• Education of Primary School Mathematics
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• v.19 no.4
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• pp.277-289
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• 2016

Fraction is one of the concepts which are difficult to elementary school students. So, many researches about fraction were performed in mathematics education research. In special, fraction has so many subordinative concepts-proper fraction, improper fraction, mixed number. We have to concentrate on the conceptual understanding in teaching of fraction. In this case, a mixed number and improper fraction are concepts which can convert respectively. And there are methods that a mixed number and improper fraction can be converted. So, it's needed to analyze the converting methods in textbooks for getting the implication of teaching in this areas. In this study, I analyzed the Korean and foreign's textbooks. I certified the methods-using addition expression, using part-whole model in the textbooks. For the conceptual understanding, I suggested to use the fusion of the various part-whole fraction models and addition expression more than the algorithm in converting between a mixed number and improper fraction. It's reason that the use of models in converting between a mixed number and improper fraction is important for the relational understanding.

• 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.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.3
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• pp.425-439
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• 2014

The concept of a fraction as division is a core idea which serves as a axiom in the process of a extension of the natural number system to rational number system. Also, it has necessary position in elementary mathematics. Nevertheless, the timing and method of the introduction of this concept in Korean elementary math textbooks is not well established. In this thesis, I suggested a solution of a various topics which is related to this problem, that is, transforming improper fraction to mixed number, the usage of quotient as a term, explaining the algorithm of division of fraction, transforming fraction to decimal.

• Journal for History of Mathematics
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• v.16 no.3
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• pp.101-108
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• 2003

The RSA cryptosystem is most commonly used for providing privacy and ensuring authenticity of digital data. 1'his system is based on the difficulty of integer factoring. Many attacks had been done, but none of them devastating. They mostly illustrate the dangers of improper use of RSA. Improper use implies many aspects, but here we imply the misuse of the parameters of RSA. Specially, sizes of parameters give strong effects on the efficiency and the security of the system. Parameters are also related each other. We analyze the relation of them. Recently many researchers are interested in side-channel attacks. We also investigate partial key exposure attacks, which was motivated by side-channel attacks. If a fraction of tile secret key bits is revealed, the private key will be reconstructed. We also study mathematical background of these attacks, solving modular multivariate polynomial equations.

• The Mathematical Education
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• v.58 no.1
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• pp.1-20
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• 2019

As the importance of algebraic thinking in elementary school has been emphasized, the links between fraction knowledge and algebraic thinking have been highlighted. In this study, we analyzed the solution methods and characteristics of thinking by fifth graders who have not yet learned fraction division when they solved 'reverse fraction problems' (Pearn & Stephens, 2018). In doing so, the contexts of problems were extended from the prior study to include the following cases: (a) the partial quantity with a natural number is discrete or continuous; (b) the partial quantity is a natural number or a fraction; (c) the equivalent fraction of partial quantity is a proper fraction or an improper fraction; and (d) the diagram is presented or not. The analytic framework was elaborated to look closely at students' solution methods according to the different contexts of problems. The most prevalent method students used was a multiplicative method by which students divided the partial quantity by the numerator of the given fraction and then multiplied it by the denominator. Some students were able to use a multiplicative method regardless of the given problem contexts. The results of this study showed that students were able to understand equivalence, transform using equivalence, and use generalizable methods. This study is expected to highlight the close connection between fraction and algebraic thinking, and to suggest implications for developing algebraic thinking when to deal with fraction operations.

• Education of Primary School Mathematics
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• v.23 no.3
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• pp.117-134
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• 2020

Based on the current curriculum, students learn the concept of fraction in the 3rd grade for the first time. At that time, fraction is introduced as whole-part relationship. But as the idea of fraction expands to improper fraction and so on, fraction as measurement would be naturally appeared. In that situation where fraction as whole-part relationship and fraction as measurement are dealt together, it is necessary for students to get experiences of understanding and exploring unit and whole adequately in order to fully understand the concept of fractions. Therefore, the purpose of this study is to analyze how to deal with unit fractions, how to implement activities to find the standard of reference from the part, and what visual representations were used to help students to understand the concept of fractions in elementary mathematics textbooks from the 7th to the 2015 revised curriculum. And we analyzed 60 3rd graders' understanding of finding and drawing the whole by looking at the part. Several didactical implications for teaching the concept of fractions were derived from the discussion according to the analysis results.

• Education of Primary School Mathematics
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• v.23 no.2
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• pp.87-116
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• 2020

The purpose of this study is to explore how units-coordination ability is related to understanding fraction concepts. For this purpose, a teaching experiment was conducted with one fourth grade student, Eunseo for four months(2019.3. ~ 2019.6.). We analyzed in details how Eunseo's units-coordinating operations related to her understanding of fraction changed during the teaching experiment. At an early stage, Eunseo with a partitive fraction scheme recognized fractions as another kind of natural numbers by manipulating fractions within a two-levels-of-units structure. As she simultaneously recognized proper fraction and a referent whole unit as a multiple of the unit fraction, she became to distinguish fractions from natural numbers in manipulating proper fractions. Eunseo with a reversible partitive fraction scheme constructed a natural number greater than 1, as having an interiorized three-levels-of-units structure and established an improper fraction with three levels of units in activity. Based on the results of this study, conclusions and pedagogical implications were presented.

• Journal of Elementary Mathematics Education in Korea
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• v.23 no.3
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• pp.323-345
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• 2019

This study examines the introduction of fractions in the third grade mathematics textbooks focusing on stages of units coordination and suggests alternative activities to help students develop their understanding of fractions. As results, the sessions of introduction units in textbooks was well organized to allow students to construct more extensive fraction schemes (i.e., Part-whole fraction scheme → Partitive unit fraction scheme → Partitive fraction scheme). However, most of the activities in textbooks were related to stages 1 and 2 of units coordination. In particular, the operations and partitioning schemes (i.e., equi-partitioning and splitting schemes), which are key to the development of students' fraction knowledge, were not explicitly revealed. Fraction schemes also did not extend to the Iterative fraction scheme, which is central to the construction of improper fractions. Based on these results, this study is expected to provide implications for the introduction of fractions in textbooks focusing on stages of units coordination to teachers and textbook developers.

• Education of Primary School Mathematics
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• v.21 no.2
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• pp.111-130
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• 2018

In this study, I compared and analyzed the contents of Korean, Japanese, Singapore, American, and Finnish textbooks about fraction which is one of the important and difficult concepts in elementary school mathematics. This is aimed to get the implications for meaningful fractional teaching and learning by analyzing the advantages and disadvantages of the methods and time of introducing the concept because fraction has the diversity of the sub-concepts and the introducing methods or process. As a result of the analysis, the fraction was introduced as part-whole(area) in all five countries' textbooks, but the use of number line, conversion between improper fraction and mixed number, whether to deal with part-whole(set) model. Furthermore, there are differences in the methods in obtaining of the equivalent fraction and the order of arrangement in comparison of fraction. Through this analysis, we discussed the reconsideration of the introducing contexts of fractions, the use of number line when introducing fractions, and the problem of segmentation and classification of contents.

• School Mathematics
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• v.9 no.3
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• pp.427-445
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• 2007

This study presents the results of a case study that investigated a seventh grader's fractional reasoning related to multiplicative reasoning. In addition, by employing domain analysis and taxonomic analysis for analyzing qualitative data, I show how a qualitative methodology was used for the data collected by teaching experiment methodology. The study identifies three distinct issues that emerged as the student engaged in solving fraction problems: a view of fractions as operations vs. results, the issue of units, and mixed numbers vs. improper fractions. These three issues have instructional implications in that each of them is critical in developing multiplicative reasoning and investigating how they relate to each other suggests a way to improve multiplicative reasoning in fractional contexts.