This book contributes to a growing body of research relevant to efforts to make algebra an integral part of early mathematics instruction, an area of studies that has come to be known as Early Algebra. It provides both a rationale for promoting algebraic reasoning in the elementary school curriculum and empirical data to support it. The authors…

The use of digital elevation models, which represent the surface of the earth by a matrix of heights, has proved an ideal introductory topic for developing both the matrix-handling and computational skills of undergraduate mathematicians. This article looks at the solving of an easily understandable, but not necessarily simple problem: given a…

This article explores how conceptualization of absolute value can start long before it is introduced. The manner in which absolute value is introduced to students in middle school has far-reaching consequences for their future mathematical understanding. It begins to lay the foundation for students' understanding of algebra, which can change…

In this article, the author presents a paper folding activity that can be used for teaching fractions. This activity can be used to describe areas of folded polygons in terms of a standard unit of measure. A paper folding fractions worksheet and its corresponding solutions are also presented in this article. (Contains 2 figures.)

In this article, we deal with students' algebraic generalizations set in the context of elementary geometric-numeric patterns. Drawing from Vygotsky's psychology, Leont'ev's Activity Theory, and Husserl's phenomenology, we focus on the various semiotic resources mobilized by students in their passage from the particular to the general. Two small…

This paper, the second of a two part article, expands on an idea that appeared in literature in the 1950s to show that by restricting the domain to those complex numbers that map onto real numbers, representations of functions other than the ones in the real plane are obtained. In other words, the well-known curves in the real plane only depict…

Lectures remain the lynchpin of mathematics teaching at university even with advances in information technology and access to the internet. This paper examines the requirements for learning mathematics and shows how important it is for lecturers to be aware of the different modes of presentation they are using. Ways to assist students to make the…

Algebra teachers find themselves overwhelmed by the sheer volume of information that they have to teach their students. One way to address this issue is to have teachers teach for understanding by focusing on concepts. In this article, I share how language can help uncover the core ideas that define the concept of function and share activities…

In this article, the author reflects on her session at the 2006 conference, including the learning styles and strategies she observed. In her session, they explored some of the more unusual aspects of convex polyhedra (with regular faces), including the notions of "valence" and "species". Although the session was about shape and space, there was…

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers' conceptions of "beginning algebra" and their conceptions of their "own" teaching practices. The teachers' understandings of their teaching practices were explored with a view to unravelling their…

With Simson's 1753 paper as a starting point, the current paper reports investigations of Simson's identity (also known as Cassini's) for the Fibonacci sequence as a means to explore some fundamental ideas about recursion. Simple algebraic operations allow one to reduce the standard linear Fibonacci recursion to the nonlinear Simon's recursion…

This article presents a rich geometry problem and discusses its solution. A variety of mathematically correct justifications for this answer are provided. (Contains 2 figures.)

The Math Forum is an online resource center for pre-algebra, algebra, geometry and pre-calculus. Its Virtual Math Teams (VMT) service provides an integrated web-based environment for small teams of people to discuss math and to work collaboratively on math problems or explore interesting mathematical micro-worlds together. The VMT Project studies…

As expectations for students to meet high academic standards have risen over the past two decades, so have the expectations for students to complete, and excel in, more rigorous mathematics courses. Once a course reserved only for the college-bound, algebra is now a graduation requirement for all California students as well as an early…

This paper addresses methodological issues arising from an experimental study of North Carolina's Early College High School Initiative, a four-year longitudinal experimental study funded by Institute for Education Sciences. North Carolina implemented the Early College High School (ECHS) Initiative in response to low high school graduation rates.…