• 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.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.521-539
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• 2016

The structures of partitive and quotitive division of fractions are dealt with differently, and this led to using partitive division context for helping develop invert-multiply algorithm and quotitive division for common denominator algorithm. This approach is unlikely to provide children with an opportunity to develop an understanding of common structure involved in solving different types of division. In this study, I propose two approaches, measurement approach and isomorphism approach, to develop a unifying understanding of fraction division. From each of two approaches of solving quotitive division based on proportional reasoning, I discuss an idea of constructing a measure space, unit of which is a quantity of divisor, and another idea of constructing an isomorphic relationship between the measure spaces of dividend and divisor. These ideas support invert-multiply algorithm for quotitive as well as partitive division and bring proportional reasoning into the context of fraction division. I also discuss some curriculum issues regarding fraction division and proportion in order to promote the proposed unifying understanding of partitive and quotitive division of fractions.

• Journal of Educational Research in Mathematics
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• v.24 no.3
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• pp.467-480
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• 2014

Concepts related to the fraction should be taught with formative thinking activities as well as concrete operational activities. Teaching improper fraction should follow the concept of fraction as a relation of two natural numbers. This concept is also important not to be skipped before teaching the fraction such as "4 is a third of 12". Mixed number should be taught as a sum of a natural number and a proper fraction. Fraction as a quotient of a division is a hard concept to be taught since it requires very high abstractive thinking process. Learning the transformation of division into multiplication of fractions should precede that of fraction as a quotient of a division.

• Communications of Mathematical Education
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• v.19 no.4 s.24
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• pp.715-731
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• 2005

There are five contexts of division algorithm of fractions such as measurement division, determination of a unit rate, reduction of the quantities in the same measure, division as the inverse of multiplication and analogy with multiplication algorithm of fractions. The division algorithm, however, should be taught by 'dividing by using reciprocals' via 'measurement division' because dividing a fraction by a fraction results in 'multiplying the dividend by the reciprocal of the divisor'. If a fraction is divided by a large fraction, then we can teach the division algorithm of fractions by analogy with 'dividing by using reciprocals'. To achieve the teaching-learning methods above in elementary school, it is essential for children to use the maniplatives. As Piaget has suggested, Cuisenaire color rods is the most efficient maniplative for teaching fractions. The instruction, therefore, of division algorithm of fractions should be focused on 'dividing by using reciprocals' via 'measurement division' using Cuisenaire color rods through analogy if necessary.

• Korean Journal of Soil Science and Fertilizer
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• v.41 no.3
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• pp.184-189
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• 2008

To assess the relationship between Cd fraction in paddy soils and Cd in polished rice, soils and rice were analyzed at the 3 Cd contaminated paddy fields near closed mines. Major Cd fractions of A field were organically bound (62.6%) and Fe-Mn oxide bound (25.3%) forms. In case of B field, major Cd fractions of B1 field were carbonate bound (46.3%) and Fe-Mn oxide bound (31.6%) form whereas B2 field were residual (54.3%) and carbonate bound (21.8%) form, respectively. It showed a huge difference of Cd fraction each other. 0.1M HCl extractable Cd in soil was positively correlated with Cd in rice. Specially, the ratios of 0.1M HCl extractable Cd against total Cd content in soils were 13.7%, 2.6%, and 0.45% in A, B1, and B2 fields, respectively. These ratio were largely affected with Cd uptake to rice grain. Also, exchangable, Fe-Mn oxide bound, and carbonate bound form, which are partially bioavailable Cd fraction to the plant, were positively correlated with Cd in rice while organically bound and residual form was not correlated. Multiple regression equation was developed with Rice Cd = -0.02861 + 0.07456 FR 1(exchangeable) + 0.00252 FR 2(carbonate bound) + 0.001075 FR 3(Fe Mn oxide bound) - 0.00095 FR 4(organically bound) - 0.00348 FR 5(residual) ($R^2=0.7893^{***}$) considering Cd fraction in soils.

• Journal of the Korean School Mathematics Society
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• v.12 no.2
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• pp.151-170
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• 2009

The purpose of the study is to analyze the sixth graders' understanding of concepts and operation about fraction. The test was administered and analyzed to 707 sixth graders' performance on fractions after the fraction instructions in elementary schools in Seoul, Korea. The participants are asked to answer two sets of questions for 40 minutes. First, they are asked to answer to 16 problems about the concepts of fraction with respect to part-whole, ratio, operator, measure, quotient, equivalent, and operations. Second, specially, to investigate sixth graders' ability of drawing and describing the situation of division including fraction, the descriptive problem asked students (1) to describe $3\;{\div}\;\frac{1}{2}$ into pictorial representation and (2) to write the solving process. The participants of this study didn't show deep understandings about the concepts and operation of fraction. The degree of understanding of subconstructs of fraction shows that their knowledge of ratio concept with respect to fraction was highest while their understanding of measure with respect to fraction was lowest. Considering their wrong answers, about 59% of participants showed misconception to the question of naming one fraction that appears between $\frac{1}{5}$ and $\frac{1}{6}$. Further, they didn't explain their understanding with drawing about the division of fraction ($3\;{\div}\;\frac{1}{2}$).

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.3
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• pp.431-455
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• 2013

In this thesis, I classified various meanings of fraction into two categories, i.e concept(rate, operator, division) and model(whole-part, measurement, allotment), and surveyed appearances which is shown in Korea elementary mathematics textbook. Based on this results, I derived several implications on learning-teaching of fraction in elementary education. Firstly, we have to pursuit a unified formation of fraction concept through a complementary advantage of various concepts and models Secondly, by clarifying the time which concepts and models of fraction are imported, we have to overcome a ambiguity or tacit usage of that. Thirdly, the present Korea's textbook need to be improved in usage of measurement model. It must be defined more explicitly and must be used in explanation of multiplication and division algorithm of fraction.

• School Mathematics
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• v.12 no.3
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• pp.439-454
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• 2010

This study presents how two eighth grade students generated their own algorithms in the context of fraction measurement division situations by modifications of unit-segmenting schemes. Teaching experiment was adopted as a research methodology and part of data from a year-long teaching experiment were used for this report. The present study indicates that the two participating students' construction of reciprocal relationship between the referent whole [one] and the divisor by using their unit- segmenting schemes and its strategic use finally led the students to establish an algorithm for fraction measurement division problems, which was on par with the traditional invert-and-multi- ply algorithm for fraction division. The results of the study imply that teachers' instruction based on understanding student-generated algorithms needs to be accounted as one of the crucial characteristics of good mathematics teaching.

• Journal of Gifted/Talented Education
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• v.24 no.3
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• pp.339-358
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• 2014

On the subjects of elementary 3-rd grade three child prodigies who learned primary concept of division, this study explored how they could compose schema and transformed schema through recognition of precise concepts and linking with the contents of division of fraction. That is to say, this study examined in depth what schema and transformed schema as primary concept of division they composed to get relational understanding of division of fraction, and how they used the schema and transformed schema composed by themselves to approach problem solving as well as how they transformed the schema in their concept composition and problem solving competence. As a result, it was found that learning of primary concept of division played a key role of composing schema and transformed schema needed for coping with division of fraction, and that at this time, composition of the transformed schema and transformed schema derived from the recognition of primary concept of division could play the inevitable role of problem solving for division of fraction.

• Toxicological Research
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• v.26 no.3
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• pp.203-208
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• 2010

Bronchoalveolar lavage (BAL) is a useful tool in researches and in clinical medicine of lung diseases because the BAL fluid contains biochemical and cytological indicators of the cellular response to infection, drugs, or toxicants. However, the variability among laboratories regarding the technique and the processing of the BAL material limits clinical research. The aim of this study was to determine the suction frequency and lavage fraction number necessary to reduce the variability in lavage using male Sprague-Dawley rats. We compared the total cell number and protein level of each lavage fraction and concluded that more cells and protein can be obtained by repetitive lavage with a suction frequency of 2 or 3 than by lavage with a single suction. On the basis of total cell recovery, approximately 70% of cells were obtained from fractions 1~3. The first lavage fraction should be used for evaluation of protein concentration because fractions 2~5 of lavage fluid were diluted in manifolds. These observations were confirmed in bleomycin-induced inflamed lungs of rats. We further compared the BAL data from the whole lobes with data from the right lobes and concluded that BAL data of the right lobes represented data of the whole lobes. However, this conclusion can only be applied to general lung diseases. At the end, this study provides an insight into the technical or analytical problems of lavage study in vivo.