Most of studies on algebraic area in school mathematics are focused on the understanding of meaning or operation of algebraic symbols, notations or equation. They hardly pay attention to the subtraction of a negative number and algebraic sum. In this study our interest is mainly devoted into two sorts of understanding about positive and negative numbers in order to clarify the transition from arithmetic to mathematics. Arithmetic has no necessity to regard subtraction as addition anywhere. Even if mathematics has necessity to develop it, mathematics teaching could not apply the idea of opposite number and inverse operation to understand the subtraction of a negative number as the addition of a positive number before going to algebra. And there is little room to explain it in arithmetic either. Therefore we use a metaphor to understand the change of subtraction to addition under the virtual reality and the language game. Moreover this idea must be formalized clearly and systematically in the algebraic sum. In this article we showed that the metaphorical understanding of the subtraction of a negative number and the systematic formalization of calculation in algebraic sum were a bridge to introduce mathematics teaching from arithmetic teaching.
