대학생의 반례 생성과 참 명제 제기 과정에 대한 연구
The study on the process of undergraduate students’ generating counter examples and proposing the true statements
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참고문헌
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초록
수학교육에서 증명에 대한 연구는 Lakatos의 인식론을 기반으로 이어져왔으며[3], 최근 반례가 수학교육의 새로운 초점이 되고 있으나, 사범대학에 재학 중인 대학생들의 반례를 바탕으로 한 참 명제 제기를연구한 수학교육연구는 부재한 실정이다. 이에 본 연구에서는 대학생 6명을 사례연구방법으로 주어진명제가 거짓임을 보이고 이를 바탕으로 참 명제를 제기하는 과정을 분석하였다. 연구 결과, 주어진명제가 거짓임을 보이기 위해 반례를 많이 생성할수록 생성한 반례를 바탕으로 정확한 참 명제를 제기하였으며, 주어진 명제를 이해하거나 다양한 반례 생성 과정 및 참 명제 제기에 필요한 수학적 지식에대해 깊은 이해를 가지지 못하고 암기한 경우 참 명제를 제안하지 못하였다. 또한 수학적 명제의 가정부분을 유지한 채 반례를 찾거나 주어진 명제를 변형해야 함에도 불구하고, 가정 부분을 변형하거나무시하는 등의 어려움을 보였다. 대학생들의 반례 생성 및 명제와 증명 이해에 대한 어려움은 이들이교사가 되면, 학교 수학 수업에서 중등학생들의 증명교육에까지 영향을 미칠 수도 있으므로, 이와관련된 예비교사 교육, 학교 수학의 교수.학습자료 개발, 교과서 내용의 강화 등이 필요할 것이다.
Abstract
There has been increasing interest in recent years in the pedagogical importance of counter-examples that focuses on pedagogical perspectives. But there is no research that undergraduate students’ generating counter-examples and proposing the true statements. This study analyze 6 undergraduate students’ response to interview tasks and the process of their generating counter-examples and proposing true statements. The results of interviews are that the more undergraduate students generate various counter-examples, the more valid they propose true statements. If undergraduate students have invalid understanding of logical implication and generate only one counter-example, they would not propose true statements that modify the given statement, preserving the antecedent. In pre-service teacher’s education and school mathematics class, we need to develop materials and textbooks about counter-examples and false statements.
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