Research on the Mutual Construction Process of Operational and Formal Proofs of Algebraic Expression in 8th Grade

Abstract

　　The purpose of this research is to clarify the process of mutually constructing operational and formal proofs in the proof learning of literal expressions involving negative numbers and subtraction. Operational proofs in the previous studies have been limited to the mathematical phenomena involving additions and natural numbers that can be expressed as quantities. Therefore, we examined the proof learning of expressions involving the concept of subtraction and negative numbers cannot be explored. However, considering the actual situation of the students, it is not a little difficult to proceed with the proof in an abstract manner immediately away from the concrete, and it is difficult to accompany the concrete image and operation with the understanding of the proof of the character expression. It is thought to contribute. We propose two things to make this clear.

　　First, as the meaning of operational proofs, “the process of deriving formal proofs based on operational proofs” and “the process of exploring operational proofs based on formal proofs” have profound implications for formal proofs. Recall a learning process that encourages understanding.

　　Second, through the process of constructing operational proofs and formal proofs, it is possible to perform operative proofs on proofs of continuous character expressions involving the concept of negative numbers. It turned out to be promoted. The problem of the sum of two numbers is pushed up to concreteness by analogy of the operation, and the problem of the difference of two numbers can be regarded as a higher-order concrete example of formal proof. It has been clarified that the mutual construction of operational and formal proofs in these two problems can lead to the search for proofs of character expressions involving the concept of negative numbers, such as the problem of the difference between two numbers.