Learning about Sets as a Foundation of Concept Formation for Numbers

Abstract

An accurate formation of the concept of number related to quantity should precede any other learning in mathematics. Taking the following three steps is essential to the formation of the concept; (1) making various sets based on various properties, (2)eliminating all the properties except number pieces and making supersets based on it, (3)comparing and estimating the size of several supersets. In this research, the authors constructed a learning course that meets (1) and (2) as preconditions for (3). In the construction of the course, we gave careful consideration to the identification of the laws for the formation of the three concepts (set making, set re-making within hierarchy, set re-making between hierarchy) and to making a systematic combination of three factors (term-manipulation, experience of scientist, experience of everyday life). We taught 13 classes to first graders immediately after the entrance into public elementary school. Whereas in the pretest the students were unable to solve almost any of the problems, in the post-test all of them were able to give the correct answers to all the problems within several difficulty levels. Many spontaneous hypothesis-verification activities also occurred, and we confirmed the formation of cardinal number concept in students at quite an early stage.