Two major challenges for collaborative filtering problems are scalability and sparseness. Some powerful approaches have been developed to resolve these challenges. Two of them are Matrix Factorization (MF) and Fuzzy C-means (FCM). In this paper we combine the ideas of MF and FCM, and propose a new clustering model --- Modified Fuzzy C-means (MFCM). MFCM has better interpretability than MF, and better accuracy than FCM. MFCM also supplies a new perspective on MF models. Two new algorithms are developed to solve this new model. They are applied to the Netflix Prize data set and acquire comparable accuracy with that of MF.