Learning Sequences: An Efficient Data Structure for Learning Spaces

Learning spaces form the basis of a combinatorial theory of the possible states of knowledge of a human learner that has been successfully deployed in computerized assessment and learning systems such as ALEKS (Falmagne and Doignon, 2011). Until recently, however, both the computational efficiency of these systems and their ability to accurately assess the knowledge of their users have been hampered by a mismatch between theory and practice: they used a simplified version of learning space theory based on partially ordered sets and quasi-ordinal spaces, leading both to computational inefficiencies and to inaccurate assessments. In this chapter we present more recent developments in algorithms and data structures that have allowed learning systems to use the full theory of learning spaces. Our methods are based on learning sequences, sequences of steps through which a student, starting with no knowledge, could learn all the concepts in the space. We show how to define learning spaces by their learning sequences and how to use learning sequences to efficiently perform the steps of an assessment algorithm