• Title/Summary/Keyword: formal proofs

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시각화를 이용한 증명교육

  • Kang, Mee-Kwang;Kim, Myung-Jee
    • East Asian mathematical journal
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    • v.24 no.5
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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.

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

  • Chang, Hyewon
    • Journal for History of Mathematics
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    • v.34 no.1
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    • pp.1-20
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    • 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.

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

  • Shin, BoMi
    • Journal of the Korean School Mathematics Society
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    • v.16 no.4
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    • pp.841-862
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    • 2013
  • This study analyzed issues in the mathematics curriculum concerning the cognitive development of the vector and inner product concepts in the light of Tall's and Watson's research(Tall, 2004a; Tall, 2004b; Watson et al., 2003; Watson, 2002). Some suggestions in teaching the vector and inner product concepts were elaborated in the terms of these analyses. First, the position vector needs to be represented by an arrow on the coordinate system in order to introduce the component form of a vector represented by a directed line segment. Second, proofs of the vector operation law should be carried out by symbolic manipulations based on the algebraic concept of a vector in the symbolic world. Third, it is appropriate that the inner product is defined as $\vec{a}{\cdot}\vec{b}=a_1b_1+a_2b_2$ (when, $\vec{a}=(a_1,a_2)$, $\vec{b}=(b_1,b_2)$) when it comes to considering the meaning of the inner product relevant to vector space in the formal world. Cognitive growth of concepts of the vector and inner product can be properly induced through revising explanation methods about the concepts in the curriculum in the basis of the above suggestions.

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

  • 이원일
    • Journal of the Korea Institute of Information Security & Cryptology
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    • v.14 no.1
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    • pp.101-111
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    • 2004
  • Preneel, Govaerts, and Vandewalle considered the 64 basic ways to construct a collision resistant hash function from a block cipher. They regarded 12 of these 64 schemes as secure, though no proofs or formal claims were given. Black, Rogaway, and Shrimpton presented a more proof-centric look at the schemes from PGV. They proved that, in the black box model of block cipher, 12 of 64 compression functions are CRHFs and 20 of 64 extended hash functions are CRHFs. In this paper, we present 64 schemes of block-cipher-based universal one way hash functions using the main idea of PGV and analyze these schemes in the black box model. We will show that 30 of 64 compression function families UOWHF and 42 of 64 extended hash function families are UOWHF. One of the important results is that, in this black box model, we don't need the mask keys for the security of UOWHF in contrast with the results in general security model of UOWHF. Our results also support the assertion that building an efficient and secure UOWHF is easier than building an efficient and secure CRHF.

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

  • Kim, Tae-Kyung
    • Hwankyungkyoyuk
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    • v.18 no.3 s.28
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    • pp.59-74
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    • 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.

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