In this paper, the author gives a further simple generalization of a power series evaluation of an integral using Taylor series to derive the result. The author encourages readers to consider numerical methods to evaluate the integrals and sums. Such methods are suitable for use in courses in advanced calculus and numerical analysis.

The use of the computer, and specifically the mathematics software package Maple, has played a central role in the authors' abstract algebra course because it provides their students with a way to see realistic examples of the topics they discuss without having to struggle with extensive computations. However, Maple does not provide the computer…

The author discusses the definition of the ordinary points and the regular singular points of a homogeneous linear ordinary differential equation (ODE). The material of this note can find classroom use as enrichment material in courses on ODEs, in particular, to reinforce the unit on the Existence-Uniqueness Theorem for solutions of initial value…

When dealing with a complex problem, solving it by reduction to simpler problems, or problems for which the solution is already known, is a common method in mathematics and other scientific disciplines, as in computer science and, specifically, in the field of computability. However, when teaching computational models (as part of computability)…

As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…

This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…

The chain rule is one of the hardest ideas to convey to students in Calculus I. It is difficult to motivate, so that most students do not really see where it comes from; it is difficult to express in symbols even after it is developed; and it is awkward to put it into words, so that many students can not remember it and so can not apply it…

The subject matter presented in this article can be used in the classroom to enrich the learning experience of students taking a course that includes a unit on combinatorics, such as discrete mathematics, graph theory, or probability. In order to provide such students with the background needed to appreciate the significance of the generalization…

Many counting problems can be modeled as "colorings" and solved by considering symmetries and Polya's cycle index polynomial. This paper presents a "Maple 7" program link http://users.tamuk.edu/kfdrc00/ that, given Polya's cycle index polynomial, determines all possible associated colorings and their partitioning into equivalence classes. These…

Teaching is a scholarly activity and a life-long learning process with no single method or pedagogy that is always most effective. Applications of teaching pedagogies properly vary from content-based to discipline-based areas. In this paper, various teaching pedagogies, including scaffolding, concept mapping, constructivism, and learning…

An individualized project is introduced to accompany a differential equations course with different modules ranging from standard to advanced levels with "Matlab" scripts. (Contains 1 figure.)

A sequence of number pairs can be used to generate many interesting examples in teaching mathematics subjects at various levels. It is often used in elementary or middle school mathematics classes to illustrate the concept of "patterns." In this paper the authors present a few interesting ways of using this sequence to form examples for high…

The purpose of this study was to explore the effectiveness of using ALEKS (Assessment and Learning in Knowledge Spaces [3]), an online interactive learning system, in teaching a college algebra course at a university. This system was used to replace traditional assignments. The study was conducted in eight college algebra courses in the Fall 2003…

Pascal's Triangle is, without question, the most well-known triangular array of numbers in all of mathematics. A well-known algorithm for constructing Pascal's Triangle is based on the following two observations. The outer edges of the triangle consist of all 1's. Each number not lying on the outer edges is the sum of the two numbers above it in…

If larger and larger samples are successively drawn from a population and a running average calculated after each sample has been drawn, the sequence of averages will converge to the mean, [mu], of the population. This remarkable fact, known as the law of large numbers, holds true if samples are drawn from a population of discrete or continuous…