Instructors are constantly struggling to help students understand mathematical concepts as well as the relevance of mathematics to the real world. In calculus, students possess misconceptions of the limit concept. "Pushing the Limit" refers to a semester-long calculus class project that required students to read about, interview calculus…

Most instructors desire to create a learning environment conducive to deep learning. However, in courses where grades are based upon the accumulation of points, students may focus on collecting points rather than on learning and achieving the course objectives. We discuss an alternative points-free course design based on the development of clear…

In this paper we discuss an interesting class of noncommutative rings which can be constructed using finite directed graphs. This construction also creates a vector space. These structures provide undergraduate students connections between ring theory and graph theory and, among other things, allow them to see a ring unity element that looks quite…

Calculus instructors struggle to teach infinite series, and students have difficulty understanding series and related concepts. Four instructional strategies, prominently used during the calculus reform movement, were implemented during a 3-week unit on infinite series in one class of second-semester calculus students. A description of each…

In this study, one section of undergraduate Precalculus was taught using a modified Moore method, a student centered inquiry-based approach, and two control sections were taught in a traditional lecture format. A survey of attitudes, beliefs, and efficacy toward mathematics and Precalculus was administered at the beginning and end of the semester…

The combination of classroom voting system (clicker) questions and peer instruction has been shown to increase student learning. While implementations in large lectures have been around for a while, mathematics has been increasingly using clickers in classes of a smaller size. In Fall 2008, I conducted an experiment to measure the effect of…

Spreadsheets lend themselves naturally to recursive computations, since a formula can be defined as a function of one of more preceding cells. A hypothesized closed form for the "n"th term of a recursive sequence can be tested easily by using a spreadsheet to compute a large number of the terms. Similarly, a conjecture about the limit of a series…

This article describes a multi-class project that employs statistical computing and writing in a statistics class. Three courses, General Ecology, Meteorology, and Introductory Statistics, cooperated on a project for the EPA's Student Design Competition. The continuing investigation has also spawned several undergraduate research projects in…

This article provides an overview of the scholarship of teaching and learning (SoTL) in mathematics. It describes the origins of SoTL in higher education and distinguishes SoTL from good teaching, scholarly teaching, and mathematics education research. It includes a widely adopted taxonomy of SoTL questions and presents several examples of SoTL…

Students and mathematicians alike have long struggled to understand the nature of probability. This article explores the use of gambling activities as a basis for defining probabilities. (Contains 1 table and 1 figure.)

Data analysis methods, both numerical and visual, are used to discover a variety of surprising patterns in the errors associated with successive approximations to the derivatives of sinusoidal and exponential functions based on the Newton difference-quotient. L'Hopital's rule and Taylor polynomial approximations are then used to explain why these…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

This paper describes an active learning idea that I have used in my applied statistics class as a first lesson in correlation and regression. Students propel spitballs from various standing distances from the target and use the recorded data to determine if the spitball accuracy is associated with standing distance and review the algebra of lines…

The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form y = u[subscript 1]y[subscript 1] + u[subscript 2]y[subscript 2], a sum of function products using solutions to the homogeneous equation y[subscript 1] and…

Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…