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The purpose of this study is to clarify for what students do generalize something in learning mathematics. In this study, we make a distinction between generalization and extension, and focus on the function of generalization in terms of its meaning, purpose, and usefulness. Through reviewing literature on generalization and philosophical considerations, six functions with their examples are identified; variablization, purification, unification, discovery, association, and socialization. We propose a new theoretical framework for the function of generalization in learning mathematics, suggesting that the framework has possibility of a principle of didactics for teachers and a guideline in forming mental habit for students. [For the complete proceedings, see ED597799.]