Deep learning and knowledge transfer techniques have permeated the field of medical imaging and are considered as key approaches for revolutionizing diagnostic imaging practices. However, there are still challenges for the successful integration of deep learning into medical imaging tasks due to a lack of large annotated imaging data. To address this issue, we propose a teacher-student learning framework to transfer knowledge from a carefully pre-trained convolutional neural network (CNN) teacher to a student CNN. In this study, we explore the performance of knowledge transfer in the medical imaging setting. We investigate the proposed network's performance when the student network is trained on a small dataset (target dataset) as well as when teacher's and student's domains are distinct. The performances of the CNN models are evaluated on three medical imaging datasets including Diabetic Retinopathy, CheXpert, and ChestX-ray8. Our results indicate that the teacher-student learning framework outperforms transfer learning for small imaging datasets. Particularly, the teacher-student learning framework improves the area under the ROC Curve (AUC) of the CNN model on a small sample of CheXpert (n=5k) by 4% and on ChestX-ray8 (n=5.6k) by 9%. In addition to small training data size, we also demonstrate a clear advantage of the teacher-student learning framework in the medical imaging setting compared to transfer learning. We observe that the teacher-student network holds a great promise not only to improve the performance of diagnosis but also to reduce overfitting when the dataset is small.