Progress in secondary education today is measured primarily through high stakes testing administered on a state-by-state basis. While states may require a common assessment instrument, how the objectives are to be taught, however, is generally up to the schools. This results in debates among educators as to the best curricula for all students.…

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…

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…

This article details the use of tablet PCs in a mathematics content course for future Mathematics Specialists. Instructors used tablet PCs instead of a traditional whiteboard to capture demonstration and discussion. Students were grouped for collaborative problem solving and exploration exercises. Each group was provided with a tablet PC for…

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'…

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…

A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

By selecting certain special triangles, students can learn about the laws of sines and cosines without wrestling with long decimal representations or irrational numbers. Since the law of cosines requires only one of the three angles of a triangle, there are many examples of triangles with integral sides and a cosine that can be represented exactly…

Mastery approaches with online Internet platforms have been shown to alleviate many students' deficiencies and open the door to higher mathematics. This paper details some current programs using online learning for precalculus courses, and detail how the research affected the design, development, and implementation of a new online approach…

We examine the behavior of the curvature function associated with most common families of functions and curves, with the focus on establishing where maximum curvature occurs. Many examples are included for student illustrations. (Contains 18 figures.)

"How can this Abstract Algebra course help me to teach Algebra in high school?" To professors of prospective mathematics teachers, this type of question has a resounding familiarity. Because students are unclear as to the relationship between upper division mathematics courses and the high school curriculum, answering the opening question to the…

Although the phase plane can be plotted and analyzed using an appropriate software package, the author found it worthwhile to engage the students with the theorem and the two proofs. The theorem is a powerful tool that provides insight into the rotational behavior of the phase plane diagram in a simple way: just check the signs of c and [alpha].…

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…

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)…