  Machine learning offers novel ways and means to design personalized learning
systems wherein each student's educational experience is customized in real
time depending on their background, learning goals, and performance to date.
SPARse Factor Analysis (SPARFA) is a novel framework for machine learning-based
learning analytics, which estimates a learner's knowledge of the concepts
underlying a domain, and content analytics, which estimates the relationships
among a collection of questions and those concepts. SPARFA jointly learns the
associations among the questions and the concepts, learner concept knowledge
profiles, and the underlying question difficulties, solely based on the
correct/incorrect graded responses of a population of learners to a collection
of questions. In this paper, we extend the SPARFA framework significantly to
enable: (i) the analysis of graded responses on an ordinal scale (partial
credit) rather than a binary scale (correct/incorrect); (ii) the exploitation
of tags/labels for questions that partially describe the question{concept
associations. The resulting Ordinal SPARFA-Tag framework greatly enhances the
interpretability of the estimated concepts. We demonstrate using real
educational data that Ordinal SPARFA-Tag outperforms both SPARFA and existing
collaborative filtering techniques in predicting missing learner responses.
