• 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.

• East Asian mathematical journal
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• v.34 no.2
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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.

• Journal of Elementary Mathematics Education in Korea
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• v.23 no.1
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• pp.1-17
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• 2019

In this study, prospective teachers were involved in arranging the teaching sequence of multiplication and division of fractions and decimal numbers based on their experience and knowledge of school mathematics. As a result, these activities provided an opportunity to demonstrate the prospective teachers' perception. Prospective teachers were able to learn the knowledge they needed by identifying the differences between their perceptions and curriculum. In other words, prospective teachers were able to understand the mathematical relationships inherent in the teaching sequence of multiplication and division of fractions and decimal numbers and the importance and difficulty of identifying students' prior knowledge and the effects of productive failures as teaching methods.

• School Mathematics
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• v.11 no.4
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• pp.685-706
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• 2009

This dissertation is aimed to investigate the reason why a contextualization is needed to help the meaningful teaching-learning concerning multiplications and divisions of fractions, the way to make the contextualization possible, and the methods which enable us to use it effectively. For this reason, this study intends to examine the differences of situations multiplying or dividing of fractions comparing to that of natural numbers, to recognize the changes in units by contextualization of multiplication of fractions, the context is set which helps to understand the role of operator that is a multiplier. As for the contextualization of division of fractions, the measurement division would have the left quantity if the quotient is discrete quantity, while the quotient of the measurement division should be presented as fractions if it is continuous quantity. The context of partitive division is connected with partitive division of natural number and 3 effective learning steps of formalization from division of natural number to division of fraction are presented. This research is expected to help teachers and students to acquire meaningful algorithm in the process of teaching and learning.

• Communications of Mathematical Education
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• v.19 no.4 s.24
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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.

• School Mathematics
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• v.11 no.4
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• pp.723-743
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• 2009

This paper analyzed the main contents of multiplication and division of fractions in elementary mathematics textbooks and workbooks aligned to the national mathematics curriculum. This paper first explored the adequacy of when to teach the contents, the connection of instructional flow across grades, and the method of constructing a unit to teach multiplication or division of fractions. This paper then analyzed in detail the contents with regard to the types and frequency of word problems, the types of visual models and frequency, and the process of formalizing the calculation methods and principles. It is expected that the issues and suggestions stemming from this analysis of current textbooks and workbooks are informative in developing new instructional materials aligned to the recently revised mathematics curriculum.

• The Mathematical Education
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• v.52 no.2
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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.

• The Mathematical Education
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• v.57 no.4
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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.

• The Mathematical Education
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• v.51 no.1
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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.

• Research in Mathematical Education
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• v.15 no.2
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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.