A Framework for Computer-Aided Design of Educational Domain Models

Abstract

Many educational applications, from tutoring to problem generation, are built on a formal model of the operational knowledge for a given domain. These domain models consist of rewrite rules that experts apply to solve problems in the domain; e.g., factoring, $ax + bx \rightarrow (a+b)x$, is one such rule for K-12 algebra. Domain models currently take hundreds of hours to create, and they differ widely in how well they meet educational objectives such as maximizing problem-solving efficiency. Rapid, objective-driven creation of domain models is a key challenge in the development of personalized educational tools.

This paper presents $\textsc {RuleSy}_{}$, a new framework for computer-aided authoring of domain models for educational applications. $\textsc {RuleSy}_{}$ takes as input a set of example problems (e.g., $x + 1 = 2$), a set of basic axiom rules for solving these problems (e.g., factoring), and a function expressing the desired educational objective. Given these inputs, it first synthesizes a set of sound tactic rules (e.g., combining like terms) that integrate multiple axioms into advanced problem-solving strategies. The axioms and tactics are then searched for a domain model that optimizes the objective. $\textsc {RuleSy}_{}$ is based on new algorithms for mining tactic specifications from examples and axioms, synthesizing tactic rules from these specifications, and selecting an optimal domain model from the axioms and tactics.

We evaluate $\textsc {RuleSy}_{}$ on the domain of K-12 algebra, finding that it recovers textbook tactics and domain models, discovers new tactics and models, and outperforms a prior tool for this domain by orders of magnitude. But $\textsc {RuleSy}_{}$ generalizes beyond K-12 algebra: we also use it to (re)discover proof tactics for propositional logic, demonstrating its potential to aid in designing models for a variety of educational domains.