A Formulation of Auxiliary Problems (<Survey Papers> Special Issue : "Computer Support for Human Learning")

Abstract

Auxiliary problems are very useful to assist students in problem solving. To utilize the auxiliary problems in ITS (Intelligent Tutoring System), a framework for formulation and categorization of them are indispensable. In this paper, an auxiliary problem is defined as a simplifid problem from original one by dividing or specializing its problem solving process. First, we propose a framework of problem description which consists of surface structure, solution structure and constraint structure. Surface structure describes surface features of a problem with objects, their configuration and each object's attributes given or required. This represents a problem formulation process. Solution structure is described by a sequence of operational relations which computes required attributes from given attributes. This represents a computation process. A network composed of operational relations among attributes included in the situation specific to the problem, is called constraint structure. A solution structure is part of a constraint structure. The auxiliary problems are defined and categorized based on this description and the two simplifying operations. In this categorization, there are three types of auxiliary problems, (1) auxiliary problems with divided surface structure, which are simplified in the formulation process, (2) auxiliary problems with divided solution structure, which are simplified in the computation process,and (3) auxiliary problems with specialized constraint structure, which are simplified in the situation. We also present two experiments for justification of this definition in mechanics problems. One is the generation experiment of auxiliary problems by persons who have experience of tutoring, and the other is analysis of the problems in exercise books which seem to be used for assisting in solving another problem. Because most of the problems are adequately categorized based on our definition, it contributes to utilize the auxiliary problems in ITS.