Automated feedback for mathematical learning environments

Digital learning environments that offer well-designed feedback have the potential to enhance mathematics education. Building such a system is typically a huge and complex undertaking. Generating informative feedback at the level of steps a student takes requires the encoding of expert knowledge about the problem domain in software. The software component that processes this knowledge is traditionally called a domain reasoner. Such a reasoner can produce various types of feedback, for example about the correctness of a step, common errors, hints about how to proceed, or complete worked-out solutions. In this paper, we highlight the main domain reasoner components that are responsible for generating feedback: rules, problem-solving procedures, normal forms, buggy rules, and constraints. Examples are drawn from the Digital Mathematics Environment (DME), which uses feedback generated by specialized domain reasoners for solving equations and structuring hypothesis tests. Similar techniques have also been used in tutoring systems for domains outside mathematics.