• The Mathematical Education
• /
• v.54 no.1
• /
• pp.13-30
• /
• 2015

The mathematical object is conceptual. Thus we can not prove the property of mathematical object in experimental viewpoint but in conceptual viewpoint. We performed the experiment for 28 middle school students to investigate whether they understand this. As a result, the majority of student didn't cognize the limit of experimental method. We had also individual interviews with four students. As results, one student was exactly cognizing the limit of experimental method, but he couldn't prove logically. The others didn't cognize the limit of experimental method. They thought that the proposition was already true regardless of the error. And one of them even thought that to be equal approximately was the same of to be equal exactly. Also, one student has confused between the experimental viewpoint and the conceptual viewpoint. This implies that it is necessary to help students understand the limit of experimental method.

• Communications of Mathematical Education
• /
• v.24 no.1
• /
• pp.221-234
• /
• 2010

So far, around 370 various verification of Pythagorean Theorem have been introduced and many studies for the analysis of the method of verification are being conducted based on these now. However, we are in short of the research for the study of the generalization for Pythagorean Theorem. Therefore, by abstracting mathematical materials that is, data(lengths of sides, areas, degree of an angle, etc) which is based on Euclid's elements Vol 1 proposition 47, various methods for the generalization for Pythagorean Theorem have been found in this study through scrutinizing the school mathematics and documentations previously studied.

• The Mathematical Education
• /
• v.44 no.4 s.111
• /
• pp.535-546
• /
• 2005

We study on intuitive verification and rigor proof which are in geometry of Korean and Russian $7\~8$ grade's mathematics textbooks. We compare contents of mathematics textbooks of Korea and Russia laying stress on geometry. We extract 4 proposition explained in Korean mathematics textbooks by intuitive verification, analyze these verification method, and compare these with rigor proof in Russian mathematics textbooks.

• Research in Mathematical Education
• /
• v.14 no.3
• /
• pp.219-231
• /
• 2010

Mathematically, a proof is to create a convincing argument through logical reasoning towards a given proposition or a given statement. Mathematics educators have been working diligently to create environments that will assist students to perform proofs. One of such environments is the use of dynamic-geometry-software in the classroom. This paper reports on a case study and intends to probe into students' own thinking, patterns they used in completing certain tasks, and the extent to which they have utilized technology. Their tasks were to explore the shape-to-shape, shape-to-part, and part-to-part interrelationships of geometric objects when dealing with certain geometric problem-solving situations utilizing dissection-motion-operation (DMO).

• Bulletin of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.93-97
• /
• 1994

Previously T.Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one .tau.-closed maximal ideal (by his notation Ma $x_{\tau}$(R).neq..phi.). In this short paper we drop his overall hypothesis that Ma $x_{\tau}$(R).neq..phi. and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let .tau. be a given hereditarty torsion theory for left R-module category R-Mod. The class of all .tau.-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all .tau.-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies ([2], Proposition 1.7 and 1.10).

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.5
• /
• pp.1013-1018
• /
• 2009

Let * be an e.a.b. star operation on an integrally closed domain D, and let $K\gamma$(D, *) be the Kronecker function ring of D. We show that if D is a P*MD, then the mapping $D_{\alpha}{\mapsto}K{\gamma}(D_{\alpha},\;{\upsilon})$ is a bijection from the set {$D_{\alpha}$} of *-linked overrings of D into the set of overrings of $K{\gamma}(D,\;{\upsilon})$. This is a generalization of [5, Proposition 32.19] that if D is a Pr$\ddot{u}$fer domain, then the mapping $D_{\alpha}{\mapsto}K_{\gamma}(D_{\alpha},\;b)$ is a one-to-one mapping from the set {$D_{\alpha}$} of overrings of D onto the set of overrings of $K_{\gamma}$(D, b).

• Bulletin of the Korean Mathematical Society
• /
• v.22 no.1
• /
• pp.23-29
• /
• 1985

Let H be an arbitrary complex Hilbert space. We constructed an extension K of H by means of weakly convergent sequences in H and the Banach limit. Let .phi. be the faithful *-representation of L(H) on K. In this note, we investigated the relations between spectra of T in L(H) and .phi.(T) in L(K) and we obtained the following results: 1) If T is a compact operator on H, then .phi.(T) is also a compact operator on K (Proposition 6), 2) .sigma.$_{l}$ (.phi.(T)).contnd..sigma.$_{l}$ (T) for any operator T.mem.L(H) (Corollary 10), 3) For every operator T.mem.L(H), .sigma.$_{ap}$ (.phi.(T))=.sigma.$_{ap}$ (T))=.sigma.$_{ap}$ (T)=.sigma.$_{p}$(.phi.(T)) (Lemma 12, 13) and .sigma.$_{c}$(.phi.(T))=.sigma.(Theorem 15).15).

• Lingua Humanitatis
• /
• v.1 no.1
• /
• pp.221-233
• /
• 2001

The purpose of this paper is twofold. On the one hand it takes issue with Engstrom's claim that conceptual metaphors are propositional; on the other, it aims to demonstrate that the mathematical term 'mapping' is inappropriate for the analysis of metaphors. To my mind, the propositional analysis of metaphors, a wrong analysis for that matter, originates in the notion 'mapping' I argue that partial 'mapping' between propositional meanings and metaphorical meanings is either mental or psychological, with no concomitant 'truth' value. When concept metaphors represent propositionality, they lose metaphoricity; when they obtain metaphoricity, they are free of propositionality. The mathematical terms 'mapping' and 'proposition,' it is stressed, should be avoided in the analysis of concept metaphors like 'A is B' because they are confusing when applied to linguistic expression. 1 suggest that the term 'mapping' be replaced by phrases such as 'interaction between two domains,' projection from source-domain to target domain,' or 'understanding the properties of two domains between A and B,' etc. This would amount to proposing a pragmatic or cognitive theory of metaphor.

• The Mathematical Education
• /
• v.55 no.1
• /
• pp.91-106
• /
• 2016

The aim of this study was to analyze characteristics of teachers' knowledge about reductio ad absurdum. In order to achieve the aim, this study conducted didactical analysis about reductio ad absurdum through examining previous researches and developed a questionnaire with reference to the results of the analysis. The questionnaire was given to 34 high school teachers and qualitative methods were used to analyze the data obtained from the written responses by the participants. This study also elaborated the framework descriptors for interpreting the teachers' responses in the light of the didactical analysis and the data was elucidated in terms of this framework. The specific features of teachers' knowledge about reductio ad absurdum were categorized into five types as a result. This study raised several implications for teachers' professional development for effective mathematics instruction related to reductio ad absurdum.

• Honam Mathematical Journal
• /
• v.3 no.1
• /
• pp.101-108
• /
• 1981

본(本) 논문(論文)에서는 연속수렴구조(連續收斂構造)의 중요(重要)한 성질(性質)(proposition)을 규명(糾明)하고 c-embedded 공간(空間) X가 완전정칙(完全正則)일 때는 연속수렴구조(連續收斂構造)를 갖는 위상(位相) T는 Compact 수렴위상(收斂位相) K보다는 일반적(一般的)으로 강(强)한 위상(位相)이지만 (6), X가 국소(局所) Compact 공간(空間)일 때는 이들 위상(位相)은 일치(一致)함을 보이고(Theorom 1), 또한 근래(近來) A, V, Skorokhod (7)가 확률과정론(確率過程論)의 극한문제(極限問題)와 관련(關聯)하여 새로 정의(定義)한 $J_{1}$- 수렴위상(收斂位相)과 비교(比較)하여 일치(一致)함을 밝혔다. 그리고 이들 관계(關係)를 요약(要約)하면 다음과 같다(Corollary 2).