• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.585-599
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• 2004

본 연구에서는 학교수학에서 증명지도의 문제점을 정당화의 측면에서 분석하고, 정당화의 한 방법으로서 확률론적 정당화를 제시하며, 학교수학에서 정당화 지도의 교육적 가치, 정당화 지도의 방향, 정당화 지도의 예와 지도 방법에 대해 논의한다. 이러한 논의에 근거하여 학교수학에서의 정당화 지도의 필요성 및 가능성에 관하여 살펴본다. 본 연구에서 '증명'은 고전적인 의미에서의 증명, 즉 엄밀한(rigorous) 증명, 수학적(mathematical) 증명이고, '정당화'는 기존의 수학적 증명 개념은 물론, 다양한 논증 기법을 포함하는 넓은 의미이다.

• Journal of Korean Philosophical Society
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• v.147
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• pp.215-237
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• 2018

The purpose of this study is to illuminate the precise nature and the central line of Kant's proof of the causal principle stated in the Second Analogy of the 2nd. edition of the Critique of Pure Reason. The study argues for the following thesis: 1. The proof of the Second Analogy concerns only the causal principle called the "every-event-some-cause" principle, and not the causal law(s) called the "same-cause-same-event" principle. 2. The goal of the proof is to establish the possibility of knowledge of an temporal order of successive states of an object. 3. The proof is broadly an single transcendental argument in two steps. The 1st. step is an analytic argument that infers from the given perceptions of an oder of successive states of an objects to the conclusion that the causal principle is the necessary condition for the objectivity of dies perceived order. The 2nd. step is a synthetic argument that infers from the formal nature of time to the conclusion that the causal principle is a necessary condition for die possibility of objective alterations and of empirical knowledge of these alterations. 4. The poof involves not the 'non sequitur' assumed by P. F. Strawson, that is, Kant infers not directly from a feature of our perceptions to a conclusion regarding the causal relations of distinct states of affairs that supposedly correspond to these perceptions.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.173-192
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• 2013

It may be replacement proofs with understanding and explaining geometrical properties that was a remarkable change in school geometry of 2009 revised national curriculum for mathematics. That comes from the difficulties which students have experienced in learning proofs. This study focuses on one of those difficulties which are caused by the forms of proofs: using letters for designating some sides or angles in writing proofs and understanding some long sentences of proofs. To overcome it, this study aims to investigate the applicability of Byrne's method which uses coloured diagrams instead of letters. For this purpose, the proofs of three geometrical properties were taught to middle school students by Byrne's visual method using the original source, dynamic representations, and the teacher's manual drawing, respectively. Consequently, the applicability of Byrne's method was discussed based on its strengths and its weaknesses by analysing the results of students' worksheets and interviews and their teacher's interview. This analysis shows that Byrne's method may be helpful for students' understanding of given geometrical proofs rather than writing proofs.

• Communications of Mathematical Education
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• v.10
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• pp.143-154
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• 2000

현재, 중학교에서 사용 가능한 수학 교과서는 8종류인데, 이처럼 다양한 종류의 교과서가 필요한 이유들 중의 하나는 수학적 개념이나 정리 등에 대한 다각적인 접근 방법들을 모색할 수 있는 가능성을 보장한다는 것이다 그러나, 현재의 교과서들은, 예를 들어, 정리의 증명에 있어 비슷한 증명 방법을 제시하고 있기 때문에, 학습자들에게 수학에 대한 폭넓은 시각과 다양한 수학적 아이디어를 제공할 수 있는 기회를 효과적으로 살리지 못하고 있다. 본 연구에서는 평면 기하학의 중요한 정리들 중의 하나인 ‘삼각형의 세 중선은 한 점에서 만나고, 각각의 중선은 교점에 의해 2:1로 나뉜다.’에 대한 다양한 증명들을 살펴보고, 각각의 증명들이 가지는 수학 교육적 의의를 고찰할 것이다.

• Korean Journal of Logic
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• v.8 no.2
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• pp.3-32
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• 2005

Hilbert's rational ambition to establish consistency in Number theory and mathematics in general was frustrated by the fact that the statement itself claiming consistency is undecidable within its formal system by $G\ddot{o}del's$ second theorem. Hilbert's optimism that a mathematician should not say "Ignorabimus" ("We don't know") in any mathematical problem also collapses, due to the presence of a undecidable statement that is neither provable nor refutable. The failure of his program receives more shock, because his system excludes any ambiguity and is based on only mechanical operations concerning signs and strings of signs. Above all, $G\ddot{o}del's$ theorem demonstrates the limits of formalization. Now, the notion of provability in the dimension of syntax comes to have priority over that of semantic truth in mathematics. In spite of his failure, the notion of algorithm(mechanical processe) made a direct contribution to the emergence of programming languages. Consequently, we believe that his program is failure, but a great one.

• Communications of Mathematical Education
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• v.19 no.3 s.23
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• pp.485-502
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• 2005

한 가지 문제에 대한 다양한 풀이 방법을 탐색하는 것은 수학적 대상의 성질을 발명, 일반화하는 것 뿐만 아니라, 학생들의 지적인 유창성 및 유연성 계발, 수학에 대한 심미적 가치의 함양을 위한 의미 있는 교수학적 경험을 제공할 수 있을 것이다. 본 연구에서는 고등학교 '미분과 적분'에 제시된 사인의 덧셈정리에 대한 다양한 증명 방법을 제시하고, 이를 분석하여 수학교수학적으로 의미로운 시사점을 도출하였다. 이를 통해, 사인의 덧셈정리에 대한 새로운 증명 방법의 탐색, 사인의 덧셈정리의 수학교수학적 활용의 다양한 가능성을 모색할 수 있는 기초자료를 제공할 것이며, 제시된 증명 방법들은 '미분과 적분'의 지도에서 심화학습 자료로도 활용할 수 있을 것이다.

• Communications of Mathematical Education
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• v.24 no.2
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• pp.325-344
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• 2010

The purpose of this study is to investigate what influences students' preferences on empirical and deductive proofs and find their relations. Although empirical and deductive proofs have been seen as a significant aspect of school mathematics, literatures have indicated that students tend to have a preference for empirical proof when they are convinced a mathematical statement. Several studies highlighted students'views about empirical and deductive proof. However, there are few attempts to find the relations of their views about these two proofs. The study was conducted to 47 students in 7~9 grades in the transition from empirical proof to deductive proof according to their mathematics curriculum. The data was collected on the written questionnaire asking students to choose one between empirical and deductive proofs in verifying that the sum of angles in any triangles is $180^{\circ}$. Further, they were asked to provide explanations for their preferences. Students' responses were coded and these codes were categorized to find the relations. As a result, students' responses could be categorized by 3 factors; accuracy of measurement, representative of triangles, and mathematics principles. First, the preferences on empirical proof were derived from considering the measurement as an accurate method, while conceiving the possibility of errors in measurement derived the preferences on deductive proof. Second, a number of students thought that verifying the statement for three different types of triangles -acute, right, obtuse triangles - in empirical proof was enough to convince the statement, while other students regarded these different types of triangles merely as partial examples of triangles and so they preferred deductive proof. Finally, students preferring empirical proof thought that using mathematical principles such as the properties of alternate or corresponding angles made proof more difficult to understand. Students preferring deductive proof, on the other hand, explained roles of these mathematical principles as verification, explanation, and application to other problems. The results indicated that students' preferences were due to their different perceptions of these common factors.

• Korean Journal of Logic
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• v.18 no.3
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• pp.307-335
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• 2015

An important component in Prawitz's and Dummett's proof-theoretic accounts of validity is the condition for validity of open arguments. According to their accounts, roughly, an open argument is valid if there is an effective method for transforming valid arguments for its premises into a valid argument for its conclusion. Although their conditions look similar to the proof condition for implication in the BHK explanation, their conditions differ from the BHK account in an important respect. If the premises of an open argument are undecidable in an appropriate sense, then that argument is trivially valid according to Prawitz's and Dummett's definitions. I call this 'the triviality problem'. After a brief exposition of their accounts of proof-theoretic validity, I discuss triviality problems raised by undecidable atomic sentences and by Godel sentence. On this basis, I suggest an emendation of Prawitz's definition of validity of argument.

• The Magazine of the IEIE
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• v.31 no.6
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• pp.40-51
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• 2004

최근 크게 흥행에 성공하고 있는 국산영화들은 국내 영화팬들을 열광케 하고 이제는 해외에서까지 인기를 크게 누리고 있다. 이러한 국산영화의 성공은 심지어 영화에 무관심했던 사람까지도 영화관으로 끌어들이고 관심을 갖게 하고 있으며 영화산업의 발전가능성이 다대함을 증명해 주고 있다.(중략)

• 주택과사람들
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• s.214
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• pp.38-39
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• 2008

지형.생태.자원 등 자연 환경과 정치.경제.사회.문화 등 인문 환경을 총망라해서 지도화하는 작업의 중요성은 역사적으로도 증명되었다. 이와 같은 연구와 조사는 국가의 통치와 민생 안정에 상당한 역할을 한다. 새 정부에서 국토 환경에 대한 연구와 조사, 기록의 중요성을 인식해 적극 실행해주길 바란다.