What exactly do students learn when they practice equation solving?: refining knowledge components with the additive factors model

Accurately modeling individual students' knowledge growth is important in many applications of learning analytics. A key step is to decompose the knowledge targeted in the instruction into detailed knowledge components (KCs). We search for an accurate KC model for basic equation solving skills, using data from an intelligent tutoring system (ITS), Lynnette. Key criteria are data fit and predictive accuracy based on a standard logistic model called the Additive Factors Model (AFM). We focus on three difficulty factors for equation solving: understanding of variables, the negative sign, and the complexity of the equation. Fine-grained KC models were found to have greater fit and predictive accuracy than an "ideal," more abstract model, indicating that there is substantial under-generalization in students' equation-solving skill related to all three difficulty factors. The work enhances scientific understanding of the challenges students face in learning equation solving. It illustrates how learning analytics could inform the improvement of technology-enhanced learning environments.