An analysis on middle school students’ understanding and concept image on the inclusion relation of quadrangle: Focusing on the opposite direction inclusion relation

In this study, the understanding and concept image of the inclusion relationship of quadrangles according to the grade and gender of middle school students were analyzed.To this end, a test tool was developed based on previous studies.For middle school students, first, the understanding of inclusion relation of quadrangles (i.e., the inclusion relation of the example set of quadrangles) and the inclusion relation in the opposite direction of a quadrangle (i.e., the inclusion relation of the property set of quadrangles) did not increase even if the grade went up.Second, there was no significant gender difference in the understanding of the quadrangle's inclusion relation, but there was a significant gender difference in the understanding of the opposite direction inclusion relation.Third, students who understand the inclusion relation in the opposite direction of a quadrangle generally explained using critical attributes such as 'The two diagonals have the same length.' and 'One angle is a right angle or all angles are a right angle.' Fourth when the inclusion relation in the opposite direction of the quadrangle was not understood, an error appeared by partially paying attention to the concept image of the quadrangle in the process of comparing the two squares.Based on this, this study considered that teaching and learning based on the use of critical properties and the difference in perspective according to the direction of the inclusion relation of quadrangles from a structural point of view can promote middle school students' understanding of the inclusion relation of quadrangles.