• 대한수학교육학회지:수학교육학연구
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• 제16권2호
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• pp.95-113
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• 2006

본 연구는 학교 수학에서 추론 지도의 수준을 보다 상세히 구분해 보고자 한 것이다. 수학의 특징으로부터, 대상에서 분리된 순수한 형식적 관점은 새로운 지식의 창안에서 한계를 지닌다는 점을 알 수 있으며, 수학교육에서도 이를 반영할 필요가 있다고 본다. 이런 점에서 귀납 추론과 형식적 연역 추론의 매개 단계로서 구체적 조작이나 감각 경험과 관련된 직관적 증명의 수준을 설정하는 것이 적절할 것으로 생각되며, 이 수준의 핵심적인 활동은 경험으로부터 일반성을 통찰하는 것이다. 이 수준은 낮은 수준의 귀납 추론보다는 대상과 분리되며 보다 형식적인 논리의 개입을 필요로 하는 과정에 있다. 이와 같이 보다 점진적으로 대상으로부터 분리되고 형식적 논리를 학습할 수 있도록 추론 지도 수준을 구분하고, 이에 따라 수학적 추론을 지도하는 것이 필요할 것이다.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제25권1호
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• pp.65-89
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• 2022

Despite its recognition in the field of mathematics education and mathematics, students' understanding about proof and performance on proof tasks have been far from promising. Research has documented that teachers tend to accept empirical arguments as proofs. In this study, an online survey was administered to examine how Korean secondary mathematic teachers make judgements on mathematical arguments varied along representations. The results indicate that, when asked to judge how convincing to their students the given arguments would be, the teachers tended to consider how likely students understand the given arguments and this surfaces as a controversial matter with the algebraic argument being both most and least convincing for their students. The teachers' judgements on the algebraic argument were shown to have statistically significant difference with respect to convincingness to them, convincingness to their students, and validity as mathematical proof.

• 한국수학교육학회지시리즈C:초등수학교육
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• 제7권2호
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• pp.85-99
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• 2003

In this paper, firstly examined various proofs types that cover informal empirical justifications by Balacheff, Miyazaki, and Harel ＆ Sowder and Tall. Using these theoretical frameworks, justification activities by 5th graders were analyzed and several conclusions were drawn as follow: 1) Children in 5th grade could justify using various proofs types and method ranged from external proofs schemes by Harel & Sowder to thought experiment by Balacheff This implies that children in elementary school can justify various mathematical statements of ideas for themselves. To improve children's proving abilities, rich experience for justifying should be provided. 2) Activities that make conjectures from cases then justify should be given to students in order to develop a sense of necessity of formal proof. 3) Children have to understand the meaning and usage of mathematical symbol to advance to formal deductive proofs. 4) New theoretical framework is needed to be established to provide a framework for research on elementary school children's justification activities. Research on proof mainly focused on the type of proof in terms of reasoning and activities involved. But proof types are also influenced by the tasks given. In elementary school, tasks that require physical activities or examples are provided. To develop students'various proof types, tasks that require various justification methods should be provided. 5) Children's justification type were influenced not only by development level but also by the concept they had. 6) Justification activities provide useful situation that assess students'mathematical understanding. 7) Teachers understanding toward role of proof(verification, explanation, communication, discovery, systematization) should be the starting point of proof activities.

• East Asian mathematical journal
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• 제25권3호
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• pp.299-320
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• 2009

In mathematics, the education of the geometry proof has been playing an important role in promoting the ability for logical thinking by means of developing the deductive reasoning. However, despite of those importance mentioned above, considering the present condition for the education of the geometry proof in middle schools, it is still found that most of classes are led mainly by teachers, operating the cramming system of eduction, and students in those classes have many difficulties in learning the geometry proof course. Accordingly this thesis suggests the other method that is distinguished from previous proof educations. The thesis of Kai-Lin Yang and Fou-Lai Lin on 'A Model of Reading Comprehension of Geometry Proof (RCGP)', which was published in 2007, have various practical examples based on the model. After composing classes based on those examples and instructing the geometry proof, found out a problem. And then advance a new teaching model that amendment and supplementation However, it is considered to have limitation because subjects were minority and classes were operated by man-to-man method. Hopefully, the method of proof education will be more developed through performing more active researches on this in the nearest future.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제13권3호
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• pp.217-233
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• 2009

The purpose of this study is to describe how dynamic geometry systems can be useful in proof activity; teaching sequences based on the use of dynamic geometry systems and to analyze the possible roles of dynamic geometry systems in both teaching and learning of proof. And also dynamic geometry environments can generate powerful interplay between empirical explorations and formal proofs. The point of this study was to show that how using dynamic geometry software can provide an opportunity to link between empirical and deductive reasoning, and how such software can be utilized to gain insight into a deductive argument.

• 대한수학교육학회지:학교수학
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• 제18권4호
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• pp.749-771
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• 2016

이 논문에서는 우리나라와 미국 수학 교과서에서 다루고 있는 평행사변형이 되기 위한 조건 관련 과제를 과제의 구조, 증명과 추론 유형, 그리고 인지적 노력 수준에 따라 비교 분석하였다. 이를 통해 두 나라 교과서 과제의 공통점과 차이점을 분석하였다. 그 결과는 다음과 같다. 첫째, 과제 구조와 관련하여, 우리나라 교과서에 비해 미국 교과서에 제시된 과제의 구조가 더 다양하다. 둘째, 증명과 추론 유형과 관련하여, 우리나라와 미국 교과서 모두 IC 과제와 DA 과제의 구성 비율이 높으며, 우리나라 교과서에 비해 미국 교과서에 제시된 과제의 유형이 더 다양하다. 셋째, 과제의 인지적 노력 수준과 관련하여, 우리나라와 미국 교과서 모두 PNC 과제와 PWC 과제가 대부분을 차지하며, 우리나라의 경우 미국에 비해 구체적인 알고리즘적 절차를 이용하는 수학 과제를 제시하는 비율이 높다. 차이점을 토대로 우리나라 교과서 재구성에 필요한 다음과 같은 시사점을 얻을 수 있었다. 첫째, 과제의 구조 및 증명과 추론 유형과 관련하여, 구성의 다양성을 높여야 한다. 둘째, 과제의 인지적 노력 수준과 관련하여, PNC 과제에 대한 편중현상을 완화해야 하며, 과제 유형별 인지적 노력 수준에 대한 재고가 필요하다. 셋째, 과제의 주제 또는 소재와 관련하여, 수학 내적, 외적인 상황과의 연결성이 강화된 과제를 도입할 수 있는 방안의 재고가 필요하다.

• 한국수학사학회지
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• 제21권3호
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• pp.143-156
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• 2008

본 연구에서는 중학교 2학년 도형의 성질 단원의 증명학습을 세 단계로 나누어 설문을 통하여 학생들이 증명학습에서 겪는 어려움을 조사하였다. 설문 분석 결과 학생들은 증명학습에서 증명의 의미를 이해하지 못해 명제의 참을 판단하는 정도의 간단한 추론도 하지 못할 뿐만 아니라 제시된 증명을 읽고 그것이 증명하려는 명제의 가정과 결론을 파악하지 못한다. 이는 학생들이 명제의 가정과 결론의 의미와 역할을 명확히 이해하지 못하는데서 비롯된다. 따라서 학생들에게 명제의 가정과 결론의 의미와 역할에 대한 지도에 좀 더 역점을 두는것이 필요하다.

• East Asian mathematical journal
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• 제24권5호
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• pp.527-545
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• 2008

One of the education purpose of the section "Figures" in the eighth grade is to develop students' deductive reasoning ability, which is basic and essential for living in a democratic society. However, most or middle school students feel much more difficulty or even frustration in the study of formal arguments for geometric situations than any other mathematical fields. It is owing to the big gap between inductive reasoning in elementary school education and deductive reasoning, which is not intuitive, in middle school education. Also, it is very burden for students to describe geometric statements exactly by using various appropriate symbols. Moreover, Usage of the same symbols for angle and angle measurement or segments and segments measurement makes students more confused. Since geometric relations is mainly determined by the measurements of geometric objects, students should be able to interpret the geometric properties to the algebraic properties, and vice verse. In this paper, we first compare and contrast inductive and deductive reasoning approaches to justify geometric facts and relations in school curricula. Convincing arguments are based on experiment and experience, then are developed from inductive reasoning to deductive proofs. We introduce teaching methods to help students's understanding for deductive reasoning in the textbook by using stepwise visualization materials. It is desirable that an effective proof instruction should be able to provide teaching methods and visual materials suitable for students' intellectual level and their own intuition.

• 한국수학교육학회지시리즈A:수학교육
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• 제49권2호
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• pp.161-174
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• 2010

The purpose of the study is to analyze various types of errors appeared in true-false proof problems of matrix and to find out correction method. In order to achieve this purpose, error test was conducted to the subject of 87 second grade students who were chosen from D high schoool. It was shown from this test that the most frequent error type was caused by the lack of understanding about concepts and essential facts of matrix(35.3%), and then caused by the invalid logically reasoning (27.4%), and then caused by the misusing conditions(18.7%). Through three hours of correction lessons with 5 students, the following correction teaching method was proposed. First, it is stressed that the operation rules and properties satisfied in real number system can not be applied in matrix. Second, it is taught that the analytical proof method and the reductio ad absurdum method are useful in the proof problem of matrix. Third, it is explained that the counter example of E=$\begin{pmatrix}1\;0\\0\;1 \end{pmatrix}$, -E should be found in proof of the false statement. Fourth, it is taught that the determinant condition should be checked for the existence of the inverse matrix.

• Journal of the Korean Data and Information Science Society
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• 제12권1호
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• pp.61-69
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• 2001

본 논문은 논리문장으로 표현된 지식을 처리하는 정리증명 과정에서 증명이 완료되기 전에 잠정적 결론을 유도하는 확률추론 기법을 제시한다. 정리증명 과정 중에 베이지안 해석을 이용하여 지식을 갱신하는 방법을 제시하고, 의사결정 방법을 사용하여 시간에 민감한 사안에 대해 신속하게 대처할 것인지 아니면 고의로 미룰 것인지를 결정하는 방법을 밝힌다.