Many exercises in mathematical induction require the student to prove a divisibility property of a function of the integers. Such problems are generally presented as being independent of each other. However, many of these problems can be presented in terms of difference equations, and the theory of difference equations can be used to provide a…

This article describes how to multiply binomials so that middle school students produce the equivalence (a + b)[superscript 2] = a[superscript 2] + 2ab + b[superscript 2] after reasoning about and representing problems. First, a two-dimensional array representation is used so that students can become comfortable with multiplication embedded in…

In this note, we present an improved Heaviside approach to compute the partial fraction expansions of proper rational functions. This method uses synthetic divisions to determine the unknown partial fraction coefficients successively, without the need to use differentiation or to solve a system of linear equations. Examples of its applications in…

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of…

Although it is difficult to find any current literature that does not encourage use of multiple representations in mathematics classrooms, there has been very limited research that compared such practice to student achievement level on standardized tests. This study examined the associations between students' achievement levels and their (a)…

Multiple-choice testing is an educational reality. Rather than complain about the negative impact these tests may have on teaching and learning, why not use them to better understand your students' true mathematical knowledge and comprehension? Maryann Wickett and Eunice Hendrix-Martin show teachers how to move beyond the student's answer--right…

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students' RF in learning environments that utilize both computer algebra systems…

The new Common Core State Standards for Mathematics have been formulated to provide students with instruction that will help them acquire a thorough knowledge of math at their grade level, which will in turn enable them to move on to higher mathematics with competence and confidence. "Hands-on Activities for Teaching the Common Core Math…

This paper explores how curriculum and classroom conceptual and procedural emphases affect the learning of algebra for students of color. Using data from a longitudinal study of the Connected Mathematics Program (CMP), we apply cross-sectional HLM to lend explanatory power to the longitudinal analysis afforded by Growth Curve Modeling that we have…

This article is the second in a series of two papers which suggest some practical, spreadsheet-based ideas for helping students to make appropriate connections between particular algebraic concepts. Solving equations has traditionally been taught as a pen-and-paper process. Spreadsheets, such as that of Excel, provide a contemporary, and powerful…

This study reports how 24 grade 4-6 students in one elementary and middle school interpreted formal and informal representations of variables. While interpretations for variables represented as letters (e.g., x and y) have been well established for students in algebra classes and beyond, little research into elementary school students'…

The classic Chickens and Pigs problem is considered to be an algebraic problem with two equations and two unknowns. In this article, the authors describe how third-grade teacher Maria is using it to develop a problem-based lesson because she is looking to her students' future needs. As Maria plans, she considers how a series of problems with the…

This study examined how literal symbols affect students' understanding of algebraic expressions. Middle school students (N = 322) were randomly assigned to 1 of 3 conditions in which they were asked to interpret an expression (e.g., 4c + 3b) in a story problem. Each literal symbol represented the price of an item. In the c-and-b condition, the…

The National Council of Teachers of Mathematics (NCTM) Algebra Standard states that instructional programs at the middle grades should enable students to "represent and analyze mathematical situations and structures using algebraic symbols" (2000, p. 222). Guess and check is a powerful problem-solving strategy that can connect a conceptual…

Systems of linear equations are used in a variety of fields. Exposure to the concept of systems of equations initially occurs at the high school level and continues through college. Attempts to unearth what students understand about the solutions of linear systems have been limited. Gaps exist in our knowledge of how students understand systems…