Pedagogical training of mathematics graduate students is often disconnected from the actual experience of teaching. Even with the support offered by standard instructor training models, mathematics graduate students find it hard to connect with students, mature pedagogically and contribute meaningfully, especially under the overwhelming conditions of teaching for the first time. In this paper, we discuss features of instructor training that support mathematics graduate students' pedagogical development. Using self-reported experiences of mathematics graduate students who taught in a team teaching model, we single out three features of the model and advocate integrating these features into existing instructor training.