• 대한수학교육학회지:수학교육학연구
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• 제8권1호
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• pp.351-364
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• 1998

This thesis analyzes the nature of proof in the perspective on the philosophy of mathematics. such as absolutism, quasi-empiricism and social constructivism. And this thesis searches for the improvement of teaching proof in the light of the result of those analyses of the nature of proof. Though the analyses of the nature of proof in the perspective on the philosophy of mathematics, it is revealed that proof is a dynamic reasoning process unifying the way of analytical thought and the way of synthetical thought, and plays remarkably important roles such as justification, discovery and conviction. Hence we should teach proof as a dynamic reasoning process unifying the way of analytic thought and the way of synthetic thought, avoiding the mistake of dealing with proof as a unilaterally synthetic method. At the same time, we should make students have the needs of proof in a natural way by providing them with the contexts of both justification and discovery simultaneously. Finally, we should introduce the aspect of proof that can be represented as conviction, understanding, explanation and communication to school mathematics.

• 한국수학교육학회지시리즈A:수학교육
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• 제58권2호
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• pp.161-185
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• 2019

이 연구에서는 중등 수학 교사와 예비교사들이 추론과 증명을 어떻게 이해하여 구성하는지를 탐색하였다. 연구 참여자들은 대부분 대수적인 증명을 시도하는데 이미 알고 있는 공식이나 식을 적용한 대수적 조작으로 답을 구하는 것에 그치며 주어진 문제에 내재된 수학적 구조를 통해 증명을 구성하지는 못하였다. 또한 참여자의 상당수가 대수적 식을 통한 증명만을 완전한 증명으로 판단하였으며 대부분은 기존에 접하지 못했던 새로운 문제유형에서 추론 및 증명구성을 완성하지 못하는 것으로 나타났다.

• 한국컴퓨터정보학회논문지
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• 제27권7호
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• pp.35-48
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• 2022

본 연구에서는 멀티에이전트 환경에서 지식을 표현하고 추론함에 있어서 증명 이론적 방법을 제안한다. 이 방법은 논리적 결과를 기계적 방법으로 결정하므로 초기 인공지능 연구부터 핵심분야로 발전해 왔다. 하지만 임의의 닫힌 문장들의 집합에서 항상 명제가 증명할 수 있지 않기에 논리적 결과가 결정할 수 있어지려면 절 형식의 문장으로 그 표현 범위를 제한한다. 그리고 절 형식의 문장들에서만 적용 가능한, 단순하면서도 강력한 추론 규칙인 비교흡수 원리(Resolution principle)를 적용한다. 또한 증명이론을 메타술어로 표현할 수 있으므로 증명이론의 메타논리로 확장 가능하다. 메타논리가 모델 이론의 인식 논리(epistemic logic)보다 향상된 표현력을 기반으로 실용적인 면과 효율면에서 우월할 수 있다. 이를 입증하기 위해 인식 논리의 의미론과 증명이론의 메타논리 방식으로 각각 Muddy Children 문제에 적용한다. 그 결과 협력적 멀티에이전트 환경에서 메타논리를 사용하여 지식과 공통지식을 표현하고 추론한 방법이 더 효율적임을 증명한다.

• 한국학교수학회논문집
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• 제11권1호
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• pp.55-68
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• 2008

이 연구에서는 중학생들이 '원의 성질' 단원의 증명학습 과정에서 어떤 어려움을 겪는지를 조사하여 GSP를 활용한 증명학습이 학생들의 어려움을 어떻게 완화시키는지를 탐구하였다. 진단검사를 통해, 학생들은 가정과 결론의 이해, 기호의 사용, 추론 과정 등에서 어려움을 겪고 있음을 확인하였다. 한편, 학생들은 GSP를 활용한 증명학습을 통해 자신의 추측이나 추론에 대한 피드백을 받을 수 있고, 구체적인 사례를 일반화하거나 증명에 필요한 아이디어를 능동적으로 찾는 탐구 태도를 형성할 수 있다는 것을 확인하였다.

• 대한수학교육학회지:수학교육학연구
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• 제9권1호
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• pp.245-261
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• 1999

Proof is an essential characteristic of mathematics and as such should be a key component in mathematics education. But, teaching proof in school mathematics have been unsuccessful for many students. The traditional approach to proofs stresses formal logic and rigorous proof. Thus, most students have difficulties of the concept of proof and students' experiences with proof do not seem meaningful to them. However, different views of proof were asserted in the reassessment of the foundations of mathematics and the nature of mathematical truth. These different views of justification need to be reflected in demonstrative geometry classes. The purpose of this study is to characterize the types of students' justification in demonstrative geometry classes taught using dynamic software. The types of justification can be organized into three categories : empirical justification, deductive justification, and authoritarian justification. Empirical justification are based on evidence from examples, whereas deductive justification are based logical reasoning. If we assume that a strong understanding of demonstrative geometry is shown when empirical justification and deductive justification coexist and benefit from each other, then students' justification should not only some empirical basis but also use chains of deductive reasoning. Thus, interaction between empirical and deductive justification is important. Dynamic geometry software can be used to design the approach to justification that can be successful in moving students toward meaningful justification of ideas. Interactive geometry software can connect visual and empirical justification to higher levels of geometric justification with logical arguments in formal proof.

• ETRI Journal
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• 제9권1호
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• pp.84-96
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• 1987

The equality relation is very important in mechanical theorem proving procedures. A proposed inference rule called RHU-resolution is intended to extend the hyperparamodulation[23, 9] by introducing a bidirectional proof search that simultaneously employs a forward reasoning and a backward reasoning, and generalize it by incorporating beneflts of extended hyper steps with a preprocessing process, that includes a subsumption check in an equality graph and a high level planning. The forward reasoning in RHU-resolution may replace the role of the function substitution link.[9] That is, RHU-deduction without the function substitution link gets a proof. In order to control explosive generation of positive equalities by the forward reasoning, we haue put some restrictions on input clauses and k-pd links, and also have included a control strategy for a positive-positive linkage, like the set-of-support concept, A linking path between two end terms can be found by simple checking of linked unifiability using the concept of a linked unification. We tried to prevent redundant resolvents from generating by preprocessing using a subsumption check in the subsumption based eauality graph(SPD-Graph)so that the search space for possible RHU-resolution may be reduced.

• 대한수학교육학회지:수학교육학연구
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• 제14권1호
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• pp.111-127
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• 2004

본 연구는 기하 부분의 내용이 초등학교에서 중학교로 발전 전개되는 과정에서 일부 용어의 정의와 평행선과 각의 취급에 대하여 알아보았고, 교육과정에 제시된 관련내용을 분석하고 그에 따른 현행 교과서를 살펴보았다. 다음에 관련된 분야의 일부 외국교과서를 비교 분석하여 그 상황을 알아보았다 그 결과 현행 교과서 보다 바람직한 내용의 취급을 위해서는 우선 체계적인 학습을 할 수 있도록 교육과정에 보다 적합한 학습내용과 그 취급 방법을 제시해야만 하고, 그에 따라 교과서도 보다 적합하게 집필되어야 함을 제시하였다. 이를테면 용어의 정의는 반복하여 충분히 이해하도록 하고, 특히 교육의 다양성을 위해서 평행선의 성질에 관한 내용은 공준으로 도입하여 활용할 수도 있고, 우수한 학생들은 증명을 하여 활용할 수도 있도록 다양한 취급을 하는 것이 바람직함을 제시하였다. 그리고 특히 현행 교과서에서는 <7-나 단계에서 취급되고 있는 맞꼭지각의 성질과 평행선의 성질과 같은 연역적 추론에 의해서 증명될 수 있는 내용들은 18- 나 단계>로 이동을 하여 학습하는 것이 학습 체계의 연계에 바람직함을 제시하였다.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제24권3호
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• pp.255-278
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• 2021

Proof serves a critical role in mathematical practices as well as in fostering student's mathematical understanding. However, the research literature accumulates results that there are not many opportunities available for students to engage with proving-related activities and that students' understanding about proof is not promising. This unpromising state of instruction of proof calls for a novel approach to address the aforementioned issues. This study investigated an instruction of proof to explore a pedagogy to teach how to prove. The teacher utilized the way of problem posing to make proving a routine part of everyday lesson and changed the classroom culture to support student proving. The study identified the teacher's support for student proving, the key pedagogical changes that embraced proving as part of everyday lesson, and what changes the teacher made to cultivate the classroom culture to be better suited for establishing a supportive community for student proving. The results indicate that problem posing has a potential to embrace proof into everyday lesson.

• 대한수학교육학회지:학교수학
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• 제4권1호
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• pp.1-13
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• 2002

The purpose of this study is to investigate the reasons and backgrounds of drawing auxiliary lines in the proof of geometry. In most of proofs in geometry, drawing auxiliary lines provide important clues, thus they play a key role in deductive proof. However, many student tend to have difficulties of drawing auxiliary lines because there seems to be no general rule to produce auxiliary lines. To alleviate such difficulties, informal activities need to be encouraged prior to draw auxiliary lines in rigorous deductive proof. Informal activities are considered to be contrasting to deductive proof, but at the same time they are connected to deductive proof because each in formal activity can be mathematically represented. For example, the informal activities such as fliping and superimposing can be mathematically translated into bisecting line and congruence. To elaborate this idea, some examples from the middle school mathematics were chosen to corroborate the relation between informal activities and deductive proof. This attempt could be a stepping stone to the discussion of how to teach auxiliary lines and deductive reasoning.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제19권4호
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• pp.229-245
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• 2015

In this study, I investigated how pre-service teachers (PSTs) proved three geometric problems by using Geometer's SketchPad (GSP) software. Based on observations in class and results from a test of geometric reasoning, eight PSTs were sorted into four of the five van Hiele levels of geometric reasoning, which were then used to predict the PSTs' levels of reasoning on three tasks involving proofs using GSP. Findings suggested that the ways the PSTs justified their geometric reasoning across the three questions demonstrated their different uses of GSP depending on their van Hiele levels. These findings also led to the insight that the notion of "proof" had somewhat different meanings for students at different van Hiele levels of thought. Implications for the effective integration of technology into pre-service teacher education programs are discussed.