• Journal of the Korean School Mathematics Society
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• v.25 no.2
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• pp.105-124
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• 2022

The contents of fraction division in textbooks are important because there were changes in situations and problem solving methods in textbooks according to the revision of the curriculum and the contents of textbooks affect students' learning directly. So, this study analyzed the achievement standards of the curriculum and formula types and situations, and the introduction process of non-standard and standard algorithms presented in Korean mathematics textbooks. The results are follows: there was little difference in the achievement standards of the curriculum, but there was a difference in the arrangement of contents by grades in textbooks. There was a difference in the types of formula according to textbooks. And the situation became more diverse; recent textbooks have changed to the direction of using the non-standard and the standard algorithm in parallel. In conclusion, I proposed categorizing rather than splitting the types of fraction division, the connection of non-standard and standard algorithm, and the need to prepare methods to pursue generalization and justification according to the common characteristics in the process of introducing standard algorithm.

• Proceedings of the IEEK Conference
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• 2003.07d
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• pp.1391-1394
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• 2003

본 논문은 ARM7 TDMI 마이크로프로세서의 연산기능 중 구현되지 알은 나눗셈 연산 기능을 추가로 구현하였다. 이를 위해 ARM ISA(Instruction Set Architecture)에 부호를 고려한 나눗셈 명령어인 'SDIV' 명령어를 추가로 정의하였으며, 나눗셈 알고리즘 Signed Nonrestoring Division을 수행할 수 있도록 ARM7 TDMI 마이크로프로세서의 Data Path를 재 설계하였다. 제안된 방법의 타당성을 검증하기 위하여 현재 ARM7 TDMI 마이크로프로세서의 정수 나눗셈 연산처리 방법과 제안된 구조에서의 정수 나눗셈 연산 처리 방법을 비교하였으며, 그 겉과 수행 cycle의 수가 40％로 감소되는 것을 확인하였다

• Communications of Mathematical Education
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• v.15
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• pp.71-76
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• 2003

학생들의 분수 나눗셈에 대한 이해는 개념적 이해를 바탕으로 수행되어야 함에도 불구하고 분수 나눗셈은 많은 학생들이 기계적인 절차적 지식으로 획득할 가능성이 높은 내용이다. 이것은 학생들이 학교에서 분수 나눗셈을 학습할 때에 일상생활에서의 경험과 선행 학습과의 연결이 잘 이루어지지 못하고 있는 것에 큰 원인이 있다고 본다. 본 연구에서는 학생들의 분수 나눗셈의 개념적 이해를 돕기 위하여 경험적 지식과의 연결 관계를 활용한 교수 방안을 실험 교수를 통해 조사하였다. 결과로서 번분수를 활용한 수업은 분수 나눗셈의 표준 알고리즘이 수행되는 이유를 알 수 있게 하는데 도움이 되나 여러 가지 절차적 지식이 뒷받침되어야 하며 분수 막대를 직접 잘라 보는 활동을 통한 수업은 분수 나눗셈에서의 나머지를 이해하는데 효과가 있다는 것을 알았다. 결론적으로, 학생들의 경험과 학교에서 이미 학습한 분수 나눗셈들의 관련 지식들을 적절히 연결하도록 한다면 수학적 연결을 통해 분수 나눗셈의 개념적 이해를 이끌 수 있다.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.29 no.5C
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• pp.726-736
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• 2004

In this paper, we proposed a new scalar multiplier structure needed for an elliptic curve cryptosystem(ECC) over the standard basis in GF(2$^{163}$ ). It consists of a bit-serial multiplier and a divider with control logics, and the divider consumes most of the processing time. To speed up the division processing, we developed a new division algorithm based on the extended Euclid algorithm. Dynamic data dependency of the Euclid algorithm has been transformed to static and fixed data flow by a localization technique, to make it independent of the input and field polynomial. Compared to other existing scalar multipliers, the new scalar multiplier requires smaller gate counts with improved processor performance. It has been synthesized using Samsung 0.18 um CMOS technology, and the maximum operating frequency is estimated 250 MHz. The resulting performance is 148 kbps, that is, it takes 1.1 msec to process a 163-bit data frame. We assure that this performance is enough to be used for digital signature, encryption/decryption, and key exchanges in real time environments.

• School Mathematics
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• v.9 no.1
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• pp.13-28
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• 2007

Division of fractions can be categorized as measurement division, partitive or sharing division, the inverse of multiplication, and the inverse of Cartesian product. Division algorithm for fractions has been interpreted with manipulative aids or models mainly in the contexts of measurement division and partitive division. On the contrary, there are few interpretations for the context of the inverse of a Cartesian product. In this paper the significance and the limits of existing interpretations of division of fractions in the context of the inverse of a Cartesian product were discussed. And some new easier interpretations of division algorithm in the context of a Cartesian product are developed. The problem to determine the length of a rectangle where the area and the width of it are known can be solved by various approaches: making the width of a rectangle be equal to one, making the width of a rectangle be equal to some natural number, making the area of a rectangle be equal to 1. These approaches may help students to understand the meaning of division of fractions and the meaning of the inverse of the divisor. These approaches make the inverse of a Cartesian product have many merits as an introductory context of division algorithm for fractions.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.1
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• pp.105-122
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• 2014

In this paper, fraction division algorithms in Korean elementary mathematics textbooks are analyzed as a part of the groundwork to improve teaching methods for fraction division algorithms. There are seemingly six fraction division algorithms in ${\ll}Math\;5-2{\gg}$, ${\ll}Math\;6-1{\gg}$ textbooks according to the 2006 curriculum. Four of them are standard algorithms which show the multiplication by the reciprocal of the divisors modally. Two non-standard algorithms are independent algorithms, and they have weakness in that the integration to the algorithms 8 is not easy. There is a need to reconsider the introduction of the algorithm 4 in that it is difficult to think algorithm 4 is more efficient than algorithm 3. Because (natural number)${\div}$(natural number)=(natural number)${\times}$(the reciprocal of a natural number) is dealt with in algorithm 2, it can be considered to change algorithm 7 to algorithm 2 alike. In textbooks, by converting fraction division expressions into fraction multiplication expressions through indirect methods, the principles of calculation which guarantee the algorithms are explained. Method of using the transitivity, method of using the models such as number bars or rectangles, method of using the equivalence are those. Direct conversion from fraction division expression to fraction multiplication expression by handling the expression is possible, too, but this is beyond the scope of the curriculum. In textbook, when dealing with (natural number)${\div}$(proper fraction) and converting natural numbers to improper fractions, converting natural numbers to proper fractions is used, but it has been never treated officially.

• 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.7 no.2
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• pp.103-121
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• 2005

This article compares and analyzes mathematics textbooks of North Korea, South Korea, China and Japan and draws meaningful ways for introducing the division algorithm of fractions. The analysis is based on the five contexts: 'measurement division', 'determination of a unit rate', 'reduction of the quantities in the same measure', 'division as the inverse of multiplication or Cartesian product', 'analogy with multiplication algorithm of fractions'. The main focus of the analysis is what context is used to introduce the algorithm and how much it can appeal to students. This analysis supports that there is a few differences of introducing methods the division algorithm of fractions among those countries and more meaningful way can be considered than ours. It finally suggests that we teach the algorithm in a way which can have students easily see the reason of multiplying the reciprocal of a divisor when they divide with fractions. For this, we need to teach the meaning of a reciprocal of fraction and consider to use the context of determination of a unit rate.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.44 no.3 s.357
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• pp.35-42
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• 2007

The finite-field division can be applied to the elliptic curve cryptosystems. However, an efficient algorithm and the hardware design are required since the finite-field division takes much time to compute. In this paper, we propose a radix-4 systolic divider on $GF(2^m)$ with comparative area and performance. The algorithm of the proposed divide, is mathematically developed and new counter structure is proposed to map on low-cost systolic cells, so that the proposed systolic architecture is suitable for YLSI design. Compared to the bit-parallel, bit-serial and digit-serial dividers, the proposed divider has relatively effective high performance and low cost. We design and synthesis $GF(2^{193})$ finite-field divider using Dongbuanam $0.18{\mu}m$ standard cell library and the maximum clock frequency is 400MHz.

• Journal of Science Education
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• v.46 no.1
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• pp.109-120
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• 2022

Fraction division is the content that is important but difficult to learn because it includes the process of finding a numerical expression in the real-world context, the process of making a context that matches a numerical expression, how to solve division, and the justification of standard algorithm. This study analyzes the word problems and problem solving methods about fraction division which elementary pre-service teachers represented. Pre-service teachers have more difficulty in making word problem where the dividend is less than the divisor and they also show typical errors in making the word problems. There were differences in the methods presented according to the contexts of division in problem solving. Through this study, it is necessary to rethink the teaching methods for fraction division instruction in the curriculum for pre-service teachers and analyze the formation process of 'knowledge for content and teaching' because of the differences in responses between grades.