Abstract
This study investigates the process of children's understanding in a classroom focusing on the role of figurative representation in mathematical learning. Cobb & McClain describe the importance of folding back to a prior activity and taken-as-shared imagery in supporting the development of mathematical understanding. It is important for Cobb et al. that imagery represents the specific situation in the problem statement. And the imagery should be shared in the classroom community. In this study, "a mathematical tool" means figurative representation for conceptual understanding and problem-solving and plays a role of imagery in "folding back". Focusing on the conceptual framework of "folding back", I analyzed the classroom discourse and children's description to clarify the process of understanding development in the classroom in which the teacher and children repeatedly use this "mathematical tool". It developed that the activity "folding back" to the mathematical tool not only supported the children's development of conceptual understanding but also developed the understanding of the mathematical tool itself in using them repeatedly. In addition, both developments mutually promoted the understanding.
