• 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.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.253-267
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• 2013

The newly revised mathematics curriculum of 2007 speaks of ultimate goal to develop ability to think and communicate mathematically, in order to develop ability to rationally deal with problems arising from the life around, which puts emphasize on mathematical communication. In this study, analysis on mathematical communication and analogical thinking process of group of students with similar level of academic achievement and that with different level, and thus analyzed if such communication has affected analogical thinking process in any way. This study contains following subjects: 1. Forms of mathematical communication took placed at the two groups based on achievement level were analyzed. 2. Analogical thinking process was observed through trapezoid's area obtaining activity and analyzed if communication within groups has affected such process anyhow. A framework to analyze analogical thinking process was developed with reference of problem solving procedure based on analogy, suggested by Rattermann(1997). 15 from 24 students of year 5 form of N elementary school at Gunpo Uiwang, Syeonggi-do, were selected and 3 groups (group A, B and C) of students sharing the same achievement level and 2 groups (group D and E) of different level were made. The students were led to obtain areas of parallelogram and trapezoid for twice, and communication process and analogical thinking process was observed, recorded and analyzed. The results of this study are as follow: 1. The more significant mathematical communication was observed at groups sharing medium and low level of achievement than other groups. 2. Despite of individual and group differences, there is overall improvement in students' analogical thinking: activities of obtaining areas of parallelogram and trapezoid showed that discussion within subgroups could induce analogical thinking thus expand students' analogical thinking stage.

• 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.

• Journal of Educational Research in Mathematics
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• v.19 no.4
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• pp.479-491
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• 2009

This paper researches the possibility of introducing Descartes' theorem to mathematically gifted students. Not only is Descartes' theorem logically equivalent to Euler's theorem but is hierarchically connected with Gauss-Bonnet theorem which is the core concept on differential geometry. It is possible to teach mathematically gifted students Descartes' theorem by generalizing mathematical property in solid geometry through analogical reasoning, that is, so in a polyhedrons the sum of the deficient angles is $720^\circ$ as in an polygon the sum of the exterior angles is $360^\circ$. This study introduces an alternative method of instruction that we enable mathematically gifted students to reinvent Descartes' theorem through analogical reasoning instead of deductive reasoning.

• 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 School Mathematics Society
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• v.16 no.4
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• pp.927-941
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• 2013

In this paper, we investigate and analysis high school students' generalization of cosine rule using analogy, and we study teaching and learning methods improving students' analogical thinking ability to improve mathematical thinking process. When students can reproduce what they have learned through inductive reasoning process or analogical thinking process and when they can justify their own mathematical knowledge through logical inference or deductive reasoning process, they can truly internalize what they learn and have an ability to use it in various situations.

• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.51-67
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• 2017

Analogical reasoning is a mathematically useful way of thinking. By analogy reasoning, students can improve problem solving, inductive reasoning, heuristic methods and creativity. The purpose of this study is to analyze the analogical reasoning of preservice mathematics teachers while constructing quadratic curves defined by eccentricity. To do this, we produced tasks and 28 preservice mathematics teachers solved. The result findings are as follows. First, students could not solve a target problem because of the absence of the mathematical knowledge of the base problem. Second, although student could solve a base problem, students could not solve a target problem because of the absence of the mathematical knowledge of the target problem which corresponded the mathematical knowledge of the base problem. Third, the various solutions of the base problem helped the students solve the target problem. Fourth, students used an algebraic method to construct a quadratic curve. Fifth, the analysis method and potential similarity helped the students solve the target problem.

• Communications of Mathematical Education
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• v.18 no.3 s.20
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• pp.139-147
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• 2004

최근 수학 창의성 개발과 관련되어 문제설정에 대한 많은 연구가 진행되고 있으나 문제설정의 기법과 지도방법에 대한 연구는 실제적인 연구는 미비한 실정이다. 이 연구에서는 문제설정의 유형과 수준을 논의함으로서 문제설정 지도에 대한 시사점을 주고자 한다. 문제설정의 유형으로는 다음과 같이 분류될 수 있다. 첫째, 문제를 구성하는 요인들을 다른 것으로 대체하여 만들 수 있는 대치적 수준의 문제설정, 둘째 유추적 사고에 의해 만들 수 있는 유추적 수준의 문제설정, 셋째는 개념이나 또는 해를 구하는 방법이나 절차를 다른 형태로 바꾸는, 즉 문제를 재구성, 재정의 및 재조직하여 문제를 만드는 재구성 수준의 문제설정, 넷째는 출판되는 논문의 주제 선정과 같은 전문가 수준의 문제설정으로 분류하였다.