The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating…

Textbooks, like many other resources teachers have at hand, are meant to be an aid for instruction; however there is little research with textbooks or on their potential to develop metacognitive knowledge. Metacognitive knowledge has received substantial attention in the literature, in particular for its relationship with problem-solving in…

Computer graphics are used to display the sum of the first few terms of the series solution for the problem of the vibrating string frequently discussed in introductory courses on differential equations. (MNS)

Argues that proving something begins with an assumption and proceeds logically to a conclusion, thus convincing by offering arguments. Describes some pitfalls involved in proving through examining several case histories. Offers suggestions for teaching the proof process. For example, sometimes an extreme example or counterexample will do more to…

How min-max problems can be solved with trigonometry and without calculus is described. (MNS)

A discussion of the mathematics of rugby is related to an earlier article about mathematics and the physical world. (MP)

Compared tutoring episodes where tutoring did and did not cause learning in university physics students to inform design of intelligent tutoring systems. Found that when students were not at an impasse, learning was uncommon regardless of the tutorial explanations employed. When students were at an impasse, tutorial explanations were sometimes…

Discussed is the problem of inscribing a rectangle of maximum area in a given right triangle. Answered is whether there is an advantageous orientation and whether a corner of the rectangle of maximum area can lie on the midpoint of a leg of the triangle. (KR)

This workbook was designed to be used as a study aid in any course covering the various techniques of indefinite integration. Many students are able to master each individual technique, but upon encountering an integral on an exam, they often have difficulties in determining which technique to use. By working through all of the exercices in this…

Some appearances of pi in a wide variety of problems are presented. Sections focus on some history, the first analytical expressions for pi, Euler's summation formula, Euler and Bernoulli, approximations to pi, two and three series for the arctangent, more analytical expressions for pi, and arctangent formulas for calculating pi. (MNS)

Discusses solutions to the problem of maximizing the range of a projectile. Presents three references that solve the problem with and without the use of calculus. Offers a fourth solution suitable for introductory physics courses that relies more on trigonometry and the geometry of the problem. (MDH)

This article, which is organized around a single, well-known algorithm for root extraction, presents a way of incorporating dynamical systems into the teaching of mathematics. Included are sample exercises using complex numbers and the computer where students have the opportunity to do some analysis on this algorithm. (KR)

This paper discusses the competency space approach to diagnosing misconceptions, skill, and knowledge acquisition. In some approaches that combine misconceptions, skill, and knowledge acquisition, the latent ability theta is used more or less as an insignificant element, but in the competency space approach, a multidimensional latent space is…

This document contains the course syllabus and 12 independent practice modules for a college level mathematics course designed to provide the necessary foundation for success in calculus, develop logical thinking skills, and enhance analytic skills through problem solving. Topics include relations and functions; inequalities; complex numbers;…