Have you ever wondered what's going on in someone's mind when they're doing math? This talk will explore relationships between calculus students' conceptual structures for reasoning quantitatively. First we will discuss the ways in which students reason about co-varying quantities and the ways in which students coordinate static units. Then we will discuss our work investigating relationships between the two through a set of interview studies with introductory calculus students.

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as "FEAST", has gained considerable attention in recent years. In this talk, we will discuss the results we obtained when FEAST and the Discontinuous Petrov Galerkin (DPG) method are applied to compute part of the spectrum of an unbounded partial differential operator. The theory also provides bounds on the discretization errors in the spectral approximations. We show the results of numerical experiments for simple operators, such as the laplacian, to illustrate the theory. The utility of the algorithm is illustrated by applying it to a reaction-diffusion operator to compute guided transverse core modes of a realistic optical fiber.