• 한국수학교육학회지시리즈A:수학교육
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• 제50권3호
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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.

• East Asian mathematical journal
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• 제34권2호
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• pp.203-217
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• 2018

This study examined preservice elementary teachers' patterns and tendencies in thinking of drawn models of multiplication with fractions. In particular, it investigated preservice elementary teachers' work in a context where they were asked to select among drawn models for symbolic expressions illustrating multiplication with non-whole number fractions including a mixed number. Preservice teachers' interpretations of fraction multiplication used in interpreting different types of drawn models were analysed-both quantitatively and qualitatively. Findings and implications are discussed and further research is suggested.

• 한국초등수학교육학회지
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• 제23권1호
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• pp.1-17
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• 2019

본 연구에서 예비교사들은 자신들의 학교수학에 대한 경험과 지식을 토대로 분수의 곱셈과 나눗셈 지도 순서를 배치하는 활동을 하였다. 그 결과 교육과정의 제시된 순서와 일치한 경우는 없었지만 이러한 활동은 예비교사의 인식을 드러낼 수 있는 기회가 되었고 예비교사들은 자신의 인식과 교육과정의 차이 그리고 서로 간의 인식이 다름을 확인하면서 교사에게 필요한 지식을 배울 수 있었다. 즉, 예비교사들은 분수와 소수의 곱셈과 나눗셈 지도 순서에 내재된 수학적 관계를 알 수 있었고, 연산 지도에서 학생들의 사전 지식과 생각을 파악하는 것의 중요성과 어려움을 알 수 있었으며, 교수학습 방법으로서 생산적 실패의 효과를 체감할 수 있었다.

• 대한수학교육학회지:학교수학
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• 제11권4호
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• pp.685-706
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• 2009

이 연구는 분수의 곱셈 나눗셈에 관련한 교수-학습을 의미 있게 도울 수 있는 맥락화가 왜 필요하며, 어떻게 가능한지, 또한 효과적인 맥락화의 활용 방안은 무엇인지를 탐구하는 것을 목적으로 한다. 이를 위해 자연수에 대하여 분수의 곱셈 나눗셈 상황의 차이는 무엇인지를 살펴보고, 그 차이에 따라 분수의 곱셈에서는 승수인 연산자의 역할을 이해할 수 이는 맥락을 설정하여, 단위의 변화에 대한 인식을 하도록 하였다. 분수의 나눗셈에서 포함제는 그 몫이 이산량인 경우이면 남은 양이 생길 수 있고, 연속량인 경우에는 분수로 그 몫을 표현해야 하는 맥락으로 구분지었다. 그리고 등분제의 맥락은 자연수의 등분제의 맥락과 연결시켜 새롭게 제시하여, 자연수의 나눗셈에서 분수의 나눗셈으로 형식화되는 3단계의 효과적인 학습 방법을 제안하였다. 이로써 교사와 학생들의 분수의 곱셈과 나눗셈의 교수-학습 과정에 있어서 유의미한 알고리즘의 습득에 도움을 줄 수 있을 것으로 기대한다.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제19권4호
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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.

• 대한수학교육학회지:학교수학
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• 제11권4호
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• pp.723-743
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• 2009

본 논문은 제7차 및 개정 수학과 교육과정에서 제시한 분수의 곱셈과 나눗셈 지도 내용을 바탕으로 관련 내용을 다루는 현행 수학교과서와 익힘책을 상세하게 분석하였다. 우선 전반적인 지도 내용과 관련하여 지도시기의 적절성, 지도계열의 연계성, 차시구성의 적절성을 탐색한 후, 구체적으로 교과서의 내용 전개를 감안하여 각 연산별로 제시된 문장제의 유형과 빈도, 활용된 시각적 모델의 유형과 빈도, 계산방법과 원리의 형식화 과정을 세부적으로 분석하였다. 이를 통해 현재 개발 중인 수학교과용 도서의 기초 자료 및 구체적인 시사점을 제공하고자 한다.

• 한국수학교육학회지시리즈A:수학교육
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• 제52권2호
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• pp.217-230
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• 2013

This study attempts to propose the construction of textbook contents of fraction division and to suggest a method to strengthen the connection among problem context, manipulation activities and symbols by proposing an algorithm of dividing fractions based on problem contexts. As showing the suitable algorithm to problem context, it is able to understand meaningfully that the algorithm of fractions division is that of multiplication of a reciprocal. It also shows how to deal with remainder in the division of fractions. The results of this study are expected to make a meaningful contribution to textbook development for primary students.

• 한국수학교육학회지시리즈A:수학교육
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• 제57권4호
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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.

• 한국수학교육학회지시리즈A:수학교육
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• 제51권1호
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• pp.21-45
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• 2012

The purpose of the study was to investigate how students understand each operations with whole numbers and fractions, and the relationship between their knowledge of operations with whole numbers and conceptual understanding of operations on fractions. Researchers categorized students' understanding of operations with whole numbers and fractions based on their semantic structure of these operations, and analyzed the relationship between students' understanding of operations with whole numbers and fractions. As the results, some students who understood multiplications with whole numbers as only situations of "equal groups" did not properly conceptualize multiplications of fractions as they interpreted wrongly multiplying two fractions as adding two fractions. On the other hand, some students who understood multiplications with whole numbers as situations of "multiplicative comparison" appropriately conceptualize multiplications of fractions. They naturally constructed knowledge of fractions as they build on their prior knowledge of whole numbers compared to other students. In the case of division, we found that some students who understood divisions with whole numbers as only situations of "sharing" had difficulty in constructing division knowledge of fractions from previous division knowledge of whole numbers.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제15권2호
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• pp.197-208
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• 2011

By using the recording and quantitative analysis of two videos about "The multiplication and division of the Fractions" and the "Flanders Interaction Analysis System," we classified the teachers' language of instruction in algebra classroom and also analysis the language of instruction in the different teaching process. The results after the analysis as follows: (1) The proportion of time was taken in teachers' language of instruction is high and vary in types, most of the teachers' language is teachers' question; (2) In the different teaching process, the proportion of time was taken in teachers' language of instruction is different; (3) Teachers attached importance to explain the example and had the similar teaching strategy, but the teachers' language is different; (4) In the practice process, teachers placed importance on exploring the tough question and its teaching strategies are different. The teachers' questions are the main teachers' language of instruction.