두 직사각형 넓이의 합 또는 차로의 변형을 통한 삼각형 넓이 구하기에 관한 예비 초등교사의 도형 조작 능력 분석 (Analysis of pre-service elementary teachers' geometric manipulation skills in calculating the area of a triangle through transformations involving the sum or difference of two rectangles' areas)

서지정보

· 발행기관 : 한국수학교육학회
· 수록지 정보 : 수학교육 논문집 / 38권 / 4호 / 487 ~ 512페이지
· 저자명 : 이호수, 최근배

초록

이 연구에서는 고정된 직사각형을 기준으로 내접 삼각형과 비-내접 삼각형의 넓이를 두 직사각형의 넓이의 합 또는차를 이용한등적변형 문제를통해예비초등교사들의응답유형과정답률등을비교분석하였다. 먼저, 내접 삼각형의 넓이 변형문항에대한학생반응결과는다음과 같다. 학년이 다른 두 집단을비교한결과, 배적변형, 추론, 카발리에리 원리사용률및정답률에유의미한차이가없었다. 반면, 학년이같은 두집단간비교에서는문제해결을돕기 위해제시된 예시의유형에따라응답유형비율과 정답률에서유의미한차이가나타났다. 둘째, 비-내접 삼각형의 넓이 변형문항에대한학생반응결과는내접 삼각형의변형과유사한 패턴을보였다. 셋째, 특수한 예시가 제공된 집단에서는 학생들이 직각삼각형의 배적변형 개념을 도출해 조작 활동을수행할 것으로예상되었다. 그러나 많은학생이 배적변형 대신 자신이 이미 내재화한 카발리에리 원리를 사용해 문제를 해결했다는 점이 확인되었다. 넷째, 문항 간 비교에 대한 학생반응 결과는 다음과 같다. 학년은 다르지만 동일한 예시를 받은 두 집단에서는 내접 삼각형 변형문항이비-내접 삼각형변형문항보다 성공적으로해결되는경향이있었으며, 두 문항간에는강한양의상관관계가 확인되었다. 반면, 특수 예시가 제공된 집단에서는 두 문항 간 성공률의 통계적 차이는 없었으나, 여전히두 문항간강한연관성이나타났다. 끝으로, 배적변형과 카발리에리 원리를 활용한 변형에서 학생들의 접근 방식은다양하지 못했다. 이러한 결과로부터 예비 초등교사의 도형과 관련된 조작활동의 경험이 중요하다는 교육적 시사점을 얻었다.

영어초록

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.

참고자료

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