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In this paper, we argue in favor of creating robots that both teach and learn. We propose a methodology for building robots that can learn a skill from an expert, perform the skill independently or collaboratively with the expert, and then teach the same skill to a novice. This requires combining insights from learning from demonstration, human-robot collaboration, and intelligent tutoring systems to develop knowledge representations that can be shared across all three components. As a case study for our methodology, we developed a glockenspiel-playing robot. The robot begins as a novice, learns how to play musical harmonies from an expert, collaborates with the expert to complete harmonies, and then teaches the harmonies to novice users. This methodology allows for new evaluation metrics that provide a thorough understanding of how well the robot has learned and enables a robot to act as an efficient facilitator for teaching across temporal and geographic separation.