• Education of Primary School Mathematics
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• v.3 no.1
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• pp.63-77
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• 1999

We introduce what is the meaning of problem and problem solving and also different type of problems and problem-solving strategies were discussed in this paper, with suggestions for teaching both Polya's four-step strategy and specific problem solving strategies. Many real and concrete examples of routine and nonroutine problems in elementary school mathematics are introduced. Especially, we have researched on the actual condition how children in elementary school think about multiplication of fraction for the routine problem. As a result, we have noticed that children have diverse thinking in their own way and also concrete expressions are much better effective than algorithm showing in textbook.

• Communications of Mathematical Education
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• v.15
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• pp.207-214
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• 2003

수학교육에서 학생들이 학습을 통하여 습득하여할 중요한 주제는 수학 지식과 수학을 다루는 인지적 조작 기술일 것이다. 특히 수학지식과 지식의 활용은 문제해결을 통한 학습에서 의미 있게 학생에게 나타나며 이를 통하여 수학 학습 동기를 강화하고 수학의 가치를 느끼게 한다는 점에서 중요한 의의를 갖는다. 대학수준의 수학교육과정에서도 문제해결은 중요한 수학교육의 중심 수단으로서 목적으로서 선언되어 있지만 실제 수업에서 잘 다루고 있지 못하다. 문제해결 지도에 대한 접근 방식으로 1950년대의 문제해결전략을 다룬 Polya, 1990년대의 메타인지적 접근을 강조한 Schoenfeld 및 최근의 여러 연구자들의 활발한 연구가 이어지고 있다. 본 논문에서 대학 수준의 문제해결 수업의 접근 방법을 소개함으로 문제해결 수업을 구현할 수 있는 지식을 제공한다. 특히 Schoenfeld의 문제해결 수업 모델은 수학 교육의 교실 수업으로의 구현 측면에서 갖는 다양한 함의를 제시한다.

• The Mathematical Education
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• v.53 no.1
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• pp.75-92
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• 2014

In this paper, we summarize briefly some of the most salient features of Repkina & Zaika's theory of learning action formation levels. We concretize Repkina & Zaika's theory by comparing various points of view of Uoo, Polya, Krutetskii, and Davydov et al. In this study we are able to diagnose students' learning action formation levels in the process of mathematics problem solving. In addition we use interview method to collect various information about students' levels. As a result we suggest data related with each level of learning action formation, and characteristics of students who belong to each level of learning action formation.

• Proceedings of the Korean Society of Computer Information Conference
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• 2011.06a
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• pp.225-228
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• 2011

창의성이 강조되는 시대에 영재 교육의 중요성은 점차 높아지고 있다. 그러나 정보 영재를 위한 연구는 수학이나 과학 영재에 비해 미미한 수준이며, 특히 초등 정보영재를 위한 프로그래밍 교육은 창의적 알고리즘을 개발하는 능력을 기르는 것보다 학습자의 수준에 맞지 않는 특정 프로그래밍 언어의 사용법이나 문법 위주의 교육에 치중하고 있다는 우려의 목소리가 높았다. 이에 본 논문에서는 초등 정보영재의 알고리즘적 사고력을 향상시키기 위한 실생활 중심의 컨텐츠를 제안하고자 한다. 초등학생의 생활과 밀접하게 연관된 주제를 선정하여 학습 동기를 유발하고, Polya의 문제해결모형을 토대로 스스로 이야기를 만들고 그 안에서 알고리즘을 찾아가는 과정을 통해 알고리즘적 사고력을 향상시킬 수 있도록 컨텐츠를 설계하였다.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.789-804
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• 2022

Striking result of Vybíral [51] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C^*-algebras and derive the results of Schur and Vybíral. As an application, we state C^*-algebraic version of Novak's conjecture and solve it for commutative unital C^*-algebras. We formulate Pólya-Szegő-Rudin question for the C^*-algebraic Schur product of positive matrices.

• Journal of the Korean School Mathematics Society
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• v.9 no.1
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• pp.105-120
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• 2006

The Seventh Curriculum makes sure that those students who don't have a proper understanding of contents required at a certain stage take a remedial course. But a trend contrary to the intention is formed since there is no systematic education for such a course and thus more students get to fall into the group of low achievement. In particular, solving a simultaneous equation in a rote way without understanding influences negatively students' achievement. Schoenfeld introduced the basic elements of one's own mathematical problem solving process and behavior, referred to Polya's. Employing Schoenfeld's strategy, this study aimed to induce students' active participation in math classes, as well as to focus on a mathematical problem solving process during the study. Two students were selected from a remedial course at 00 Middle School and administered with a qualitative case study method over 17 lessons, each of which lasted for 30 minutes. In the beginning, they used such knowledge as facts and definitions a lot. There was a tendency of their resorting to intuitive knowledge more when they lacked basic knowledge or met with a difficult question. As the lessons were given, however, they improved their ability to implement algorithm procedures and used more familiar ones with the developed common procedures in the area of resources.

• Journal of the Korean School Mathematics Society
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• v.22 no.3
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• pp.329-350
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• 2019

Productive struggle is a student's persevering effort to understand mathematical concepts and solve challenging problems that are not easily solved, but the problem can lead to curiosity. Productive struggle is a key component of students' learning mathematics with a conceptual understanding, and supporting it in learning mathematics is one of the most effective mathematics teaching practices. In comparison to research on students' productive struggles, there is little research on preservice mathematics teachers' productive struggles. Thus, this study focused on the productive struggles that preservice mathematics teachers face in solving a non-routine mathematics problem. Polya's four-step problem-solving process was used to analyze the collected data. Examples of preservice teachers' productive struggles were analyzed in terms of each stage of the problem-solving process. The analysis showed that limited prior knowledge of the preservice teachers caused productive struggle in the stages of understanding, planning, and carrying out, and it had a significant influence on the problem-solving process overall. Moreover, preservice teachers' experiences of the pleasure of learning by going through productive struggle in solving problems encouraged them to support the use of productive struggle for effective mathematics learning for students, in the future. Therefore, the study's results are expected to help preservice teachers develop their professional expertise by taking the opportunity to engage in learning mathematics through productive struggle.

• School Mathematics
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• v.7 no.4
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• pp.353-373
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• 2005

The purpose of this paper is to analysis the basic network problem which can be applied to the mathematically gifted students in elementary school. Mainly, we discuss didactic transpositions of the double counting principle, the game of sprouts, Eulerian graph problem, and the minimum connector problem. Here the double counting principle is related to the handshaking lemma; in any graph, the sum of all the vertex-degree is equal to the number of edges. The selection of these subjects are based on the viewpoint; to familiar to graph theory, to raise algorithmic thinking, to apply to the real-world problem. The theoretical background of didactic transpositions of these subjects are based on the Polya's mathematical heuristics and Lakatos's philosophy of mathematics; quasi-empirical, proofs and refutations as a logic of mathematical discovery.

• Journal of Educational Research in Mathematics
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• v.13 no.3
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• pp.365-382
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• 2003

This study intends to develope and apply mathematical activities for gifted students. According to the Polya's research and Krutetskii's study, mathematical activities were developed and observed. The activities were aimed at discovery of Euler's theorem through exploration of soccer ball at first. After the repeated application and reflection, the aim and the main activities were changed to the exploration of soccer ball itself and about related mathematical facts. All the students actively participated in the activities, proposed questions need to be proved, disproved by counter examples during the fourth program. Also observation, conjectures, inductive arguments played a prominent role.

• The Mathematical Education
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• v.49 no.2
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• pp.223-246
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• 2010

Goos(2004) introduced educational researchers' demand for change on the way that mathematics is taught in schools and the series of curriculum documents produced by the National council of Teachers of Mathematics. The documents have placed emphasis on the processes of problem solving, reasoning, and communication. In Korea, the national curriculum documents have also placed increased emphasis on mathematical activities such as reasoning and communication(1997, 2007).The purpose of this study is to analyze the impact of inquiry-oriented instruction with guided reinvention on students' mathematical activities containing communication and reasoning for science high school students. In this paper, we introduce an inquiry-oriented instruction containing Polya's plausible reasoning, Freudenthal's guided reinvention, Forman's sociocultural approach of learning, and Vygotsky's zone of proximal development. We analyze the impact of mathematical findings from inquiry-oriented instruction on students' mathematical activities containing communication and reasoning.