• 제목/요약/키워드: formal proofs

검색결과 25건 처리시간 0.021초

시각화를 이용한 증명교육

  • 강미광;김명지
    • East Asian mathematical journal
    • /
    • 제24권5호
    • /
    • pp.527-545
    • /
    • 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.

  • PDF

Euclid 원론과 Clairaut 원론의 비교를 통한 기하 교육에서 논리와 직관의 고찰 (Revisiting Logic and Intuition in Teaching Geometry: Comparing Euclid's Elements and Clairaut's Elements)

  • 장혜원
    • 한국수학사학회지
    • /
    • 제34권1호
    • /
    • pp.1-20
    • /
    • 2021
  • Logic and intuition are considered as the opposite extremes of teaching geometry, and any teaching method of geometry is to be placed between these extremes. The purpose of this study is to identify the characteristics of logical and intuitive approaches for teaching geometry and to derive didactical implications by taking Euclid's Elements and Clairaut's Elements respectively representing the extremes. To this end, comparing the composition and contents of each book, we analyze which propositions Clairaut chose from Euclid's Elements, how their approaches differ in definitions, proofs, and geometrical constructions, and what unique approaches Clairaut took. The results reveal that Clairaut mainly chose propositions from Euclid's books 1, 3, 6, 11, and 12 to provide the contexts that show why such ideas were needed, rather than the sudden appearance of abstract and formal propositions, and omitted or modified the process of justification according to learners' levels. These propose a variety of intuitive strategies in line with trends of teaching geometry towards emphasis on conceptual understanding and different levels of justification. Specifically, such as the general principle of similarity and the infinite geometric approach shown in Clairaut's Elements, we could confirm that intuition-based geometry does not necessarily aim for tasks with low cognitive demand, but must be taught in a way that learners can understand.

'기하와 벡터' 교육과정의 벡터와 내적 개념 분석 (An Analysis of the Vector and Inner Product Concepts in Geometry and Vector Curriculum)

  • 신보미
    • 한국학교수학회논문집
    • /
    • 제16권4호
    • /
    • pp.841-862
    • /
    • 2013
  • 이 연구는 2007 개정 교육과정의 '기하와 벡터' 교과에서 다루어지는 벡터와 내적 개념을 분석하여 그 특징을 기술함으로써 벡터와 내적 개념 지도의 교수학적 시사점을 얻는데 목적을 두었다. 이를 위해 '기하와 벡터' 교육과정에서 다루어지는 벡터와 내적 개념 분석을 위한 세부 관점을 Tall(2002a; Tall, 2004b)과 Watson et al.(2003; Watson, 2002)에 기초하여 5가지로 추출하고, 이렇게 추출된 세부 관점을 토대로 '기하와 벡터' 교육과정 및 교육과정해설서, '기하와 벡터' 교과서 10종 모두에서 다루어지는 벡터와 내적 개념의 특징을 분석하였다. 이로부터 벡터와 내적 개념 형성과 관련된 교육과정상의 이슈를 구체화하였으며 이에 비추어 '기하와 벡터' 교과서에서 벡터 단원의 내용을 전개하는 방식과 관련된 시사점을 논의하였다.

  • PDF

유니버설 일방향 해쉬 함수에 대한 블록 암호 기반 구성 방법 (Construction of UOWHF based on Block Cipher)

  • 이원일
    • 정보보호학회논문지
    • /
    • 제14권1호
    • /
    • pp.101-111
    • /
    • 2004
  • Preneel, Govaerts, Vandewalle은 1994년 [13]에서 블록 암호를 이용한 충돌 저항 해쉬 함수의 구성에 관한 64 가지 방법을 고려하였다. 그들은 64 가지 방법 중에서 12 가지가 안전하다고 주장하였으나 이에 대한 엄밀한 증명은 제시하지 않았다. Black, Rogaway, Shrimpton은 2002년 [2]에서 위의 64 가지 방법에 대하여 수학적 정의를 토대로 엄밀하게 분석한 결과를 제시하였다 그들은 블랙 박스 안전성 모델 하에서 타 가지의 압축 함수들 중 12 가지가 충돌 저항 해쉬 함수가 되며 이에 대응하는 64 가지의 화장된 왜쉬 함수들 중에서 20 가지가 충돌 저항 해쉬 함수가 됨을 증명하였다. 본 논문에서는 Preneel, Govaerts, Vandewalle이 제시한 64가지 방법을 기초로 하여 블록 암호를 이용한 UOWHF의 구성 방법에 대한 최초의 결과를 제시한다. 본 논문에서는 Black, Rogaway, Shrimpton이 사용한 블랙 박스 안전성 모델을 통하여 64 가지의 압축 함수 집합들 중에서 30 가지의 압축 함수 집합들이 UOWHF가 되며 이에 대응하는 64가지의 확장된 해쉬 함수 집합들 중에서 42 가지가 UOWHF가 됨을 보일 것이다. 이러한 결과는 또한 CRHF를 구성하는 것보다 UOWHF를 구성하는 것이 용이하다는 것을 간접적으로 시사한다. 또한 본 논문의 중요 결과들 중 하나는 블랙 박스 모델 하에서는 UOWHF의 안전성을 위하여 일반적으로 널리 알려진 안전성 모델 하에서는 필수적인 마스크 키들이 필요하지 않다는 것이다. 이는 UOWHF의 효율성을 매우 향상시킴을 의미한다.

환경관리주의 환경교육에 대한 비판적 고찰 - 원인규명에서 해결기제로의 전환을 위하여 - (A Critical Approach on Environmental Education Biased to Environmental Possibilism - From Clearing up the Cause to Problem-Solving Mechanism -)

  • 김태경
    • 한국환경교육학회지:환경교육
    • /
    • 제18권3호
    • /
    • pp.59-74
    • /
    • 2005
  • We can't deny Korean EE has basically developed on the basis of Environmental Possibilism (Environmental management or Reformism) in lots of aspects. I would show three representative proofs here, the first, the philosophy of Korean EE has been mainly focused on dichotomy of human-techno centrism and eco-centrism with no considering other alternative environmentalism since 4th Formal Curriculum, 1981. The second, simultaneously, the concept of EE has not distinguished from it of Science education. (Furthermore, unfortunately some says EE has been a part of Science education, although there should be many differences on its contextual aspect.) And the third one is that the limit of possibilism which market economists have worried, has scarcely mentioned in various kinds of EE-related teaching materials. Possibilism is basically likely to be accompanied by science and economics-oriented approach, and in this aspect this dichotomy, human-techno centrism and eco-centrism, has come from perspectives of Economical development process and over-addicted belief to Science. So it is enough to say that Korean EE has basically developed with biased to Environmental possibilism, in other words, biased to preference to it. And I'll critically focus on these two axes of possibilism, Science and Economics and its dichotomy. Of course, we should accept there are so many same parts in its contents between EE and Science, but we should know its contextual differences for triangular position of environmentalism suitable to EE and also overcome science-dependant approach to EE. Although science-dependant approach to EE and dichotomy could provide some tools for clearing up the causes of environmental problem, especially always it has insisted fundamental causes of environmental problem originated in human faults and over-use of eco-source or over-economic development, but now it is old-fashioned discourse, furthermore it come to have unavoidable limits in the debates of problem-solving mechanism to environmental problems. The paramount important thing is to supply the ways or thoughtful mechanism for solving or coordinating the Environmental problems, not just searching for cause of it. But scientific approach and its dichotomy based on possibilism have continuously born cause & effect in EE-related discourse. So there are so much needs to transfer from continuous bearing of cause & effect to constructive alternatives at least in environmentalism of EE. Traditionally, dichotomical division in EE Environmentalism, human-techno centrism and eco-centrism, couldn't have Provided any answers to our real society, it just gives us only cause & effects of Environmental problems. And also we can't find the description on the limits of capitalism market approach to Environmental problems especially in Korean EE text books, other teaching materials and its teaching-learning process, although market approach economist has been proved its fault beyond its functional merits as Environmental management tools. So we should introduce other alternative Environmental philosophy instead of Possibilism such as eco-socialism insisted by Schmacher M. and Boochin etc, or marxist-environmentalism for relative and comparative views to market-thought such as commodification. In this aspect we need to accept Oriental philosophy based on moderation(中庸) as new another alternatives with the reflection that we have recognized monism as representative Oriental philosophical environmentalism. Fundamentally monism has done its role with providing relative concepts to Dichotomy Enlightenment, but we can't say it has been core concept for understanding of oriental environmentalism, and we can't distinguish monism from oriental philosophy itself, just because oriental thought itself was basically monism. So conceptual difference should be recognized between EE and Science education in teaching-learning process on the basis of life-philosophy(Philosophie des Lebens) from epistemology. For this transformation, we should introduce existentialism in Science education, in other words, only existential Science education based on phenomenology or interpretivism can be EE. And simultaneously we need some ways for overcoming of scientific foundationalism which has been tradition making science not stand on existentialism, formulating and featuring of almost all of natural things and its phenomenon from after enlightenment in western world, but it has malfunctioned in fixing conception of science just into essentialism itself. And we also introduce integrated approach to science and society for EE like STS. Those are ways for overcoming of Environmental possibilism in EE.

  • PDF