Like many mathematics teachers, the authors often find that students who struggle with a difficult concept may be assisted by the use of a well-chosen graph or other visual representation. While one should not rely solely on such tools, they can suggest possible theorems which then might be proved with the proper rigor. Even when a picture…

Calculators and computers make new modes of instruction possible; yet, at the same time they pose hardships for school districts and mathematics educators trying to incorporate technology with limited monetary resources. In the "Standards," a recommended classroom is one in which calculators, computers, courseware, and manipulative materials are…

As anyone who has taught or taken a statistics course knows, statistical calculations can be tedious and error-prone, with the details of a calculation sometimes distracting students from understanding the larger concepts. Traditional statistics courses typically use scientific calculators, which can relieve some of the tedium and errors but…

One of the main problems in undergraduate research in pure mathematics is that of determining a problem that is, at once, interesting to and capable of solution by a student who has completed only the calculus sequence. It is also desirable that the problem should present something new, since novelty and originality greatly increase the enthusiasm…

It is in the field of numerical analysis that this "easy-looking" function, also known as the Runge function, exhibits a behavior so idiosyncratic that it is mentioned even in most undergraduate textbooks. In spite of the fact that the function is infinitely differentiable, the common procedure of (uniformly) interpolating it with polynomials that…

One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…

Uses the visual and programming capabilities of the graphing calculator to discern both differences and similarities between two independent collections of sample data. (MKR)

Description of a technique in Maple programming language that automatically prints all paths of any desired length along with the name of each vertex, proceeding in order from the beginning vertex to the ending vertex for a given graph. (Author/MM)

Explores an unexpected connection between a function, its inverse, and the arithmetic mean, algebraically and graphically. (MM)

In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian…

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

For those instructors lacking artistic skills, teaching 3-dimensional calculus can be a challenge. Although some instructors spend a great deal of time working on their illustrations, trying to get them just right, students nevertheless often have a difficult time understanding some of them. To address this problem, the author has written a series…

Sophisticated simulations using computer graphics can lead to students deducing virtually all conditions of the Central Limit Theorem. Eight graphs illustrate the discussion. (MNS)

The focus is on how line graphs can be used to approximate solutions to rate problems and to suggest equations that offer exact algebraic solutions to the problem. Four problems requiring progressively greater graphing sophistication are presented plus four exercises. (MNS)