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…

Triangular numbers are used to unravel a new sequence of natural numbers here-to-fore not appearing on the Encyclopedia of Integer Sequences website. Insight is provided on the construction of the sequence using "buildings" as a viewable model of the sequence entries. A step-by-step analysis of the sequence pattern reveals a method for generating…

This article describes a unique supplementary program that teaches early childhood mathematics (Pre-K to Grade 2), through a series of six problem-solving adventure stories. The mathematics concepts are taught to the children through the medium of oral storytelling sagas in an integrated approach that addresses language arts as well as early…

This article focuses on the kinds of conceptual changes that occur when students have to deal with negative numbers in elementary algebraic operations. Interviews were carried out with twelve 8th-grade level students who were selected on the basis of their results in a test where they were required to reduce polynomials. The questions applied to…

This article reports work to assist learners in acquiring insights into certain grammatical and lexical areas of English by drawing on their awareness of fairly elementary mathematical concepts--and shows how often English structures can behave in ways parallel to numbers. For learners with an appropriate background and learning style, it is…

Offering mathematics majors the opportunity to engage in current, real-world applications can be an important enhancement to their undergraduate course curriculum. Instead of focusing on the traditional topic areas in pure and/or applied mathematics, one may structure a seminar course for senior mathematics majors by concentrating on a specific…

This paper reflects on the use of didactic material in mathematics classes. It focuses on the mathematical activities of students and the didactical activities of teachers. Its point of departure is a critique of technical-managerial approaches to teaching, learning, and innovation. Based on this critique, fundamental tensions between the…

The focus is on the metacognitive awareness of ten high-achieving high school pupils in mathematics in Denmark and England and their understanding of their cognitive learning processes and strategies. Mainly unstructured focus group interviews investigate how they explain that they learn a mathematical concept that is new to them. I develop the…