• Journal of Educational Research in Mathematics
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• v.27 no.3
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• pp.351-373
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• 2017

The meanings of division don't change and rather are connected from whole numbers to rational numbers. In this respect, connecting division of natural numbers, division of fractions, and division of decimal numbers could help for students to study division in meaningful ways. Against this background, the units of division of fractions and division of decimal numbers in fifth grade were redesigned in a way for students to connect meanings of division and procedures of division. The results showed that most students were able to understand the division meanings and build correct expressions. In addition, the students were able to make appropriate division situations when given only division expressions. On the other hand, some students had difficulties in understanding division situations with fractions or decimal numbers and tended to use specific procedures without applying diverse principles. This study is expected to suggest implications for how to connect division throughout mathematics in elementary school.

• Education of Primary School Mathematics
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• v.19 no.3
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• pp.193-210
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• 2016

In the $10{\div}2.4$ problem situation, we could find that curious upper and middle level students' solution. They solved $10{\div}2.4$ and wrote the result as quotient 4, remainder 4. In this curious response, we researched how students realize quotient and remainder in division of decimal. As a result, many students make errors in division of decimal especially in remainder. From these response, we constructed fraction based teaching method about division of decimal. This method provides new aspects about quotient and remainder in division of decimal, so we can compare each aspects' strong points and weak points.

• Journal of Educational Research in Mathematics
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• v.17 no.3
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• pp.233-251
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• 2007

The purpose of this study was to propose instructional methods using base-ten blocks in teaching the division of decimal for 5th grade students by analyzing the process of their conceptual comprehension. The students in this study were found to understand the two main meanings of the division of decimal, distribution and area, by modeling them with base-ten blocks. They were able to identify the algorithm through the use of base-ten blocks and to understand the principle of calculations by connecting the manipulative activities to each stage of algorithm. The students were also able to determine using base-ten blocks whether the results of division of decimal might be reasonable. This study suggests that the appropriate use of base-ten blocks promotes the conceptual understanding of the division of decimal.

• Journal of Educational Research in Mathematics
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• v.22 no.4
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• pp.445-458
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• 2012

Current textbooks may provide students and teachers with three improper notions related to the quotient and the remainder in division for decimal numbers as in the following. First, only the calculated results in (natural numbers)${\div}$(natural numbers) is the quotient. Second, when the quotient and the remainder are obtained in division for decimal numbers, the quotient is natural number and the remainder is unique. Third, only when the quotient cannot be divided exactly, the quotient can be rounded off. These can affect students and teachers on their notions of division for decimal numbers, so improvements are needed for to break it. For these improvements, the following measures are required. First, in the curriculum guidebook, the meaning of the quotient and the remainder in division for decimal numbers should be presented clearly, for preventing the possibility of the construction of such improper notions. Second, examples, problems, and the like should be presented in the textbooks enough to break such improper notions. Third, the didactical intention should be presented clearly with respect to the quotient and the remainder in division for decimal numbers in teacher's manual.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.3
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• pp.533-557
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• 2011

The purpose of this study was to understand PCK to improve professionalism of teachers and derive implications about proper teachings methods. For achieving these research purposes, different PCK and teaching methods in class of three teachers(A, B, C) were compared and analyzed targeting division of decimals for 6th grade. For this study, criteria of PCK analysis of teachers was set, PCK questionnaires were produced and distributed, teachers had interviews, PCK of teachers were analyzed, division of decimals class for 6th grade was observed and analyzed, and PCK of teachers and their classes were compared. The implications deriving from comparative analyzing PCK and classes are as follows. First of all, there was a close relation between PCK and classes, leading to a need for efforts of increasing PCK of teachers in every field in order to realize effective classes. Secondly, self study and in-service training are needed to enhance PCK of teachers. Thirdly, more of expertises and materials have to be provided on the instruction manual for teachers.

• School Mathematics
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• v.9 no.1
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• pp.99-117
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• 2007

In this paper it is discussed how children develop their logical reasoning beyond difficulties in the process of making sense of division with decimals in the classroom setting. When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter levels, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school should be clarified. This study focuses on the teaching and learning of division with decimals in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals. It is suggested that children begin to conceive division as the relationship between the equivalent expressions at the hypothetical-deductive level detached from the concrete one, and that children's explanation based on a reversibility of reciprocity are effective in overcoming the difficulties related to division with decimals. It enables children to conceive multiplication and division as a system of operations.

• Education of Primary School Mathematics
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• v.24 no.2
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• pp.103-114
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• 2021

In this study I recognized the problems with the use of the terms 'quotient' and 'reminder' in the division of decimal and explored ways to improve them. The prior studies and current textbooks critically analyzed because each researcher has different views on the use of the terms 'quotient' and 'reminder' because of the same view of the values in the division calculation. As a result of this study, I proposed to view the result 'q' and 'r' of division of decimals by division algorithms b=a×q+r as 'quotient' and 'reminder', and the amount equal to or smaller to q the problem context as a final 'result value' and the residual value as 'remained value'. It was also proposed that the approximate value represented by rounding the quotient should not be referred to as 'quotient'.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.39 no.7
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• pp.66-73
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• 2002

As floating-point operations become widely used in various applications such as computer graphics and high-definition DSP, the needs for fast division become increased. However, conventional floating-point dividers occupy a large hardware area, and bring bottle-becks to the entire floating-point operations. In this paper, a high-performance and small-area floating-point divider, which is suitable for embedded processors, is designed using he series expansion algorithm. The algorithm is selected to utilize two MAC(Multiply-ACcumulate) units for quadratic convergence to the correct quotient. The two MAC units for SIMD-DSP features are shared and the additional area for the division only is very small. The proposed divider supports all rounding modes defined by IEEE 754 standard, and error estimations are performed for appropriate precision.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.19 no.10
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• pp.2341-2349
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• 2015

The commonly used Goldschmidt's floating-point divider algorithm performs two multiplications in one iteration. In this paper, a tentative error corrected K'th Goldschmidt's floating-point number divider algorithm which performs K times multiplications in one iteration is proposed. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation in single precision and double precision divider is derived from many reciprocal tables with varying sizes. In addition, an error correction algorithm, which consists of one multiplication and a decision, to get exact result in divider is proposed. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a divider unit. Also, it can be used to construct optimized approximate reciprocal tables.

• Journal of Elementary Mathematics Education in Korea
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• v.21 no.1
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• pp.115-134
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• 2017

It is necessary to understand the characteristics of each type of division problems in other to help students develop a rich understanding when they learn each type of division problems. This study focuses on a specific type of division problems; a quotitive division as finding a scale factor in enlargement context. First, this study investigated via survey how 4th-6th graders and preservice and inservice elementary teachers solved a quotitive division relating to scaling problem. And semi-structured interviews with preservice and inservice elementary teachers were conducted to explore what knowledge they brought when they tried to solve enlargement quotitive division problems. Most of participants solved the given quotitive division problem in the same way. Only a few preservice and inservice teachers interpreted it as a proportion problem and solved in a different way. From the interviews, it was found that different conceptions of context and decontextualization, and different conceptions of times (as repeated addition or as a multiplicative operator) were connected to different solutions. Finally, three issues relating to teaching enlargement quotitive division were discussed; visual representation of two solutions, conceptions connected each solution, and integrating quotitive division and proportion in math textbooks.