• 대한수학교육학회지:학교수학
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• 제18권3호
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• pp.527-539
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• 2016

"평행사변형은 사다리꼴이다."에서 '이다'는 애매하고 그 의미가 매우 풍부한 기호이다. 이 연구는 일상적 언어 '이다'가 문맥과 상황에 따라 다양하게 해석되는 의미원소임을 밝히고 수학에서 사용되는 '이다'의 의미를 구분하여 논의한다. 그리고 '동일성'의 관념에 주목하여, 수학적으로 '같음'을 나타내기 위해 사용되기도 하는 '이다'를 동치관계의 개념과 Van Hieles의 기하 사고 수준 이론으로 재해석하여 살펴본다. 수학적 기호로서 '이다'에 대한 분석 결과 '이다'는 수학적 아이디어를 의미 있게 생성하는 데 중요한 의의가 있다고 판단된다.

• 한국수학교육학회지시리즈C:초등수학교육
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• 제14권1호
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• pp.13-26
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• 2011

본 연구의 목적은 초등수학 영재 교육 대상 학생들의 기하 인지 수준과 그들이 증명을 전개하는 과정에서 논리적인 정당화의 특성을 분석하고 이를 기반으로 수학 영재 교육을 위한 시사점을 제시하는 것이다. 이를 위하여 서울특별시 A영재교육원에 재학 중인 5, 6학년 학생 18명을 대상으로 그들의 기하 수준을 확인하고 그들이 기하문제를 증명을 하고 설명하는 과정에서 어떤 논리적인 정당화를 해 가는지 분석하였다. 연구 결과 이들은 van Hieles의 기하 사고의 0수준부터 4수준 중에서 대부분 2∼3수준에 있었다. 그리고 증명의 정당화 과정에서 이 영재 교육 대상 학생들은 잘라 붙이기와 수치적 접근을 사용하려는 시도와 이미 선행으로 학습한 내용의 기억을 되살려 사용하는 예가 많았고, 독창적이고 일반적인 증명으로 이끌어가는 데는 어려움을 가지고 있었다. 따라서 초등수학 영재 교육 대상자들을 위한 교육은 이들의 수준에 맞는 보다 정교화된 과제로 이들이 자신들의 증명의 정당화 과정을 인지하면서 보다 창의적이고 연역적 사고의 수준으로 이끌어 줄 필요가 있다.

• 대한수학교육학회지:학교수학
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• 제4권1호
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• pp.15-27
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• 2002

The purpose of this thesis is, through analysing the characteristics of the definitions in Korean school mathematics textbooks, to explore the levels of them. Definitions use din academic mathematics are rigorous. But they should be transformed into various types, which are presented in school mathematics textbooks, with didactical purposes. In this thesis we investigated such types of transformation. With the result of this investigation we tried to identify the levels of the definitions in school mathematics textbooks. We tried to construct, with consideration about methods of definition, frame for analysing the types of the definitions in school mathematics. Methods of definition are classified as connotative method, denotative method, and synonymous method. Especially we identified that connotative method contains logical definition, genetic definition, relational definition, operational definition, and axiomatic definition. With these analyses we made a frame for investigating the characteristics of the definitions in school mathematics textbooks. With this frame we identified concrete types of transformations of methods of definition. We tried to analyse this result with van Hieles' theory about let·els of geometry learning and the mathematical language levels described by Freudenthal, and identify the levels of definitions in school mathematics. We showed the levels of definitions in the geometry area of the Korean school mathematics.

• 대한수학교육학회지:수학교육학연구
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• 제11권2호
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• pp.363-384
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• 2001

The purpose of this thesis is, through analysing the characteristics of the definitions in Korean school mathematics textbooks, to explore the levels of them and to make suggestions for definition - teaching as a mathematising activity, Definitions used in academic mathematics are rigorous. But they should be transformed into various types, which are presented in school mathematics textbooks, with didactical purposes. In this thesis we investigated such types of transformation. With the result of this investigation we tried to identify the levels of the definitions in school mathematics textbooks. And in school mathematics textbooks there are definitions which carry out special functions in mathematical contexts or situations. We can say that we understand those definitions, only if we also understand the functions of definitions in those contexts or situations. In this thesis we investigated the cases in school mathematics textbooks, when such functions of definition are accompanied. With the result of this investigation we tried to make suggestions for definition-teaching as an intellectual activity. To begin with we considered definition from two aspects, methods of definition and functions of definition. We tried to construct, with consideration about methods of definition, frame for analysing the types of the definitions in school mathematics and search for a method for definition-teaching through mathematization. Methods of definition are classified as connotative method, denotative method, and synonymous method. Especially we identified that connotative method contains logical definition, genetic definition, relational definition, operational definition, and axiomatic definition. Functions of definition are classified as, description-function, stipulation-function, discrimination-function, analysis-function, demonstration-function, improvement-function. With these analyses we made a frame for investigating the characteristics of the definitions in school mathematics textbooks. With this frame we identified concrete types of transformations of methods of definition. We tried to analyse this result with van Hieles' theory about levels of geometry learning and the mathematical language levels described by Freudenthal, and identify the levels of definitions in school mathematics. We showed the levels of definitions in the geometry area of the Korean school mathematics. And as a result of analysing functions of definition we found that functions of definition appear more often in geometry than in algebra or analysis and that improvement-function, demonstration-function appear regularly after demonstrative geometry while other functions appear before demonstrative geometry. Also, we found that generally speaking, the functions of definition are not explained adequately in school mathematics textbooks. So it is required that the textbook authors should be careful not to miss an opportunity for the functional understanding. And the mathematics teachers should be aware of the functions of definitions. As mentioned above, in this thesis we analysed definitions in school mathematics, identified various types of didactical transformations of definitions, and presented a basis for future researches on definition teaching in school mathematics.