• Journal of the Korean School Mathematics Society
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• v.14 no.4
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• pp.477-490
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• 2011

The concept of pedagogical content knowledge (PCK) has been developed and expanded to identify essential components of mathematical knowledge for teaching (MKT) by Ball and her colleagues (2008). This study proposes an alternative perspective to view MKT focusing on key developmental understandings (KDUs) that carry through an instructional sequence, that are foundational for learning other ideas. In this study we provide constructive components of KDUs in fraction multiplication by focusing on the constructs of 'three-level-of-units structure' and 'recursive partitioning operation'. Expecially, our participating preservice elementary teacher, Jane, demonstrated that recursive partitioning operations with her length model played a significant role as a KDU in fraction multiplication.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2008.10a
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• pp.732-735
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• 2008

In order for fast calculation for the modular multiplication which plays an essential role in RSA cryptography algorithm, the Montgomery algorithm has been studed and developed in varous ways. Since there is no division operation in the algorithm, it is able to perform a fast modular multiplication. However, the Montgomery algorithm requires a few extra operations in the progress of which transformation from/to ordinary modular form to/from Montgomery form should be made. Concept of high radix operation can be considered by splitting the key size into word-defined units in the RSA cryptosystems which use longer than 1024 key bits. In this paper, We adopted the concept of operand scanning methods to enhance the traditional Montgomery algorithm. The methods consider issues of optimization, memory usage, and calculation time.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.1
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• pp.39-56
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• 2011

In 7th grade textbooks, the distributive property is generalized as in algebraic forms, and it seems that the students have not so good grip on this property. To get a good stock of knowledge on that generalized property, full understanding of it in concrete context should take precedence. This study would aim to propose some educational implications for better understanding of that property, through analysing the contents of it comparatively in Korean and Japanese elementary textbooks.

• Journal of Elementary Mathematics Education in Korea
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• v.19 no.4
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• pp.563-588
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• 2015

In elementary school, didactical transposition is inevitable due to several reasons. In mathematics, addition and multiplication are taught as binary operations, subtraction and division are taught as unary operations. But in elementary school, we try to teach all the four operations as binary operations by didactical transposition. In 'Mastering' the concepts of the four operations, the way of concept introduction is dealt importantantly. So it is different from understanding the four operations. In this study, we analyzed the four operations of natural numbers and fractions from two perspectives: concept understanding (how to introduce concepts and how to choose an operation) and connection between the operations. As a result, following implications were obtained. In division of fractions, students attempted a connection with multiplication of fractions right away without choosing an operation, based on the situation. Also, to understand division of fractions itself, integrate division of fractions presented from the second semester of the fifth grade to the first semester of the sixth grade are needed. In addition, this result can be useful in the future textbook development.

• Journal of Educational Research in Mathematics
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• v.18 no.4
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• pp.483-497
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• 2008

This study investigated what kind of informal knowledge is emergent and what role informal knowledge play in process of solving 2-digit by 2-digit multiplication task. The data come from 4 times interviews with a 3th grade student who had not yet received regular school education regarding 2-digit by 2-digit multiplication. And the data involves the student's activity paper, the characteristics of action and the clue of thinking process. Findings from these interviews clarify the child's informal knowledge to modeling strategy, doubling strategy, distributive property, associative property. The child formed informal knowledge to justify and modify her conjecture of the algorithm.

• Education of Primary School Mathematics
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• v.25 no.1
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• pp.57-79
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• 2022

Current mathematics It is necessary to ensure that ratio and proportion concept is not distorted or broken while being treated as if they were easy to teach and learn in school. Therefore, the purpose of this study is to analyze the activities presented in the textbook. Based on prior work, this study reinterpreted the proportional reasoning task from the proportional perspective of Beckmann and Izsak(2015) to the multiplicative structure of Vergnaud(1996) in four ways. This compared how they interpreted the multiplicative structure and relationships between two measurement spaces of ratio and rate units and proportional expression and proportional distribution units presented in the revised textbooks of 2007, 2009, and 2015 curriculum. First, the study found that the proportional reasoning task presented in the ratio and rate section varied by increasing both the ratio structure type and the proportional reasoning activity during the 2009 curriculum, but simplified the content by decreasing both the percentage structure type and the proportional reasoning activity. In addition, during the 2015 curriculum, the content was simplified by decreasing both the type of multiplicative structure of ratio and rate and the type of proportional reasoning, but both the type of multiplicative structure of percentage and the content varied. Second, the study found that, the proportional reasoning task presented in the proportional expression and proportional distribute sections was similar to the previous one, as both the type of multiplicative structure and the type of proportional reasoning strategy increased during the 2009 curriculum. In addition, during the 2015 curriculum, both the type of multiplicative structure and the activity of proportional reasoning increased, but the proportional distribution were similar to the previous one as there was no significant change in the type of multiplicative structure and proportional reasoning. Therefore, teachers need to make efforts to analyze the multiplicative structure and proportional reasoning strategies of the activities presented in the textbook and reconstruct them according to the concepts to teach them so that students can experience proportional reasoning in various situations.

• Journal for History of Mathematics
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• v.17 no.3
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• pp.23-32
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• 2004

In this paper, Ive investigated algebraic concepts which are contained in Euclid's Elements. In the Books II, V, and VII∼X of Elements, there are concepts of quadratic equation, ratio, irrational numbers, and so on. We also analyzed them for mathematical meaning with modem symbols and terms. From this, we can find the essence of the genesis of algebra, and the implications for students' mathematization through the experience of the situation where mathematics was made at first.

• Proceedings of the IEEK Conference
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• 2003.07b
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• pp.867-870
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• 2003

본 논문은 멀티미디어 데이터 처리를 위한 효율적인 RISC 프로세서 유닛의 설계를 목표로 Vector 프로세서의 SIMD(Single Instruction Multiple Data) 개념을 바탕으로 고정된 연산기 데이터 비트 수에 비해 상대적으로 작은 비트수의 데이터 연산의 부분 병렬화를 통하여 멀티미디어 데이터 연산의 기본이 되는 곱셈누적(MAC : Multiply and Accumulate) 연산의 성능을 향상 시킨다. 또한 기존의 MMX나 VIS 등과 같은 범용 프로세서들의 부분 병렬화를 위해 전 처리 과정의 필요충분조건인 데이터의 연속성을 위해 서로 다른 길이의 데이터 흑은 비트 수가 작은 멀티미디어의 데이터를 하나의 데이터로 재처리 하는 재정렬 혹은 Packing/Unpacking 과정이 성능 전체적인 성능 저하에 작용하게 되므로 본 논문에서는 기존의 프로세서의 연산기 구조를 재이용하여 병렬 곱셈을 위한 연산기 구조를 구현하고 이를 위한 데이터 정렬 연산 구조를 제안한다.

• The Mathematical Education
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• v.47 no.4
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• pp.405-419
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• 2008

The purpose of this study is to find oui the effects of teaching mathematical concepts by designing and applying teaching and learning programs that takes into consideration the students' strong intelligence, through the teaching and learning strategies based on the multiple intelligences theory. For this study, developmental and experimental research was conducted. In the developmental research part of the study, teaching and learning programs for teaching the concept of multiplication were designed and the activities based on the multiple intelligences were chosen. On the other hand, in the experimental research part, the data acquired from the application of nonequivalent control group pretest-posttest design in the actual classes was processed and analyzed. The results above indicate that the teaching and learning program based on the multiple intelligences theory improved the students' overall understanding of mathematical concepts by providing various types of activities. In addition, this program helped students to increase their confidence and generate a positive attitude towards learning math.

• Journal of the Korean School Mathematics Society
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• v.13 no.3
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• pp.345-356
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• 2010

Variables and expressions are essential for doing mathematics. Especially abbreviations of the multiplication sign ${\times}$ and the division sign ${\div}$ are current rules that we usually follow. In this paper, we devised a Card-Game for exercising abbreviations of the multiplication sign ${\times}$ and the division sign ${\div}$ in calculating expressions, designed a teaching unit for the calculation of expressions using the Card-Game in the variables and expressions strand, and discussed the implications of using the Card-Game for motivating students, cooperative learning, diagnosis and correction of errors, and so on.