The Hurwitz space is a moduli space parametrizing branched covers of curves. It admits an "admissible covers" compactification, in which the target and source curves of a cover degenerate into nodal curves when branch points come together. The boundary of the Hurwitz space is then stratified by lower-dimensional Hurwitz spaces. This structure is strikingly similar to the stratification of the stable curves compactification of the moduli space of curves. Building on this analogy, we take inspiration from the well-studied intersection theory of the moduli space of curves to develop new techniques for tackling computations on the Hurwitz space. In this talk, I will present recent work on computing the first cohomology group of the Hurwitz space of degree-3 covers.