Do Skills Combine Additively to Predict Task Di culty in Eighth-grade Mathematics?

During the 2004–2005 school year, over 900 eighthgrade students used an online intelligent tutoring system, the Assistment System of He ernan, et al. (2001), to prepare for the mathematics portion of the Massachusetts Comprehensive Assessment System (MCAS) end-of-year exam. A transfer model, identifying the skills that each tutoring task and exam problem depends upon, was developed to help align tutoring tasks with exam problems. We use a Bayesian form of item response theory (IRT) modeling to attempt to model the di culty of tutoring tasks and exam items additively in terms of these component skills: the more skills, the more di cult the task or test item. Our goal is to directly examine the alignment between tutoring tasks and assessment items and to use the transfer model to build more e cient functions for predicting end-of-year exam performance from student activity with the online tutor. However, our analysis shows that the additive skills model (the Linear Logistic Test Model, LLTM) does not adequately account for task-totask or item-to-item variation in di culty.