Intelligent tutoring systems can support students in solving multi-step tasks by providing hints regarding what to do next. However, engineering such next-step hints manually or via an expert model becomes infeasible if the space of possible states is too large. Therefore, several approaches have emerged to infer next-step hints automatically, relying on past students' data. In particular, the Hint Factory (Barnes and Stamper, 2008) recommends edits that are most likely to guide students from their current state towards a correct solution, based on what successful students in the past have done in the same situation. Still, the Hint Factory relies on student data being available for any state a student might visit while solving the task, which is not the case for some learning tasks, such as open-ended programming tasks. In this contribution we provide a mathematical framework for edit-based hint policies and, based on this theory, propose a novel hint policy to provide edit hints in vast and sparsely populated state spaces. In particular, we extend the Hint Factory by considering data of past students in all states which are similar to the student's current state and creating hints approximating the weighted average of all these reference states. Because the space of possible weighted averages is continuous, we call this approach the Continuous Hint Factory. In our experimental evaluation, we demonstrate that the Continuous Hint Factory can predict more accurately what capable students would do compared to existing prediction schemes on two learning tasks, especially in an open-ended programming task, and that the Continuous Hint Factory is comparable to existing hint policies at reproducing tutor hints on a simple UML diagram task.
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