• 한국수학교육학회지시리즈A:수학교육
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• 제51권1호
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• pp.21-45
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• 2012

The purpose of the study was to investigate how students understand each operations with whole numbers and fractions, and the relationship between their knowledge of operations with whole numbers and conceptual understanding of operations on fractions. Researchers categorized students' understanding of operations with whole numbers and fractions based on their semantic structure of these operations, and analyzed the relationship between students' understanding of operations with whole numbers and fractions. As the results, some students who understood multiplications with whole numbers as only situations of "equal groups" did not properly conceptualize multiplications of fractions as they interpreted wrongly multiplying two fractions as adding two fractions. On the other hand, some students who understood multiplications with whole numbers as situations of "multiplicative comparison" appropriately conceptualize multiplications of fractions. They naturally constructed knowledge of fractions as they build on their prior knowledge of whole numbers compared to other students. In the case of division, we found that some students who understood divisions with whole numbers as only situations of "sharing" had difficulty in constructing division knowledge of fractions from previous division knowledge of whole numbers.

• 한국수학교육학회지시리즈A:수학교육
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• 제45권1호
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• pp.61-74
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• 2006

The ability of understanding the number and number systems, grasping the properties of number systems, and manipulating number systems is the foundation to understand algebra. It is useful to deepen students' mathematical understanding of number systems and operations. The authentic understanding of numbers and operations can make it possible for the students to manipulate algebraic symbols, to represent relationship among sets of numbers, and to use variables to investigate the properties of sets of numbers. The high school students need to understand the number systems from more abstract perspective. The purpose of this study is to study on understanding and application ability of eleventh graders of basic properties of operations with real numbers.

• 대한수학교육학회지:학교수학
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• 제4권2호
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• pp.205-222
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• 2002

In school mathematics, the negative numbers have been instructed using the intuitive models such as the number line model, the counting model, and inductive-extrapolation on the additionand multiplication and using inverse operation on the subtraction and division. Theseinstructions on the negative numbers did not present their formal nature and caused the difficulty for students to understand their operations because of the incomplete function of the intuitive models. In this study, we tried to improve such problems of the instructions of the negative numbers on the basis of the didactical phenomenological analysis. First of all, we analysed the nature of the negative numbers and the cognitive obstructions through the examination about the historic process of them. Second, we examined hew the nature of the negative numbers were analysed and described in mathematics. Third, we explored the improving directions for them on the ground of the didactical phenomenological analysis. In school mathematics, the rules of operations using the intuitive models of the negative numbers have been Instructed rather than approaching toward the nature of them. The negative numbers have been developed from the necessity to find the general solution of equations. The study tries to approach the operations instructions of the negative numbers formative]y to overcome the problems of those that are using the intuitive models and to reflect the formative Furthermore of the negative numbers. Furthermore, we examine the way of the instruction of the negative numbers in real context so that the algebraic feature and the real context should be Interactive.

• 대한수학교육학회지:학교수학
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• 제8권1호
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• pp.1-25
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• 2006

양수와 음수의 취급이 처음으로 시작되는 제 7-단계 교과서들을 살펴본 결과 양의 부호, 음의 부호가 붙은 수를 읽는 방법과 유리수에 대한 일부 교과서의 정의는 재고해야 하며, 반대의 수 곧 반수를 정의하여 활용하는 것과 양수와 음수의 도입은 대소 관계를 다루는 곳에서 그 정의를 하는 것이 바람직함을 알 수 있었다. 연산에서는 양의 부호와 음의 부호가 붙은 수에 대한 가시적인 표현을 충분히 익히게 하여 초등학교에서의 연산 도입을 구체적이고 가시적으로 처리한 것과 같이 양수, 음수의 연산에도 그 방법을 연계하여 활용 할 수 있도록 하는 것이 바람직하다고 생각되어 화살표(유향선분)를 사용하여 양수 음수를 가시적으로 도입한 후에 이들을 사용하여 초등학교에서의 계산 방법을 양수 음수까지 확장된 수에까지 그대로 적용한 학습 내용을 제시하였다. 그리고 제시한 학습 내용으로 지도를 하여본 결과 이와 같이 연계된 학습내용이 보다 바람직한 것임을 알 수 있었다.

• Journal of the Korean Data and Information Science Society
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• 제14권1호
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• pp.93-101
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• 2003

Computation with fuzzy numbers is a prospective branch of a fuzzy set theory regarding the data processing applications. In this paper we consider an open problem about distributivity of fuzzy quantities based on the extension principle suggested by Mare (1997). Indeed, we show that the distributivity on the class of fuzzy numbers holds and min-norm is the only continuous t-norm which holds the distributivity under t-norm based fuzzy arithmetic operations.

• 한국지능시스템학회:학술대회논문집
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• 한국퍼지및지능시스템학회 2002년도 추계학술대회 및 정기총회
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• pp.1-4
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• 2002

There have been several tipical methods being used to measure the fuzziness (entropy) of fuzzy sets. Pedrycz is the original motivation of this paper. This paper studies the entropy variation on the fuzzy numbers with arithmetic operations(addition, subtraction, multiplication) and the relationship between entropy and information energy. It is shown that through the arithmetic operations, the entropy of the resultant fuzzy number has the arithmetic relation with the entropy of each original fuzzy number. Moreover, the information energy variation on the fuzzy numbers is also discussed. The results generalize earlier results of Pedrycz [FSS 64(1994) 21-30] and Wang and Chiu [FSS 103(1999) 443-455].

• 대한수학회논문집
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• 제28권3호
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• pp.635-642
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• 2013

There are many results on the extended operations of two fuzzy numbers based on the Zadeh's extension principle. For the calculation, we have to use existing operations between two ${\alpha}$-cuts. In this paper, we define parametric operations between two ${\alpha}$-cuts which are different from the existing operations. But we have the same results as the extended operations of Zadeh's principle.

• 한국지능시스템학회논문지
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• 제15권6호
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• pp.754-758
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• 2005

There have been several tipical methods being used tomeasure the fuzziness (entropy) of fuzzy sets. Pedrycz is the original motivation of this paper. Recently, Wang and Chiu [FSS103(1999) 443-455] and Pedrycz [FSS 64(1994) 21-30] showed the relationship(addition, subtraction, multiplication) between the entropies of the resultant fuzzy number and the original fuzzy numbers of same type. In this paper, using Lebesgue-Stieltjes integral, we generalize results of Wang and Chiu [FSS 103(1999) 443-455] concerning entropy arithmetic operations without the condition of same types of fuzzy numbers. And using this results and trade-off relationship between information energy and entropy, we study more properties of information energy of fuzzy numbers.

• 특수아동교육연구
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• 제20권3호
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• pp.23-44
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• 2018

연구목적: 본 연구는 초등학교 2학년 느린 학습자가 제 학년에서 요구하는 수와 연산 수준을 도달하기 위하여 수와 연산 영역을 소집단 직접교수 프로그램으로 구성하여 중재를 실시하고 그 효과를 알아보고자 하였다. 연구방법: 이를 위하여 J시에 소재하고 있는 J와 C초등학교 2학년을 212명을 대상으로 16명의 느린 학습자를 선별하였다. 이들에게 47회기의 소집단 직접교수 프로그램을 적용하였다. 결과 처리는 효과치 검증을 통해 분석하였고 그래프로 시각화하여 그 변화추이를 살펴보았다. 연구결과: 연구 결과는 다음과 같다. 첫째, 소집단 직접교수가 느린 학습자의 수와 연산 능력 향상에 효과적이었다. 둘째, 중재 후 수와 연산 능력의 향상 정도에 따라 느린 학습자를 3유형으로 분류할 수 있었다. 결론: 따라서 소집단 직접교수가 느린 학습자가 제 학년에서 요구하는 수와 연산 수준을 도달하도록 하는데 효과가 있음을 알 수 있었다. 한편 제 학년 수준에 도달하지 못한 유형의 느린 학습자에게는 더욱 집중적이고 장기간의 2단계 중재가 필요함을 확인할 수 있었다.

• 충청수학회지
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• 제31권4호
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• pp.379-386
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• 2018

We generalized the quadratic fuzzy numbers on ${\mathbb{R}}$ to ${\mathbb{R}}_2$. By defining parametric operations between two regions valued ${\alpha}-cuts$, we got the parametric operations for two triangular fuzzy numbers defined on ${\mathbb{R}}_2$. The results for the parametric operations are the generalization of Zadeh's extended algebraic operations. We generalize the 2-dimensional quadratic fuzzy numbers on ${\mathbb{R}}_2$ that may have maximum value h < 1. We calculate the algebraic operations for two generalized 2-dimensional quadratic fuzzy sets.