This article shows how ideas of maximum or minimum values of fundamental quantities have developed through the ages. (Author/JG)

The typical precalculus book contains the obscure trigonometric identities known as the product-to-sum formulas. They usually get short treatment (or none) in a precalculus course because they are so rarely used. This is unfortunate since they have an interesting history. Before the invention of logarithms they were used to perform multiplications…

This essay investigates the rhetoric surrounding the appearance of the concept of the infinitesimal in the seventeenth-century Calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Although historians often have positioned rhetoric as a supplemental discipline, this essay shows that rhetoric is the "material" out of which a new and powerful…

A study of first-year undergraduate students' interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function…

The paper considers a class of real integrals performed by using a convenient integral in the complex plane. A complex integral containing a multi-valued function with two branch points is transformed into another integral containing a pole and a unique branch point. As a by-product we obtain a new class of integrals which can be calculated in a…

The purpose of this study was to assess algebra and geometric prerequisites skills as incorporated into the Applied Calculus Optimization Problem (ACOP) solution. The difficulties that students encounter in applying algebraic and geometric prerequisites at the early stages of the ACOP solution were identified. The study analyzes errors related to…

This study investigates the role of graphing calculators in multiple representations for knowledge transfer and the omission of oversimplification in complex function graphs. The main aim is to examine whether graphing calculators were used efficiently to see different cases and multiple perspectives among complex function graphs, or whether…

In this note we develop a framework that makes explicit the inherent dynamic structure of certain mathematical definitions by means of the four facets of context-entity-process-object. These facets and their interrelations are then used to capture and interpret specific aspects of student constructions of the concept of solution to first order…

During the last two years we have developed a precalculus course customized around biology by using the scientific method as a framework to engage and motivate biology students. Historically, the precalculus and calculus courses required for the Suffolk County Community College biology curriculum were designed using examples from the physical…

Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…

Recent events have led to an increased interest in emerging infectious diseases. This article applies various deterministic models to the SARS epidemic of 2003 and a measles outbreak in the Netherlands in 1999-2000. We take a historical approach beginning with the well-known logistic curve and a lesser-known extension popularized by Pearl and Reed…

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students' progressively sophisticated ways of reasoning and…

This paper is essentially a story that can be used to help students understand some of the definitions found in a standard introductory real analysis course.

This paper deals with a brief introduction to major remarkable discoveries of the "soliton" and the "inverse scattering transform" in the 1960s. The discovery of the soliton (or the solitary waves) began with the famous physical experiments of the Scottish Engineer and Naval Architect John Scott Russell in the Glasgow-Edinburgh…