Machine teaching (MT) studies the task of designing a training set. Specifically, given a learner (e.g., an artificial neural network or a human) and a target model, a teacher aims to create a training set which results in the target model being learned. MT applications include optimal education design for human learners and computer security where adversaries aim to attack learning-based systems. In this work, we formulate pool-based MT as a state space search problem. We discuss the properties and challenges of the resulting problem and highlight opportunities for novel search techniques. In our preliminary study we use a beam search approach, and find that training and evaluating empirical risk of models dominate the run time of the search. Toward the goal of better search techniques for future work, we develop optimizations ranging from implementation details for specific learners to algorithm changes applicable to general blackbox learners. We conclude with a discussion of open problems and research directions.