• School Mathematics
• /
• v.9 no.1
• /
• pp.119-139
• /
• 2007

The purpose of this study is to analyze students' misconception in the teaming of the conic sections with the cognitive and pedagogical point of view. The conics sections is very important concept in the high school geometry. High school students approach the conic sections only with algebraic perspective or analytic geometry perspective. So they have various misconception in the conic sections. To achieve the purpose of this study, the research on the following questions is conducted: First, what types of misconceptions do the students have in the loaming of conic sections? Second, what types of errors appear in the problem-solving process related to the conic sections? With the preliminary research, the testing worksheet and the student interviews, the cause of error and the misconception of conic sections were analyzed: First, students lacked the experience in the constructing and manipulating of the conic sections. Second, students didn't link the process of constructing and the application of conic sections with the equation of tangent line of the conic sections. The conclusion of this study ls: First, students should have the experience to manipulate and construct the conic sections to understand mathematical formula instead of rote memorization. Second, as the process of mathematising about the conic sections, students should use the dynamic geometry and the process of constructing in learning conic sections. And the process of constructing should be linked with the equation of tangent line of the conic sections. Third, the mathematical misconception is not the conception to be corrected but the basic conception to be developed toward the precise one.

• East Asian mathematical journal
• /
• v.28 no.4
• /
• pp.453-474
• /
• 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.

• Honam Mathematical Journal
• /
• v.33 no.3
• /
• pp.335-340
• /
• 2011

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

• Journal for History of Mathematics
• /
• v.15 no.1
• /
• pp.69-82
• /
• 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.

• Journal of Science Education
• /
• v.47 no.3
• /
• pp.263-272
• /
• 2023

We consider how a pre-service teacher can understand and utilize various concepts of Euclidean geometry by learning conic sections using mathematical definitions in non-Euclidean geometry. In a third-grade class of D University, we used mathematical definitions to demonstrate that learning conic sections in non-Euclidean space, such as taxicab geometry and Minkowski distance space, can aid pre-service teachers by enhancing their ability to acquire and accept new geometric concepts. As a result, learning conic sections using mathematical definitions in taxicab geometry and Minkowski distance space is expected to contribute to enhancing the education of pre-service teachers for Euclidean geometry expertise by fostering creative and flexible thinking.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.28 no.1
• /
• pp.22-32
• /
• 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.

• The Mathematical Education
• /
• v.46 no.3
• /
• pp.331-349
• /
• 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.

• Journal for History of Mathematics
• /
• v.25 no.4
• /
• pp.83-99
• /
• 2012

The conic sections are defined as algebraic expressions using the focus and the directrix in the high school curriculum. However it is difficult that students understand the conic sections without environment which they can manipulate the conic sections. To make up for this weak point, we have found the evidence for generating method of a conic section through a sundial and investigated the history of terms 'focus', 'directrix' and the tool of drawing them continuously.

• The Pure and Applied Mathematics
• /
• v.19 no.4
• /
• pp.315-325
• /
• 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.

• Korean Journal of Computational Design and Engineering
• /
• v.5 no.4
• /
• pp.336-346
• /
• 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.