  While computer and communication technologies have provided effective means
to scale up many aspects of education, the submission and grading of
assessments such as homework assignments and tests remains a weak link. In this
paper, we study the problem of automatically grading the kinds of open response
mathematical questions that figure prominently in STEM (science, technology,
engineering, and mathematics) courses. Our data-driven framework for
mathematical language processing (MLP) leverages solution data from a large
number of learners to evaluate the correctness of their solutions, assign
partial-credit scores, and provide feedback to each learner on the likely
locations of any errors. MLP takes inspiration from the success of natural
language processing for text data and comprises three main steps. First, we
convert each solution to an open response mathematical question into a series
of numerical features. Second, we cluster the features from several solutions
to uncover the structures of correct, partially correct, and incorrect
solutions. We develop two different clustering approaches, one that leverages
generic clustering algorithms and one based on Bayesian nonparametrics. Third,
we automatically grade the remaining (potentially large number of) solutions
based on their assigned cluster and one instructor-provided grade per cluster.
As a bonus, we can track the cluster assignment of each step of a multistep
solution and determine when it departs from a cluster of correct solutions,
which enables us to indicate the likely locations of errors to learners. We
test and validate MLP on real-world MOOC data to demonstrate how it can
substantially reduce the human effort required in large-scale educational
platforms.
