• Journal of Educational Research in Mathematics
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• v.26 no.4
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• pp.663-682
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• 2016

This study conducted an analysis of strategies that the 3rd to 6th grade elementary students used when they were solving problems of comparing the size of the fractions with like and unlike denominators, and unit fractions. Although there were slight differences in the students' use of strategies according to the problem types, students were found to use the 'part-whole strategy', 'transforming strategy', and 'between fractions strategy' frequently. But 'pieces strategy', 'unit fraction strategy', 'within fraction strategy', and 'equivalent fraction strategy' were not used frequently. In regard to the strategy use that is appropriate to the problem condition, it was found that students needed to use the 'unit fraction strategy', and the 'within fraction strategy', whereas there were many errors in their use of the 'between fractions strategy'. Based on the results, the study attempted to provide pedagogical implications in teaching and learning for comparing the size of the fractions.

• School Mathematics
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• v.16 no.4
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• pp.783-802
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• 2014

The purpose of this study was to explore in detail students' understanding of fraction as quotient. A total of 158 sixth graders in 6 elementary schools were surveyed by 8 tasks in relation to fraction as quotient. As a result, students used various partitioning strategies to solve the given sharing tasks such as partitioning the singleton unit, the composite unit, or the whole unit of the dividend. They also used incorrect partitioning strategies that were not appropriate to the given context. Students' partitioning strategies and performance of fraction as quotient varied depending on the given contexts and models. This study suggests that students should have rich experience to partition various units and reinterpret the context based on the singleton unit of the dividend.

• School Mathematics
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• v.14 no.1
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• pp.135-150
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• 2012

This paper analyzes the activities for (fraction) ${\times}$(fraction) in Korean elementary textbooks focusing on the connection between visual models and the algorithm. New Korean textbook attempts a new approach to use length model (as well as rectangular area model) for developing the standard algorithm for the multiplication of fractions, $\frac{a}{b}{\times}\frac{d}{c}=\frac{a{\times}d}{b{\times}c}$. However, activities with visual models in the textbook are not well connected to the algorithm. To bridge the gap between activities with models and the algorithm, distributive strategy should be emphasized. A wealth of experience of solving problems of fraction multiplication using the distributive strategy with visual models can serve as a strong basis for developing the algorithm for the multiplication of fractions.

• The Mathematical Education
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• v.60 no.1
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• pp.1-19
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• 2021

This study investigates students' conceptions of fractions from a measurement approach while providing a technological environment designed to support students' understanding of the relationships between quantities and adjustable units. 13 third-graders participated in this study and they were involved in a series of measurement tasks through task-based interviews. The tasks were devised to investigate the relationship between units and quantity through manipulations. Screencasting videos were collected including verbal explanations and manipulations. Drawing upon the theory of semiotic mediation, students' constructed concepts during interviews were coded as mathematical words and visual mediators to identify conceptual profiles using a fine-grained analysis. Two students changed their strategies to solve the tasks were selected as a representative case of the two profiles: from guessing to recursive partitioning; from using random units to making a relation to the given unit. Dragging mathematical objects plays a critical role to mediate and formulate fraction understandings such as unitizing and partitioning. In addition, static and dynamic representations influence the development of unit concepts in measurement situations. The findings will contribute to the field's understanding of how students come to understand the concept of fraction as measure and the role of technology, which result in a theory-driven, empirically-tested set of tasks that can be used to introduce fractions as an alternative way.

• School Mathematics
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• v.7 no.2
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• pp.139-168
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• 2005

The purpose of this study is to investigate children's informal knowledge of the fractional multiplication and to develop a teaching material connecting the informal and the formal knowledge. Six lessons of the pre-teaching material are developed based on literature reviews and administered to the 7 students of the 4th grade in an elementary school. It is shown in these teaching experiments that children's informal knowledge of the fractional multiplication are the direct modeling of using diagram, mathematical thought by informal language, and the representation with operational expression. Further, teaching and learning methods of formalizing children's informal knowledge are obtained as follows. First, the informal knowledge of the repeated sum of the same numbers might be used in (fractional number)$\times$((natural number) and the repeated sum could be expressed simply as in the multiplication of the natural numbers. Second, the semantic meaning of multiplication operator should be understood in (natural number)$\times$((fractional number). Third, the repartitioned units by multiplier have to be recognized as a new units in (unit fractional number)$\times$((unit fractional number). Fourth, the partitioned units should be reconceptualized and the case of disjoint between the denominator in multiplier and the numerator in multiplicand have to be formalized first in (proper fractional number)$\times$(proper fractional number). The above teaching and learning methods are melted in the teaching meterial which is made with corrections and revisions of the pre-teaching meterial.

• Journal of the Korean School Mathematics Society
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• v.21 no.4
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• pp.303-326
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• 2018

Fraction has difficulties in learning because of the diversity of meanings, the ways of presenting contents and teaching methods in elementary school mathematics. Therefore, the various strategies of teaching of fraction concept is proposed as an alternative. The problem of equal sharing problem is that children can experience the concept of fractions naturally in the context of everyday distribution. Even before learning formal fractions, children can solve them in various ways based on their own experiences. The purpose of this study is to investigate the degree of problem solving and problem solving strategies for children in 2nd, 4th, and 6th grades in elementary school. As a result of the research, the percentage of correct answers increased as the grade increased, but the grade levels showed a difference depending on the numbers given to the problems. Also, there were differences in the problem solving strategies according to the grade levels. Also, according to the numbers presented in the problem, the percentage of correct answers was high in items that were easy to divide, and the percentage of correct answers was low in items that were difficult to divide. When children solved the problems, they were affected by the strategies they could use immediately according to the number presented in the problem, and their learning experiences were also affected.

• School Mathematics
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• v.14 no.1
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• pp.29-43
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• 2012

This paper investigated the conceptual schemes in which four children constructed a strategy representing the situation as a figure and partitioning it related to the work which they quantify the result of partitioning to various types of fractions when an equal sharing situation was given to them in contextual or an abstract symbolic form of division. Also, the paper researched how the relationship of factors and multiples between the numerator and denominator, or between the divisor and dividend affected the construction. The children's partitioning strategies were developed such as: repeated halving stage ${\rightarrow}$ consuming all quantity stage ${\rightarrow}$ whole number objects leftover stage ${\rightarrow}$ singleton object analysis/multiple objects analysis ${\rightarrow}$ direct mapping stage. When children connected the singleton object analysis with multiple object analysis, they finally became able to conceptualize division as fractions and fractions as division.

• School Mathematics
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• v.10 no.4
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• pp.603-623
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• 2008

The purpose of this research was to gain a better understanding of the characteristics of students' mathematical representations using additive strategy in ratio and proportion tasks. The additive strategy is the erroneous one used most often among the strategies reported in solving ratio and proportion tasks. It is a problem solving strategy that preserves the difference from one ratio to another. Students' additive strategies were categorized into four parts: subtracting without considering units of quantities, comparing the numbers that represent the whole subtracted from the part and same part, adding the difference, and subtracting the difference. In order to change from additive strategy to multiplicative strategy, the researcher asked to find out the unit quantity and found the characteristics of students' mathematical notations in the following: Firstly, the students made the number which they wanted by multiplying and adding same numbers. Secondly, they represented the mid-points between natural numbers. Thirdly, they related $a{\div}b$ to decimal number, not $\frac{a}{b}$. Fourthly, they were inclined to divide the larger number with the smaller number without understanding the context of the problem. These results are interpreted as showing that lower level of performance in the dividing operation with the notations of fraction hinders the transformation from additive strategy to multiplicative strategy.

• Journal of Educational Research in Mathematics
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• v.16 no.2
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• pp.157-177
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• 2006

The concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this development process. One fourth student, Kyungsu, was encourage to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Kyungsu made during the teaching experiment and analyzes the challenges Kyungsu faced in attempting to schematize his method. Finally, the mathematical knowledge Kyungsu needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.