• 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.

• Communications of Mathematical Education
• /
• v.24 no.3
• /
• pp.731-744
• /
• 2010

Nowadays in school mathematics, the skill and method for solving problems are often emphasized in preference to the theoretical principles of mathematics. Students pay attention to how to make an equation mechanically before even understanding the meaning of the given problem. Furthermore they do not get to really know about the principle or theorem that were used to solve the problem, or the meaning of the answer that they have obtained. In contemporary textbooks the conic section such as circle, ellipse, parabola and hyperbola are introduced as the cross section of a cone. But they do not mention how conic section are connected with the quadratic equation or how these curves are related mutually. Students learn the quadratic equations of the conic sections introduced geometrically and are used to manipulating it algebraically through finding a focal point, vertex, and directrix of the cross section of a cone. But they are not familiar with relating these equations with the cross section of a cone. In this paper, we try to understand the quadratic curves better through the analysis of the discussion made in the process of the discovery and eventual development of the conic section and then seek for way to improve the teaching and learning methods of quadratic curves.

• Communications of Mathematical Education
• /
• v.27 no.4
• /
• pp.567-580
• /
• 2013

As integrated essay writing is performed in university entrance examinations, teachers and students recognize the importance of integrated essay, but teachers have still difficulties of teaching methods. The purpose of this study is to derive educational implications through case of mathematics instruction for integrated essay education to pre-service mathematics teachers. The content knowledge of this class is a definition of conic section in mathematics and properties of conic section in an antenna reflector. The students have to discover them using the history of math, manipulative material, paper-folding and computer simulation. In this teaching and learning process the students can realize mathematical knowledge invented by humans through history of mathematics. The students can evaluate the validity of that as create and justify a mathematical proposition. Also, the students can explain the relation between them logically and descript cause or basis convincingly in the process of justifying. We should keep our study to instructional materials and teaching methods in integrated essay education.

• Communications of Mathematical Education
• /
• v.12
• /
• pp.479-488
• /
• 2001

• The Mathematical Education
• /
• v.52 no.1
• /
• pp.83-95
• /
• 2013

The purpose of this paper is to explain and reinterprets Apollonius' Symptoms on conic sections based on the current secondary curriculum of mathematics, present the historical background of Apollonius' Symptoms to teachers and students and introduce visualization proof of Apollonius' symptoms on a parabola, a hyperbola and an ellipse by a new method using dynamic geometry software(GSP) respectively.

• 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.

• Communications of Mathematical Education
• /
• v.31 no.2
• /
• pp.103-121
• /
• 2017

Taxicab geometry was a typical non-Euclid geometry for mathematically gifted. Most educational material related quadratic curves on taxicab geometry for mathematically gifted served them to inquire the graph of the curves defined by focis and constant. In this study, we provide a shape of quadratic curves on taxicab geometry by applying three definitions(geometric algebraic definition, eccentricity definition, conic section definition).

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.33 no.9
• /
• pp.859-869
• /
• 2009

The sharp indenters such as Berkovich and conical indenters have a geometrical self-similarity in theory, but different materials have the same load-depth curve in case of single indentation. In this study, we analyze the load-depth curves of conical indenter with angles of indenter via finite element method. From FE analyses of dual-conical indentation test, we investigate the relationships between indentation parameters and load-deflection curves. With numerical regressions of obtained data, we finally propose indentation formulae for material properties evaluation. The proposed approach provides stress-strain curve and the values of elastic modulus, yield strength and strain-hardening exponent with an average error of less than 2%. It is also discussed that the method is valid for any elastically deforming indenters made of tungsten carbide and diamond for instance. The proposed indentation approach provides a substantial enhancement in accuracy compared with the prior methods.

• The Mathematical Education
• /
• v.50 no.2
• /
• pp.233-246
• /
• 2011

The geometry field in the current high school curriculum deals mainly with analytic geometry and the reference to logic geometry leaves much to be desired. This study investigated the construction on a heptagon by using conic sections as one of measures for achieving harmony between analytic geometry and logic geometry in the high school curriculum with the Geometer's Sketchpad(GSP), which is a specialized software prevalent in mathematics education field and is intended to draw an educational suggestion on it.

• 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.