• Journal of Educational Research in Mathematics
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• v.14 no.4
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• pp.367-385
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• 2004

This study intends to compare the way of introducing fractions in elementary mathematics textbooks of south and those of north Korea. After thorough investigations of the seven differences were identified. First, the mathematics textbooks of south Korea use concrete materials like apples when they introduce equal partition context, while those of north Korea do not use that kind of concrete materials. Second, in the textbooks of south Korea, equal partition of discrete quantities are considered after continuous ones are introduced. This is different from the approach of the north Korean text-books in which both quantities are regarded at the same time. Third, the quantitative fraction which refers to the rational number with unit of measure at the end of it, is hardly used in the textbooks of south. However, the textbooks of north Korea use it as the main representations of fractions. Fourth, in the textbooks of south Korea, vanous activities related to fractions are more emphasized, while in the textbooks of north Korea, various meanings of fractions textbooks from south and north Korea focused on the ways of introducing partition approach and equivalence relation as operational schemes of fractions, the following play an important role before defining fraction. Fifth, the textbooks of south Korea introduce equivalent fractions with number one using number bar, and do not consider the reason why that sort of fractions are regarded. On the contrary, the textbooks of north Korea introduce structural equivalence relation by using various contexts including length measure and volume measure situations. Sixth, whereas real-life contexts are provided for introducing equivalent fractions in the textbooks of south Korea, visual explanations and mathematical representations play an important role in the textbooks of north Korea. Seventh, the means of finding equivalent fractions are provided directly in the textbooks of south Korea, whereas the nature of equivalent fractions and the methods of making equivalent fractions are considered in the textbooks of north Korea.

• Communications of Mathematical Education
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• v.37 no.2
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• pp.135-157
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• 2023

Double scale models have been introduced in elementary mathematics textbooks under the 2015 revised mathematics curriculum. However, few studies have examined in detail how students understand or utilize such models. In this study, we analyzed how 154 sixth-grade students who had learned the division of fractions from textbooks containing double scale models understood such models. The results showed that the students tended to identify the components of the model relatively well, but had difficulties exploring the unit or the meaning of the bottom number line of a model. They also had a lot of difficulties using the double scale model to complete the computation process and explain the computation principle. Based on these findings, we discuss the implications of teaching double scale models.

• 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 the Korean School Mathematics Society
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• v.23 no.2
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• pp.203-222
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• 2020

Because the role of the teacher is important for the education to actualize the goals of the curriculum, the interest about the teacher's knowledges has been addressed as an important research topic. Among them, the pedagogical content knowledge is the knowledge that can emphasize the professionalism of the teacher. In this study, I analyzed the elementary pre-service teachers' the problem solving strategies that they imagined the methods that elementary school students can think about fraction division. Pre-service teachers who participated in this study were completed all of the mathematics education courses in the pre-service teachers' education courses. The research was conducted using the four type-problems of fraction division. The results showed that elementary pre-service teachers responded in the order of equal sharing problem-measurement division-partitive division-context of determination of a unit rate problem. They presented significant responses not only with typical algorithms but also with pictures or expressions. On the basis of this research, we have to take an interest in the necessity of sharing and recognizing various methods of fraction division in pre-service teachers education.

• International Area Studies Review
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• v.21 no.2
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• pp.3-20
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• 2017

Traditional co-integration analysis method is known to be difficult to clearly determine the relationship between the cointegrated variables. This study utilizes a fractional cointegation method and a causal analysis of time and frequency domain among the exchange rates of Korea, China and Japan. The results show that even though traditional cointegration methods did not clarify the existence of cointegration, exchange rates were fractionally cointegrated. Causal analysis of time domain and frequency domain provided somewhat different results, but the yen/dollar was useful for forecasting won/dollar and yuan/dollar. Proper use of causal analysis of frequency domain and fractional cointegration emthods may provide useful information that can not be explained from the traditional method.

• Education of Primary School Mathematics
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• v.25 no.2
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• pp.179-195
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• 2022

Mathematical communication competency, one of the six mathematical competencies emphasized in the latest mathematics curriculum, plays an important role both as a means and as a goal for students to learn mathematics. Therefore, it is meaningful to find instructional methods to improve students' mathematical communication competency and analyze their communication competency in detail. Given this background, this study analyzed 64 sixth graders' mathematical communication competency after they participated in the lessons of fraction division emphasizing mathematical communication. A written assessment for this study was developed with a focus on the four sub-elements of mathematical communication (i.e., understanding mathematical representations, developing and transforming mathematical representations, representing one's ideas, and understanding others' ideas). The results of this study showed that students could understand and represent the principle of fraction division in various mathematical representations. The students were more proficient in representing their ideas with mathematical expressions and solving them than doing with visual models. They could use appropriate mathematical terms and symbols in representing their ideas and understanding others' ideas. This paper closes with some implications on how to foster students' mathematical communication competency while teaching elementary mathematics.

• Education of Primary School Mathematics
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• v.27 no.1
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• pp.91-110
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• 2024

The purpose of this study is to explore visual models for deriving the fractional division algorithm, to see how students understand this integrated model, the rectangular partition model, when taught in elementary school classrooms, and how they structure relationships between fractional division situations. The conclusions obtained through this study are as follows. First, in order to remind the reason for multiplying the reciprocal of the divisor or the meaning of the reciprocal, it is necessary to explain the calculation process by interpreting the fraction division formula as the context of a measurement division or the context of the determination of a unit rate. Second, the rectangular partition model can complement the detour or inappropriate parts that appear in the existing model when interpreting the fraction division formula as the context of a measurement division, and can be said to be an appropriate model for deriving the standard algorithm from the problem of the context of the inverse of a Cartesian product. Third, in the context the inverse of a Cartesian product, the rectangular partition model can naturally reveal the calculation process in the context of a measurement division and the context of the determination of a unit rate, and can show why one division formula can have two interpretations, so it can be used as an integrated model.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.428-428
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• 2004

• The Mathematical Education
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• v.50 no.3
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• pp.337-354
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• 2011

The purpose of the study was to investigate how students understand multiplication and division of fractions and how their understanding influences the solutions of fractional word problems. Thirteen students from 5th to 6th grades were involved in the study. Students' understanding of operations with fractions was categorized into "a part of the parts", "multiplicative comparison", "equal groups", "area of a rectangular", and "computational procedures of fractional multiplication (e.g., multiply the numerators and denominators separately)" for multiplications, and "sharing", "measuring", "multiplicative inverse", and "computational procedures of fractional division (e.g., multiply by the reciprocal)" for divisions. Most students understood multiplications as a situation of multiplicative comparison, and divisions as a situation of measuring. In addition, some students understood operations of fractions as computational procedures without associating these operations with the particular situations (e.g., equal groups, sharing). Most students tended to solve the word problems based on their semantic structure of these operations. Students with the same understanding of multiplication and division of fractions showed some commonalities during solving word problems. Particularly, some students who understood operations on fractions as computational procedures without assigning meanings could not solve word problems with fractions successfully compared to other students.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.385-403
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• 2018

Fraction division can be categorized as partitive division, measurement division, and the inverse of a Cartesian product. In the contexts of quotitive division and the inverse of a Cartesian product, the multiply-by-the-reciprocal algorithm is drawn well out. In this study, I analyze the potential and significance of the method of using $1{\div}$(divisor) as an alternative way of developing the multiply-by-the-reciprocal algorithm in the context of quotitive division. The method of using $1{\div}$(divisor) in quotitive division has the following advantages. First, by this method we can draw the multiply-by-the-reciprocal algorithm keeping connection with the context of quotitive division. Second, as in other contexts, this method focuses on the multiplicative relationship between the divisor and 1. Third, as in other contexts, this method investigates the multiplicative relationship between the divisor and 1 by two kinds of reasoning that use either ${\frac{1}{the\;denominator\;of\;the\;divisor}}$ or the numerator of the divisor as a stepping stone. These advantages indicates the potential of this method in understanding the multiply-by-the-reciprocal algorithm as the common structure of fraction division. This method is based on the dual meaning of a fraction as a quantity and the composition of times which the current elementary mathematics textbook does not focus on. It is necessary to pay attention to how to form this basis when developing teaching materials for fraction division.