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Students in Algebra I classrooms typically learn at different rates andstruggle at different points in the curriculum---a common challenge for mathteachers. Cognitive Tutor Algebra I (CTA1), educational computer program,addresses such student heterogeneity via what they term "mastery learning,"where students progress from one section of the curriculum to the next bydemonstrating appropriate "mastery" at each stage. However, when students areunable to master a section's skills even after trying many problems, they areautomatically promoted to the next section anyway. Does promotion withoutmastery impair the program's effectiveness?  At least in certain domains, CTA1 was recently shown to improve studentlearning on average in a randomized effectiveness study. This paper usesstudent log data from that study in a continuous principal stratification modelto estimate the relationship between students' potential mastery and the CTA1treatment effect. In contrast to extant principal stratification applications,a student's propensity to master worked sections here is never directlyobserved. Consequently we embed an item-response model, which measuresstudents' potential mastery, within the larger principal stratification model.We find that the tutor may, in fact, be more effective for students who aremore frequently promoted (despite unsuccessfully completing sections of thematerial). However, since these students are distinctive in their educationalstrength (as well as in other respects), it remains unclear whether thisenhanced effectiveness can be directly attributed to aspects of the masterylearning program.

The Role of Mastery Learning in Intelligent Tutoring Systems: Principal Stratification on a Latent Variable