• East Asian mathematical journal
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• 제33권4호
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• pp.353-380
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• 2017

The purpose of this study is to analyze the multiplication units in the elementary school mathematics. In the 2015 revised curriculum, students learn multiplication in $2^{nd}$ grade. The multiplication units is divided into two: multiplication and multiplication facts. In these two units, we mainly analyze situations involving multiplication, models for teaching multiplication, and multiplication strategies for teaching multiplication facts in relation to Subject Matter Knowledge. We called these contents Multiplication Matter Knowledge. We examined the precedent study with regard to multiplication at the elementary mathematics. As results, we prepared an analysis framework for this study. This study was conducted according to qualitative research methods, expecially 'qualitative contents analysis'. The contents here refer to Multiplication Matter Knowledge that can be found in the elementary mathematics textbooks and working books etc. As results of analysis, We can confirm that various multiplication situations and multiplication models are presented in the textbooks. And it has been examined that various multiplication properties are presented in the textbook according to the multiplication strategy levels. We insist elementary school teachers should be aware of these Multiplication Matter Knowledge. This study aims to provide elementary school teachers with basic data in these contexts.

• East Asian mathematical journal
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• 제32권4호
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• pp.443-464
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• 2016

The purpose of ths study is to compare and analyze the sequence of teaching multiplication facts in the elementary school mathematics. Generally, the multiplication in the elementary school mathematics is composed of the followings; concepts of multiplication, situations involving multiplication, didactical models for multiplication, and multiplication strategies for teaching multiplication facts. This study is focusing to multiplication facts, especially to the sequence of teaching and multiplication strategies. 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~2009 revised curriculum) and the 9 foreign elementary mathematics textbooks(Japan, China, Germany, Finland, Hongkong etc.). As results of comparative investigation, the sequence of teaching multiplication facts is reconsidered on a basis of elementary students' mathematical thinking. And the connectivity of multiplication facts is strengthened in comparison with the foreign elementary mathematics textbooks. Finally multiplication strategies for teaching multiplication facts are discussed for more understanding and reasoning the principles of multiplication facts in the elementary school mathematics.

• East Asian mathematical journal
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• 제35권4호
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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.

• 대한수학교육학회지:학교수학
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• 제15권4호
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• pp.889-920
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• 2013

본 연구는 초등학교 수학에서 곱셈 개념의 도입과 곱셈 구구 지도를 위한 교수학적 배경을 알아보고, 앞으로의 곱셈 지도 개선을 위한 시사점을 제공하는 데 그 목적이 있다. 이를 위해 여러 곱셈 연구에 대한 이론적 고찰을 통해 곱셈 지도의 교수학적 배경의 핵심 내용으로 곱셈 개념, 곱셈 상황, 곱셈 지도 모델, 곱셈 전략을 추출하고 이에 대해 살펴보았고, 이를 기초로 미국, 핀란드, 네덜란드, 독일과 우리나라 교과서를 비교 분석하였다. 이런 이론적 고찰과 분석 결과를 바탕으로 이후의 우리나라 초등학교 수학의 곱셈 지도 개선을 위한 시사점으로 곱셈 상황에서 묶음 상황의 다양화와 곱셈적 비교 상황의 강조 및 데카르트 곱의 재고, 곱셈 지도 모델에서 묶음 모델, 배열 모델, 직선 모델의 균형과 구조화와 형식화로의 이행, 곱셈 구구 지도에서 곱셈 전략과 곱셈 성질의 강조와 곱셈 구구 사이의 연결을 제안하였다.

• 대한수학회지
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• 제60권2호
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• pp.327-339
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• 2023

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

• 한국학교수학회논문집
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• 제22권3호
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• pp.303-327
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• 2019

본 연구는 초등학교 수학에서 곱셈에 대한 학생들의 이해를 돕는 하나의 방안으로 곱셈의 통합적 접근을 제안하였다. 곱셈의 통합적 접근이란 수학 수업에서 학생들이 하나의 곱셈 상황을 다양한 방법으로 해결하고 서로의 방법에 대해 탐색하고 논의하면서 곱셈에 대해 폭넓은 이해를 하도록 하는 것이다. 곱셈의 통합적 접근은 곱셈에 대한 다양한 접근, 일관적 접근, 특정한 접근을 강조한 여러 선행 연구를 기반으로 도출되었다. 연구 결과, 곱셈의 통합적 접근은 하나의 곱셈 상황을 크게 4가지 방법으로 해석할 수 있으며 각각의 방법은 선행 연구에서 강조한 곱셈의 중요한 특성과 모두 연결된다. 또한, 곱셈의 통합적 접근은 곱셈뿐만 아니라 나눗셈, 분수 및 분수의 연산, 비와 비율, 비례 등으로 자연스럽게 확장되는 데 중요하다는 것을 이론적으로 확인하였다. 이를 통해 초등학교 수학에서 다루는 곱셈과 관련하여 실제 수업을 진행하는 교사에게 시사점을 제공하고자 한다.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제24권1호
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• pp.1-27
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• 2021

For elementary school children, learning the standard multiplication algorithm with accuracy, clarity, consistency, and efficiency is a daunting task. Nonetheless, what should be our expectation in procedural fluency, for example, in finding the product of 25 and 37 among fifth grade students? Collectively, has the mathematics education community emphasized the value of conceptual understanding to the detriment of procedural fluency? In addition to examining these questions, we survey multiplication algorithms throughout history and in textbooks and reconceptualize the standard multiplication algorithm by using a new tool called the Multiplication Aid Template.

• 한국인터넷방송통신학회논문지
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• 제14권2호
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• pp.121-126
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• 2014

큰 -자리수의 2개 10진수에 대한 곱셈을 보다 빠르게 수행하는 방법은 존재하는가? 이 문제는 수학과 컴퓨터공학 분야에서 미해결 문제로 남아 있다. 이 문제에 대해 곱셈 횟수를 줄이는 연구로는 Karatsuba와 Toom-Kook 알고리즘이 있다. 본 논문은 곱셈 횟수를 줄이는 방법과는 완전히 별개로, 10진수 곱셈을 전적으로 덧셈만으로 효율적으로 수행하는 방법을 제안하였다. 제안된 방법은 2진수의 자리이동-덧셈법만으로도 RSA-100과 같이 컴퓨터로 수행이 불가한 매우 큰 자리수의 10진수 곱셈을 수행할 수 있음을 보였다. 제안된 방법은 수행 복잡도 (n) 의 덧셈으로 곱셈을 수행한다.

• 충청수학회지
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• 제25권4호
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• pp.717-723
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• 2012

In this paper, we investigate when gr-multiplication modules are finitely generated and show that if M is a finitely generated gr-multiplication R-module then there is a lattice isomorphism between the lattice of all graded ideals I of R containing ann(M) and the lattice of all graded submodules of M.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.315-321
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• 2012

In this paper, we provide the necessary and sufficient conditions for a faithful graded module to be a graded multiplication module and for a graded submodule of a faithful gr-multiplication to be gr-essential.