• The Mathematical Education
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• v.44 no.1 s.108
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• pp.103-112
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• 2005

This paper is designed to characterize the concept of flexibility of mind and analyze relationship between flexibility of mind and divergent thinking in view of mathematical problem solving. This study shows that flexibility of mind is characterized by two constructs, ability to overcome fixed mind in stage of problem understanding and ability to shift a viewpoint in stage of problem solving process, Through the analysis of writing test, we come to the conclusion that students who overcome fixed mind surpass others in divergent thinking and so do students who are able to shift a viewpoint.

• Communications of Mathematical Education
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• v.19 no.1 s.21
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• pp.307-307
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• 2005

• East Asian mathematical journal
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• v.27 no.4
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• pp.381-389
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• 2011

The flexible mathematical thinking - the ability to generate and connect various representations of concepts - is useful in understanding mathematical structure and variation in problem solving. In particular, the flexible mathematical thinking with the inventive mathematical thinking, the original mathematical problem solving ability and the mathematical invention is a core concept, which must be emphasized in all branches of mathematical education. In this paper, the author considered a case of flexible mathematical thinking with an inventive problem solving ability shown by his student via real analysis courses. The case is on the proofs of the equivalences of three different definitions on the concept of limit superior shown in three different real analysis books. Proving the equivalences of the three definitions, the student tried to keep the flexible mathematical thinking steadily.

• Communications of Mathematical Education
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• v.15
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• pp.153-159
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• 2003

수학 학습의 목표를 수학적 사고력의 신장이라는 측면에서 보았을 때 이를 위하여 문제에 대한 다양한 해법을 찾는 활동은 중요하다. 문제에 대한 다양한 접근은 문제해결의 전략을 학습시키고 사고의 유연성을 길러줄 수 있는 방법이 된다. 문제에 대한 다양한 해법을 찾는 과정에서 이미 알고 있는 지식이 어떻게 응용되는지를 알게 된다. 특히 기하 문제에 대한 다양한 접근은 문제해결의 전략을 학습시킬 수 있는 좋은 예가 된다. 본고에서는 문제해결을 통한 수학적 일반성을 발견하기 위한 방법으로서 문제에 대한 다양한 해법을 연역과 귀납에 의하여 일반화하는 과정을 탐색하고자 한다. 특히 수학 문제에 대한 다양한 해법을 찾는 것은 문제해결 전략으로서 뿐만 아니라 창의적 사고의 신장 측면에서 시사점을 던져준다.

• Journal of the Korean School Mathematics Society
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• v.16 no.3
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• pp.539-560
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• 2013

The purpose of this study was to investigate characteristics of mathematically gifted high school students' thinking in design activities using GrafEq. Eight mathematically gifted high school students, who already learned graphs of functions and inequalities necessary for design activities, were selected to work in pairs in our experiment. Results indicate that logical thinking and mathematical abstraction, intuitive and structural insights, flexible thinking, divergent thinking and originality, generalization and inductive reasoning emerged in the design activities. Nonetheless, fine-grained analysis of their mathematical activities also implies that teachers for gifted students need to emphasize both geometric and algebraic aspects of mathematical subjects, especially, algebraic expressions, and the tasks for the students are to be rich enough to provide a variety of ways to simplify the expressions.

• Journal of Educational Research in Mathematics
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• v.26 no.1
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• pp.1-22
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• 2016

This study aims to design an exemplification task to facilitate the students' creative thinking, and to investigate mathematical creativity which emerges from exemplification. In particular, we aim to identify the ways to design exemplification tasks which encourage creative thinking, and characterize mathematical creativity fostered by exemplification. The findings showed that the students' creative thinking related to fluency, flexibility, elaboration, and originality emerged through exemplification.

• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.69-88
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• 2017

One of the most important activities in the process of mathematical modeling is to build models by conjecturing mathematical rules and principles in the real phenomena and to validate the models by considering its validity. Due to uncertainty and ambiguity inherent real-contexts, various strategies and solutions for mathematical modeling can be available. This characteristic of mathematical modeling can offer a proper environment in which creativity could intervene in the process and the product of modeling. In this study, first we analyze the process and the product of mathematical modeling, especially focusing on the students' models and validating way, to find evidences about whether modeling can facilitate students'creative thinking. The findings showed that the students' creative thinking related to fluency, flexibility, elaboration, and originality emerged through mathematical modeling.

• Journal of Educational Research in Mathematics
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• v.19 no.1
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• pp.45-61
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• 2009

The powerful role of analogical reasoning in discovering mathematics is well substantiated in the history of mathematics. Mathematically gifted students, thus, are encouraged to learn via in-depth exploration on their own based on analogical reasoning. In this study, 57 gifted students (31in the 7th and 26 8th grade) were asked to formulate or clarify analogy. Students produced fruitful constructs led by analogical reasoning. Participants in this study appeared to experience the deep thinking that is necessary to solve problems made with analogies, a process equivalent to the one that mathematicians undertake. The subjects had to reflect on prior knowledge and develop new concepts such as an orthogonal projection and a point of intersection of perpendicular lines based on analogical reasoning. All subjects were found adept at making meaningful analogues of a triangle since they all made use of meta-cognition when searching relations for analogies. In the future, methodologies including the development of tasks and teaching settings, measures to evaluate the depth of mathematic exploration through analogy, and research on how to promote education related to analogy for gifted students will enhance gifted student mathematics education.

• Journal of Educational Research in Mathematics
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• v.20 no.4
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• pp.459-476
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• 2010

The purpose of this study is to suggest a new direction in using LOGO as a gifted education program and to seek an effective approach for LOGO teaching and learning, by analyzing the strategic thinking of mathematically gifted elementary students. This research is exploratory and inquisitive qualitative inquiry, involving observations and analyses of the LOGO Project learning process. Four elementary students were selected and over 12 periods utilizing LOGO programming, data were collected, including screen captures from real learning situations, audio recordings, observation data from lessons involving experiments, and interviews with students. The findings from this research are as follows: First, in LOGO Project Learning, the mathematically gifted elementary students were found to utilize such strategic ways of thinking as inferential thinking in use of prior knowledge and thinking procedures, generalization in use of variables, integrated thinking in use of the integration of various commands, critical thinking involving evaluation of prior commands for problem-solving, progressive thinking involving understanding, and applying the current situation with new viewpoints, and flexible thinking involving the devising of various problem solving skills. Second, the students' debugging in LOGO programming included comparing and constrasting grammatical information of commands, graphic and procedures according to programming types and students' abilities, analytical thinking by breaking down procedures, geometry-analysis reasoning involving analyzing diagrams with errors, visualizing diagrams drawn following procedures, and the empirical reasoning on the relationships between the whole and specifics. In conclusion, the LOGO Project Learning was found to be a program for gifted students set apart from other programs, and an effective way to promote gifted students' higher-level thinking abilities.

• Journal of the Korean School Mathematics Society
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• v.3 no.1
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• pp.211-217
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• 2000

본 논문의 목적은 G. Cantor 에 의하여 출발된 집합론을 보통집합 이론이라고 구별하여 부를 때, 보통 집합 이론이 그 바탕에 깔고 있는 논리적 제한 점들 곧, 배중률이라든지 모순의 법칙 등을 어떻게 보완할 수 있을 것인가\ulcorner 하는 점과 그러한 점을 보완하여야 할 필요성에 대하여도 생각하고자 한다. 그런 관점에서 보통집합 이론과 퍼지집합 이론의 기본개념을 상호 비교함으로써 앞서 제기한 문제의 보완 요소를 찾아보려고 한다. 실제에 있어 인간의 사고 가운데에서는 중간을 배제하는 일이 없음에도 불구하고 이를 수학적으로 접근하고 표현하는 수단이 부족함으로 인하여 부자연스러운 논리의 법칙을 받아들일 수밖에 없었던 것도 사실이다. 특히, 논리적 응용력이 부족한 중등과정의 학생들에게 있어서 수학이 전적으로 2가 논리에 의하여 지배되고 있다는 방식으로만 지도하는 것은 여러 가지 측면에서 그 내용의 보완이 요구된다. 보다 다양한 수학적 표현의 여지를 열어주는 지도법은 쉼없이 연구되어야 할 것이다. 무엇보다도 배우는 학생들이 보다 폭 넓은 사고의 영역을 소유하고, 그를 바탕으로 창의적이고 자유로운 발상이 이어 질 수 있도록 하기 위하여는 교사의 수학적 시야가 보다 넓고 유연해져야 한다함은 재론할 필요가 없을 것이다. 그런 의미에서 본 논문이 작은 역할을 할 수 있기를 바란다.