Conceptualizing Procedural Knowledge Targeted at Students with Different Skill Levels

Conceptualizing procedural knowledge is one of the most challenging tasks of building systems for intelligent tutoring. We present a novel algorithm that enables teachers to accomplish this task (semi)automatically. Furthermore, it is desired to adapt the level of conceptualization to the skill level of particular students. We argue that our algorithm facilitates such adaptation in a straightforward fashion. We demonstrate this feature of the algorithm with a case study. 1 Conceptualization of Procedural Knowledge In symbolic problem solving domains (like physics, mathematics, or games like chess), a particular domain is defined with a basic domain theory and a solution to be achieved. The task is to find a sequence of steps that bring us from the beginning state of the problem (definition of the problem) to the goal state (the solution). The basic domain theory (or basic declarative knowledge of the domain) is usually simple and easy to remember and, in principle, sufficient for solving problems; e.g. knowing rules of chess could in theory enable optimal play. However, finding a solution using only declarative knowledge would require far too extensive searching. A human student is incapable of searching very deep, therefore we need to teach him also the procedural knowledge – how to solve problems. The “complete” procedural knowledge would be a function mapping from each problem state to an action that leads to the solution. For example, in chess endgames a tablebase specifies best moves for all possible positions. Tablebases can be used easily because they only require trivial amount of search. But now the problem is the space complexity – it is impossible for humans to memorize such tablebases that typically contain millions of positions. There is a way, however, that enables humans to solve problems in such chess endgames quite comfortably. Humans use some intermediate representation of the domain that lies between the basic domain theory and the ‘”complete” procedural knowledge. We call such an intermediate representation a “conceptualized domain”. We propose a goal-oriented conceptualization of domains. A goal-oriented rule has the following structure: IF preconditions THEN goal (depth). The rule's preconditions and goal are both expressed in terms of attributes used for describing states. The term preconditions specifies applicability of the rule, while a goal specifies the values of attributes in the state to be achieved. The depth property of a rule is the maximum allowed number of steps in achieving the goal. We developed an interactive algorithm that combines specialized minimax search with the ABML principle [2] for (semi) automatic construction of such rules, where the teacher and the algorithm in turns improve the model. The depth parameter is set prior to learning and can be used to dictate