Discovering and describing types of mathematical errors

Given a large number of incorrect responses to mathematical exercises, we ask, “What errors might the learner have made to arrive at their answer?” Even though our data does not contain intermediate steps, we find that we are able to infer well over 50% and sometimes over 90% of the types of errors learners make on an exercise when they only supply final answers. Our approach capitalizes on the sheer volume of data to highlight patterns and the fact that these exercises come from item banks of mathematical templates. Since items generated from mathematical templates deliver different parameters to different learners (e.g., one learner might see y = 2x+ 3 while another learner might see y = 3x+ 5), misconceptions and mechanical errors are more easily recognized. We enumerated different errors for simpler-stated problems and utilized other forms of signal analysis in other cases to uncover error types. Our results show that there are many types of errors even for seemingly simple problems, and we can quantify their relative degrees of prevalence. We can also determine bias in the templates that make a problem easier or more difficult depending on which parameters are used. Since error categories correlate with knowledge components, our work highlights the relative degree of knowledge components embedded within a problem and exposes some knowledge components that may otherwise remain unconsidered.