The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes…

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

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 Capstone project studied a pilot of the web-based homework system "Hawkes" in developmental mathematics classes at a mid-sized community college. The purpose of the study was to investigate how three instructors of developmental mathematics courses incorporated "Hawkes" in their classes, what obstacles they encountered,…

In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…

Little is known about the variety of ways students conceptualize matrix multiplication, yet this is a fundamental part of most introductory linear algebra courses. My dissertation follows a three-paper format, with the three papers exploring conceptualizations of matrix multiplication from a variety of viewpoints. In these papers, I explore (1)…