The popular fuzzy c-means algorithm based on Euclidean distance function converges to a local minimum of the objective function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, Gustafson-Kessel clustering algorithm needs added constraint of fuzzy covariance matrix, Gath-Geva clustering algorithm can only be used for the data with multivariate Gaussian distribution. In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. In this paper, an improved Normalized Supervised Clustering Algorithm Based on FCM by taking a new threshold value and a new convergent process is proposed. The experimental results of real data sets show that our proposed new algorithm has the best performance. Not only replacing the common covariance matrix with the correlation matrix in the objective function in the Normalized Supervised Clustering Algorithm.

The purpose of this study was to cooperate student-problem chart (S-P Chart) and ordering theory (OT) as an integrated method of cognition diagnosis. S-P chart was used to classify students into proper learning styles, In order to use OT to determine existing hierarchies among items with specifically learning style. Furthermore, an empirical data of capacity concepts testing for fundamental mathematics learning was analyzed based on the integrated method. The results showed that cognition diagnosis would be feasible for remedial teaching and design of teaching materials.

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. The purpose of this study is improved Fuzzy C-Means algorithm based on Mahalanobis distance to identify concept structure for Linear Algebra. In addition, Concept structure analysis (CSA) could provide individualized knowledge structure. CSA algorithm is the major methodology and it is based on fuzzy logic model of perception (FLMP) and interpretive structural modeling (ISM). CSA could display individualized knowledge structure and clearly represent hierarchies and linkage among concepts for each examinee. Each cluster of data can easily describe features of knowledge structures. The results show that there are five clusters and each cluster has its own cognitive characteristics. In this study, the author provide the empirical data for concepts of linear algebra from university students. To sum up, the methodology can improve knowledge management in classroom more feasible. Finally, the result shows that Algorithm based on Mahalanobis distance has better performance than Fuzzy C-Means algorithm.

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters by employing Mahalanobis distance in objective function, however, both of them need to add some constrains for Mahalanobis distance. In this paper, the authors’ improved Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is used to identify the mastery concepts in linear algebra, for comparing the performances with other four partition algorithms; FCM-M, GG, GK, and FCM. The result shows that FCM-CM has better performance than others.

The purpose of this study is to integrate two methodological approaches to explore concept structures. One approach is student-problem chart (S-P chart) and ordering theory (OT) and the other is fuzzy clustering and item relational structure (IRS). S-P chart is adopted to classify all students into proper learning styles. OT is to determine hierarchies of concept structures. Fuzzy clustering is soft computation to cluster features of students and combine IRS so as to determine the precondition and ordering relationship among items. The empirical data is statistics concepts test of university students. The results show that integration of these two approaches is feasible for cognition diagnosis and would be helpful for remedial instruction. Finally, some suggestions and recommendations for future research and educational research are provided.