Canonical Forms in Interactive Exercise Assistants

Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes, fine control over the symbolic simplification procedures of the underlying mathematical machinery is needed. In this paper, we introduce views for mathematical expressions. A view defines an equivalence relation by choosing a canonical form of mathematical expressions. We use views to track and recognize intermediate answers, to help in presenting expressions to a user, and to control the granularity of the steps in worked-out examples. We develop the concept of a view, discuss the laws it satisfies, and show how views are composed, which means that they can be used for multiple exercise classes.