear expressions, mathematical modeling, solving systems with linear equations, problem solving using proportional reasoning, and powers and exponents. Students work at their own pace to develop problem-solving skills. The…

I identify and discuss ways in which different types of connections are described in the South African mathematics National Curriculum Statement and its related documents, particularly at the Further Education and Training (FET) level. I argue that connections are central to the way the discipline of mathematics, its learning outcomes, and…

New Mexico Mathematics Content Standards, Benchmarks, and Performance Standards identify what students should know and be able to do across all grade levels, forming a spiraling framework in the sense that many skills, once introduced, develop over time. While the Performance Standards are set forth at grade-specific levels, they do not exist as…