• School Mathematics
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• v.5 no.4
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• pp.421-440
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• 2003

It is difficult for the learner to understand completely the ratio concept which forms a basis of proportional reasoning. And proportional reasoning is, on the one hand, the capstone of children's elementary school arithmetic and, the other hand, it is the cornerstone of all that is to follow. But school mathematics has centered on the teachings of algorithm without dealing with its essence and meaning. The purpose of this study is to analyze the essence of ratio concept from multidimensional viewpoint. In addition, this study will show the direction for improvement of ratio concept. For this purpose, I tried to analyze the historical development of ratio concept. Most mathematicians today consider ratio as fraction and, in effect, identify ratios with what mathematicians called the denominations of ratios. But Euclid did not. In line with Euclid's theory, ratio should not have been represented in the same way as fraction, and proportion should not have been represented as equation, but in line with the other's theory they might be. The two theories of ratios were running alongside each other, but the differences between them were not always clearly stated. Ratio can be interpreted as a function of an ordered pair of numbers or magnitude values. A ratio is a numerical expression of how much there is of one quantity in relation to another quantity. So ratio can be interpreted as a binary vector which differentiates between the absolute aspect of a vector -its size- and the comparative aspect-its slope. Analysis on ratio concept shows that its basic structure implies 'proportionality' and it is formalized through transmission from the understanding of the invariance of internal ratio to the understanding of constancy of external ratio. In the study, a fittingness(or comparison) and a covariation were examined as the intuitive origins of proportion and proportional reasoning. These form the basis of the protoquantitative knowledge. The development of sequences of proportional reasoning was examined. The first attempts at quantifying the relationships are usually additive reasoning. Additive reasoning appears as a precursor to proportional reasoning. Preproportions are followed by logical proportions which refer to the understanding of the logical relationships between the four terms of a proportion. Even though developmental psychologists often speak of proportional reasoning as though it were a global ability, other psychologists insist that the evolution of proportional reasoning is characterized by a gradual increase in local competence.

• Journal of the Korean School Mathematics Society
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• v.10 no.2
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• pp.149-171
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• 2007

This study is aimed at development of pedagogical content knowledge on fraction in the elementary school mathematics. Elementary students regard fraction as the difficult topic in school mathematics. Furthermore, fraction is the fundamentally important concept in studying mathematics. So it is important to develop the pedagogical content knowledge on fraction. The reason of attention to the pedagogical content knowledge is that improving the quality of teaching is the central focus of a high quality mathematics education. Shulman suggested that various knowledges are required for teacher to improve their classes. Of course, pedagogical content knowledge is the most valuable in teaching mathematics. Pedagogical content knowledge is related to the promotion of students' understanding about the learning. Pedagogical content knowledges are categorized by five factors in this study. These are understanding about curriculum, understanding about students and students' knowledge, understanding about teachers and teachers' knowledge, understanding about the methods, contents, and management of class, and understanding about methods of assessments. I develop the pedagogical content knowledge on fraction according to the these categories. I concentrate on the two types of pedagogical content knowledges in developing. That is, I present knowledges which teachers have to know for teaching fraction effectively and materials which teachers can use during the teaching fraction. Pedagogical content knowledges guarantee teachers as the professionalists. Teachers should not teach only content knowledges but teach various knowledges including the meta-knowledges which have relation to fraction.

• Communications of Mathematical Education
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• v.23 no.1
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• pp.145-174
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• 2009

Based on the thinking that people can understand more clearly when the problem is related with their prior knowledge, the Purpose of this study was to analysis students' informal knowledge, which is constructed through their mathematical experience in the context of real-world situations. According to this purpose, the following research questions were. 1) What is the characteristics of students' informal knowledge about fraction before formal fraction instruction in school? 2) What is the difference of informal knowledge of fraction according to reasoning ability and grade. To investigate these questions, 18 children of first, second and third grade(6 children per each grade) in C elementary school were selected. Among the various concept of fraction, part-whole fraction, quotient fraction, ratio fraction and measure fraction were selected for the interview. I recorded the interview on digital camera, drew up a protocol about interview contents, and analyzed and discussed them after numbering and comment. The conclusions are as follows: First, students already constructed informal knowledge before they learned formal knowledge about fraction. Among students' informal knowledge they knew correct concepts based on formal knowledge, but they also have ideas that would lead to misconceptions. Second, the informal knowledge constructed by children were different according to grade. This is because the informal knowledge is influenced by various experience on learning and everyday life. And the students having higher reasoning ability represented higher levels of knowledge. Third, because children are using informal knowledge from everyday life to learn formal knowledge, we should use these informal knowledge to instruct more efficiently.

• Geomechanics and Engineering
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• v.8 no.3
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• pp.305-321
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• 2015

In this paper, a new hyperbolic shear deformation plate theory based on neutral surface position is developed for the static analysis of functionally graded plates (FGPs). The theory accounts for hyperbolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Based on the present new hyperbolic shear deformation plate theory and the neutral surface concept, the governing equations of equilibrium are derived from the principle of virtual displacements. Numerical illustrations concern flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions.

• Bulletin of the Korean Chemical Society
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• v.35 no.6
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• pp.1654-1658
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• 2014

High value-added aromatic aldehydes (e.g. vanillin and syringaldehyde) were produced from heavy fraction of bio-oil (HFBO) via catalytic oxidation. The concept is based on the use of metalloporphyin as catalyst and hydrogen peroxide ($H_2O_2$) as oxidant under alkaline condition. The biomimetic catalyst cobalt(II)-sulfonated tetraphenylporphyrin ($Co(TPPS_4)$) was prepared and characterized. It exhibited relative high activity in the catalytic oxidation of HFBO. 4.57 wt % vanillin and 1.58 wt % syringaldehyde were obtained from catalytic oxidation of HFBO, compared to 2.6 wt % vanillin and 0.86 wt % syringaldehyde without $Co(TPPS_4)$. Moreover, a possible mechanism of HFBO oxidation using $Co(TPPS_4)/H_2O_2$ was proposed by the research of model compounds. The results showed that this is a promising and environmentally friendly method for production of aromatic aldehydes from HFBO under $Co(TPPS_4)/H_2O_2$ system.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.9
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• pp.2770-2781
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• 1996

Longitudinal shear modulus of continuous fiber reinforced 2-phase composites is predicted by theoretical and numerical analysis methods. In this paper, circular, hexagonal and rectangular shapes of reinforced fiber are considered using unit cell concept. And fiber array is regular rectangular and hexagonal fiber arrangement. Longitudinal shear modulus is a function of fiber distribution pattern and fiber volume change. It is found that the rectangular array has a higher longitudinal shear modulus than the hexagonal one. Also, the rectangular fiber shape in lower fiber volume fraction and the circular fiber shape in higher fiber volume fraction show the higher longitudinal shear modulus. And it has been found that the theoretical and numerical predictions of the longitudinal shear modulus give a good agreement with the experimental data at lower fiber volume fraction. Both the distance and stress transfer between the fibers are discussed as the major determing factors.

• Medical Lasers
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• v.8 no.2
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• pp.90-93
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• 2019

The treatment of keloid and hypertrophic scars (HTSs) remains one of the most difficult challenges, with a high recurrence rate regardless of the method of treatment. The latest trend in scar management is a combined approach using multiple modalities that are individualized to the patient and that would provide successful results for keloid and HTSs. There are previous reports that stromal vascular fraction (SVF) is effective for scar remodeling. Based on these reports, we introduced the concept of a combination treatment using SVF injection and fractional ablative CO₂ laser. In this report, we present a 21-year-old woman who was involved in a car accident. A defect on her foot was covered with a skin graft, but the scars became elevated, which turned out to be HTSs. She was treated with a fractional ablative CO₂ laser for five sessions. A month later, SVF injection and fractional ablative CO₂ laser were conducted simultaneously. The result of a year's follow-up showed a flattened scar with resolution of pigment deposition. In conclusion, the combination treatment for HTSs with SVF injection and ablative fractional CO₂ laser is one of the modalities to achieve an excellent outcome for treating HTS.

• Transactions of Materials Processing
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• v.10 no.5
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• pp.396-401
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• 2001

The concept of the mixtures can be used not only in the composites but also in the materials with precipitates and matrix. In this investigation, the finite element method of axisymmetric unit cell models and the rule of mixtures of the Voigt and the Reuss models are used to analyze the overall mechanical response of composites with homogeneously distributed particles. The calculations have been cameo out by taking the materials as i) hardening and ii) perfect plastic materials. The Plastic properties are predicted for various volume fractions of the soft and hard particles. The computational results are compared with the results of the rule of mixtures. It is found that the plastic flow curves agree well with the Voigt model when the volume fraction of the particles is high. On the other hand, the calculated flow curves exist between the Voigt model and the Reuss model when the volume fraction of the particles is low.

• Transactions of Materials Processing
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• v.25 no.6
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• pp.396-401
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• 2016

Recently, Multi-strength hot stamping process has been widely used to achieve lightweight and crashworthiness in automotive industry. In concept of multi-strength hot stamping process, process design of tailor rolled blank(TRB) in partial heating is difficult because of thickness and temperature variation of blank. In this study, springback prediction of TRB in partial heating process was performed considering its thickness and temperature variation. In partial heating process, TRB was heated up to $900^{\circ}C$ for thicker side and below $Ac_3$ transformation temperature for thinner side, respectively. Johnson-Mehl-Avrami-Kolmogorov(JMAK) equation was applied to calculate austenite fraction according to heating temperature. Calculated austenite fraction was applied to FE-simulation for the prediction of springback. Experiment for partial heating process of TRB was also performed to verify prediction accuracy of FE-simulation coupled with JMAK equation.

• Research in Mathematical Education
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• v.10 no.3 s.27
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• pp.189-203
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• 2006

Teaching of fractions is very challengeable for elementary and middle school teachers. Many teachers feel uncomfortable teaching the subject. How shall we introduce the concept of fractions? Shall we focus more on concept development or computational procedures and skills? What methodology can be used in teaching those important topics? These are the kind of questions many teachers and researchers try to answer. In this article, the authors are to look at this issue from a cross nation perspective, by examining how the Chinese mathematics curriculum and the Chinese teachers deal with this subject.