As mathematics educators we want our students to develop a natural curiosity that will lead them on the path toward solving problems in a changing world, in fields that perhaps do not even exist today. Here we present student projects, adaptable for several mid- and upper-level mathematics courses, that require students to formulate their own…

We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.

For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.

The efforts to improve the quantitative reasoning (quantitative literacy) skills of college students in the United States have been gaining momentum in recent years. At the same time, the blended learning approach to course delivery has gained in popularity, promising better learning with flexible modalities and pace. This paper presents the…

We propose an introduction to the Laplace transform in which Riemann sums are used to approximate the expected net change in a function, assuming that it quantifies a process that can terminate at random. We assume only a basic understanding of probability.

Linear algebra students are typically introduced to the problem of how to convert from one coordinate system to another in a very abstract way. Often, two bases for a given vector space are provided, and students are taught how to determine a transition matrix to be used for changing coordinates. If students are successful in memorizing this…

We present a first-hand, longer-term account of one student's (Christine's) experiences in and after a mathematics inquiry course. In this course, students actively posed problems, conducted their own mathematical explorations, and wrote journal entries about their experiences. During the course, Christine found that inquiry helped her develop…

We describe a case study of a problem-solving section, using the "Harkness" discussion method, of an honors multivariable calculus course. Students in the problem-solving section had equivalent outcomes on exams, reported higher ratings in self-assessments of skills, and took more math classes in the following year, compared with…

This article describes various courses designed to incorporate mathematical proofs into courses for non-math and non-science majors. These courses, nicknamed "math beauty" courses, are designed to discuss one topic in-depth rather than to introduce many topics at a superficial level. A variety of courses, each requiring students to…

In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor…

Many colleges are finding that the use of acceleration in developmental education is a promising direction for improved student progress toward a degree or certificate. Acceleration has been defined in the literature as the reorganization of curricula and instruction in ways that facilitate the completion of educational requirements in an…

Open-ended questions that can be solved using different strategies help students learn and integrate content, and provide teachers with greater insights into students' unique capabilities and levels of understanding. This article provides a problem that was modified to allow for multiple approaches. Students tended to employ high-powered, complex,…

In order to improve problem-solving dispositions, a section of an inquiry-based math sequence for first-year business students was linked with a section of our general education English sequence. We describe how the linked classes worked and compare some preliminary results from linked and unlinked sections of the math sequence.

The Applied and Industrial Mathematics program partners applied mathematics students with businesses, governments, laboratories, and agencies to solve real problems. We will discuss the logistics of the program and give advice on starting and running similar programs. We also give a detailed example of a specific project, namely our work with the…

There is a long-standing tradition in mathematics education to look to history to inform instruction. An historical analysis of the genesis of a mathematical idea offers insight into: (i) the contexts that give rise to a need for a mathematical construct; (ii) the ways in which available tools might shape the development of that mathematical idea;…