A Study on Operative Proof in Mathematics Education : Research on Children's Operation with Counters and Place Value Table

Abstract

The purpose of this research is to consider the concept of "Operative proof" that E. Ch. Wittmann proposed as a mathematical learning activity in 1996 and to construct foundations for developing the Substantial Learning Environments with "Operative proof". Especially, for developing the Substantial Learning Environment with "Operative proof" that used counters and place value table, in this paper, I investigated how children operate counters and place value table. I interviewed children to examine their reaction for counters and place value table. This interview is composed of the following questions. 1. Reading the representations of numbers with counters and place value table, and expressing the number on the place value table with counters. 2. Understanding the meaning of the operation of counters, and explaining it. 3. Using the operation of counters to solve arithmetic problems. And this was executed for 45 children of the fifth grader at an elementary school. The following suggestions were obtained from results of the interview for the design of the Substantial Learning Environment. ・Children don't need special guidance to understand the representations of numbers with counters on place value table, but they need some guidance to understand the meaning of the operation of counters. ・At the first step of the learning activity, teacher should guide children how to use counters and place value table. ・For the learning activity with operative proof, children need not only formal computation practice like calculation by writing but also basic learning activity to see number flexibly. Future tasks are to design the concrete Substantial Learning Environments that used operative proof by making the best use of these suggestions.