• Research in Mathematical Education
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• v.24 no.3
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• pp.255-278
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• 2021

Proof serves a critical role in mathematical practices as well as in fostering student's mathematical understanding. However, the research literature accumulates results that there are not many opportunities available for students to engage with proving-related activities and that students' understanding about proof is not promising. This unpromising state of instruction of proof calls for a novel approach to address the aforementioned issues. This study investigated an instruction of proof to explore a pedagogy to teach how to prove. The teacher utilized the way of problem posing to make proving a routine part of everyday lesson and changed the classroom culture to support student proving. The study identified the teacher's support for student proving, the key pedagogical changes that embraced proving as part of everyday lesson, and what changes the teacher made to cultivate the classroom culture to be better suited for establishing a supportive community for student proving. The results indicate that problem posing has a potential to embrace proof into everyday lesson.

• The Mathematical Education
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• v.50 no.4
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• pp.441-466
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• 2011

In this article, we study on the teaching design, focused on the geometric methods, of 2-D isoperimetric problem for the elementary mathematically gifted students. For our teaching design, we discussed the ideals of Zenodorus's polygon proof, Steiner's four-hinge proof, Steiner's mean boundary proof, Steiner's snowball-packing proof, Edler's finite existence proof and Lawlor's dissection proof, and then the ideals achieved were modified with the theoretical backgrounds-the theory of Freudenthal's mathematisation, the method of analysis-synthesis. We expect that this article would contribute to the elementary mathematically gifted students to acquire and to improve spatial sense.

• Communications of Mathematical Education
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• v.20 no.3 s.27
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• pp.373-389
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• 2006

The purpose of this study is to compare the way of proving using combinational proof with the way of proving presented in the existing math textbook in the proof of combinational equation and to classify the problem-solving into some categories using combinational proof in combinational equation. Corresponding with these, this study suggests the application of combinational equation using combinational proof and the fundamental material to develop material for advanced study.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.22 no.6
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• pp.1265-1270
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• 2012

In this paper, we propose a new closest vector problem based interactive proof that is useful for authentication. Contribution of this paper is that the proposed protocol does not use a special form of a lattice, but a general lattice, which makes the protocol design very simple and easy to be proved. We prove its security in terms of completeness, soundness, simulatability.

• Journal of Korea Technical Association of The Pulp and Paper Industry
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• v.33 no.4
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• pp.15-20
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• 2001

The function of the moisture-proof paper is to prevent moisture from adsorbing into the packed goods. Water-vapor transmission rate of the moisture-proof paper should be less than 100g/$m^2$.24hr and the optimum rate would be less than 50g/$m^2$.24hr. In general the moisture-proof paper has been made by laminating polyethylene or polypropylene on top of the base paper. However this kind of moisture-proof paper has a problem in recycling so that it brings about environmental pollution. In general the moisture-proof paper has been made by laminating polyethylene or polypropylene on top of the base paper. However this kind of moisture-proof paper has a problem in recycling so that it brings about environmental pollution. The purpose of this paper was to make moisture-proof paper using the mixture of SB latex and wax emulsion which was recyclable and environmentally friendly. Water vapor transmission rate showed less than 50g/$m^2$.24hr in mixture ratio of 85:15, 87:13, 90:10. Especially the mixture ratio of 87:13 showed the most favorable water-vapor transmission rate. However, the moisture-proof layer was destroyed slightly by folding in packing. It has been observed that there was no close relationship between water-vapor transmission rate of the moisture-proof paper and grammage of the base paper, but the density of base paper had influenced on water vapor transmission rate. It was also observed that the moisture-proof paper could be recycled. The moisture-proof paper was similar to base paper in degree of the pulping, and there was no significant difference in dispersion between moisture-proof paper and base paper. Most of wax particles which caused the spots during drying process could be removed by flotation process. Tensile strength and tear strength of both moisture-proof paper and base paper after pulping were measured to examine the fiber bonding, and no significant difference in physical properties was observed.

• Communications of Mathematical Education
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• v.22 no.4
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• pp.471-487
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• 2008

Inventive mathematical thinking, original mathematical problem solving ability, mathematical invention and so on are core concepts, which must be emphasized in all branches of mathematical education. In particular, Polya(1981) insisted that inventive thinking must be emphasized in a suitable level of university mathematical courses. In this paper, the author considered two cases of inventive problem solving ability shown by his many students via real analysis courses. The first case is about the proof of the problem "what is the derived set of the integers Z?" Nearly all books on mathematical analysis sent the question without the proof but some books said that the answer is "empty". Only one book written by Noh, Y. S.(2006) showed the proof by using the definition of accumulation points. But the proof process has some mistakes. But our student Kang, D. S. showed the perfect proof by using The Completeness Axiom, which is very useful in mathematical analysis. The second case is to show the infinite countability of NxN, which is shown by informal proof in many mathematical analysis books with formal proofs. Some students who argued the informal proof as an unreasonable proof were asked to join with us in finding the one-to-one correspondences between NxN and N. Many students worked hard and find two singled-valued mappings and one set-valued mapping covering eight diagrams in the paper. The problems are not easy and the proofs are a little complicated. All the proofs shown in this paper are original and right, so the proofs are deserving of inventive mathematical thoughts, original mathematical problem solving abilities and mathematical inventions. From the inventive proofs of his students, the author confirmed that any students can develope their mathematical abilities by their professors' encouragements.

• Journal for History of Mathematics
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• v.11 no.2
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• pp.17-28
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• 1998

The problem of mathematical rigor is that of giving an objective definition of a rigorous proof. But standards of rigor have changed in mathematics and the notion of proof is not absolute. There are different versions of proof or rigor, depending on time, place, and other things. In this paper we will briefly trace that evolution.

• Journal of Educational Research in Mathematics
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• v.8 no.1
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• pp.291-298
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• 1998

This study deals with the problem of proof education in the middle school geometry bby examining van Hiele#s geometric thought level theory and Freudenthal#s mathematization teaching theory. The implications that have been revealed by examining the theory of van Hie이 and Freudenthal are as follows. First of all, the proof education at present that follows the order of #definition-theorem-proof#should be reconsidered. This order of proof-teaching may have the danger that fix the proof education poorly and formally by imposing the ready-made mathematics as the mere record of proof on students rather than suggesting the proof as the real thought activity. Hence we should encourage students in reinventing #proving#as the means of organization and mathematization. Second, proof-learning can not start by introducing the term of proof only. We should recognize proof-learning as a gradual process which forms with understanding the meaning of proof on the basic of the various activities, such as observation of geometric figures, analysis of the properties of geometric figures and construction of the relationship among those properties. Moreover students should be given this natural ground of proof.

• Korean Journal of Logic
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• v.18 no.3
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• pp.307-335
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• 2015

An important component in Prawitz's and Dummett's proof-theoretic accounts of validity is the condition for validity of open arguments. According to their accounts, roughly, an open argument is valid if there is an effective method for transforming valid arguments for its premises into a valid argument for its conclusion. Although their conditions look similar to the proof condition for implication in the BHK explanation, their conditions differ from the BHK account in an important respect. If the premises of an open argument are undecidable in an appropriate sense, then that argument is trivially valid according to Prawitz's and Dummett's definitions. I call this 'the triviality problem'. After a brief exposition of their accounts of proof-theoretic validity, I discuss triviality problems raised by undecidable atomic sentences and by Godel sentence. On this basis, I suggest an emendation of Prawitz's definition of validity of argument.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.547-552
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• 2007

In this note, we extends some of the results of Liu [Fuzzy Sets and systems 157 (2006) 869-878]. This extension consists of a simple proof involving weighted functions and their preference index. We also give an elementary simple proof of the maximum entropy weighting function problem with a given preference index value without using any advanced theory like variational principles or without using Lagrangian multiplier methods.