Nonlinear wave equations of power-type serve as excellent toy models for geometric PDEs such as the Yang-Mills and wave maps equations. Of great interest in the energy supercritical setting is that of threshold phenomena. In this setting, unstable self-similar blowup solutions are believed to play an essential role in describing the threshold of singularity formation. We will discuss the stability of an explicitly known, unstable self-similar blowup solution of the energy supercritical quadratic wave equation in a region of spacetime which extends beyond the time of blowup. To overcome this instability, we introduce a new canonical method to investigate unstable self-similar solutions. This work represents the first steps toward an understanding of threshold phenomena in the energy supercritical setting.