Polytomous Model Integrating S–P Chart and Ordering Theory: Application to Cognitive Diagnosis of Fraction Calculation

This study proposes an integrated model for cognition diagnosis applicable to polytomous data and conducts an empirical analysis using data on fraction addition and subtraction performance collected from elementary school students in Taiwan and Japan. This study develops computational procedures that extend the dichotomous student–problem (S–P) chart and ordering theory (OT) to polytomous response formats. The integrated polytomous student problem–ordering theory (SP–OT) model consists of a three-step procedure: (1) calculation of the caution indices of test-takers based on the polytomous S–P chart; (2) implementation of a two-stage cluster analysis using both the caution indices and score as clustering indicators; (3) construction of polytomous OT item structures for each cluster derived from the two-stage cluster analysis. The polytomous SP–OT model captured varying levels of student understanding and provided richer diagnostic insights. Furthermore, this model enhanced the accuracy and effectiveness of educational assessment. This study provided valuable methodological insights by developing a polytomous model that integrated the S–P chart and OT. This new model overcame the limitations of traditional dichotomous data by introducing a visualized and clustered approach to cognitive diagnosis. Future research may apply this polytomous SP–OT model to other disciplines.