We analyze naturally occurring datasets from student use of educational technologies to explore a long‐standing question of the scope of transfer of learning. We contrast a faculty theory of broad transfer with a component theory of more constrained transfer. To test these theories, we develop statistical models of them. These models use latent variables to represent mental functions that are changed while learning to cause a reduction in error rates for new tasks. Strong versions of these models provide a common explanation for the variance in task difficulty and transfer. Weak versions decouple difficulty and transfer explanations by describing task difficulty with parameters for each unique task. We evaluate these models in terms of both their prediction accuracy on held‐out data and their power in explaining task difficulty and learning transfer. In comparisons across eight datasets, we find that the component models provide both better predictions and better explanations than the faculty models. Weak model variations tend to improve generalization across students, but hurt generalization across items and make a sacrifice to explanatory power. More generally, the approach could be used to identify malleable components of cognitive functions, such as spatial reasoning or executive functions.