Accurate segmentation of breast masses is an essential step in computer aided diagnosis of breast cancer. The scarcity of annotated training data greatly hinders the model's generalization ability, especially for the deep learning based methods. However, high-quality image-level annotations are time-consuming and cumbersome in medical image analysis scenarios. In addition, a large amount of weak annotations is under-utilized which comprise common anatomy features. To this end, inspired by teacher-student networks, we propose an Anatomy-Aware Weakly-Supervised learning Network (AAWS-Net) for extracting useful information from mammograms with weak annotations for efficient and accurate breast mass segmentation. Specifically, we adopt a weakly-supervised learning strategy in the Teacher to extract anatomy structure from mammograms with weak annotations by reconstructing the original image. Besides, knowledge distillation is used to suggest morphological differences between benign and malignant masses. Moreover, the prior knowledge learned from the Teacher is introduced to the Student in an end-to-end way, which improves the ability of the student network to locate and segment masses. Experiments on CBIS-DDSM have shown that our method yields promising performance compared with state-of-the-art alternative models for breast mass segmentation in terms of segmentation accuracy and IoU.