Abstract
It has been noted that there is an important aspect in an emergence of mathematical knowledge and concepts for all children in a mathematics classroom. It is, however, difficult for an observer to visualize the abstract concept, and whether the emergence is for children or for all participants including the teacher in the classroom. The purpose of this study is to clarify the concept of the emergence in mathematics education in order to develop lessons that activate students’ and/or teachers’ emergence. Based on the review of previous works on emergence and findings of our study, we define emergence, and proposed five required elements of the emergence: (E1) fundamental property, (E2) subjects, (E3) methods, (E4) mathematical ideas, (E5) newness/valueness of mathematical ideas. We then demonstrate the appropriateness of these five requirements with a reported case and two additional cases of elementary school mathematics classes, and created a framework with four different types of emergences identified in terms of who and what, i.e. the difference between emergences for children and for all participants including the teacher, and the difference between the mathematical knowledge and concepts and the mathematical view and thinking. Finally, we proposed to define the emergence in mathematics education in terms of characteristic, requirements, and types of emergence. The results will be useful for an observer to define emergence, and describe and analyze the dynamic function of social interactions leading to the emergence in school mathematics classrooms.
