While many methods purport to explain predictions by highlighting salient features, what aims these explanations serve and how they ought to be evaluated often go unstated. In this work, we introduce a framework to quantify the value of explanations via the accuracy gains that they confer on a student model trained to simulate a teacher model. Crucially, the explanations are available to the student during training, but are not available at test time. Compared to prior proposals, our approach is less easily gamed, enabling principled, automatic, model-agnostic evaluation of attributions. Using our framework, we compare numerous attribution methods for text classification and question answering, and observe quantitative differences that are consistent (to a moderate to high degree) across different student model architectures and learning strategies.