Chapter 6: Prerequisite Structure Finding using the Conjunctive Root Causes Model

In a learning process, some prerequisite skills must be mastered before learning higher level skills. For example, in most curricula, addition is considered as a prerequisite of multiplication and is usually taught prior to multiplication. The structure of prerequisite relationships between skills can be encoded as a graph, which can show the suggested order of learning each skill. Skill prerequisite structure information is necessary for designing curricula. Also, it can help educator and tutoring system locate the root cause(s) when a student has a problem (e.g., answering an item incorrectly), and offer remediation.

The prerequisite structures are traditionally designed by experts. Some researchers criticize that it can be time consuming and unreliable (Chen, González-Brenes, & Tian, 2016). Recently, some studies have tried to recover the skill prerequisite structures using educational data mining techniques. Brunskill (2011) and Chen, Wuillemin, & Labat, (2015) study pairwise relationships between skills by comparing data likelihood and doing association rule mining, respectively. Chen, et al. (2016) estimates the prerequisite structure globally using Bayesian network (Mislevy, Almond, Yan, & Steinberg, 1999), which can distinguish between direct and indirect relationship between skills. Han, Yoon, & Yoo (2017) uses pseudo-Bayes factor to do model selection between the strict model (advanced skill cannot be learned without its prerequisite skills) and the full model (advanced skill can be learned without the prerequisite skills). In these studies, skill-to-item mappings (i.e., Q-matrix, Tatsuoka [1983]) are needed. With a Q-matrix, the relationships between items and skills are specified, and thus only the prerequisite structure between skills need to be estimated. Desmarais, Meshkinfam, & Gagnon (2006) discovers the prerequisite structure among items (without Q-matrix) using Bayesian network, while the focus of the study is to predict item outcome with a subset of evidence. The prerequisite relationship between items are not the author’s interest. In fact, it is not easy to make interpretation for item-to-item Bayesian network since the edges might represent different types of causal relationship (other than prerequisite). One must check the conditional probability tables to make appropriate interpretation.