Polytomous IRS with Application in Concepts Diagnosis and Clustering on Fraction Subtraction for Pupils

The purpose of this study is to provide the polytomous item relational structure (PIRS) analysis and utilize it in fraction subtraction concepts diagnosis with classification on pupils by fuzzy clustering. Beyond the limitations of dichotomous item relational structure (IRS), PIRS is well efficient in the analysis of polytomous data. Most IRS application aims to analyze item hierarchies, not concept hierarchies. However, hierarchies of conceptual attributes will be meaningful for pedagogy and PIRS could provide graphical concept hierarchies according to response data matrix and item-concept matrix. PIRS analysis based on results of clustering on subjects would be more effective for remedial instruction. In this study, empirical data of fraction subtraction testing for sixth graders was analyzed by fuzzy clustering and PIRS is applied to display concept hierarchies for each group. The results show that the integration of PIRS and fuzzy clustering are feasible for cognition diagnosis and remedial instruction. Finally, some suggestions and recommendations are discussed.