• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.593-607
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• 2021

The aim of the present paper is to establish an intrinsic generalization of Cartan connection in Finsler geometry. This connection is called the vertical recurrent Finsler connection. An intrinsic proof of the existence and uniqueness theorem for such connection is investigated. Moreover, it is shown that for such connection, the associated semi-spray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Explicit intrinsic expression relating this connection and Cartan connection is deduced. We also investigate some applications concerning the fundamental geometric objects associated with this connection. Finally, three important results concerning the curvature tensors associated to a special vertical recurrent Finsler connection are studied.

• Communications of Mathematical Education
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• v.34 no.4
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• pp.587-603
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• 2020

This study is a basic study to effectively develop a mathematics experience object, an important tool and educational tool in mathematics education. Recently, as mathematics education based on action theory is emphasized, various mathematics experience objects are being developed. It is also used through various after-school activities in the school. However, there are insufficient cases in which a mathematics experience teaching tools is developed and used as a tool for explaining mathematics concepts in mathematics classrooms. Also, the mathematical background of the mathematics experience teaching tools used by students is unclear. For this reason, the mathematical understanding of the toolst for mathematics experience is also very insufficient. Therefore, in this study, a development model is proposed as a systematic method for developing a mathematics experience teaching tools. Also, in this study, we developed 'the Great Common Divisor' mathematics experience teaching tool according to the development model. Through the model proposed through this study and the actual mathematics experience teaching tool, the development of various tools for mathematical experience will be practically implemented. In addition, it is expected that various tools for experiencing mathematics based on mathematical foundations will be developed.

• The Mathematical Education
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• v.49 no.1
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• pp.39-51
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• 2010

The purpose of this article is to study students' responses to non-routine problems which are presented by using solely numerals or symbolic figures. Such figures have no mathematical meaning but just symbolical meaning. Most students understand geometric figures more concrete objects than numerals because geometric figures such as circles and squares can be visualized by the manipulatives in real life. And since students need not consider (unvisible) any operational structure of numerals when they deal with (visible) figures, problems proposed using figures are considered relatively easier to them than those proposed using numerals. Under this assumption, we analyze students' problem solving processes of numeral problems and figural problems, and then find out when students' difficulties arise in the problem solving process and how they response when they feel difficulties. From this experiment, we will suggest several comments which would be considered in the development and application of both numerical and figural problems.

• East Asian mathematical journal
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• v.24 no.5
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• pp.527-545
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• 2008

One of the education purpose of the section "Figures" in the eighth grade is to develop students' deductive reasoning ability, which is basic and essential for living in a democratic society. However, most or middle school students feel much more difficulty or even frustration in the study of formal arguments for geometric situations than any other mathematical fields. It is owing to the big gap between inductive reasoning in elementary school education and deductive reasoning, which is not intuitive, in middle school education. Also, it is very burden for students to describe geometric statements exactly by using various appropriate symbols. Moreover, Usage of the same symbols for angle and angle measurement or segments and segments measurement makes students more confused. Since geometric relations is mainly determined by the measurements of geometric objects, students should be able to interpret the geometric properties to the algebraic properties, and vice verse. In this paper, we first compare and contrast inductive and deductive reasoning approaches to justify geometric facts and relations in school curricula. Convincing arguments are based on experiment and experience, then are developed from inductive reasoning to deductive proofs. We introduce teaching methods to help students's understanding for deductive reasoning in the textbook by using stepwise visualization materials. It is desirable that an effective proof instruction should be able to provide teaching methods and visual materials suitable for students' intellectual level and their own intuition.

• School Mathematics
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• v.2 no.1
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• pp.29-51
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• 2000

In this study, we tried to suggest an example of the analysis on the use of mathematical manipulatives. Taking algebra tiles as an example of mathematical manipulatives, we analysed several effects resulted from the use of algebra tiles. The algebra tiles make it possible to do activities that are needed to introduce and explain the distributive law and factoring. The algebra tiles have a several advantages; First of all, This model is simple. Even though they cannot make algebra easy, this model can play an important role in the transition to a new algebra course. This model provides access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense. This model provides a geometric interpretation of symbol manipulation, thereby enriching students' understanding, This model supports cooperative learning, and help improve discourse in the algebra class by giving students objects to think with and talk about. On the other hand, The disadvantages of this model are as follows; the model reinforces the misconception that -x is negative, and x is positive; the area model of multiplication is not geometrically sound when minus is involved; only the simplest expressions involving minus can be represented; It is ineffective when be used the learning of already known concept. Mathematics teachers must have a correct understanding about these advantages and disadvantages of manipulatives. Therefore, they have to plan classroom work that be maximized the positive effect of manipulatives and minimized the negative effect of manipulatives.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.189-196
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• 1990

In this paper, we try to generalize ultraproducts in the category of locally convex spaces. To do so, we introduce D-ultracolimits. It is known [7] that the topology on a non-trivial ultraproduct in the category T $V^{ec}$ of topological vector spaces and continuous linear maps is trivial. To generalize the category Ba $n_{1}$ of Banach spaces and linear contractions, we introduce the category L $C_{1}$ of vector spaces endowed with families of semi-norms closed underfinite joints and linear contractions (see Definition 1.1) and its subcategory, L $C_{2}$ determined by Hausdorff objects of L $C_{1}$. It is shown that L $C_{1}$ contains the category LC of locally convex spaces and continuous linear maps as a coreflective subcategory and that L $C_{2}$ contains the category Nor $m_{1}$ of normed linear spaces and linear contractions as a coreflective subcategory. Thus L $C_{1}$ is a suitable category for the study of locally convex spaces. In L $C_{2}$, we introduce $l_{\infty}$(I. $E_{i}$ ) for a family ( $E_{i}$ )$_{i.mem.I}$ of objects in L $C_{2}$ and then for an ultrafilter u on I. we have a closed subspace $N_{u}$ . Using this, we construct ultraproducts in L $C_{2}$. Using the relationship between Nor $m_{1}$ and L $C_{2}$ and that between Nor $m_{1}$ and Ba $n_{1}$, we show thatour ultraproducts in Nor $m_{1}$ and Ba $n_{1}$ are exactly those in the literatures. For the terminology, we refer to [6] for the category theory and to [8] for ultraproducts in Ba $n_{1}$..

• The Mathematical Education
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• v.46 no.3
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• pp.245-261
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• 2007

This paper deals with the possible problems which may arise when students learn the names of elementary geometric figures in the languages of Korean, Chinese, English. The names of some simple geometric figures in these languages are analyzed, and a specially designed test was administered to some grade eight students from the three language groups to explore the possible influence of the characteristics of the languages on students' capability in identifying the figures, the way students define the figures, and students' understanding of the inclusive relationship among figures. It was found that the usage of the terms to describe geometric figures may indeed have affected students' understanding of the figures. The names of geometric figures borrowed from those of everyday life objects may cause students to fix on some attributes of the objects which may not be consistent with the definition of the figures. Even when the names of the geometric figures depict the features of the figures, the words used in the naming of the figures may still affect students' understanding of the inclusive relations. If there is discrepancy between the definition of a geometric figure and the features that the name depicts, it may affect students' understanding of the definition of the figure, and if there is inconsistency in the classification of figures, it may affect students' understanding of the inclusive relationship involving those figures. Some implications of the study are then discussed.

• Publications of The Korean Astronomical Society
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• v.26 no.2
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• pp.71-87
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• 2011

Gravitational waves are predicted by the Einstein's theory of General Relativity. The direct detection of gravitational waves is one of the most challenging tasks in modern science and engineering due to the 'weak' nature of gravity. Recent development of the laser interferometer technology, however, makes it possible to build a detector on Earth that is sensitive up to 100-1000 Mpc for strong sources. It implies an expected detection rate of neutron star mergers, which are one of the most important targets for ground-based detectors, ranges between a few to a few hundred per year. Therefore, we expect that the gravitational-wave observation will be routine within several years. Strongest gravitational-wave sources include tight binaries composed of compact objects, supernova explosions, gamma-ray bursts, mergers of supermassive black holes, etc. Together with the electromagnetic waves, the gravitational wave observation will allow us to explore the most exotic nature of astrophysical objects as well as the very early evolution of the universe. This review provides a comprehensive overview of the theory of gravitational waves, principles of detections, gravitational-wave detectors, astrophysical sources of gravitational waves, and future prospects.

• 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.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.2
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• pp.215-238
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• 2016

The 2015 national revised curriculum was notified officially the last year. The intent and direction of the revision caused more or less change for mathematical contents to be taught and is expected to cause a considerable change in math class. In the level of elementary school mathematics, it turned that several contents were deleted or moved to the upper grades because the revision focused especially both on reducing students' burden of learning and on fostering the mathematical key competences. This study aims to examine the relevance of the change through investigation of the national curriculums for elementary school mathematics since 1946. The mathematical contents to be analyzed in this study were mixed calculation of natural numbers, mixed calculation of fractions and decimal fractions, position and direction of objects, are/hectare and ton, the range of numbers and estimating, surface and volume of cylinders, pattern and correspondence, and direct/inverse proportionality, which were changed in any aspect relative to 2009 national revised curriculum. Based on the results of these analyses, the discussion will provide some suggestions for setting the direction of elementary mathematics curriculum.