• The Mathematical Education
• /
• v.46 no.1 s.116
• /
• pp.123-134
• /
• 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
• /
• v.51 no.6
• /
• pp.1177-1187
• /
• 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
• /
• v.44 no.4 s.111
• /
• pp.627-651
• /
• 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.

• The Mathematical Education
• /
• v.47 no.2
• /
• pp.155-168
• /
• 2008

The theory of representation, which has been an important topic of epistemology, has long history of study. But it has diverse meaning according to the fields of argument. In this paper the author set the meaning of mathematical representation as the interrelation of internal and external representations. With this concept, the following items were studied. 1. Survey on the concepts of mathematical representations. 2. Investigation of pedagogical significance of the mathematical representations, taking into account the characteristics of school mathematics. 3. Recommendation of principles for teaching representation to cope with the problems that are related with cause of disliking each domain of the secondary school mathematics. This study is expected to enable the development of teaching methods to help students strengthening their ability to comprehend mathematical sentences.

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

• Journal of the Korean Mathematical Society
• /
• v.20 no.2
• /
• pp.123-132
• /
• 1983

• Research in Mathematical Education
• /
• v.18 no.1
• /
• pp.31-40
• /
• 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
• /
• v.34 no.3
• /
• pp.673-693
• /
• 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
• /
• v.36 no.6
• /
• pp.1021-1031
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.177-182
• /
• 2003

A metacyclic group is a group that has a cyclic normal subgroup with cyclic quotient. We characterize finite metacyclic groups that have faithful irreducible representations over a given field.