• School Mathematics
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• v.16 no.3
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• pp.491-502
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• 2014

This paper analyzed pre-service teachers' justification of $0.99{\cdots}$=1 from their disproof of $0.99{\cdots}$ <1. Some pre-service teachers thought of the difference between $0.99{\cdots}$ and 1 as an infinitesimal. On the contrary, the others claimed that the difference between $0.99{\cdots}$ and 1 was zero as the standard real, but were content with their intuitive justifications. The pre-service teachers' limitation revealed in the process of disproving $0.99{\cdots}$ <1 can be closely related to the orthodox view: the standard real number system is the only absolutely true number system. The existence of nonstandard real number system in which $0.99{\cdots}$ is less than 1, shows that the plain question of whether or not $0.99{\cdots}$ equals 1, cannot be properly answered by common explanations of textbooks or teachers' intuitive justification.

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

• Korean Journal of Logic
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• v.14 no.2
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• pp.39-76
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• 2011

We explain some basic elements of Martin-L$\ddot{o}$f's type theory and examine the distinction between propositions and judgments. In section 1, we introduce the problem. In section 2, we explain the concept of proposition in the intuitionistic type theory as a development of the intuitionistic conception of proposition. In section 3, we explain the concept of judgment in the intuitionistic type theory. In section 4, we explain some basic inference rules and examine a particular derivation in the theory. In section 5, we examine one route from the Fregean distinction between propositions and judgments to the distinction between them in the intuitionistic type theory, paying attention to the alleged necessity for introducing different forms of judgments.

• Korean Journal of Logic
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• v.13 no.2
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• pp.117-134
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• 2010

In this paper, I attempted to show that 'Sorensen's Sorites' is not a successful argument for the vagueness of 'vague'. There are a lot of debates about it, but the central issue is whether Sorensen's Sorites is just small sorites; whether the vagueness certified by Sorensen's Sorites is just the vagueness of 'small'. Deas and Hull thought it was and rejected Sorensen's proof based on his sorites. But their rejection was rebutted by Varzi. The basis of his argument is that the subject of Sorensen's sentences - 'n-small' is vague - is not used but mentioned. I tried to reply on behalf of Deas and Hull and to show that the predicate 'vague' has not any effect on determining the truth value of "'n-small' is vague." Then it can be removed from the sentence. Of course I approve 'vague' is a homological term. What I do not agree with is only Sorensen's argument.

• Communications of Mathematical Education
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• v.35 no.4
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• pp.425-444
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• 2021

In this study, under the assumption that the goal pursued in measurement area can be reached through the composition of the measurement activity considering the mathematical process, the method of summing the interior angles of a triangle using the measurement error was applied to the 4th grade class of the elementary school. Results of the study, first, students were able to recognize the possibility of measurement error by learning the sum of the interior angles of a triangle using the measurement error. Second, the discussion process based on the measurement error became the basis for students to attempt mathematical justification. Third, the manipulation activity using the semicircle was recognized as a natural and intuitive way of mathematical justification by the students and led to generalization. Fourth, the method of guiding the sum of the interior angles of a triangle using the measurement error contributed to the development of students' mathematical communication skills and positive attitudes toward mathematics.

• Journal of The Korean Association For Science Education
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• v.37 no.3
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• pp.523-537
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• 2017

The purpose of this study is to investigate the features of elementary students' intuitive thinking emerged during problem solving activities as it related to thermal phenomena, focusing on when intuitive thinking appears and how it is related to logical thinking. For this, we presented a problem related to thermal phenomena to nine 5th-grade students, and examined how students' thinking emerged in the activities. We conducted clinical interviews to investigate the thinking process of students. The results of this study are as follows. First, students made their own solutions and justified it later during the emergence process of intuitive thinking. It was also found that students connected concrete materials and abstract concepts intuitively. They solved the problem by making predictions even when information is insufficient. Second, it was shown that intuitive thinking can emerge through the intended strategies such as drawing a mental image, thinking from a different perspective, and integrating methods. These results, which are related to the students' intuitive thinking has received little attention and will be the basis for helping students in the context of discovery of their problem solving activities.

• School Mathematics
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• v.15 no.2
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• pp.389-404
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• 2013

Researchers have suggested that students should be experienced in progress of geometric thinking set out in naive and intuitive level and deduced throughout gradual formalization rather than completed mathematics are conveyed to students for students' understanding. This study examined naive and intuitive thinking of students by investigating students' geometric problem solving without diagrams. The students showed these naive thinking: lack of recognition of relation between problem and conditions, use of intuitive judgement depending on diagrams, lacking in understanding of role of specific case, and use of unjustified assumption. This study suggests implication for instruction in geometry.

• School Mathematics
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• v.13 no.1
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• pp.19-36
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• 2011

This study provides middle school students with an opportunity to construct elementary functions with dynamic geometry based on the proportion between lengths of triangle to activate students' intuition in handling elementary algebraic functions and their geometric properties. In addition, this study emphasizes the process of justification about the choice of students' construction method to improve students' deductive reasoning ability. As a result of the pilot lesson study, this paper shows the characteristics of the students' construction process of elementary functions and the roles the teacher plays in the process.

• Journal of the Korean School Mathematics Society
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• v.19 no.4
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• pp.373-394
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• 2016

The "Figure Equation" chapter of current high school curriculum prevents students from relating the concept with what they studied in middle school Euclidean geometry. Woo(1998) concerns that the curriculum introduces the concept merely in algebraic ways without providing students with opportunities to relate it with their prior understanding of geometry, which is based on Euclidean one. In the present study, a sequence of GeoGebra-embedded-geometry lessons was designed so that students could be introduced to and solve problems of the Analytic Geometry by triggering their prior understanding of the Euclidean Geometry which they had learnt in middle school. The study contributes to the field of mathematics education by suggesting a sequence of geometry lessons where students could introduce to the coordinate geometry meaningfully and conceptually in high school.

• (The)Study of the Eastern Classic
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• no.34
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• pp.363-385
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• 2009

I criticize Mou's moral metaphysics and propose the alternative depending on Kant's analysis on the sublime. (1) Mou's moral metaphysics conflicts with the scientific world view. (2) Kant's 'aesthetic space' does not conflict with the scientific world view. Futhermore, Kant's 'aesthetic space' satisfies the conditions which Mou's moral metaphysics requires. (3) Mencius's autonomous morality is not sufficient for justifying the moral law or categorical imperative. (4) At this point, the sublime plays the important role in bridging between nature and morality. (5) In Kant's context, the possibility of the autonomous moral action is achieved on the basis of the educated feeling of the sublime.