This article describes how three algebra problems involving systems of equations can be modeled and solved with manipulative materials or pictorial representation, then the solutions are compared with standard solutions using algebraic symbols. As students begin to use manipulatives and represent their pictures with number sentences, they are…

This article focuses on the dynamics of language in the context of mathematics. The interaction of everyday words and specialized mathematics vocabulary impacts students' development of mathematical understanding. (Contains 4 tables and 1 figure.)

This article describes using area and ratio concepts to examine why some animals, or people wearing different types of shoes, sink into the surface on which they are standing. Students compute "sinking values" to explain these differences. (Contains 2 figures.)

The purpose of this article is to describe a middle school mathematics teacher's model of his students' responses to algebraic tasks involving equivalent expressions and the distributive property. The teacher engaged in two model-eliciting activities designed for teachers by creating a library of his students' work and an accompanying "Ways…

The development of conceptual understanding in arithmetic is a gradual process and children may make use of a concept in some situations before others. Previous research has demonstrated that when children are given arithmetic problems with an inverse relationship they can infer that the initial and final quantities are the same. However, we do…

P. A. Arbisi and Y. S. Ben-Porath (1995) originally proposed that the Infrequency Psychopathology scale, F(p), be used as the final step in an algorithm to determine the validity of a Minnesota Multiphasic Personality Inventory-2 (MMPI-2) protocol. The current study used taxometric procedures to determine the latent structure of F(p) among…

This study was conducted to reveal how students at university level justify their solutions to tasks with various degrees of difficulty. The study is part of a larger study of students' concept formation of limits. The mathematical area is limits of functions. The study was carried out at a Swedish university at the first level of mathematics. The…

In set theory, one comes across the notion of "vacuous truth." A statement is vacuously true if it is true but does not quite say anything. The structure of a vacuously true statement is typically of the form: everything with property A also has property B, with the caveat being that there is nothing in property A. For instance one could…

What happens when children pour water through a funnel? They begin to understand science and math concepts such as flow, force, gravity, and volume. What happens when children mold sand to create a tunnel? They develop skills in areas such as problem solving and predicting. They also gain knowledge about absorption and the properties of sand and…

Cyberchase is an animated television series airing on public broadcasting stations throughout the United States. It focuses on mathematics for elementary-level students ages 8-11 and is designed to engage young people with math concepts in a highly stimulating multimedia format. As part of continuing program improvement, a website for parents and…

It seems self-evident that there is a significant gap between secondary school and university mathematics, even though the gap may take different forms, which vary with different education systems in different places and at different times. This paper attempts to capture some common forms of this gap, and in particular, it discusses some core…

The United States Military Academy (USMA) has a four course core mathematics curriculum that is studied by all students. The third course is MA205, Calculus II; a multivariate calculus course filled with practical applications. During a Problem Solving Lab (PSL), students participated in a hands-on exercise with multiple vector operations,…

The interconnected world of actors and movies is a familiar, rich example for graph theory. This paper gives the history of the "Kevin Bacon Game" and makes extensive use of a Web site to analyze the underlying graph. The main content is the classroom development of the weighted average to determine the best choice of "center" for the graph. The…

This paper suggests examples that may be used to better integrate modern technology into the calculus I curriculum, and at the same time extend the student's understanding of the underlying concepts. Examples are chosen from the usual topics considered in most courses and not limited to any specific form of the technology.

Based on the properties of a Primitive Pythagorean Triple (PPT), a computer program was written to generate, print, and count all PPTs greater than or equal to I[subscript x], where I[subscript x] is an arbitrarily chosen integer. The Density of Primitive Pythagorean Triples may be defined as the ratio of the number of PPTs whose hypotenuse is…