• 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.

• School Mathematics
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• v.11 no.1
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• pp.71-91
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• 2009

The purpose of this study was to analyze operation sense in detail with regard to division of fraction. For this purpose, two sixth grade students who were good at calculation were clinically interviewed three times. The analysis was focused on (a) how the students would understand the multiple meanings and models of division of fraction, (b) how they would recognize the meaning of algorithm related to division of fraction, and (c) how they would employ the meanings and properties of operation in order to translate them into different modes of representation as well as to develop their own strategies. This paper includes several episodes which reveal students' qualitative difference in terms of various dimensions of operation sense. The need to develop operation sense is suggested specifically for upper grades of elementary school.

• East Asian mathematical journal
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• v.32 no.4
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• pp.565-587
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• 2016

The purpose of this study was to analyze the understanding of the meaning of fraction division and fraction division algorithm of elementary mathematical gifted students through the process of problem posing and solving activities. For this goal, students were asked to pose more than two real-world problems with respect to the fraction division of ${\frac{3}{4}}{\div}{\frac{2}{3}}$, and to explain the validity of the operation ${\frac{3}{4}}{\div}{\frac{2}{3}}={\frac{3}{4}}{\times}{\frac{3}{2}}$ in the process of solving the posed problems. As the results, although the gifted students posed more word problems in the 'inverse of multiplication' and 'inverse of a cartesian product' situations compared to the general students and pre-service elementary teachers in the previous researches, most of them also preferred to understanding the meaning of fractional division in the 'measurement division' situation. Handling the fractional division by converting it into the division of natural numbers through reduction to a common denominator in the 'measurement division', they showed the poor understanding of the meaning of multiplication by the reciprocal of divisor in the fraction division algorithm. So we suggest following: First, instruction on fraction division based on various problem situations is necessary. Second, eliciting fractional division algorithm in partitive division situation is strongly recommended for helping students understand the meaning of the reciprocal of divisor. Third, it is necessary to incorporate real-world problem posing tasks into elementary mathematics classroom for fostering mathematical creativity as well as problem solving ability.

• The Mathematical Education
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• v.60 no.4
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• pp.409-429
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• 2021

This study investigated instructional methods for fractional division emphasizing algebraic thinking with sixth graders. Specifically, instructional elements for fractional division emphasizing algebraic thinking were derived from literature reviews, and the fractional division instruction was reorganized on the basis of key elements. The instructional elements were as follows: (a) exploring the relationship between a dividend and a divisor; (b) generalizing and representing solution methods; and (c) justifying solution methods. The instruction was analyzed in terms of how the key elements were implemented in the classroom. This paper focused on the fractional division instruction with problem contexts to calculate the quantity of a dividend corresponding to the divisor 1. The students in the study could explore the relationship between the two quantities that make the divisor 1 with different problem contexts: partitive division, determination of a unit rate, and inverse of multiplication. They also could generalize, represent, and justify the solution methods of dividing the dividend by the numerator of the divisor and multiplying it by the denominator. However, some students who did not explore the relationship between the two quantities and used only the algorithm of fraction division had difficulties in generalizing, representing, and justifying the solution methods. This study would provide detailed and substantive understandings in implementing the fractional division instruction emphasizing algebraic thinking and help promote the follow-up studies related to the instruction of fractional operations emphasizing algebraic thinking.

• 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.

• School Mathematics
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• v.14 no.2
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• pp.191-212
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• 2012

In the field of arithmetic in mathematics education, there has been lack of fine-grained investigations addressing the relationship between students' construction of division knowledge with fractional quantities and their whole number division knowledge. This study, through the analysis of part of collected data from a year-long teaching experiment, presents a possible constructive itinerary as to how a student could modify her unit-segmenting scheme to deal with various fraction measurement division situations: 1) unit-segmenting scheme with a remainder, 2) fractional unit-segmenting scheme. Thus, this study provides a clue for curing a fragmentary approach to teaching whole number division and fraction division and preventing students' fragmentary understanding of the same arithmetical operation in different number systems.