두 직사각형 넓이의 합 또는 차로의 변형을 통한 삼각형 넓이 구하기에 관한 예비 초등교사의 도형 조작 능력 분석

Abstract

This study analyzed the response types and accuracy rates of pre-service elementary teachers in solving problems involving the areas of inscribed and non-inscribed triangles within a fixed rectangle. These problems were based on equiareal transformations using the sum or difference of two rectangles' areas. The findings are as follows: First, the analysis of student responses to questions about the area transformation of inscribed triangles revealed that there was no significant difference in the use of double-area transformations, reasoning, the Cavalieri principle, or accuracy rates between two groups of different grade levels. In contrast, when comparing two groups of the same grade level, significant differences in response types and accuracy rates were observed depending on the type of examples provided to assist problem-solving. Second, student responses to questions about the area transformation of non-inscribed triangles exhibited patterns similar to those for the transformation of inscribed triangles. Third, in the group provided with specific examples, it was expected that students would derive the concept of double-area transformations for right triangles and engage in manipulative activities. However, it was found that a significant number of students relied on the Cavalieri principle, which they had already internalized, rather than using double-area transformations. Fourth, the analysis of student responses to comparisons between questions revealed that groups of different grade levels but provided with the same examples showed a tendency to solve the inscribed triangle transformation questions more successfully than the non-inscribed triangle questions. A strong positive correlation was found between the two types of questions. On the other hand, in the group provided with specific examples, there was no statistically significant difference in success rates between the two question types, though a strong correlation between them was still evident. Finally, students' approaches to transformations using double-area methods and the Cavalieri principle lacked diversity. These findings suggest important educational implications, emphasizing the need for pre-service elementary teachers to gain experience in manipulative activities related to geometric education.