• Journal of Elementary Mathematics Education in Korea
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• v.18 no.1
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• pp.105-122
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• 2014

In this paper, fraction division algorithms in Korean elementary mathematics textbooks are analyzed as a part of the groundwork to improve teaching methods for fraction division algorithms. There are seemingly six fraction division algorithms in ${\ll}Math\;5-2{\gg}$, ${\ll}Math\;6-1{\gg}$ textbooks according to the 2006 curriculum. Four of them are standard algorithms which show the multiplication by the reciprocal of the divisors modally. Two non-standard algorithms are independent algorithms, and they have weakness in that the integration to the algorithms 8 is not easy. There is a need to reconsider the introduction of the algorithm 4 in that it is difficult to think algorithm 4 is more efficient than algorithm 3. Because (natural number)${\div}$(natural number)=(natural number)${\times}$(the reciprocal of a natural number) is dealt with in algorithm 2, it can be considered to change algorithm 7 to algorithm 2 alike. In textbooks, by converting fraction division expressions into fraction multiplication expressions through indirect methods, the principles of calculation which guarantee the algorithms are explained. Method of using the transitivity, method of using the models such as number bars or rectangles, method of using the equivalence are those. Direct conversion from fraction division expression to fraction multiplication expression by handling the expression is possible, too, but this is beyond the scope of the curriculum. In textbook, when dealing with (natural number)${\div}$(proper fraction) and converting natural numbers to improper fractions, converting natural numbers to proper fractions is used, but it has been never treated officially.

• The Mathematical Education
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• v.62 no.1
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• pp.1-21
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• 2023

The purpose of this study is to explore how the student, who interiorized three levels of units, constructed fractions as multipliers by analyzing her ways of conceiving improper fractions with three levels of units and coordinating two three-levels-of-units structures. Among the data collected from our teaching experiment with two 4th grade students meeting 13 times for three months, we focus on how Seyeon, one of the participating students, wrote numerical expressions in the form of "× fraction" for the given situations using her splitting operation for composite units. Given the importance of splitting operation for composite units for the construction of fractions as multipliers, implications for further research are discussed.

• Proceedings of the Korean Society for Technology of Plasticity Conference
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• 2003.10a
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• pp.60-63
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• 2003

Semi-solid forming is the process of stirring alloy during solidification, making the mixture of liquid and solid, solidifying it, reheating it to the solid-liquid coexistent temperature, and then injecting this semi solid slurry into dies. In the semi-solid die casting process, it is very important to find out the correlation of injection condition, microstructure and mechanical properties. Especially, an improper injection condition is the main cause of liquid segregation and non-homogeneous mechanical properties due to the difference of solid fraction according to the position of the products. To ensure the database requisite to the semi-solid die casting product, it is essential to acquire the mechanical properties considering liquid segregation to the injection condition. In this study, the effect of injection condition on liquid segregation, formability, microstructure and mechanical properties in a thin product was investigated.

• Journal of the Korean Statistical Society
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• v.29 no.4
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• pp.443-454
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• 2000

In Bayesian model selection or testing problems of different dimensions, the conventional Bayes factors with improper noninformative priors are not well defined. The intrinsic Bayes factor and the fractional Bayes factor are used to overcome such problems by using a data-splitting idea and fraction, respectively. This article addresses a Bayesian testing for the comparison of two normal means with unknown variance. We derive proper intrinsic priors, whose Bayes factors are asymptotically equivalent to the corresponding fractional Bayes factor. We demonstrate our results with two examples.

• Journal of Educational Research in Mathematics
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• v.22 no.1
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• pp.53-68
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• 2012

This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching / semi-structured interviews, and a final post-test clinical interview. Results showed that for equal sharing situations, children conceptualized division in two ways: For mixed numbers, division generated a whole number portion of quotient and a fractional portion of quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. Results suggest that first, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Second, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems. If not, their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator.

• Applied Chemistry for Engineering
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• v.19 no.5
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• pp.539-546
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• 2008

A study on the selection of an optimal entrainer as the third component among water, aniline, 1,3-diethylbenzene, furfural, and MEK, for the extractive distillation of an azeotropic acetone/methanol system was performed using both the entrainer effect vapor-liquid equilibrium (VLE) and the relative volatility. In the case of water as the entrainer, a VLE curve without azeotropic point in the range of water composition from 0.3 up to 0.7 mole fraction could be obtained by both the experiment and the calculation using modified-UNIFAC model. For aniline and 1,3-diethylbenzene, however, VLE curve without azeotropic point could be obtained only at compositions above 0.7 mole fraction, which exhibited that they could be hardly utilized as the entrainer. Moreover, both furfural and MEK were verified to be improper entrainer since they formed an azeotropic phase. Relative volatility of water showed greater than 1.0 and increased with compositions, while those of the others decreased non-linearly, exhibiting that only water could be utilized as the proper entrainer for the extractive distillation of azeotropic acetone/methanol system.

• The Mathematical Education
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• v.61 no.1
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• pp.1-28
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• 2022

The purpose of this study is to explore how students' fractional knowledge is related to their solving of proportion problems. To this end, 28 clinical interviews with four middle-grade students, each lasting about 30~50 minutes, were carried out from May 2021 to August 2021. The present study focuses on two 7th grade students who exhibited their ability to coordinate three levels of units prior to solving whole number problems. Although the students showed interiorization of three levels of units in solving whole number problems, how they coordinated three levels of units were different in solving proportion problems depending on whether the problems required reasoning with whole numbers or fractions. The students could coordinate three levels of units prior to solving the problems involving whole numbers, they coordinated three levels of units in activity for the problems involving fractions. In particular, the ways the two students employed partitioning operations and how they coordinated quantitative unit structures were different in solving proportion problems involving improper fractions. The study contributes to the field by adding empirical data corroborating the hypotheses that students' ability to transform one three levels of units structure into another one may not only be related to their interiorization of recursive partitioning operations, but it is an important foundation for their construction of splitting operations for composite units.

• Nuclear Engineering and Technology
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• v.47 no.1
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• pp.11-25
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• 2015

The accident at Japan's Fukushima Daiichi nuclear power plant in March 2011, caused by an earthquake and a subsequent tsunami, resulted in a failure of the power systems that are needed to cool the reactors at the plant. The accident progression in the absence of heat removal systems caused Units 1-3 to undergo fuel melting. Containment pressurization and hydrogen explosions ultimately resulted in the escape of radioactivity from reactor containments into the atmosphere and ocean. Problems in containment venting operation, leakage from primary containment boundary to the reactor building, improper functioning of standby gas treatment system (SGTS), unmitigated hydrogen accumulation in the reactor building were identified as some of the reasons those added-up in the severity of the accident. The Fukushima accident not only initiated worldwide demand for installation of adequate control and mitigation measures to minimize the potential source term to the environment but also advocated assessment of the existing mitigation systems performance behavior under a wide range of postulated accident scenarios. The uncertainty in estimating the released fraction of the radionuclides due to the Fukushima accident also underlined the need for comprehensive understanding of fission product behavior as a function of the thermal hydraulic conditions and the type of gaseous, aqueous, and solid materials available for interaction, e.g., gas components, decontamination paint, aerosols, and water pools. In the light of the Fukushima accident, additional experimental needs identified for hydrogen and fission product issues need to be investigated in an integrated and optimized way. Additionally, as more and more passive safety systems, such as passive autocatalytic recombiners and filtered containment venting systems are being retrofitted in current reactors and also planned for future reactors, identified hydrogen and fission product issues will need to be coupled with the operation of passive safety systems in phenomena oriented and coupled effects experiments. In the present paper, potential hydrogen and fission product issues raised by the Fukushima accident are discussed. The discussion focuses on hydrogen and fission product behavior inside nuclear power plant containments under severe accident conditions. The relevant experimental investigations conducted in the technical scale containment THAI (thermal hydraulics, hydrogen, aerosols, and iodine) test facility (9.2 m high, 3.2 m in diameter, and $60m^3$ volume) are discussed in the light of the Fukushima accident.

• The Mathematical Education
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• v.40 no.2
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• pp.195-215
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• 2001

The purpose of this paper is to present a specific set of home teaching methods in hopes to prevent slow learner of the elementary mathematics. This paper deals with the number and operations, one of five topics in the elementary mathematics A survey of two hundred elementary school teachers was made to see the teacher's opinions of the role of home studying and to concretize the contents of the research topics. There were asked which is the most essential contents for the concrete loaming and which is the most difficult monad that might cause slow leaner. And those were found to be; counting, and arithmetic operations(addition and subtraction) of one or two-digit numbers and multiplication and their concepts representations and operations(addition and subtraction) of fractions. The home teaching methods are based on the situated learning about problem solving in real life situations and on the active teaming which induces children's participation in the process of teaching and learning. Those activities in teaching each contents are designed to deal with real objects and situations. Most teaching methods are presented in the order of school curriculum. To teach the concepts of numbers and the place value, useful activities using manipulative materials (Base ten blocks, Unifix, etc.) or real objects are also proposed. Natural number's operations such as addition, subtraction and multiplication are subdivided into small steps depending upon current curriculum, then for understanding of operational meaning and generalization, games and activities related to the calculation of changes are suggested. For fractions, this paper suggest 10 learning steps, say equivalent partition, fractional pattern, fractional size, relationship between the mixed fractions and the improper fraction, identifying fractions on the number line, 1 as a unit, discrete view point of fractions, comparison of fractional sizes, addition and subtraction, quantitative concepts. This research basically centers on the informal activities of kids under the real-life situation because such experiences are believed to be useful to prevent slow learner. All activities and learnings in this paper assume children's active participation and we believe that such active and informal learning would be more effective for learning transfer and generalization.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.241-262
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• 2010

This study was carried out to identify the cognitive obstacles while using addition and subtraction with fractions, and to analyze the sources of cognitive obstacles. For this purpose, the following research questions were established : 1. What errors do elementary students make while performing the operations with fractions, and what cognitive obstacles do they have? 2. What sources cause the cognitive obstacles to occur? The results obtained in this study were as follows : First, the student's cognitive obstacles were classified as those operating with same denominators, different denominators, and both. Some common cognitive obstacles that occurred when operating with same denominators and with different denominators were: the students would use division instead of addition and subtraction to solve their problems, when adding fractions, the students would make a natural number as their answer, the students incorporated different solving methods when working with improper fractions, as well as, making errors when reducing fractions. Cognitive obstacles in operating with same denominators were: adding the natural number to the numerator, subtracting the small number from the big number without carrying over, and making errors when doing so. Cognitive obstacles while operating with different denominators were their understanding of how to work with the denominators and numerators, and they made errors when reducing fractions to common denominators. Second, the factors that affected these cognitive obstacles were classified as epistemological factors, psychological factors, and didactical factors. The epistemological factors that affected the cognitive obstacles when using addition and subtraction with fractions were focused on hasty generalizations, intuition, linguistic representation, portions. The psychological factors that affected the cognitive obstacles were focused on instrumental understanding, notion image, obsession with operation of natural numbers, and constraint satisfaction.