Self-stabilization guarantees convergence to a legitimate configuration in every execution starting from any initial configuration. However, during convergence, most self-stabilizing protocols make unnecessary output changes that do not directly contribute to the progress of convergence. We define and study monotonic stabilization, where every output change is a step toward convergence. That is, any output change at a process p gives the final output of p in the legitimate configuration to be reached. It turns out that monotonic stabilization requires additional information exchange between processes, and we present task dependent tradeoff results with respect to the locality of exchanged information.