Formulated in 1984, Green's Conjecture predicts that one can recognize the intrinsic complexity of an algebraic curve from the syzygies of its canonical embedding. Green's Conjecture for a general curve has been resolved using geometric methods in two landmark papers by Voisin in the early 00s. I will explain how the theory of Koszul modules provides an alternative solution to this problem, by relating it via Hermite reciprocity to the study of the syzygies of the tangent developable surface to a rational normal curve. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.