In educational HRI, it is generally believed that a robots behavior has a direct effect on the engagement of a user with the robot, the task at hand and also their partner in case of a collaborative activity. Increasing this engagement is then held responsible for increased learning and productivity. The state of the art usually investigates the relationship between the behaviors of the robot and the engagement state of the user while assuming a linear relationship between engagement and the end goal: learning. However, is it correct to assume that to maximise learning, one needs to maximise engagement? Furthermore, conventional supervised models of engagement require human annotators to get labels. This is not only laborious but also introduces further subjectivity in an already subjective construct of engagement. Can we have machine-learning models for engagement detection where annotations do not rely on human annotators? Looking deeper at the behavioral patterns and the learning outcomes and a performance metric in a multi-modal data set collected in an educational human-human-robot setup with $68$ students, we observe a hidden link that we term as Productive Engagement. We theorize a robot incorporating this knowledge will 1) distinguish teams based on engagement that is conducive of learning; and 2) adopt behaviors that eventually lead the users to increased learning by means of being productively engaged. Furthermore, this seminal link paves way for machine-learning models in educational HRI with automatic labeling based on the data.