• Title, Summary, Keyword: 통약불가능

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Inducing Irrational Numbers in Junior High School (중학교에서의 무리수 지도에 관하여)

  • Kim, Boo-Yoon;Chung, Young-Woo
    • Journal for History of Mathematics
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    • v.21 no.1
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    • pp.139-156
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    • 2008
  • We investigate the inducing method of irrational numbers in junior high school, under algebraic as well as geometric point of view. Also we study the treatment of irrational numbers in the 7th national curriculum. In fact, we discover that i) incommensurability as essential factor of concept of irrational numbers is not treated, and ii) the concept of irrational numbers is not smoothly interconnected to that of rational numbers. In order to understand relationally the incommensurability, we suggest the method for inducing irrational numbers using construction in junior high school.

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A study on the relation between the real number system of Dedekind and the Eudoxus theory of proportion (에우독소스의 비례론과 데데킨트의 실수계에 관한 고찰)

  • Kang, Dae-Won;Kim, Kwon-Wook
    • Journal for History of Mathematics
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    • v.22 no.3
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    • pp.131-152
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    • 2009
  • The Eudoxean theory of Proportion is correlated with 'Dedekind cut' with which Dedekind defined the real number system in modern usage. Dedekind established a firm foundation for the real number system by retracing some of Eudoxus' steps of over two thousand years earlier. Thus it should be quite worthy that we separate Greek inheritance from the definition of Dedekind, However, there is a fundamental difference between Eudoxean theory of proportion and Dedekind cut. Basically, it seems impossible for Greeks to distinguish between the distinction between number and magnitude. In this paper, we will consider how the Eudoxean theory of proportion was related to Dedekind cut introduced to prove the Dedekind's real number completion and how it influenced Dedekind cut by looking at the relation between Eudoxos's explication of the notion of ratio and Dedekind's well-known construction of the real numbers.

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The Meaning of the Definition of the Real Number by the Decimal Fractions (소수에 의한 실수 정의의 의미)

  • Byun Hee-Hyun
    • Journal for History of Mathematics
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    • v.18 no.3
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    • pp.55-66
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    • 2005
  • In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.

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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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A Construction of 'Decimal Fraction' Unit of Elementary Mathematics Textbook and Analysis of Students' State of Understanding Based on Measurement Activity (초등수학에서 측정활동에 기반한 소수의 학습.지도 방안 및 학생의 이해 실태 분석)

  • Kim, Eun Jung;Kang, Heung Kyu
    • Journal of Elementary Mathematics Education in Korea
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    • v.18 no.1
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    • pp.37-62
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    • 2014
  • In this thesis, we inquire into teaching method of decimal fraction concept in elementary mathematics education based on measurement activity. For this purpose, our research tasks are as follows: First, we design a experimental learning-teaching plan of 'decimal fraction' unit in 4th grade textbook and verify its effect. Second, after teaching experiment using experimental learning-teaching plan, we analyze the student's status of understanding about decimal fraction concept. As stated above, we have performed teaching experiment which is ruled by new lesson design and analysed the effects of teaching experiment. Through this study, we obtained the following results. First, introduction of decimal fraction based on measurement activity promotes student's understanding of measuring number and decimal notation. Second, operator concept of decimal fraction is widely used in daily life. Its usage is suitable for elementary mathematics education within the decimal notation system. Third, a teaching method of times concepts using place value expansion of decimal fraction is more meaningful and has less possibility of misunderstanding than mechanical shift of decimal point. Fourth, teaching decimal fraction through the decimal expansion helps students with understanding of digit 0 contained in decimal fraction, comparing of size and place value. Fifth, students have difficulties in understanding conversion process of decimal fraction into decimal notation system using measurement activity. It can be done easily when fraction is used. However, that is breach to curriculum. It is necessary to succeed research for this.

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Teacher Knowledge Necessary to Analyze Student's Errors and Difficulties about the Concept of Irrational Numbers (무리수 개념에 관한 학생의 오류와 어려움 해석에 필요한 교사지식)

  • Kang, Hyangim;Choi, Eunah
    • School Mathematics
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    • v.19 no.2
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    • pp.319-343
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    • 2017
  • In this study, we hope to reveal specialized content knowledge(SCK) and its features necessary to analyze student's errors and difficulties about the concept of irrational numbers. The instruments and interview were administered to 3 in-service mathematics teachers with various education background and teaching experiments. The results of this study are as follows. First, specialized content knowledge(SCK) were characterized by the fixation to symbolic representation like roots when they analyzed the concentration and overlooking of the representations of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations as the standard of judgment about irrational numbers. Thirdly, In-service teachers were influenced by content of students' error when they analyzed the error and difficulties of students. Lately, we confirmed that the content knowledge about the viewpoint of procept and actual infinity of irrational numbers are most important during the analyzing process.