• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1327-1340
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• 2021

In this paper, we investigate hereditary and semihereditary representations of quivers over an arbitrary ring. As consequences hereditary and semihereditary category of representations of quivers over an arbitrary ring are characterized.

• The Mathematical Education
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• v.46 no.1 s.116
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• pp.123-134
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• 2007

In the article, representations useful in mathematics education are introduced and show how they are related in the context of mathematics education. They are classified in three categories: representations in mind, representations for understanding and problem solving, and mathematical representations.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.673-693
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• 1997

In this work, the irreducible complex representations of degree 4 of $B_4$, the braid group on 4 strings, are classified. There are 4 families of representations: A two-parameter family of representations for which the image of $P_4$, the pure braid group on 4 strings, is abelian; two families of representations which are the composition of an irreducible representation of $B_3$, the braid group on 3 strings, with a certain special homomorphism $\pi : B_4 \longrightarrow B_3$; a family of representations which are the tensor product of 2 irreducible two-dimensional representations of $B_4$.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1177-1187
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• 2014

In the current paper we study absolutely pure representations of quivers. Then over some nice quivers including linear quivers some sufficient conditions guaranteeing a representation to be absolutely pure is characterized. Furthermore some relations between atness and absolute purity is investigated. Finally it is shown that the absolutely pure covering of representations of linear quivers (including $A^-_{\infty}$, $A^+_{\infty}$ and $A^{\infty}_{\infty}$) by R-modules whenever R is a coherent ring exists.

• The Mathematical Education
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• v.44 no.4 s.111
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• pp.627-651
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• 2005

This study was conducted to attain an in-depth understanding of students' mathematical representations and to present the educational implications for teaching them. Twelve mathematical tasks were developed according to the six types of problems. A task performance was executed to 151 third graders from four classes in DaeJeon and GyeongGi. We analyzed the types and forms of representations generated by them. Then, qualitative case studies were conducted on two small-groups of five from two classes in GyeongGi. We analyzed how individuals' representations became elaborated into group representation and what patterns emerged during the collaborative small-group learning. From the results, most students used more than one representation in solving a problem, but they were not fluent enough to link them to successful problem solving or to transfer correctly among them. Students refined their representations into more meaningful group representation through peer interaction, self-reflection, etc.. Teachers need to give students opportunities to think through, and choose from, various representations in problem solving. We also need the in-depth understanding and great insights into students' representations for teaching.

• Journal of Korean Elementary Science Education
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• v.27 no.2
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• pp.158-169
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• 2008

The purpose of this study was to analyze the uses of the external representations in the $3{\sim}6th$ grade science textbooks developed under the 7th National Curriculum on the basis of the theories and the research results on learning with the multiple representations. The results showed that the frequencies of the macroscopic external representations were higher than those of the symbolic external representations. The external representations with drawing and/or writing, especially writing, were used more frequently than those without drawing and/or writing. However, the most of the external representations were rarely used according to the principles and/or the theories (e.g., personalization principle, dual coding theory, cognitive load theory, and social constructivism theory) for effective uses of the multiple external representations in the science textbooks. The present study provides the guideline to establish the effective uses of the external representations in the science textbooks that not only meet learners but also teachers.

• Journal of The Korean Association For Science Education
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• v.40 no.1
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• pp.61-75
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• 2020

In this study, external representation presented in two units, 'Property of Gas' and 'Changes of States of Matter,' in seventh grade of 2015 revised science curriculum, were analyzed to suggest educational implications. External representations presented in five science textbooks were analyzed according to the six criteria, which were 'type of representation,' 'interpretation of surface features,' 'relatedness to text,' 'existence and properties of a caption,' 'degree of correlation between representations comprising a multiple one,' and 'function of representation.' The characteristics of typical representations related to each achievement standard of two units were also analyzed. The results were as follows: The macro representations for 'type of representation', and explicit representations for 'interpretation of surface features' showed highest frequency. For 'relatedness to text' criteria, 'completely related and linked' and 'completely related and unlinked' representations showed the highest frequency. It means that most representations were properly related with the text. There were appropriate captions for most representations. The degree of correlation between representations comprising a multiple one was largely sufficiently linked with regards to the criteria 'degree of correlation between representations comprising a multiple one'. The complete representations for 'function of representation' showed the highest frequency in the aggregate, however incomplete representations showed more frequencies in the inquiry parts. The typical representations for each achievement standard differed in terms of the type, contained information, used symbols and so on. The educational implications with the use of representations presented in seventh grade textbook were discussed.

• Journal of the Korean Home Economics Association
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• v.50 no.5
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• pp.91-100
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• 2012

The purpose of this study was to examine the effects of children's representations of mothers and peer competence on preschool adjustment. 102 preschoolers (47 boys and 55 girls, aged 4-5) participated in this study. The data were analyzed using descriptive statistics and structural equation modeling analysis. The results were as follows: Firstly, children's representations of mothers had a direct effect on peer competence. Secondly, peer competence had a direct effect on preschool adjustment. Thirdly, children's representations of mothers did not directly affect preschool adjustment, but their representations of mothers did indirectly affect preschool adjustment, through peer competence. In conclusion, children's representations of mothers and peer competence influence preschool adjustment.

• Research in Mathematical Education
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• v.18 no.1
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• pp.31-40
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• 2014

Working with dynamic visual representations can help students-with-computer discover new mathematical ideas. Students translate among multiple representations as a strategy to investigate non-routine problems to explore possible solutions in mathematics classrooms. In this paper, we use the area models as new representations for our secondary students to investigate three problems related to the average speed of a particle. Students show their ideas in the process of investigating arithmetic mean, harmonic mean, and average speed through their created dynamic figures. These figures really utilize dynamic geometry software.

• Journal of the Korean Mathematical Society
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• v.36 no.6
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• pp.1021-1031
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• 1999

In this paper we introduce matrix representations of polygroups over hyperrings and show such representations induce representations of the fundamental group over the corresponding fundamental ring. We also introduce the notion of a polygroup hyperring generalizing the notion of a group ring. We establish homo-morphisms among various polygroup hyperrings.