본 연구에서는 문제해결에서 귀납적 추론의 과정을 분석하여 귀납적 추론의 단계를 0단계 문제 이해, 1단계 규칙성 인식, 2단계 자료 수집 실험 관찰, 3단계 추측(3-1단계)과 검증(3-2단계), 4단계 발전의 총 5단계로, 귀납적 추론의 흐름은 0단계에서 4단계로의 순차적인 흐름을 포함하여 자신이 찾은 규칙이나 추측에 대하여 반례를 발견하였을 때 대처하는 방식에 따라 다양하게 설정하였다. 또한 초등학교 6학년 학생 4명에 대한 사례 연구를 통하여 연구자가 설정한 귀납적 추론 단계와 흐름의 적절성을 확인하였고 귀납적 추론의 지도를 위한 시사점을 도출하였다.
본 연구에서는 문제해결에서 귀납적 추론의 과정을 분석하여 귀납적 추론의 단계를 0단계 문제 이해, 1단계 규칙성 인식, 2단계 자료 수집 실험 관찰, 3단계 추측(3-1단계)과 검증(3-2단계), 4단계 발전의 총 5단계로, 귀납적 추론의 흐름은 0단계에서 4단계로의 순차적인 흐름을 포함하여 자신이 찾은 규칙이나 추측에 대하여 반례를 발견하였을 때 대처하는 방식에 따라 다양하게 설정하였다. 또한 초등학교 6학년 학생 4명에 대한 사례 연구를 통하여 연구자가 설정한 귀납적 추론 단계와 흐름의 적절성을 확인하였고 귀납적 추론의 지도를 위한 시사점을 도출하였다.
Problem solving is important in school mathematics as the means and end of mathematics education. In elementary school, inductive reasoning is closely linked to problem solving. The purpose of this study was to examine ways of improving problem solving ability through analysis of inductive reasoning...
Problem solving is important in school mathematics as the means and end of mathematics education. In elementary school, inductive reasoning is closely linked to problem solving. The purpose of this study was to examine ways of improving problem solving ability through analysis of inductive reasoning process. After the process of inductive reasoning in problem solving was analyzed, five different stages of inductive reasoning were selected. It's assumed that the flow of inductive reasoning would begin with stage 0 and then go on to the higher stages step by step, and diverse sorts of additional inductive reasoning flow were selected depending on what students would do in case of finding counter examples to a regulation found by them or to their inference. And then a case study was implemented after four elementary school students who were in their sixth grade were selected in order to check the appropriateness of the stages and flows of inductive reasoning selected in this study, and how to teach inductive reasoning and what to teach to improve problem solving ability in terms of questioning and advising, the creation of student-centered class culture and representation were discussed to map out lesson plans. The conclusion of the study and the implications of the conclusion were as follows: First, a change of teacher roles is required in problem-solving education. Teachers should provide students with a wide variety of problem-solving strategies, serve as facilitators of their thinking and give many chances for them ide splore the given problems on their own. And they should be careful entegieto take considerations on the level of each student's understanding, the changes of their thinking during problem-solving process and their response. Second, elementary schools also should provide more intensive education on justification, and one of the best teaching methods will be by taking generic examples. Third, a student-centered classroom should be created to further the class participation of students and encourage them to explore without any restrictions. Fourth, inductive reasoning should be viewed as a crucial means to boost mathematical creativity.
Problem solving is important in school mathematics as the means and end of mathematics education. In elementary school, inductive reasoning is closely linked to problem solving. The purpose of this study was to examine ways of improving problem solving ability through analysis of inductive reasoning process. After the process of inductive reasoning in problem solving was analyzed, five different stages of inductive reasoning were selected. It's assumed that the flow of inductive reasoning would begin with stage 0 and then go on to the higher stages step by step, and diverse sorts of additional inductive reasoning flow were selected depending on what students would do in case of finding counter examples to a regulation found by them or to their inference. And then a case study was implemented after four elementary school students who were in their sixth grade were selected in order to check the appropriateness of the stages and flows of inductive reasoning selected in this study, and how to teach inductive reasoning and what to teach to improve problem solving ability in terms of questioning and advising, the creation of student-centered class culture and representation were discussed to map out lesson plans. The conclusion of the study and the implications of the conclusion were as follows: First, a change of teacher roles is required in problem-solving education. Teachers should provide students with a wide variety of problem-solving strategies, serve as facilitators of their thinking and give many chances for them ide splore the given problems on their own. And they should be careful entegieto take considerations on the level of each student's understanding, the changes of their thinking during problem-solving process and their response. Second, elementary schools also should provide more intensive education on justification, and one of the best teaching methods will be by taking generic examples. Third, a student-centered classroom should be created to further the class participation of students and encourage them to explore without any restrictions. Fourth, inductive reasoning should be viewed as a crucial means to boost mathematical creativity.
그렇다면 수학적 추론은 무엇일까? 추론이란 이미 알고 있는 판단으로부터 새로운 판단을 이끌어 내는 사유 작용이다(우정호, 2007). 수학적 추론은 연역적 추론과 귀납적 추론으로 나눌 수 있는데 고대 그리스 시대 이래로 수학의 연역적 추론에 의한 방법은 정신도야를 위한 교육 방안으로서 최선으로 생각되었으며 Polya나 Lakatos는 수학교육에서 귀납 추론 또는 귀납적 발견의 중요성을 강조하였다(서동엽, 2008).
귀납적 추론의 의미에 대해 어떤 견해들이 있는가?
첫째, 수학에서의 귀납은 연역의 반대 개념으로는 그 본질을 충분히 내포한다고 볼 수 없다. 개별적이고 특수한 사례의 관찰로부터 일반적인 사실을 도출해낸다는 일반적인 의미로는 수학적 사실을 발견해가는 역동적인 과정을 정확하고 명확하게 설명할 수 없기 때문이다.
둘째, 여러 학자들이 귀납을 바라보는 관점에 따라 귀납에 대한 생각의 차이를 발견할 수 있다. 그렇지만 새로운 수학적 사실이나 규칙 또는 법칙을 발견하는 과정에서 다양한 추측을 해보고 이를 확인해보는 일련의 과정을 통하여 새로운 수학적 지식을 발견해가는 과정을 중요하게 생각한다는 점은 공통임을 알 수 있다.
수학적 추론은 무엇으로 나뉘는가?
그렇다면 수학적 추론은 무엇일까? 추론이란 이미 알고 있는 판단으로부터 새로운 판단을 이끌어 내는 사유 작용이다(우정호, 2007). 수학적 추론은 연역적 추론과 귀납적 추론으로 나눌 수 있는데 고대 그리스 시대 이래로 수학의 연역적 추론에 의한 방법은 정신도야를 위한 교육 방안으로서 최선으로 생각되었으며 Polya나 Lakatos는 수학교육에서 귀납 추론 또는 귀납적 발견의 중요성을 강조하였다(서동엽, 2008). 근래 수학은 귀납적 추론에 의해 발견되고 연역적 추론에 의해 확립되어 간다는 점이 널리 받아들여지고 있기 때문에(Polya, 1986; 우정호, 2000; 서동엽, 2003에서 재인용), 지금까지 증명만을 강조하고 귀납적 추론의 과정을 아주 소홀히 다루었던 것은 수학적 사고의 반을 소홀히 한 것이다(우정호, 2007).
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