• The Mathematical Education
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• v.42 no.5
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• pp.637-646
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• 2003

Visual representation is very important topic in Mathematics Education since it fosters understanding of Mathematical concepts, principles and rules and helps to solve the problem. So, the purpose of this paper is to analyze and clarify the various meaning and roles about the visual representation. For this purpose, I examine the status of the visual representation. Since the visual representation has the roles of creatively mathematical activity, we emphasize the using of the visual representation in teaching and learning. Next, I examine the errors in relation to the visual representation which come from limitation of the visual representation. It suggests that students have to know conceptual meaning of the visual representation when they use the visual representation. Finally, I suggest some examples of problem solving via the visual representation. This examples clarify that the visual representation gives the clues and solution of problem solving. Students can apprehend intuitively and easily the mathematical concepts, principles and rules using the visual representation because of its properties of finiteness and concreteness. So, mathematics teachers create the various visual representations and show students them. Moreover, mathematics teachers ask students to design the visual representation and teach students to understand the conceptual meaning of the visual representation.

• Education of Primary School Mathematics
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• v.16 no.3
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• pp.285-301
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• 2013

The purpose of the study is to analyze the 4th graders' visual representation in mathematics problem solving process and to find out how to teach the visual representation in mathematics problem solving process. on the basis of the results, this study gives several pedagogical implication related to the mathematics problem solving. The following were the conclusions drawn from the results obtained in this study. First, The achievement level of students and using visual representation in the mathematics problem solving are closely connected. High achieving students used visual representation in the mathematics problem solving process more frequently. Second, high achieving students realize the usefulness of visual representation in the mathematics problem solving process and use visual representation to solve mathematical problem. But low achieving students have no conception that visual representation is one of the method to solve mathematical problem. Third, students tend to especially focus on 'setting up an equation' when they solve a mathematical problem. Because they mostly experienced mathematical problems presented by the type of 'word problem-equation-answer'. Fourth even through students tried visual representation to solve a mathematical problem, they could not solve the problem successfully in numerous instances. Because students who face a difficulty in solving a problem try to construct perfect drawing immediately. But generating visual representation 2)to represent mathematical problem cannot be constructed at one swoop.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.12 no.7
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• pp.3172-3193
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• 2018

Recently, several studies have shown that linear representation based approaches are very effective and efficient for image classification. One of these linear-representation-based approaches is the Collaborative representation (CR) method. The existing algorithms based on CR have two major problems that degrade their classification performance. First problem arises due to the limited number of available training samples. The large variations, caused by illumintion and expression changes, among query and training samples leads to poor classification performance. Second problem occurs when an image is partially noised (contiguous occlusion), as some part of the given image become corrupt the classification performance also degrades. We aim to extend the collaborative representation framework under limited training samples face recognition problem. Our proposed solution will generate virtual samples and intra-class variations from training data to model the variations effectively between query and training samples. For robust classification, the image patches have been utilized to compute representation to address partial occlusion as it leads to more accurate classification results. The proposed method computes representation based on local regions in the images as opposed to CR, which computes representation based on global solution involving entire images. Furthermore, the proposed solution also integrates the locality structure into CR, using Euclidian distance between the query and training samples. Intuitively, if the query sample can be represented by selecting its nearest neighbours, lie on a same linear subspace then the resulting representation will be more discriminate and accurately classify the query sample. Hence our proposed framework model the limited sample face recognition problem into sufficient training samples problem using virtual samples and intra-class variations, generated from training samples that will result in improved classification accuracy as evident from experimental results. Moreover, it compute representation based on local image patches for robust classification and is expected to greatly increase the classification performance for face recognition task.

• School Mathematics
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• v.5 no.3
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• pp.361-384
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• 2003

The purpose of this study was to investigate students' problem solving process based on the model of IDEAL if they learn to solve word problems of simultaneous linear equations through structure-representation instruction. The problem solving model of IDEAL is followed by stages; identifying problems(I), defining problems(D), exploring alternative approaches(E), acting on a plan(A). 160 second-grade students of middle schools participated in a study was classified into those of (a) a control group receiving no explicit instruction of structure-representation in word problem solving, and (b) a group receiving structure-representation instruction followed by IDEAL. As a result of this study, a structure-representation instruction improved word-problem solving performance and the students taught by the structure-representation approach discriminate more sharply equivalent problem, isomorphic problem and similar problem than the students of a control group. Also, students of the group instructed by structure-representation approach have less errors in understanding contexts and using data, in transferring mathematical symbol from internal learning relation of word problem and in setting up an equation than the students of a control group. Especially, this study shows that the model of direct transformation and the model of structure-schema in students' problem solving process of I and D stages.

• East Asian mathematical journal
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• v.29 no.4
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• pp.449-466
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• 2013

This paper started from a question: Can it help a student solve the problem to give supports in point of view of a teacher knowing the solution. We performed a case study to get an answer for the question. We analysed a case which students do not make full use of data in the mathematical problem from this point of view of the mental representation. We examined closely the cause for not making full use of data. We got that the wrong mental representation which the students get from data in the problem lead to not making full use of data. We knew that it is insufficient to present the data not making use of. To help a student truly, it is necessary to give a aid based on a student's mental representation. From the conclusion of study, We got that figuring out student's mental representation is important and hope that many investigation about student's mental representation for various problem occur with frequency.

• Education of Primary School Mathematics
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• v.8 no.1
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• pp.23-32
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• 2004

It is not too much to say that problem solving is still the focus of school mathematics though the trend of mathematics education for ten year from the one of 1980 is problem solving and the one of mathematics education for ten year from the one of 1990 is standards and constructivism. There are so many crucial clues or methods in good problem solving but I think that one of them is a representation. So, the purpose of this study is to investigate what is the meaning of representation in general and why representation is so important in elementary mathematics learning, Moreover, I have analyzed the gifted children's thinking of representation which is appeared in the previous internet home task of 40 gifted children who are selected through the examination of 1st, 2nd with paper and pencil and 3rd with practical skill and interview and finally I have presented some examples of children's representation how they use representation to model, investigate and understand special concept more easily in elementary school mathematics class.

• Journal of KIISE:Software and Applications
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• v.32 no.4
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• pp.268-276
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• 2005

This paper deals with optimum communication spanning tree(OCST) problem. Generally, OCST problem is known as NP-hard problem and recently, it is reveled as MAX SNP hard by Papadimitriou and Yannakakis. Nevertheless, many researchers have used polynomial approximation algorithm for solving this problem. This paper uses evolutionary algorithm. Especially, when an evolutionary algorithm is applied to tree network problem such as the OCST problem, representation and genetic operator should be considered simultaneously because they affect greatly the performance of algorithm. So, we introduce a new representation method to improve the weakness of previous representation which is proposed for solving the degree constrained minimum spanning tree problem. And we also propose a new decoding method to generate a reliable tree using the proposed representation. And then, for finding a suitable genetic operator which works well on the proposed representation, we tested three kinds of genetic operators using the information of network or the genetic information of parents. Consequently, we could confirm that the proposed method gives better results than the previous methods.

• Journal of Korean Institute of Industrial Engineers
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• v.31 no.1
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• pp.10-19
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• 2005

The minimum spanning tree (MST) problem is one of the traditional optimization problems. Unlike the MST, the degree constrained minimum spanning tree (DCMST) of a graph cannot, in general, be found using a polynomial time algorithm. So, finding the DCMST of a graph is a well-known NP-hard problem of importance in communications network design, road network design and other network-related problems. So, it seems to be natural to use evolutionary algorithms for solving DCMST. Especially, when applying an evolutionary algorithm to spanning tree problems, a representation and search operators should be considered simultaneously. This paper introduces a new tree representation scheme and a genetic operator for solving combinatorial tree problem using evolutionary algorithms. We performed empirical comparisons with other tree representations on several test instances and could confirm that the proposed method is superior to other tree representations. Even it is superior to edge set representation which is known as the best algorithm.

• Journal of The Korean Association of Information Education
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• v.5 no.2
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• pp.185-196
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• 2001

Elementary school students feel more difficult the mathematical word problems than the numberical formula. I think that this reason isn't the ability of mathematical calculation but the problems representation. It is demanded exactly understanding about the requirements of problem for improving ability of the mathematical word problem representation. It is necessary that we take multimedia data and communication for this, because web advances multimedia materialization and promotes mutual communication, then it gives us with the most environment for word problem representation learning. According to, this thesis is designed web-based system to improve ability of the mathematical word problem representation, applied the sixth grade it experimentally.

• The Mathematical Education
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• v.45 no.3 s.114
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• pp.263-273
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• 2006

The purpose of this paper is to research the factors and the effects of immediacy in mathematics teaching and learning and mathematical problem solving. The factors of immediacy are visualization, functional fixedness and representatives. In special, students can apprehend immediately the clues and solution using the visual representation because of its properties of finiteness and concreteness. But the errors sometimes originate from visual representation which come from limitation of the visual representation. It suggests that students have to know conceptual meaning of the visual representation when they use the visual representation. And this phenomenon is the same in functional fixedness and representatives which are the factors of immediacy The methods which overcome the errors of immediacy is that problem solvers notice the limitation of the factors of immediacy and develop the meta-cognitive ability. And it means we have to emphasize the logic and the intuition in mathematical teaching and learning. Clearly, we can't solve all mathematical problems using only either the logic or the intuition.