A Study on the Function of Generalization in Mathematics Learning

Abstract

　　Generalization is very important knowing process in mathematics learning. However, in daily life, knowledge about the particular is easier than knowledge about general and enough for most of our purpose, and such knowledge sometimes may be more useful than knowledge about the general. Thus, on the one hand students have ability and tendency to generalize something from very young age (Vinner, 2011), but on the other hand they are not intending to generalize something (cf. Tatsis and Tatsis, 2012). Hence, the purpose of this study is to clarify for what do students 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 for children.

　　Through reviewing literature on generalization (e.g., Harel & Tall, 1991) and philosophical considerations, six functions with their examples are identified; variablization, purification, unification, discovery, association, and socialization. Then we implied three didactical suggestions for teaching and learning mathematics in classroom. First, teacher should design didactical situations where students can discern meaning of the six functions of generalization. Second, the implied order and/or structure of the function has possibility of a principle of didactics for designing mathematics classes. Third, the function may become a guideline in forming mental habit for students through their experiencing the functions of generalization in mathematics classes.