• Journal of Educational Research in Mathematics
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• v.19 no.3
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• pp.355-369
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• 2009

Though there is no agreement on the definition of analogical reasoning, there is no doubt that analogical reasoning is the means of mathematical knowledge construction. Mathematicians generally have a tendency or desire to find similarities between new and existing Ideas, and new and existing representations. They construct appropriate links to new ideas or new representations by focusing on common relational structures of mathematical situations rather than on superficial details. This focus is analogical reasoning at work in the construction of mathematical knowledge. Since analogical reasoning is the means by which mathematicians do mathematics and is close]y linked to measures of intelligence, it should be considered important in mathematics education. This study investigates how mathematicians used analogical reasoning, what role did it flay when they construct new concept or problem solving strategy.

• 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.24 no.2
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• pp.103-124
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• 2014

The process of analogical reasoning can be conventionally summarized in five steps : Representation, Access, Mapping, Adaptation, Learning. The purpose of this study is to develop more detailed model for reason of analogies considering the distinct characteristics of the mathematical education based on the process of analogical reasoning which is already established. Ultimately, This model is designed to facilitate students to use analogical reasoning more productively. The process of developing model is divided into three steps. The frist step is to draft a hypothetical model by looking into historical example of Leonhard Euler(1707-1783), who was the great mathematician of any age and discovered mathematical knowledge through analogical reasoning. The second step is to modify and complement the model to reflect the characteristics of students' thinking response that proves and links analogically between the law of cosines and the Pythagorean theorem. The third and final step is to draw pedagogical implications from the analysis of the result of an experiment.

• Communications of Mathematical Education
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• v.16
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• pp.245-267
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• 2003

수학적 문제 해결은 수학 교육에서 중요한 이슈이고 문제 해결 전략으로서의 유추를 주제로 본 연구에서는 중학생들을 대상으로 단순히 유사한 문제를 제시하는 것만으로 문제 해결에 성공을 할 수 있는지, 문제 해결에 성공을 할 수 없다면 중학생들에게 어떤 과정을 제시해야만 문제 해결 과정에서 유추를 사용하여 문제를 해결 할 수 있는지를 알아보고자 한다. 이를 위하여 본 연구에서는 유추에 의한 문제 해결과정을 표상 형성, 인출, 사상, 적합성, 스키마 형성의 과정으로 보고, 이러한 과정 중 사상 단계에서 사상 과정의 명료화를 중심으로 학생들의 유추 추론에 의한 문제해결 과정을 탐구하였다. 연구 결과, 유추 추론 과정에서 근거 문제만을 제시하는 것은 목표 문제를 해결하는데 유추 추론의 성공을 보장한다고 할 수 없었으며, 근거 문제가 제시되었는데도 목표 문제를 해결하지 못하는 경우 사상 과정을 명료화하자 목표 문제를 성공적으로 해결하였다. 또한 학생들은 목표 문제의 성공 이후 유사한 새로운 목표문제를 푸는데 성공하였다.

• Journal of Educational Research in Mathematics
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• v.20 no.1
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• pp.57-71
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• 2010

Similarity between concept and concept, principle and principle, theory and theory is known as a strong motivation to mathematical knowledge construction. Metaphor and analogy are reasoning skills based on similarity. These two reasoning skills have been introduced as useful not only for mathematicians but also for students to make meaningful conjectures, by which mathematical knowledge is constructed. However, there has been lack of researches connecting the two reasoning skills. In particular, no research focused on the interplay between the two in didactic transposition. This study investigated the process of knowledge construction by metaphor and analogy and their roles in didactic transposition. In conclusion, three kinds of models using metaphor and analogy in didactic transposition were elaborated.

• Communications of Mathematical Education
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• v.28 no.1
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• pp.97-130
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• 2014

The purpose of the study was to investigate the aspects of analogy of high school student's thinking process revealed in the inquiry activity with synthetic division. The case study method of qualitative research was conducted with two high school 10th grade students. Structure-mapping model(SMM) of Gentner and similarity frames which were proposed by other researchers were utilized to analyze the data. Two students used analogy as a tool and they could discover synthetic division of more than 2 degrees, but they revealed different levels of mathematics discovery depending on the different degree of analogical thinking. Surface similarity in the process of inquiry activity played a vital role in analogical thinking. We asked students to explore and discover analogy based on structure similarity. Analogy based on the systematic approach made it possible to predict upper domain. Analogy based on the procedure similarity induced internalization. We could conclude that analogy has instrumental, heuristic and reflective characteristics.

• Journal of the Korean Geographical Society
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• v.46 no.4
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• pp.534-553
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• 2011

Analogical thinking is a problem-solving strategy to use a familiar problem (or base analog) to solve a novel problem of the same type (the target problem). The purpose of this study is to provide new insight into geography teaching and learning by connecting cognitive science research on analogical thinking with issues of geography education and suggest that teaching with analogies can be a productive instructional strategy for geography. In this study, using the various examples of analogical thinking used in geography we defined analogical thinking, addressed the theoretical models on analogical transfer, and discussed conditions that make an effective analogical transfer. The major research findings include the following: a) the spatial analogy, indicating skills to find places that may be far apart but have similar locations, and therefore have other similar conditions and/or connections, can provide a useful way to design contents for place learning; b) representational transfer, specifying a common representation for two problems, can play a key role in solving geographic problems requiring data visualization and spatialization processes; and c) either asking learners to compare/analyze similar examples sharing common structure or providing them examples bridging the gap between concrete, real-life phenomena and the ideas and models can contribute to learning in geographic concepts and skills. The spatial analogy requiring both geographic content knowledge and visual/spatial thinking has the potential to become a content-specific problem-solving strategy. We ended with recommendations for future research on analogy that is important in geography education.

• Proceedings of the Korea Information Processing Society Conference
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• 2014.04a
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• pp.559-562
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• 2014

일반적으로 정적 분석은 자동으로 후 조건은 유추할 수 있지만 전조건은 유추하지 못한다. 만약 전조건을 분석할 수 있다면 프로그램을 부품 별로 분석 할 수 있게 되고 이는 계산 양을 줄이면서도 유의미한 결과를 가지는 분석을 가능하게 한다. 본 연구는 정수 구간 도메인에 한해 기존 정적 분석을 변형하여 양방향 분석정보를 토대로 전조건을 유추하는 방법을 처음으로 제시한다.

• Journal of the Korean School Mathematics Society
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• v.15 no.3
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• pp.535-563
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• 2012

This study was conducted to confirm the necessity of analogical thinking and to empirically verify the effectiveness of analogical reasoning through the visual representation by analyzing the factors of problem solving depending on analogical conditions. Four conditions (a visual representation mapping condition, a conceptual mapping condition, a retrieval hint condition and no hint condition) were set up for the above purpose and 80 twelfth-grade students from C high-School in Cheong-Ju, Chung-Buk participated in the present study as subjects. They solved the same mathematical problem about sequence of complex numbers in their differed process requirements for analogical transfer. The problem solving rates for each condition were analyzed by Chi-square analysis using SPSS 12.0 program. The results of this study indicate that retrieval of base knowledge is restricted when participants do not use analogy intentionally in problem solving and the mapping of the base and target concepts through the visual representation would be closely related to successful analogical transfer. As the results of this study offer, analogical thinking is necessary while solving mathematical problems and it supports empirically the conclusion that recognition of the relational similarity between base and target concepts by the aid of visual representation is closely associated with successful problem solving.

• Communications of Mathematical Education
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• v.21 no.1 s.29
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• pp.51-64
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• 2007

In this study we generalize the law of cosine(in any triangle the square of one side is equal to the sum of the squares of the other sides minus twice their product times the cosine of their included angle), We find the following generalized law of cosine: in any polygon the square of one side is equal to the sum of the squares of the other sides minus twice their products times the cosines of their included angles, and prove it using vector.