The last ten years have seen an explosion of research in the zero-divisor graphs of commutative rings--by professional mathematicians "and" undergraduates. The objective is to find algebraic information within the geometry of these graphs. This topic is approachable by anyone with one or two semesters of abstract algebra. This article gives the…

Data suggests that a significant loss of African American students from STEM majors occur between their freshmen and sophomore year. This attrition corresponds to the time period when students encounter the calculus sequence. For this reason, calculus persists as a serious barrier preventing African American students from entering STEM fields.…

One sign of a good task is that it will work just as well with Year 7 as with Year 13. The author asks learners to use graph-drawing software to draw a picture with graphs. He uses his imagination to suggest what a particular curve might resemble and helps learners to adjust their drawings to achieve what they are trying to make. Sometimes, one…

A small-group, web-interactive approach to teaching precalculus concepts was investigated. Following an online pretest, 3 participants were given a brief (15 min) presentation on the details of reciprocal math relations and how they operate on the coordinate axes. During baseline, participants were tested regarding their ability to construct…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

The number "e" is one of those fascinating numbers whose properties are of special interest to mathematicians. In this article, the author aims to provide a method of introducing a visual concept of the number "e". These ideas are suitable for secondary school and undergraduate tertiary students. The main concept involves areas…

We present the topic of graph connectivity along with a famous theorem of Menger in the real-world setting of the national computer network infrastructure of "National LambdaRail". We include a set of exercises where students reinforce their understanding of graph connectivity by analysing the "National LambdaRail" network. Finally, we give…

This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to…

The authors explore the importance of "range" and its relationship to continuously differentiable functions that have inverses when their graphs are reflected about lines other than y = x. Some open questions are posed for the reader. (Contains 5 figures.)

Building block toys can be used to illustrate various concepts connected with functions including graphs and rates of change of linear and exponential functions, piecewise functions, and composition of functions. Five brief activities suitable for a pre-calculus course are described.

This article presents a procedure for generating a sequence of data sets which will yield exactly the same fitted simple linear regression equation y = a + bx. Unless rescaled, the generated data sets will have progressively smaller variability for the two variables, and the associated response and covariate will "regress" towards their…

The purpose of this study is to determine whether educators believe mathematics should be taught to liberal arts students, what topics are taught in liberal arts mathematics courses, and what the motivation is for teaching such topics. The study consists of two parts, a review of the opinions of educators regarding liberal arts mathematics, and a…

In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth…

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

Under the backdrop of the investigation of rational functions and their respective curved asymptotes, the reader is invited to experience the mathematical process alongside the authors and observe the application of the NCTM Process Standards and the use of multiple representations in the investigation and solution of a problem. (Contains 9…