In this classroom note, the authors present a method to solve variable coefficients ordinary differential equations of the form p(x)y([squared])(x) + q(x)y([superscript 1])(x) + r(x)y(x) = 0. They propose an iterative method as an alternate method to solve the above equation. This iterative method is accessible to an undergraduate student studying…

In this note, using the method of undetermined coefficients, we obtain the power series for exp ( f ( x )) and ln ( f ( x )) by means of a simple recursion. As applications, we show how those power series can be used to reproduce and improve some well-known results in analysis. These results may be used as enrichment material in an advanced…

The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…

Three proofs are given for the fact that the derivative of an everywhere-positive non-constant real polynomial function must change sign. This self-contained note could find classroom use in courses on calculus or abstract algebra.

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.

[Slides] presented at the American Educational Research Association (AERA) in 2009. This study investigated the presence of a general rise in high school grades without a corresponding rise in SAT scores, as well as different grading criteria between schools, as shown by differential relationships between AP course grades and exam scores. These…

This paper focuses on using technology to support mathematics instruction. Technologies such as computers and calculators are widely used in the teaching of mathematics. The educators advocate the use of these tools to reduce tiresome computations and tasks so that class time can be more effectively used for learning mathematics. The purpose of…

Progress toward equitable gender representation among faculty in higher education has been "glacial" since the early 1970s (Glazer-Raymo, 1999; Lomperis, 1990; Trower & Chait, 2002). Women, who now make up a majority of undergraduate degree earners and approximately 46% of Ph.D. earners nationwide (National Center for Education Statistics [NCES],…

Research on student achievement within the University of Chicago School Mathematics Project (UCSMP) and Core-Plus Mathematics Project (CPMP) at the secondary level is beginning to accumulate, however, much less is known about how prepared these students are for post-secondary education. Therefore this study involving students within one tracked…

The Bondi K-calculus is a delightful method that has been used to provide rich insights into relativistic kinematics. In this paper, we will try to show how several important results of relativistic kinetics can be derived simply by using this approach. In addition, we will also indicate how the K-calculus can be used to simplify certain…

During the past decade three different bridging mathematics courses have been offered at the University of Auckland. A case study approach was used to investigate the effectiveness of these courses: two larger courses and one individual study programme. A different teaching approach, by committed experienced teachers, was used in each course. The…

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…

In this article, the author talks about Project Lead the Way (PLTW), a rigorous four-year program of honors-level math and science, plus engineering, culminating in at least precalculus and advanced science classes, along with an intensive, hands-on collaborative engineering project. The curriculum is produced by Project Lead the Way Inc., a…

An online homework system created for use by beginning calculus students is described. This system was designed with the specific goal of supporting student engagement outside of class by replicating the attempt-feedback-reattempt sequence of events which often occurs in a teacher's presence. Evidence is presented which indicates that this goal…

This article focuses on the important role of applications in teaching mathematics to students with career paths other than mathematics. These include the fields as diverse as education, engineering, business, and life sciences. Particular attention is given to instructional computing as a means for concept development in mathematics education…