• Communications of Mathematical Education
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• v.28 no.4
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• pp.529-552
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• 2014

Recently, mathematical reasoning has been considered as one of the most important mathematical thinking abilities to be established in school mathematics. This study is to investigate the mathematical problems on the content of 'Set and Statement' and 'Sequences' in high school according to the four types of reasoning, namely Making Conjectures, Investigating Conjectures, Developing Arguments, and Evaluating Arguments. Those types of reasoning were reconstructed based on Johnson's six types of reasoning suggested in 2010. The content is dealt with in 'Mathematics II' textbook developed and published according to the mathematics curriculum revised in 2009. The subject of this study is nine types of textbooks and mathematical problems in the textbook are consisted of as two parts of 'general problem' and 'evaluation problem'. Finally, the results of this study can be summarized as follow: First, it is stated that students be establishing a logical justification activity, the highest reasoning activity through dealing with the 'Developing Arguments' type of problems affluently in both 'Set and Statement' and 'Sequence' chapters of Mathematics II textbook. Second, it is mentioned that students have an chance to investigate conjectures and develop logical arguments in 'Set and Statement' chapter of Mathematics II textbook. In particular, whereas they have an chance to investigate conjectures and also develop arguments in 'Statement', the 'Set' chapter is given only an opportunity of developing arguments. Third, students are offered on an opportunity of reasoning that can make conjectures and develop logical arguments in 'Sequences' chapter of Mathematics II textbook. Fourth, Mathematics II textbook are geared to do activities that could evaluate arguments while dealing with the problems relevant to 'mathematical process' included in 'general problem'.

• The Mathematical Education
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• v.40 no.2
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• pp.291-315
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• 2001

There are some problems in the introduction of proof in middle school mathematics. Among the problems, one is the use of postulates and the another is the methods of proof how to connect a statement with others. The first case has been treated mainly in this note. Since proof means to state the reason logically why the statement is true on the basis of others which have already been known as true and basic properties, in order to prove logically, it is necessary to take the basic properties and the statement known already as true. But the students don't know well what are the basic properties and the statement known already as true for proving. No use of the term postulation(or axiom) cause the confusion to distinguish postulation and theorem. So they don't know which statements are accepted without proof or not accepted without proof, To solve this problems, it is necessary to use the term postulate in middle school mathematics. In middle school mathematics, we present same model of the introduction of proof which are used the postulates needed for the proof.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.773-773
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• 2004

In our paper "Paranormal contractions and invariant subspaces" published in Journal of the Korean Mathematical Society, Volume 40 (2003), Number 6, pp.933-942, the statement to observation (1) on page 935 should read:(omitted)

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.759-769
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• 2006

Let L : $C\;\rightarrow\;C$ be a hyperbolic automorphism. Then the hyperbolic toral automorphism $L_A\;T^2\;\rightarrow\;T^2$, induced by L, is a chaotic map ([2] pg.192). We characterize hyperbolic toral automorphisms by proving the converse of the above statement.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.653-655
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• 2006

We give a simple proof of the statement that if T is a bounded linear operator on a complex HIlbert space then T is Fredholm if and only if ${\pi}(T)$ is not a TDZ, where ${\pi}(\cdot)$ is the Catkin homomorphism.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.205-210
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• 2004

According to Diriclet's Theorem, there are infinitely many primes of the form An+ 1 for any fixed positive integer A. For this, there already exists a classical simple proof using cyclotomic polynomials. In this paper, we develop another elementary proof of this statement.

• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.325-338
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• 1998

Some nonlocal boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of the solutions. We consider the Dirichlet type boundary value problem and the Neumann type boundary value problem with nonlinear boundary conditions. We also provide a regularity results for the solutions.

• The Mathematical Education
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• v.42 no.2
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• pp.137-157
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• 2003

Mathematical Problem solving has had the largest focus in the spread of mathematical topics since 1980. In Korea, most of the articles on problem solving appeared 1980s and 1990s, during which there were special concerns on this issue. And there is general acceptance of the idea that the famous statement "Problem solving must be the focus of school mathematics"(NCTM, 1980, p.1) in Agenda for Action, reflected in the curriculum of Korea. In a historical review focusing on the problem solving in the National Curriculum of Mathematics, we can infer that the primary goal of mathematics instruction should be to have students become competence problem solver. However, the practices of mathematics classroom and the trends of research in mathematical problem solving have oriented to ′teaching about problem solving′ and ′teaching for problem solving′. The issue of teaching via problem solving′ remain unsolved in the community of mathematics education and we need much more attention to this issue.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1305-1319
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• 2015

We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.691-699
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• 1997

The following statement is known as the generalized Hilbert-Smith conjecture : If G is a compact group and acts effectively on a manifold, then G is a Lie group. In this paper we prove that the generalized Hilbert-Smith conjecture is equivalent to the following : A known, but has never been published before.