A Study on Operative Proof in Mathematics Education : Operative Proof with Counters and the Place Value Table

Abstract

The purpose of this paper 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". Firstly, we surveyed Kunimoto's researches concerning "Pre-formal proofs" and Wittmann's researches concerning "Operative proof" in mathematics teaching. And we pointed out that "Operative proof" is included in "Pre-formal proof". Moreover, through considering Wittmann's papers concerning "Operative proof", we pointed out the following essential points of the concept. ・"Operative proof" is the proof integrated into one of the mathematical activities. ・"Operative proof" is the proof based on the results of the operations given to the mathematical objects expressed appropriately. Secondary, we considered "Das Zahlenbuch" from the point of view of how operative proofs are embodied in the classroom situation and showed that counters and the place value table are nicely used in "Operative proof". Finally, we considered the possibilities of "Operative proof" with counters and the place value table referring to "Das Zahlenbuch" and pointed out the following two roles of "Operative proof". ・"Operative proof" has the possibilities of supporting the formal proofs of advance levels. ・"Operative proof" has the possibilities of promoting to discover another mathematical patterns. Our further tasks are to research experimentally how children work with the "Operative proof" and to develop investigate the Substantial Learning Environments with "Operative proof" for children.