  The analysis of log data generated by online educational systems is an
important task for improving the systems, and furthering our knowledge of how
students learn. This paper uses previously unseen log data from Edulab, the
largest provider of digital learning for mathematics in Denmark, to analyse the
sessions of its users, where 1.08 million student sessions are extracted from a
subset of their data. We propose to model students as a distribution of
different underlying student behaviours, where the sequence of actions from
each session belongs to an underlying student behaviour. We model student
behaviour as Markov chains, such that a student is modelled as a distribution
of Markov chains, which are estimated using a modified k-means clustering
algorithm. The resulting Markov chains are readily interpretable, and in a
qualitative analysis around 125,000 student sessions are identified as
exhibiting unproductive student behaviour. Based on our results this student
representation is promising, especially for educational systems offering many
different learning usages, and offers an alternative to common approaches like
modelling student behaviour as a single Markov chain often done in the
literature.
