• School Mathematics
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• v.13 no.2
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• pp.267-286
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• 2011

In this paper I investigated the historical developments of the algorithms for multiplication of natural numbers. Through this analysis I tried to describe more concretely what is to understand the common algorithm for multiplication of natural numbers. I found that decomposing dividends and divisors into small numbers and multiplying these numbers is the main strategy for carrying out multiplication of large numbers, and two decomposing and multiplying processes are very important in the algorithms for multiplication. Finally I proposed some implications based on these analysis.

• 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.

• Education of Primary School Mathematics
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• v.25 no.3
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• pp.251-278
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• 2022

The purpose of this study is to obtain implications for mathematical education by analyzing how the multiplication and division of decimal numbers are presented in the elementary mathematics textbooks in Korea, Japan, Singapore, and Finland. Compared to the fact that students often have misconceptions about multiplication and division of decimal numbers, there have been not many comparative studies in recent elementary mathematics textbooks. For this study, we selected elementary mathematics textbooks those are widely used in Japan, Singapore, and Finland along with Korean elementary mathematics textbooks. We chose the textbooks because the students in the selected countries have scored high in international achievement studies such as TIMSS and PISA. The analysis was examined in terms of elementary mathematics curriculum related to multiplication and division of decimal numbers, introduction and content, real-life situations, use of visual models, and formalization methods of algorithms. As a result of the study, the mathematics curricula related to multiplication and division of decimal numbers includes estimation in Korea and Finland, while Japan and Singapore emphasize real-life connections more, and Finland completes the operations in secondary schools. The introduction and content are intensively provided in a short period of time or distributed in various grades and semesters. The real-life situations are presented in a simple sentence format in all countries, and the use of visual models or formalization of algorithms is linked to the operations of natural numbers in unit conversions. Suggestions were made for textbook development and teacher training programs.

• School Mathematics
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• v.7 no.2
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• pp.139-168
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• 2005

The purpose of this study is to investigate children's informal knowledge of the fractional multiplication and to develop a teaching material connecting the informal and the formal knowledge. Six lessons of the pre-teaching material are developed based on literature reviews and administered to the 7 students of the 4th grade in an elementary school. It is shown in these teaching experiments that children's informal knowledge of the fractional multiplication are the direct modeling of using diagram, mathematical thought by informal language, and the representation with operational expression. Further, teaching and learning methods of formalizing children's informal knowledge are obtained as follows. First, the informal knowledge of the repeated sum of the same numbers might be used in (fractional number)$\times$((natural number) and the repeated sum could be expressed simply as in the multiplication of the natural numbers. Second, the semantic meaning of multiplication operator should be understood in (natural number)$\times$((fractional number). Third, the repartitioned units by multiplier have to be recognized as a new units in (unit fractional number)$\times$((unit fractional number). Fourth, the partitioned units should be reconceptualized and the case of disjoint between the denominator in multiplier and the numerator in multiplicand have to be formalized first in (proper fractional number)$\times$(proper fractional number). The above teaching and learning methods are melted in the teaching meterial which is made with corrections and revisions of the pre-teaching meterial.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.475-496
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• 2018

Although the multiplication of decimal fractions is expected to be easy for students to understand because of the similarity to natural numbers multiplication in computing methods, students show many errors in the multiplication of decimal fractions. This is a result of the instruction focused more on skill mastery than conceptual understanding. This study is a basic study for effectively developing a unit of multiplication of decimal fractions. For this purpose, we analyzed the curriculums' performance standards, significance in teaching-learning and evaluation, contents and methods for teaching multiplication of decimal fractions from the 7th curriculum to the revised curriculum of 2015 and the textbooks' activities and lessons. Further, we analyzed preceding studies and introductory books to suggest effective directions for developing teaching unit. As a result of the analysis, three implications were obtained: First, a meaningful instruction for estimation is needed. Second, it is necessary to present a visual model suitable for understanding the meaning of decimal multiplication. Third, the process of formalizing an algorithms for multiplying decimal fractions needs to be diversified.

• 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 the Korean Mathematical Society
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• v.34 no.1
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• pp.159-165
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• 1997

In this paper, k will denote an arbitrary field. If m, n are natural numbers, then $M_{m \times n}(k)$ will denote the set of all $m \times n$ matrices with entries in k. Every k-algebras will be assumed to contain a (multiplicative) identity $1 \neq 0$. A k-subspace $R_0$ of a k-algebra R will be called a k-subalgebra of R if $R_0$ is closed under multiplication from R and $R_0$ contains the identity of R. We will assume all k-algebra homomorphisms take the identity to identity.

• School Mathematics
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• v.11 no.4
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• pp.685-706
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• 2009

This dissertation is aimed to investigate the reason why a contextualization is needed to help the meaningful teaching-learning concerning multiplications and divisions of fractions, the way to make the contextualization possible, and the methods which enable us to use it effectively. For this reason, this study intends to examine the differences of situations multiplying or dividing of fractions comparing to that of natural numbers, to recognize the changes in units by contextualization of multiplication of fractions, the context is set which helps to understand the role of operator that is a multiplier. As for the contextualization of division of fractions, the measurement division would have the left quantity if the quotient is discrete quantity, while the quotient of the measurement division should be presented as fractions if it is continuous quantity. The context of partitive division is connected with partitive division of natural number and 3 effective learning steps of formalization from division of natural number to division of fraction are presented. This research is expected to help teachers and students to acquire meaningful algorithm in the process of teaching and learning.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.1045-1073
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• 2014

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359].

• The Mathematical Education
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• v.40 no.2
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• pp.195-215
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• 2001

The purpose of this paper is to present a specific set of home teaching methods in hopes to prevent slow learner of the elementary mathematics. This paper deals with the number and operations, one of five topics in the elementary mathematics A survey of two hundred elementary school teachers was made to see the teacher's opinions of the role of home studying and to concretize the contents of the research topics. There were asked which is the most essential contents for the concrete loaming and which is the most difficult monad that might cause slow leaner. And those were found to be; counting, and arithmetic operations(addition and subtraction) of one or two-digit numbers and multiplication and their concepts representations and operations(addition and subtraction) of fractions. The home teaching methods are based on the situated learning about problem solving in real life situations and on the active teaming which induces children's participation in the process of teaching and learning. Those activities in teaching each contents are designed to deal with real objects and situations. Most teaching methods are presented in the order of school curriculum. To teach the concepts of numbers and the place value, useful activities using manipulative materials (Base ten blocks, Unifix, etc.) or real objects are also proposed. Natural number's operations such as addition, subtraction and multiplication are subdivided into small steps depending upon current curriculum, then for understanding of operational meaning and generalization, games and activities related to the calculation of changes are suggested. For fractions, this paper suggest 10 learning steps, say equivalent partition, fractional pattern, fractional size, relationship between the mixed fractions and the improper fraction, identifying fractions on the number line, 1 as a unit, discrete view point of fractions, comparison of fractional sizes, addition and subtraction, quantitative concepts. This research basically centers on the informal activities of kids under the real-life situation because such experiences are believed to be useful to prevent slow learner. All activities and learnings in this paper assume children's active participation and we believe that such active and informal learning would be more effective for learning transfer and generalization.