In both baseball and mathematics education, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, it is not always the best strategy. Sometimes an analysis of errors provides much deeper insights into mathematical ideas and, rather than something to eschew, certain types of errors…

The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

In Pre-Calculus courses, students are taught the composition and combination of functions to model physical applications. However, when combining two or more functions into a single more complicated one, students may lose sight of the physical picture which they are attempting to model. A block diagram, or flow chart, in which each block…

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…

It is common to see, in the books on calculus, primitives of functions (some authors use the word "antiderivative" instead of primitive). However, the majority of authors pay scant attention to the domains over which the primitives are valid, which could lead to errors in the evaluation of definite integrals. In the teaching of calculus, in…

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…

Everyone with a thorough knowledge of single variable calculus knows that integration can be used to find the length of a curve on a given interval, called its arc length. Fortunately, if one endeavors to pose and solve more interesting problems than simply computing lengths of various curves, there are techniques available that do not require an…

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…

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…

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…

The behavior of certain functions in advanced calculus is discussed, with the mathematics explained. (MNS)

The behavior of some functions near the point of origin is discussed. Each function oscillates, and as x approaches 0, the oscillations become increasingly more rapid; their behavior near the origin improves with increasing values of n. Examples for a calculus class to consider are given. (MNS)

Documents the experiences of 25 first-year university students with regard to the kinds of tasks calculus instructors should design in order to engage students in mathematical practices that often require the use of a graphing calculator. (MM)

In this paper, the authors evaluate the series and integrals presented by P. Glaister. The authors show that this function has the Maclauren series expansion. The authors derive the series from the integral in two ways. The first derivation uses the technique employed by Glaister. The second derivation uses a change in variable in the integral.

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…