• Title/Summary/Keyword: 대수적 개념

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Two original concepts in linear algebra (선형대수학의 두 가지 기원적 개념)

  • Pak, Hong-Kyung
    • Journal for History of Mathematics
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    • v.21 no.1
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    • pp.109-120
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    • 2008
  • Today linear algebra is one of compulsory courses for university mathematics by virtue of its theoretical fundamentals and fruitful applications. However, a mechanical computation-oriented instruction or a formal concept-oriented instruction is difficult and dull for most students. In this context, how to teach mathematical concepts successfully is a very serious problem. As a solution for this problem, we suggest establishing original concepts in linear algebra from the students' point of view. Any original concept means not only a practical beginning for the historical order and theoretical system but also plays a role of seed which can build most of all the important concepts. Indeed, linear algebra has exactly two original concepts : geometry of planes, spaces and linear equations. The former was investigated in [2], the latter in the present paper.

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On the Algebraic Concepts in Euclid's Elements (유클리드의 원론에 나타난 대수적 개념에 대하여)

  • 홍진곤;권석일
    • Journal for History of Mathematics
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    • v.17 no.3
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    • pp.23-32
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    • 2004
  • In this paper, Ive investigated algebraic concepts which are contained in Euclid's Elements. In the Books II, V, and VII∼X of Elements, there are concepts of quadratic equation, ratio, irrational numbers, and so on. We also analyzed them for mathematical meaning with modem symbols and terms. From this, we can find the essence of the genesis of algebra, and the implications for students' mathematization through the experience of the situation where mathematics was made at first.

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다원환의 보편적 미분가군

  • Han, Jae-Yeong;Yeon, Yong-Ho
    • Communications of Mathematical Education
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    • v.6
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    • pp.383-407
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    • 1997
  • 가환다원환의 대수적 미분에 관한 성질들은 많은 연구의 대상이 되어 왔다. 본 논문은 가환 다원환에서 정의된 대수적 미분의 일반화로써 가환일 필요가 없는 일반다원환의 대수적 미분의 성질을 연구한 것이다. 비가환다원환의 미분정의를 바탕으로 하여 가환다원환에서 연구되어 온 보편적 미분가군의 성질을 일반다원환 의 미분가군에 적용하려고 노력하였다. 이 논문에서 사용한 정리의 증명과정이나 기본개념은 가환다원환의 미분개념에서 나타난 성질들을 바탕으로 하였다.

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A study on the teaching of algebraic structures in school algebra (학교수학에서의 대수적 구조 지도에 대한 소고)

  • Kim, Sung-Joon
    • Journal of the Korean School Mathematics Society
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    • v.8 no.3
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    • pp.367-382
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    • 2005
  • In this paper, we deal with various contents relating to the group concept in school mathematics and teaching of algebraic structures indirectly by combining these contents. First, we consider structure of knowledge based on Bruner, and apply these discussions to the teaching of algebraic structure in school algebra. As a result of these analysis, we can verify that the essence of algebraic structure is group concept. So we investigate the previous researches about group concept: Piaget, Freudenthal, Dubinsky. In our school, the contents relating to the group concept have been taught from elementary level indirectly. Tn elementary school, the commutative law and associative law is implicitly taught in the number contexts. And in middle school, various linear equations are taught by the properties of equality which include group concept. But these algebraic contents is not related to the high school. Though we deal with identity and inverse in the binary operations in high school mathematics, we don't relate this algebraic topics with the previous learned contents. In this paper, we discussed algebraic structure focusing to the group concept to obtain a connectivity among school algebra. In conclusion, the group concept can take role in relating these algebraic contents and teaching the algebraic structures in school algebra.

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Revisiting Linear Equation and Slope in School Mathematics : an Algebraic Representation and an Invariant of Straight Line (직선의 대수적 표현과 직선성(直線性)으로서의 기울기)

  • Do, Jong-Hoon
    • Communications of Mathematical Education
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    • v.22 no.3
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    • pp.337-347
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    • 2008
  • 'Slope' is an invariant of a straight line and 'Linear Equation' is an algebraic representation of a straight line in the cartesian plane. The concept 'slope' is necessary for algebraically representing a geometrical figure, line. In this article, we investigate how those concepts are dealt with in school mathematics and suggest some improvement methods.

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선형 대수의 가르침에 고려하여야 할 사항에 관한 연구

  • Choe, Yeong-Han
    • Communications of Mathematical Education
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    • v.18 no.2 s.19
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    • pp.93-108
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    • 2004
  • Wassily Leontief가 미국 경제의 모델에 선형 대수를 적용한 이론으로 1973년에 노벨 경제학상을 받은 후로는 인문${\cdot}$사회 과학(특히 상경(商經) 분야)을 전공하는 사람에게도 선형 대수는 큰 관심 분야가 되었다. 그래서 1980년대 부터는 대학의 기초 과목으로써 선형 대수를 가르치는 것은 유행처럼 퍼졌고 또 가르침에 관한 연구도 활발하여졌다. 현행 우리나라의 초${\cdot}$${\cdot}$고등 학교의 수학과 교육과정(이른바 “제 7차 개정”) 속에는 선형대수의 내용이 어느 정도 있으나 학생들에게 확실한 개념을 갖도록 가르치고 있지 않다. 수직선, 순서 쌍, n-겹수, 직교 좌표, 벡터 등 해석기하적인 내용과 선형 방정식계의 풀이법(가우스${\cdot}$조르단 소거법을 쓰지 않는 풀이법) 등 일반 대수적인 내용은 다루지만 선형 변환, 벡터 공간의 구조 등은 다루지 않는다. m${\sim}$n 행렬은 수학II에 나와 있긴 하나 소개하는 정도에 그친다. 한편 과학 계열 고등학교 학생을 위한 "고급 수학"에는 비교적 많은 양의 선형 대수의 내용이 있다. 일반 계열 고등학교의 수학에서도 선형 대수의 내용을 확장하고 학생들에게 확실한 개념을 갖도록 가르쳐서 이들이 대학에 진학하여 전공 분야에서 아무 어려움이 없도록 하는 것이 바람직하다.

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선형대수에서의 학생들의 오개념 - 일차변환을 중심으로 -

  • Sin, Gyeong-Hui
    • Communications of Mathematical Education
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    • v.19 no.2 s.22
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    • pp.379-388
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    • 2005
  • 일차변환은 선형대수에서 가장 중요한 개념 중 하나이다. 그럼에도 많은 학생들에게서 나타나는 이 개념에 대한 오류는 무엇이며 또 어디에 근거하는가? 이 논문은 효과적인 선형대수 교수학습 연구의 일부로, 주어진 여러 함수 중에서 일차변환인 것을 찾는 과정 중에 나타난 학생들의 오류와 그 근거를 알아보았다. 본 연구 결과는 선형대수 학습에 어려움을 겪는 학생들에게 보다 효율적인 교수디자인 설계를 위한 기초 자료의 의미를 갖는다.

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The historical developments process of the representations and meanings for ratio and proportion (비와 비례 개념의 의미와 표현에 대한 역사적 발달 과정)

  • Park, Jung-Sook
    • Journal for History of Mathematics
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    • v.21 no.3
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    • pp.53-66
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    • 2008
  • The concepts of ratio and proportion are familiar with students but have difficulties in use. The purpose of this paper is to identify the meanings of the concepts of ratio and proportion through investigating the historical development process of the meanings and representations of them. The early meanings of ratio and proportion were arithmetical meanings, however, geometrical meanings had taken the place of them because of the discovery of incommensurability. After the development of algebraic representation, the meanings of ratio and proportion have been growing into algebraic meanings including arithmetical and geometrical meanings. Through the historical development process of ratio and proportion, it is observable that the meanings of mathematical concepts affect development of symbols, and the development of symbols also affect the meanings of mathematical concepts.

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Promising Advantages and Potential Pitfalls of Reliance on Technology in Learning Algebra (대수학습에서 테크놀로지 사용의 긍적적인 요소와 잠정적인 문제점)

  • Kim, Dong-Joong
    • Journal of the Korean School Mathematics Society
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    • v.13 no.1
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    • pp.89-104
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    • 2010
  • In a rapidly changing and increasingly technological society. the use of technology should not be disregarded in issues of learning algebra. The use of technology in learning algebra raises many learning and pedagogical issues. In this article, previous research on the use of technology in learning algebra is synthesized on the basis of the four issues: conceptual understanding, skills, instrumental genesis, and transparency. Finally, suggestions for future research into technological pedagogical content knowledge (TPCK) are made.

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Characteristics of Algebraic Thinking and its Errors by Mathematically Gifted Students (수학영재의 대수적 사고의 특징과 오류 유형)

  • Kim, Kyung Eun;Seo, Hae Ae;Kim, Dong Hwa
    • Journal of Gifted/Talented Education
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    • v.26 no.1
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    • pp.211-230
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    • 2016
  • The study aimed to investigate the characteristics of algebraic thinking of the mathematically gifted students and search for how to teach algebraic thinking. Research subjects in this study included 93 students who applied for a science gifted education center affiliated with a university in 2015 and previously experienced gifted education. Students' responses on an algebraic item of a creative thinking test in mathematics, which was given as screening process for admission were collected as data. A framework of algebraic thinking factors were extracted from literature review and utilized for data analysis. It was found that students showed difficulty in quantitative reasoning between two quantities and tendency to find solutions regarding equations as problem solving tools. In this process, students tended to concentrate variables on unknown place holders and to had difficulty understanding various meanings of variables. Some of students generated errors about algebraic concepts. In conclusions, it is recommended that functional thinking including such as generalizing and reasoning the relation among changing quantities is extended, procedural as well as structural aspects of algebraic expressions are emphasized, various situations to learn variables are given, and activities constructing variables on their own are strengthened for improving gifted students' learning and teaching algebra.