• East Asian mathematical journal
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• v.35 no.4
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• pp.467-488
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• 2019

The purpose of this study is to analyze multiplication facts(1st and 0th multiplication) in elementary mathematics. In the 2015 revised curriculum, students learn multiplication and multiplication facts in the 2nd grade. Many teachers experience difficulties in organizing the multiplication problem situation in multiplication facts(1st and 0th multiplication). This study aims to consider the causes of these difficulties and devise teaching methods. The method of this study is a comparative and analytic method. In order to compare textbooks, we select the Korean elementary mathematics textbooks(1st curriculum~2015 revised curriculum) and the six foreign elementary mathematics textbooks(Taiwan, Japan, Finland, Unites States, Hongkong, Singapore). As a result, the multiplication problem situation and the multiplication model assume the same bundle and bundle model. Also, we must consider the teaching timing of multiplication facts(1st and 0th multiplication) and the use of commutative law. In this study, we proposed a multiplication teaching scheme in consideration of the multiplication problem situation and teaching model, teaching period and commutative law etc.. to teach multiplication facts(1st and 0th multiplication) in elementary mathematics.

• Communications of Mathematical Education
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• v.35 no.4
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• pp.505-525
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• 2021

In this study, we investigated how multiplication by 2-digit numbers had been taught in elementary mathematics textbooks of Korea, Japan, Singapore, and USA. As a result of analysis, we found as follows. Korean textbooks do not teach the multiplication by 10 and the multiplication by power of 10, but Japanese, Singapore, and US textbooks explicitly teach related content. In the '×tens' teaching, Japanese and American textbooks teach formally the law of association of multiplication applied in the process of calculating the partial product of multiplication. The standard multiplication algorithm generally followed a standard method of recording partial product result according to the law of distribution, but the differences were confirmed in the multiplication model, the teaching method of the law of distribution, and the notation of the last digit '0'. Based upon these results, we suggested some proposals for improving the multiplication teaching.

• Journal of Educational Research in Mathematics
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• v.22 no.2
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• pp.293-309
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• 2012

The algorithm of natural number multiplication is one of the basic topics of elementary school mathematics. Mastery of algorithm and understanding of the principles are important educational aims. In this paper we analyzed elementary school mathematics textbooks of South Korea, the United States, Singapore, Japan. As a result of analysis, we found out that there are much differences in the teaching of multiplication with three numbers, '${\times}10$', and '${\times}tens$'. We suggested some implication for the teaching of algorithm of natural number multiplication.

• School Mathematics
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• v.11 no.4
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• pp.745-771
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• 2009

To acquire the hints of the development of children's multiplication strategies, this study tried to find the differences between the students who learned multiplication and the students who didn't. And we also tried to explore their acquired computational resources. As a result, we confirm that there is a certain direction on the development of children's multiplication strategies according to their grades and the level of acquirement of mathematical knowledge. Moreover, we comprehend that commutative law is an important part of the strategies on two-digit multiplication and that acquisition of the computational resources must precede the learning of multiplication strategies. In the end part, this article proposes a new taxonomy of strategies for multiplication. To support our proposal, we integrated the prior researches with our findings.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2022.05a
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• pp.246-248
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• 2022

Binary Edwards curves (BEdC), a new form of elliptic curves proposed by Bernstein, satisfy the complete addition law without exceptions. This paper describes an efficient hardware implementation of point scalar multiplication on BEdC using projective coordinates. Modified Montgomery ladder algorithm was adopted for point scalar multiplication, and binary field arithmetic operations were implemented using 257-bit binary adder, 257-bit binary squarer, and 32-bit binary multiplier. The hardware operation of the BEdC crypto-core was verified using Zynq UltraScale+ MPSoC device. It takes 521,535 clock cycles to compute point scalar multiplication.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.907-928
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• 2015

We show in many cases that the Siegel-Ramachandra invariants generate the ray class fields over imaginary quadratic fields. As its application we revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of inequality argument together with Shimura's reciprocity law.

• The Mathematical Education
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• v.48 no.3
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• pp.353-358
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• 2009

The article discusses the independence concept occurring in the learning of probability. The author does not distinguishes the independence in the events from the independence in the trials. Instead, the author suggests the physico-empirical independence and the logico-mathematical independence to distinguish between the two concepts.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.287-314
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• 2010

The algorithm is a chain of mechanical procedures, capable of solving a problem. In modern mathematics educations, the teaching algorithm is performing an important role, even though contracted than in the past. The conspicuous characteristic of current elementary mathematics textbook's manner of manipulating multiplication algorithm is exceeding converge to 'standard algorithm.' But there are many algorithm other than standard algorithm in calculating multiplication, and this diversity is important with respect to didactical dimension. In this thesis, we have reconstructed the experimental learning and teaching plan of multiplication algorithm unit by making the best use of diversity of multiplication algorithm. It's core contents are as follows. Firstly, It handled various modified algorithms in addition to standard algorithm. Secondly, It did not order children to use standard algorithm exclusively, but encouraged children to select algorithm according to his interest. As stated above, we have performed teaching experiment which is ruled by new lesson design and analysed the effects of teaching experiment. Through this study, we obtained the following results and suggestions. Firstly, the experimental learning and teaching plan was effective on understanding of the place-value principle and the distributive law. The experimental group which was learned through various modified algorithm in addition to standard algorithm displayed higher degree of understanding than the control group. Secondly, as for computational ability, the experimental group did not show better achievement than the control group. It's cause is, in my guess, that we taught the children the various modified algorithm and allowed the children to select a algorithm by preference. The experimental group was more interested in diversity of algorithm and it's application itself than correct computation. Thirdly, the lattice method was not adopted in the majority of present mathematics school textbooks, but ranked high in the children's preference. I suggest that the mathematics school textbooks which will be developed henceforth should accept the lattice method.

• School Mathematics
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• v.13 no.4
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• pp.581-598
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• 2011

Given the growing agreement that algebra should be taught in the early stage of the curriculum, considerable studies have been conducted with regard to early algebra in the elementary school. However, there has been lack of research on how to organize mathematic lessons to develop of algebraic reasoning ability of the elementary school students. This research attempted to gain specific and practical information on effective algebraic teaching and learning in the elementary school. An exploratory qualitative case study was conducted to the fourth graders. This paper focused on the associative law of the multiplication. This paper showed what kinds of activities a teacher may organize following three steps: (a) focus on the properties of numbers and operations in specific situations, (b) discovery of the properties of numbers and operations with many examples, and (c) generalization of the properties of numbers and operations in arbitrary situations. Given the steps, this paper included an analysis on how the students developed their algebraic reasoning. This study provides implications on the important factors that lead to the development of algebraic reasoning ability for elementary students.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.25 no.10
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• pp.1403-1408
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• 2021

Recently, FPGA-based AI processors are being studied actively. Deep convolutional neural networks (CNN) are basic computational structures performed by AI processors and require a very large amount of multiplication. Considering that the multiplication coefficients used in CNN inference operation are all constants and that an FPGA is easy to design a multiplier tailored to a specific coefficient, this paper proposes a methodology to optimize the multiplier. The method utilizes 2's complement and distributive law to minimize the number of bits with a value of 1 in a multiplication coefficient, and thereby reduces the number of required stacked adders. As a result of applying this method to the actual example of implementing CNN in FPGA, the logic usage is reduced by up to 30.2% and the propagation delay is also reduced by up to 22%. Even when implemented with an ASIC chip, the hardware area is reduced by up to 35% and the delay is reduced by up to 19.2%.