• Communications of Mathematical Education
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• v.23 no.3
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• pp.707-734
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• 2009

The purpose of the study was to investigate how children understand addition and subtraction of fractions and how their understanding influences the solutions of fractional word problems. Twenty students from 4th to 6th grades were involved in the study. Children's understanding of operations with fractions was categorized into "joining", "combine" and "computational procedures (of fraction addition)" for additions, "taking away", "comparison" and "computational procedures (of fraction subtraction)" for subtractions. Most children understood additions as combining two distinct sets and subtractions as removing a subset from a given set. In addition, whether fractions had common denominators or not did not affect how they interpret operations with fractions. Some children understood the meanings for addition and subtraction of fractions as computational procedures of each operation without associating these operations with the particular situations (e.g. joining, taking away). More children understood addition and subtraction of fractions as a computational procedure when two fractions had different denominators. In case of addition, children's semantic structure of fractional addition did not influence how they solve the word problems. Furthermore, we could not find any common features among children with the same understanding of fractional addition while solving the fractional word problems. In case of subtraction, on the other hand, most children revealed a tendency to solve the word problems based on their semantic structure of the fractional subtraction. Children with the same understanding of fractional subtraction showed some commonalities while solving word problems in comparison to solving word problems involving addition of fractions. Particularly, some children who understood the meaning for addition and subtraction of fractions as computational procedures of each operation could not successfully solve the word problems with fractions compared to other children.

• Education of Primary School Mathematics
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• v.27 no.2
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• pp.187-198
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• 2024

The addition and subtraction relationship and the multiplication and division relationship are explicitly dealt with in second and third grade mathematics textbooks. However, these relationships are not discussed anymore in the problem situations and activities in the 4th, 5th, and 6th grade mathematics textbooks. In this study, we investigate the calculation principles of subtraction and division in the elementary school mathematics textbooks. Based on our investigation, we justify the addition and subtraction relationship and the multiplication and division relationship at the level of children's understanding so that we discuss some problem situations and activities where the relationships can be applied to subtraction and division. In addition, we suggest educational implications that can be obtained from children's applying the relationships and the properties of equations to subtraction and division.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1995.11a
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• pp.151-162
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• 1995

보다 짧은 길이의 덧셈/뺄셈 사슬($addition_{traction-chain}$)을 찾는 문제는 정수론을 기반으로 하는 많은 암호시스템들에 있어서 중요한 문제이다. 특히, RSA에서의 모듈라멱승(modular exponentiation)이나 타원 곡선(elliptic curve)에서의 곱셈 연산시간은 덧셈사슬(addition-chain) 또는 덧셈/뺄셈 사슬의 길이와 정비례한다 본 논문에서는 덧셈/뻘셈 사슬을 구하는 새로운 알고리즘을 제안하고, 그 성능을 분석하여 기존의 방법들과 비교한다. 본 논문에서 제안하는 알고리즘은 작은윈도우(small-window) 기법을 기반으로 하고, 뺄셈을사용해서 윈도우의 개수를 최적화함으로써 덧셈/뺄셈 사슬의 길이를 짧게 한다. 본 논문에서 제안하는 알고리즘은 512비트의 정수에 대해 평균길이 595.6의 덧셈/뺄셈 사슬을 찾는다.

• Communications of Mathematical Education
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• v.16
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• pp.331-344
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• 2003

본 연구의 목적은 예비 초등교사의 덧셈과 뺄셈에 대한 교과 지식과 교수학적 지식이 어떠한가를 알아보는 것이었다. 29명의 예비 초등교사가 연구에 참여하였으며 자료는 개방형 답을 하는 질문지를 사용하여 수집하였다. 분석결과 예비 초등교사들은 문장제에서 의미론적 구성과 합병과 구차의 상황에 대한 이해에 어려움을 가지고 있는 것으로 나타났다. 교수학적 방법에서는 알고리즘에 의한 설명 방법을 주로 사용하였으며 뺄셈을 설명하는데 몇 가지 오개념을 보였다. 이 결과는 앞으로 초등교사양성대학의 프로그램 개발과 운영에 기초가 될 것이다.

• Education of Primary School Mathematics
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• v.19 no.3
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• pp.177-191
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• 2016

Students can calculate the addition and subtraction problem using informal knowledge before receiving the formal instruction. Recently, the value that a computation lesson focus on the understanding and developing the various strategies is highlighted by curriculum developers as well as in reports. Ideally, a educational setting and classroom culture reflected students' learning and problem solving strategies. So, this paper analyzed the similarity and difference with respect to the numeric sentence and word problem in the addition and subtraction. The subjects for the study were 100 third-grade Korean students and 68 third-grade American students. Researcher developed the questionnaire in the addition and subtraction and used it for the survey. The following results have been drawn from this study. The computational ability of Korean students was higher than that of American students in both the numeric sentence and word problem. And it was revealed the differences of the strategies which were used problem solving process. Korean students tended to use algorithms and numbers' characters and relations, but American students tended to use the drawings and algorithms with drawings.

• Journal of Educational Research in Mathematics
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• v.24 no.4
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• pp.623-644
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• 2014

Many researchers have supported in various aspects that elementary students should experience alternative algorithms as well as formal standard one for addition and subtraction. Korean elementary mathematics textbooks have some units for alternative algorithms for addition and subtraction. In special, the change of unit sequence in the second grade revised mathematics textbooks may cause the necessity for discussion about teaching sequence and teaching purpose between alternative algorithms and formal standard one. Therefore, this study aims to consider the purpose of teaching alternative algorithms and to induce implications for their teaching strategies and sequence. To do this, related references, curriculum and textbooks were analyzed. Four lessons were observed and three teachers were interviewed. The main content of this study is the result of analysis on students' activities and teachers'teaching approaches. This study also includes didactical implications based on the result.

• School Mathematics
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• v.10 no.4
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• pp.557-571
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• 2008

The objective of this paper is to study De Morgan's thoughts on teaching and learning negative numbers. We studied De Morgan's point of view on negative numbers, and analyzed his didactical approaches for negative numbers. De Morgan make students explore impossible subtractions, investigate the rule of the impossible subtractions, and construct the signification of the impossible subtractions in succession. In De Morgan' approach, teaching and learning negative numbers are connected with that of linear equations, the signs of impossible subtractions are used, and the concept of negative numbers is developed gradually following the historic genesis of negative numbers. Also, we analyzed the strengths and weaknesses of the De Morgan's approaches compared with the mathematics curriculum.

• Journal of Educational Research in Mathematics
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• v.19 no.2
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• pp.307-319
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• 2009

Current school mathematics introduces addition/subtraction between natural numbers, fractions, decimal fractions, and square roots, step-by-step in order. It seems that, however, school mathematics focuses too much on learning the calculation method of addition/subtraction between each stages of numbers, to lead most of students to understand the coherent principle, lying in addition/subtraction algorithm between real numbers in all. This paper raises questions on this problematic approach of current school mathematics, in learning addition/subtraction. This paper intends to clarify the fact that, if we recognize addition/subtraction between numbers from the viewpoint of 'measuring' and 'common measure', as Dewey did when he argued that the psychological origin of the concept of number was measuring, then we could find some common principles of addition/subtraction operation, beyond the superficial differences among algorithms of addition/subtraction between each stages of numbers. At the end, this paper suggests the necessity of improving the methods of learning addition/subtraction in current school mathematics.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 2003.07a
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• pp.307-311
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• 2003

Okeya-Sakurai는［12］에서 단순한 형태의 랜덤한 덧셈-뺄셈 체인의 대응방법［14］은 SPA공격에 취약함을 보였다. 그러나 그들의 분석 방법은 복잡한 형태［14］에는 적용되지 않는다. 본 논문에서는 Okeya-Sakurai의 공격 알고리듬에 두 가지 잠재된 문제가 있음을 보인다. 또한［12,15］와는 다른 강하고 견고한 새로운 공격 알고리듬을 제안한다. 본 논문에서 제안하는 공격 알고리듬을 사용하면 복잡한 형태의 랜덤한 덧셈-뺄셈 체인［14］또한 완벽하게 분석된다. 본 논문의 결과를 표준에서 제안된 163비트로 실험한 결과 단순한 형태에서는 20개의 AD수열로 대략 94%의 확률로 공격이 성공하며 30개의 AD수열로는 대략 99%의 확률로 공격이 성공한다. 또한, 복잡한 형태에서는 40개의 AD수열로 94%의 확률로 70개의 AD수열로는 99%로의 확률로 공격이 성공한다.

• KIPS Transactions on Computer and Communication Systems
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• v.2 no.2
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• pp.83-92
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• 2013

RSA-CRT is a widely used algorithm that provides high performance implementation of the RSA-signature algorithm. Many previous studies on each operation step have been published to verify the physical leakages of RSA-CRT when used in smart devices. This paper proposes SAED (subtraction algorithm analysis on equidistant data), which extracts sensitive information using the event information of the subtraction operation in a reduction algorithm. SAED is an attack method that uses algorithm-dependent power signal changes. An adversary can extract a key using differential power analysis (DPA) of the subtraction operation. This paper indicates the theoretical rationality of SAED, and shows that its results are better than those of other methods. According to our experiments, only 256 power traces are sufficient to acquire one block of data. We verify that this method is more efficient than those proposed in previously published studies.