Infinite series is a challenging topic in the undergraduate mathematics curriculum for many students. In fact, there is a vast literature in mathematics education research on convergence issues. One of the most important types of infinite series is the geometric series. Their beauty lies in the fact that they can be evaluated explicitly and that…

Despite few limitations, GeoGebra as a dynamic geometry software stood as a powerful instrument in helping university math majors understand, explore, and gain experiences in visualizing the limits of functions and the ?-d formalism. During the process of visualizing a theorem, the order mattered in the sequence of constituents. Students made use…

The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone. The analogous curve when the plane intersects a…

The Sequences and Series calculus course (S&S) can be structured to provide students with a unique opportunity to build their proofs skills prior to or concurrently with a bridge course. This article offers a framework for S&S which places logical reasoning on equal footing with content, by employing the theorems and convergence tests as axioms,…

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show…

This article reports on initial findings, including the mathematics components, of a multi-institutional Science, Technology, Engineering, and Mathematics (STEM) project, "It's part of my life: Engaging university and community to enhance science and mathematics education." This project is focussed on improving the scientific and…

In intermediate and college algebra courses there are a number of methods for factoring quadratic trinomials with integer coefficients over the integers. Some of these methods have been given names, such as trial and error, reversing FOIL, AC method, middle term splitting method and slip and slide method. The purpose of this article is to discuss…

Undergraduate students who are pre-service teachers need to make connections between the college mathematics they are learning and the pre-college mathematics they will be teaching. Spanning a broad range of undergraduate curricula, this article describes useful lesser-known connections, explorations, and original proofs involving fractions. In…

Calculus II students know that many alternating series are convergent by the Alternating Series Test. However, they know few alternating series (except geometric series and some trivial ones) for which they can find the sum. In this article, we present a method that enables the students to find sums for infinitely many alternating series in the…

We argue that mathematics majors learn little from the proofs they read in their advanced mathematics courses because these students and their teachers have different perceptions about students' responsibilities when reading a mathematical proof. We used observations from a qualitative study where 28 undergraduates were observed evaluating…

In the context of a department of applied mathematics, a program assessment was conducted to assess the departmental goal of enabling undergraduate students to recognize, appreciate, and apply the power of computational tools in solving mathematical problems that cannot be solved by hand, or would require extensive and tedious hand computation. A…

We consider a variety of writing assignments used for two general education mathematics courses. The student writing ranges from brief informal pieces to more formal assignments that address the full scope of a course, with an emphasis on encouraging students to have a richer experience of mathematical people and ideas and of their own learning.…

Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction passes from local implication (if true for "k", the theorem is true for "k" + 1) to a global conclusion in the natural numbers.

This paper tells the story of what happened when students in the authors' sophomore-level introduction-to-proof classes were given a theorem to prove with no expectation about what proof method to use. The paper discusses the nine different student-produced proofs of the statement: If "n" is an integer such that "n" is greater than or equal to 3,…

It is important to avoid ambiguity with numbers because unfortunate choices of numbers can inadvertently make it possible for students to form misconceptions or make it difficult for teachers to tell if students obtained the right answer for the right reason. Therefore, it is important to make sure when introducing basic summary statistics that…