• 대한수학교육학회지:학교수학
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• 제9권1호
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• pp.119-139
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• 2007

이차곡선은 고등학교 기하 내용의 중요한 개념의 하나이다. 그러나 교수-학습 상황에서 학생들은 단순히 대수적인 접근과 해석기하적인 접근만 시도하므로 그 본질적인 기하학적 의미를 파악하지 못하며 단순한 기계적인 계산만을 수행하여 문제를 풀어나가려 하기 때문에 여러 가지 오개념(misconception)을 가지게 된다. 이 논문은 효과적인 이차곡선 교수학습 연구의 일부로, 학생들의 오개념을 인지적 관점, 심리학적 관점, 교수학적 관점에서 분석하고 그 원인을 분석하였다. 연구 결과, 학생들의 직접적이고 다양한 작도 경험의 부재가 오개념의 주된 원인이 되었다. 이차곡선에 대한 교수-학습은 기하적인 관점으로 접근 한 후 대수적인 관점으로 연결시켜야 할 필요성과 오개념에 대한 정확한 진단은 효과적인 교수-학습의 기초가 됨을 확인 하였다.

• East Asian mathematical journal
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• 제28권4호
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• pp.453-474
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• 2012

The conic sections is an important topic in the current high school geometry. It has been recognized by many researchers that high school students often have difficulty or misconception in the learning of the conic sections because they are taught the conic sections only with algebraic perspective or analytic geometry perspective. In this research, we suggest a way of teaching the conic sections using a dynamic geometry software based on some mathematics teaching and learning theories such as Freudenthal's and Dienes'. Students have various experience of constructing and manipulating the conic sections for themselves and the experience of deriving the equations of the quadratic curves under the teacher's careful guidance. We identified this approach was a feasible way to improve the teaching and learning methods of the conic sections.

• 호남수학학술지
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• 제33권3호
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• pp.335-340
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• 2011

We study some properties of tangent lines of conic sections. As a result, we establish a characterization of conic sections.

• 한국수학사학회지
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• 제15권1호
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• pp.69-82
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• 2002

The purpose of this study is to explore historical development of conic sections and analyze formal aspects, application aspects and intuitive aspects in conic sections. We suggest implication for learning-teaching conic sections.

• 과학교육연구지
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• 제47권3호
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• pp.263-272
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• 2023

예비교사가 비유클리드 기하학에서 수학적 정의를 이용한 이차곡선의 학습으로 유클리드 기하학의 다양한 개념을 어떻게 이해하고 활용할 수 있는지를 살펴본다. 본 연구에서는 D 대학교 수학교육과 3학년 수업에서 수학적 정의를 이용하여 택시기하, 민코프스키 거리공간과 같은 비유클리드 공간의 이차곡선 학습이 예비교사들에게 새로운 기하학적 개념을 습득하고 수용하는 능력 향상에 도움을 줄 수 있음을 보였다. 이러한 결과로부터 택시기하와 민코프스키 거리공간에서의 정의를 활용한 이차곡선 학습이 창의적이고 유연한 사고를 유도하여, 예비교사들의 유클리드 기하학 교육 전문성 향상에 기여할 것으로 기대된다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제28권1호
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• pp.22-32
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• 2024

In this paper we present a method of conic section approximation by hexic Bézier curves. The hexic Bézier approximants are G³ Hermite interpolations of conic sections. We show that there exists at least one hexic Bézier approximant for each weight of the conic section The hexic Bézier approximant depends one parameter and it can be obtained by solving a quartic polynomial, which is solvable algebraically. We present the explicit upper bound of the Hausdorff distance between the conic section and the hexic Bézier approximant. We also prove that our approximation method has the maximal order of approximation. The numerical examples for conic section approximation by hexic Bézier curves are given and illustrate our assertions.

• 한국수학교육학회지시리즈A:수학교육
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• 제46권3호
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• pp.331-349
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• 2007

This study aims to improve the teaching and learning method on the conic sections. To do that the researcher analyzed the impact of dynamic geometry software on students' problem solving of the conic sections. Students often say, "I have solved this kind of problem and remember hearing the problem solving process of it before." But they often are not able to resolve the question. Previous studies suggest that one of the reasons can be students' tendency to approach the conic sections only using algebra or analytic geometry without the geometric principle. So the researcher conducted instructions based on the geometric and historico-genetic principle on the conic sections using dynamic geometry software. The instructions were intended to find out if the experimental, intuitional, mathematic problem solving is necessary for the deductive process of solving geometric problems. To achieve the purpose of this study, the researcher video taped the instruction process and converted it to digital using the computer. What students' had said and discussed with the teacher during the classes was checked and their behavior was analyzed. That analysis was based on Branford's perspective, which included three different stage of proof; experimental, intuitive, and mathematical. The researcher got the following conclusions from this study. Firstly, students preferred their own manipulation or reconstruction to deductive mathematical explanation or proving of the problem. And they showed tendency to consider it as the mathematical truth when the problem is dealt with by their own manipulation. Secondly, the manipulation environment of dynamic geometry software help students correct their mathematical misconception, which result from their cognitive obstacles, and get correct ones. Thirdly, by using dynamic geometry software the teacher could help reduce the 'zone of proximal development' of Vigotsky.

• 한국수학사학회지
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• 제25권4호
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• pp.83-99
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• 2012

고등학교 교육과정에서 학생들은 원뿔곡선 조작 환경을 제공 받지 못하고 초점과 준선을 이용하여 대수적 정의를 받아들이며, 원뿔곡선을 동적인 의미 없이 정적인 대수적 문제로 국한해서 생각하는 경향이 있다. 대수적인 표현뿐만 아니라 동적인 기하학적 표현을 보완하기 위해 원뿔곡선을 원뿔 절단으로 정의한 역사적 근거를 해시계에서 찾고 원뿔 절단으로는 설명할 수 없는 초점과 준선 개념의 역사도 살펴본다. 그리고 원뿔곡선을 연속적으로 그리기 위해 사용된 도구들에 대해서 알아보고, 학생들의 활동을 위한 공학적 도구로 컴퓨터 환경을 살펴본다.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권4호
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• pp.315-325
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• 2012

Let A and B denote a point, a line or a circle, respectively. For a positive constant $a$, we examine the locus $C_{AB}$($a$) of points P whose distances from A and B are, respectively, in a constant ratio $a$. As a result, we establish some equivalent conditions for conic sections. As a byproduct, we give an easy way to plot points of conic sections exactly by a compass and a straightedge.

• 한국CDE학회논문집
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• 제5권4호
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• pp.336-346
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• 2000

This paper presents a geometric method that can detect and compute all conic sections in the intersection of two tori. Conic sections contained in a torus must be circles. Thus, when two tori intersect in a conic section, the intersection curve must be a circle as well. Circles in a torus are classified into profile circles, cross-sectional circlet, and Yvone-Villarceau circles. Based on a geometric classification of these circles, we present a procedural method that can detect and construct all intersection circles between two tori. All computations can be carried out using simple geometric operations only: e.g., circle-circle intersections, circle-line intersections, vector additions, and inner products. Consequently, this simple structure makes our algorithm robust and efficient, which is an important advantage of our geometric approach over other conventional methods of surface intersection.