Application of Weighted Polytomous Ordering Theory in Concepts Structure Analysis of Fraction Addition for Pupils

Weighted polytomous ordering theory (WPOT) is suitable for polytomous data set. It considers the weight on precondition relations between elements. The method WPOT would help us to find the hierarchical structure among elements. The purpose of this paper is to discuss WPOT and utilize it in analyzing the mathematics measurement data. The research applies WPOT in the concept structures analysis of fraction addition for pupils. The results show the relationship and hierarchies among concepts. Further suggestions and recommendations are also discussed. Key-Words: weighted polytomous ordering theory, ordering theory, fraction addition, concepts structure, cognition diagnosis. 1 Background and Motivation In the viewpoint of learning process and psychometrics, the concept structure is an important issue which is worth discussing [9]. It is important to find the ordering relation and hierarchical structure among concepts. Ordering theory (OT) was developed by Bart and Krus and it aims to find the precondition between items based on cross-tabulation of dichotomous items [6] [7]. Bart and Krus proposed to find hierarchies among items by OT so that it is helpful to comprehend the structure among items. However, it can't deal with the polytomous data. That is, OT can only analyze the dichotomous data and this limits its application in many fields. For this reason, Lin, Bart, and Huang extended the dichotomous OT to weighted polytomous ordering theory (WPOT). Both weight of precondition and polytomous score are considered so that WPOT could extend the application of ordering theory [8]. In addition to popular application on item hierarchies, ordering theory could also be applied in the analysis of concept hierarchies. Therefore, it can describe the relation of hierarchical structures among concepts. Many researches about OT are discussed in the past [2][3][4][5]. Lin, Bart, and Huang extended OT to WPOT and implemented the computer software. This research will adopt WOPT for concept structure analysis of fraction addition for pupils. Concepts of fractional are important units in the mathematics course of elementary schools [10]. Related literatures show that pupils make quite a few misconceptions on fraction and calculation operation. They have many baffles on learning process of meaning and operations of fraction. Hence, in order to investigate the hierarchical structures among concepts for pupils, this research utilize WPOT to explore the cognition diagnosis about fractional addition. 2 Literature Review The ordering theory proposed by Bart and Krus can find the ordering relation and precondition between dichotomous items by cross-tabulation [1]. For Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007 177