Suppose that we are given a rectangular box in 3-space. Given any two points on the surface of this box, we can define the surface distance between them to be the length of the shortest path between them on the surface of the box. This paper determines the pairs of points of maximum surface distance for all boxes. It is often the case that these…

The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite…

Some infinite series are analysed on the basis of the hypergeometric function and integer structure and modular rings. The resulting generalized functions are compared with differentiation of the "mother" series. (Contains 1 table.)

The standard approach to finding antiderivatives of trigonometric expressions such as sin(ax) cos(bx) is to make use of certain trigonometric identities. The disadvantage of this technique is that it gives no insight into the problem, but relies on students using a memorized formula. This note considers a technique for finding antiderivatives of…

This paper addresses the forming of integral expressions such as [integral][superscript b][subscript a]f(x)dx by differential or infinitesimal derivations. A rigorous justification of these handy informal derivations is a major pedagogical problem in calculus. In this paper, checks are presented by which one can usually verify--with clear…

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

This article presents a problem that can help high school students develop the concept of instantaneous velocity and connect it with the slope of a tangent line to the graph of position versus time. It also gives a method for determining the tangent line to the graph of a polynomial function at any point without using calculus. (Contains 1 figure.)

This study developed a computer-assisted mathematical problem-solving system in the form of a network instruction website to help low-achieving second- and third-graders in mathematics with word-based addition and subtraction questions in Taiwan. According to Polya's problem-solving model, the system is designed to guide these low-achievers…

To say Earth systems are complex, is not the same as saying they are a complex system. A complex system, in the technical sense, is a group of -agents (individual interacting units, like birds in a flock, sand grains in a ripple, or individual units of friction along a fault zone), existing far from equilibrium, interacting through positive and…

This paper reports on three primary school students' explorations of 3D rotation in a virtual reality learning environment (VRLE) named VRMath. When asked to investigate if you would face the same direction when you turn right 45 degrees first then roll up 45 degrees, or when you roll up 45 degrees first then turn right 45 degrees, the students…

Some schools are replacing music classes with additional math classes in an effort to improve student scores on standardized tests. Fewer academic options may hinder a well-rounded education. Within a conceptual framework that included an interdisciplinary learning, constructivist, and multiple intelligences theories, this research investigated…

Ratio (and associated topics such as fractions and proportion) is known to be an area of mathematics that students find difficult. Multiplicative thinking is necessary, and students benefit from a wide range of strategies and representations for interpreting ratio. This study examined aspects of teachers' pedagogical content knowledge for teaching…

This book examines the five key elements (meaningful use of symbols, mindful manipulation, reasoned solving, connection algebra with geometry, and linking expressions and functions) identified in "Focus in High School Mathematics: Reasoning and Sense Making" in more detail and elaborates on the associated reasoning habits. This volume is one of a…

The aim of this study is to identify the effects of formalism in teaching on primary and secondary school mathematics teacher trainees' algebraic and geometric interpretations of the notions of linear dependency/independency. Quantitative research methods are drawn in order to determine differences in success levels between algebraic and geometric…

In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the…