Teaching Time Savers: Style Points

When I began as an assistant professor, I had a pretty good sense of how much time it would take for me to prepare for each class. After a few conversations with my new colleagues, I even had a good sense of how much time I should devote to tasks like office hours and committee work. Somewhere in the middle of grading my first exam, though, it became painfully clear that I had underestimated the amount of time I would need to grade exams

Teaching Time Savers: Some Advice on Giving Advice

There are always a lot of questions that need to be answered at the beginning of a course. When are office hours? What are the grading policies? How many exams will there be? Will late homework be accepted? We have all seen the answers to these sorts of questions form the bulk of a standard course syllabus, and most of us feel an obligation (and rightly so) to provide such information

Teaching Time Savers: The Exam Practically Wrote Itself!

When I first started teaching, creating an exam for my upper division courses was a genuinely exciting process. The material felt fresh and relatively unexplored (at least by me), and I remember often feeling pleasantly overwhelmed with what seemed like a vast supply of intriguing and engrossing exam-ready problems. Crafting the perfect exam, one that was noticeably inviting, exceedingly fair, and unavoidably illuminating, was a real joy

Teaching Time Savers: A Recommendation for Recommendations

I admit it — I enjoy writing recommendation letters for my students. I likelearning about their hopes and dreams, where they have been and where they want to go. A recommendation letter is an opportunity to remind myself how much my students can grow while they are in college, and how much I have grown as an instructor, advisor, and mentor

The Linear Complexity of a Graph

The linear complexity of a matrix is a measure of the number of additions, subtractions, and scalar multiplications required to multiply that matrix and an arbitrary vector. In this paper, we define the linear complexity of a graph to be the linear complexity of any one of its associated adjacency matrices. We then compute or give upper bounds for the linear complexity of several classes of graphs

Imagine Math Day: Encouraging Secondary School Students and Teachers to Engage in Authentic Mathematical Discovery

Research mathematicians and school children experience mathematics in profoundly different ways. Ask a group of mathematicians what it means to “do mathematics” and you are likely to get a myriad of responses: mathematics involves analyzing and organizing patterns and relationships, reasoning and drawing conclusions about the world, or creating languages and tools to describe and solve important problems. Students of mathematics often report “doing mathematics” as performing calculations or following rules. It’s natural that they see mathematics as monolithic rather than an evolving, growing, socially constructed body of knowledge, because most mathematical training in primary and secondary schools consists of learning how to use pre-existing mathematical tools. They rarely get to see the process by which those tools came about, let alone authentically participate in the construction of those tools

Teaching Time Savers: Is Homework Grading on Your Nerves?

You have probably heard it said that we learn mathematics best when we do mathematics, or that mathematics is not a spectator sport. For most of our students, this means that their mathematics courses will involve a fair amount of homework. This homework is often used to evaluate individual student progress, but it can also be used, for example, as a catalyst for discussion, to emphasize a point made in class, and to identify common misunderstandings throughout the class as a whole. There is, however, the matter of grading homework

Computing Isotypic Projections with the Lanczos Iteration

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric group

The Art of Teaching Mathematics

On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity