Interpreting freeform equation solving

Abstract

Learners’ step-by-step solutions can offer insight into their misunderstandings. Because of the difficulty of automatically interpreting freeform solutions, educational technologies often structure problem solving into particular patterns. Hypothesizing that structured interfaces may frustrate some learners, we conducted an experiment comparing two interfaces for solving equations: one requires users to enter steps in an efficient sequence and insists each step be mathematically correct before the user can continue, and the other allows users to enter any steps they would like. We find that practicing equation solving in either interface was associated with improved scores on a multiple choice assessment, but that users who had the freedom to make mistakes were more satisfied with the interface. In order to make inferences from these more freeform data, we develop a Bayesian inverse planning algorithm for diagnosing algebra understanding that interprets individual equation solving steps and places no restrictions on the ordering or correctness of steps. This algorithms draws inferences and exhibits similar confidence based on data from either interface. Our work shows that inverse planning can interpret freeform problem solving, and suggests the need to further investigate how structured interfaces affect learners’ motivation and engagement.