Labeling Mathematical Errors to Reveal Cognitive States

While technology can enhance learning, ironically, many online systems occlude learners' cognitive states because instructors do not directly observe students solving problems. In this paper, we show how we utilized an online mathematics homework system where students simply provided final answers to exercises. We then asked, "What can we infer about the cognitive state of the student if they gave an incorrect response?" Through data mining techniques, we found we were able to ascribe a particular type of mechanical error or misconception to 60-75% of the incorrect responses learners made on the subset of problems we analyzed. As such, we illustrate methods for extracting this data to discover knowledge components embedded in an exercise, expose item bias, and reveal learners' cognitive states.