The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation…

This case study investigates the effectiveness of a lecture in advanced mathematics. We video recorded a lecture delivered by an experienced professor. Using video recall, we then interviewed the professor to determine the content he intended to convey and we analyzed his lecture to see if and how this content was conveyed. We also interviewed six…

The aim of this study is to explore mathematics teaching department students' perceptions on the concepts of proposition, theorem, and proof which are very important for daily life, mathematical literacy and studying mathematics; the common mathematical content used in constructing these concepts; and whether these constructions and content…

The primary goal of this paper is to highlight the possibilities and benefits of incorporating games into college mathematics classrooms. This is illustrated through the personal success of using the board game "The Resistance" to teach validity and soundness of arguments in a discrete mathematics course. Along the way, we will give some…

We study situations in introductory analysis in which students affirmed false statements as true, despite simple counterexamples that they easily recognized afterwards. The study draws attention to how simple counterexamples can become hidden in plain sight, even in an active learning atmosphere where students proposed simple (as well as more…

There is a generally acknowledged need for students to be quantitatively literate in an increasingly quantitative world. This includes the ability to reason critically about data in context. We have noted that students experience difficulty with the application of certain mathematical and statistical concepts, which in turn impedes progress in the…

This paper gives a very elementary, essentially visual proof of Viete's product. We employ only the Pythagorean theorem, similarity of triangles, and exhaustion.

Alternating series have the simplest of sign patterns. What about series with more complicated patterns? By inspecting the alternating series test closely, we find a theorem that applies to more complicated sign patterns, and beyond.

If a series of real numbers converges absolutely, then it converges. The usual proof requires completeness in the form of the Cauchy criterion. Failing completeness, the result is false. We provide examples of rational series that illustrate this point. The Cantor set appears in connection with one of the examples.

A visual proof of Ptolemy's theorem.

Until recently, most colleges required students to pass a college-level algebra course in order to earn a degree. As many as 50 percent to 70 percent of community college students enter college unprepared to take these courses, and fewer than 20 percent of such students ever successfully complete a college-level math course; the rest are…

This article adopts the following classification for a Euclidean planar [triangle]ABC, purely based on angles alone. A Euclidean planar triangle is said to be acute angled if all the three angles of the Euclidean planar [triangle]ABC are acute angles. It is said to be right angled at a specific vertex, say B, if the angle ?ABC is a right angle…

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

Symbols play important roles in higher-level mathematical thinking by providing flexibility and reducing cognitive load. However, they often have a dual nature since they can signify both processes and products of mathematics. The limit notation is considered to be a visual and symbolic mediator that reflects such duality, which presents…

We give elementary proofs of formulas for the area and perimeter of a planar convex body surrounded by a band of uniform thickness. The primary tool is a integral formula for the perimeter of a convex body which describes the perimeter in terms of the projections of the body onto lines in the plane.