• Education of Primary School Mathematics
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• v.19 no.1
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• pp.19-30
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• 2016

This study examined pre-service teachers' pedagogical content knowledge of fraction division in a context where they were asked to write a story problem for a symbolic expression illustrating a whole number divided by a proper fraction. Problem-posing is an important instructional strategy with the potential to create meaningful contexts for learning mathematical concepts, especially when real-world applications are intended. In this study, story problems written by 135 elementary pre-service teachers were analyzed with respect to mathematical correctness. error types, and division models. Patterns and tendencies in elementary pre-service teachers' knowledge of fraction division were identified. Implicaitons for teaching and teacher education are discussed.

• Education of Primary School Mathematics
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• v.26 no.3
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• pp.181-197
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• 2023

The importance of reasoning processes based on fractional concepts and number senses, rather than a formalized procedural method using common denominators, has been noted in a number of studies in relation to compare the magnitudes of unlike fractions. In this study, a unlike fraction magnitudes comparison test was conducted on fourth-grade elementary school students who did not learn equivalent fractions and common denominators to analyze the reasoning perspectives of the correct and wrong answers for each of the eight problem types. As a result of the analysis, even students before learning equivalent fractions and reduction to common denominators were able to compare the unlike fractions through reasoning based on fractional sense. The perspective chosen by the most students for the comparison of the magnitudes of unlike fractions is the 'part-whole perspective', which shows that reasoning when comparing the magnitudes of fractions depends heavily on the concept of fractions itself. In addition, it was found that students who lack a conceptual understanding of fractions led to difficulties in having quantitative sense of fraction, making it difficult to compare and infer the magnitudes of unlike fractions. Based on the results of the study, some didactical implications were derived for reasoning guidance based on the concept of fractions and the sense of numbers without reduction to common denominators when comparing the magnitudes of unlike fraction.

• 한국연소학회:학술대회논문집
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• 2014.11a
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• pp.165-166
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• 2014

The application of particle reaction model in the packed bed process modeling is discussed for iron ore pellet induration process. Combustion of coke breeze in the pellet is estimated by using shrinking unreacted-core model and grain model in which the progress of chemical reaction is described in different concepts. Under the identical inlet gas and solid conditions, the calculation using shrinking core model showed deviated results in terms of temperature profile and conversion fraction, which may imply the significance of selecting proper particle reaction model in consideration of particle characteristics and process operation conditions.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.1
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• pp.105-129
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• 2016

The elements of mathematical processes include mathematical reasoning, mathematical problem-solving, and mathematical communications. Proportion reasoning is a kind of mathematical reasoning which is closely related to the ratio and percent concepts. Proportion reasoning is the essence of primary mathematics, and a basic mathematical concept required for the following more-complicated concepts. Therefore, the study aims to analyze the proportion reasoning ability of sixth graders of primary school who have already learned the ratio and percent concepts. To allow teachers to quickly recognize and help students who have difficulty solving a proportion reasoning problem, this study analyzed the characteristics and patterns of proportion reasoning of sixth graders of primary school. The purpose of this study is to provide implications for learning and teaching of future proportion reasoning of higher levels. In order to solve these study tasks, proportion reasoning problems were developed, and a total of 22 sixth graders of primary school were asked to solve these questions for a total of twice, once before and after they learned the ratio and percent concepts included in the 2009 revised mathematical curricula. Students' strategies and levels of proportional reasoning were analyzed by setting up the four different sections and classifying and analyzing the patterns of correct and wrong answers to the questions of each section. The results are followings; First, the 6th graders of primary school were able to utilize various proportion reasoning strategies depending on the conditions and patterns of mathematical assignments given to them. Second, most of the sixth graders of primary school remained at three levels of multiplicative reasoning. The most frequently adopted strategies by these sixth graders were the fraction strategy, the between-comparison strategy, and the within-comparison strategy. Third, the sixth graders of primary school often showed difficulty doing relative comparison. Fourth, the sixth graders of primary school placed the greatest concentration on the numbers given in the mathematical questions.

• Journal of Elementary Mathematics Education in Korea
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• v.12 no.1
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• pp.79-100
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• 2008

Freudenthal has advocated the reinvention method. In that method, the pupils start with a meaningful context, not ready-made concepts, and invent informative method through which he could arrive at the formative concepts progressively. In many face the reinvention method is contrary to the traditional method. In traditional method, which was named as 'concretization method' by Freudenthal, the pupils start with ready-made concepts, and applicate this concepts to various instances through which he could arrive at the understanding progressively. Through analysis, it turns out that Korea's seventh elementary mathematics textbook is based on concretization method. In this thesis, first of all, I will reconstruct probability unit of seventh elementary textbook according to Freudenthal's reinvention method. Next, I will perform teaching experiment which is ruled by new lesson design. Lastly, I analysed the effects of teaching experiment. Through this study, I obtained the following results and suggestions. First, the reinvention method is effective on the teaching of probability concept and algorithm. Second, in comparison with current textbook strand, my strand which made probability concept go ahead and combinatorics concept let behind is not deficiency. Third, tree diagram is effective matrix which contribute to formalization of combinatorics calculation. Lastly, except for fraction, diverse representation of probability, for example percentage or informal ratio expression must be introduced in teaching process.

• Microbiology and Biotechnology Letters
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• v.26 no.6
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• pp.537-544
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• 1998

The cultivation conditions of transformant Alcaligenes eutrophus AER5 harboring cloned phbC gene for mass production of poly (3-hydroxybutyrate-3-hydroxyvalerate)[P(3HB-3HV)] containing high molar fraction of 3-hydroxyvalerate (3-HV) were investigated. In two-stage batch cultivation, transformant accumulated P(3HB-3HV) containing 52.2 mol% of 3HV compared to 30 mol% of parent strain A. eutrophus H16. The increased 3-HV molar fraction was due to the amplified activity of PHB synthase participating in condensation of 3-HB and 3-HV. To increase efficiency of P(3HB-3HV) accumulation, fructose was added along with precursor compound valerate, and total cell mass and P(3HB-3HV) concentrations remarkably increased, but not 3-HV molar fraction. The effect of magnesium ion showed that P(3HB-3HV) concentration and 3-HV molar fraction were significantly increased upto 6.1 g/L and 71.3 mol% at 0.01 g/L of MgSO$_4$, respectively. The efficiency of several pH adjuster, NaOH, NaOH and (NH$_4$)$_2$SO$_4$, and NH$_4$OH, on total cell mass, p(3HB-3HV) concentration, and 3-HV molar fraction was also compared. To overcome the disadvantage of two-stage cultivation, one-stage intermittent fed-batch cultivation was attempted, such a way 10.0 g/L of fructose was supplied for cell growth at initial 36 hr and then 10.0 g/L of valerate and 5.0 g/L of fructose were applied to induce the accumulation of P(3HB-3HV), consequently, 10.4 g/L of P(3HB-3HV) with 38 mol% of 3-HV fraction could be obtained after 72 hr. These results can be used for elucidating cultivation strategy for mass production of P(3HB-3HV) containing high 3-HV molar fraction using transformant A. eutrophus AER5 harboring cloned phbC gene.

• Journal of Elementary Mathematics Education in Korea
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• v.19 no.4
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• pp.563-588
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• 2015

In elementary school, didactical transposition is inevitable due to several reasons. In mathematics, addition and multiplication are taught as binary operations, subtraction and division are taught as unary operations. But in elementary school, we try to teach all the four operations as binary operations by didactical transposition. In 'Mastering' the concepts of the four operations, the way of concept introduction is dealt importantantly. So it is different from understanding the four operations. In this study, we analyzed the four operations of natural numbers and fractions from two perspectives: concept understanding (how to introduce concepts and how to choose an operation) and connection between the operations. As a result, following implications were obtained. In division of fractions, students attempted a connection with multiplication of fractions right away without choosing an operation, based on the situation. Also, to understand division of fractions itself, integrate division of fractions presented from the second semester of the fifth grade to the first semester of the sixth grade are needed. In addition, this result can be useful in the future textbook development.

• Publications of The Korean Astronomical Society
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• v.23 no.2
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• pp.65-71
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• 2008

It has been considered that 'exhausted eclipses' (日食旣) were total eclipses. However, modern precise calculations show that a significant fraction of such records are not realized to be total. Thus we doubt that the two concepts are equivalent. Here we investigate the meaning of 'exhausted eclipses' in the east-Asian history. We first find that eclipses of magnitude greater than 0.8 were regarded as 'exhausted eclipses' by a Korean astronomer of the 18th-century Choson dynasty. His notion was based upon the definition of 'exhausted eclipses' in the ephemerides of pre-modern Chinese dynasties. According to those ephemerides, the 'exhausted eclipses', whose magnitude is greater than 0.8, have the first contact at the western part of the solar disk and the fourth contact at the eastern part of the solar disk. A simple geometrical calculation shows that such cases really occur when the magnitude of eclipse is greater than 0.7. We pointed out that such an ancient definition might not be impractical for ancient astronomers, because the uncertainty of eclipse magnitude estimated by ancient Chinese ephemerides was 10% and the human sight has a spatial resolution of 1.2 arcmin, which is approximately one twentieth of the Sun's angular diameter.

• Korea-Australia Rheology Journal
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• v.18 no.4
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• pp.171-181
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• 2006

We present a short review for authors' previous work on direct numerical simulations for inertialess hard particle suspensions formulated either with a Newtonian fluid or with viscoelastic polymeric fluids to understand the microstructural evolution and the bulk material behavior. We employ two well-defined bi-periodic domain concepts such that a single cell problem with a small number of particles may represent a large number of repeated structures: one is the sliding bi-periodic frame for simple shear flow and the other is the extensional bi-periodic frame for planar elongational flow. For implicit treatment of hydrodynamic interaction between particle and fluid, we use the finite-element/fictitious-domain method similar to the distributed Lagrangian multiplier (DLM) method together with the rigid ring description. The bi-periodic boundary conditions can be effectively incorportated as constraint equations and implemented by Lagrangian multipliers. The bulk stress can be evaluated by simple boundary integrals of stresslets on the particle boundary in such formulations. Some 2-D example results are presented to show effects of the solid fraction and the particle configuration on the shear and elongational viscosity along with the micro-structural evolution for both particles and fluid. Effects of the fluid elasticity has been also presented.

• The Mathematical Education
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• v.57 no.4
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• pp.453-475
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• 2018

This study examines how elementary students understand fractions with operations conceptually and how they perform procedures in the division of fractions. We attempted to look into students' understanding about fractions with divisions in regard to mathematical proficiency suggested by National Research Council (2001). Mathematical proficiency is identified as an intertwined and interconnected composition of 5 strands- conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. We developed an instrument to identify students' understanding of fractions with multiplication and division and conducted the survey in which 149 6th-graders participated. The findings from the data analysis suggested that overall, the 6th-graders seemed not to understand fractions conceptually; in particular, their understanding is limited to a particular model of part-whole fraction. The students showed a tendency to use memorized procedure-invert and multiply in a given problem without connecting the procedure to the concept of the division of fractions. The findings also proposed that on a given problem-solving task that suggested a pathway in order for the students to apply or follow the procedures in a new situation, they performed the computation very fluently when dividing two fractions by multiplying by a reciprocal. In doing so, however, they appeared to unable to connect the procedures with the concepts of fractions with division.