• School Mathematics
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• v.5 no.4
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• pp.401-420
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• 2003

We have many problems in the teaching and learning of proof, especially in the demonstrative geometry of middle school mathematics introducing the proof for the first time. Above all, it is the serious problem that many students do not understand the meaning of proof. In this paper we intend to show that teaching the meaning of proof in terms of historic-genetic approach will be a method to improve the way of teaching proof. We investigate the development of proof which goes through three stages such as experimental, intuitional, and scientific stage as well as the development of geometry up to the completion of Euclid's Elements as Bran-ford set out, and analyze the teaching process for the purpose of looking for the way of improving the way of teaching proof through the historic-genetic approach. We conducted lessons about the angle-sum property of triangle in accordance with these three stages to the students of seventh grade. We show that the students will understand the meaning of proof meaningfully and properly through the historic-genetic approach.

• Communications of Mathematical Education
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• v.8
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• pp.17-32
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• 1999

본 논문의 목적은 Cabri II 를 이용하여 형식적이고 연역적인 증명수업 방법의 대안을 찾는 데 있다. 형식적인 증명을 하기 전에 탐구와 추측을 통한 발견과 그 결과에 대한 비형식적인 증명 활동을 강조한다. 역동적인 기하소프트웨어인 Cabri II 는 작도가 편리하고 다양한 예를 제공하여 추측과 탐구 그리고 그 결과의 확인을 위한 풍부한 환경을 제공할 수 있으며, 끌기 기능을 이용한 삼각형의 변화과정에서 관찰할 수 있는 불변의 성질이 형식적인 증명에 중요한 역할을 한다. 또한 도형에 기호를 붙이는 활동은 형식적인 증명을 어렵게 만드는 요인 중의 하나인 명제나 정리의 기호적 표현을 보다 자연스럽게 할 수 있게 해 준다. 그러나, 학생들이 증명은 더 이상 필요 없으며, 실험을 통한 확인만으로도 추측의 정당성을 보장받을 수 있다는 그릇된 ·인식을 심어줄 수도 있다. 따라서 모든 경우에 성립하는 지를 실험과 실측으로 확인할 수는 없다는 점을 강조하여 학생들에게 형식적인 증명의 중요성과 필요성을 인식시킬 필요가 있다. 본 연구에 대한 다음과 같은 후속연구가 필요하다. 첫째, Cabri II 를 이용한 증명 수업이 학생들의 증명 수행 능력 또는 증명에 대한 이해에 어떤 영향을 끼치는지 특히, van Hiele의 기하학습 수준이론에 어떻게 작용하는 지를 연구할 필요가 있다. 둘째, 본 연구에서 제시한 Cabri II 를 이용한 증명 교수학습 방법에 대한 구체적인 사례연구가 요구되며, 특히 탐구, 추측을 통한 비형식적인 중명에서 형식적 증명으로의 전이 과정에서 나타날 수 있는 학생들의 반응에 대한 조사연구가 필요하다.

• Journal of Educational Research in Mathematics
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• v.19 no.2
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• pp.185-206
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• 2009

In the study, I conducted the teaching experiment designed to instruct proof to four 7th grade students by utilizing the analysis method. As the results of this study I could identified that it is effective to teach and learn to find proof methods using the analysis. The results of the study showed that four 7th grade students succeeded in finding the proof methods by utilizing the analysis and representing the proof after 15 hours of the teaching experiment. In addition to the difficulties that students faced in learning proof utilizing the analysis were related to the search for the light conditions for triangles to be congruent, symbolic representation of the proof methods, reinterpretation of drawings given in the proof problems.

• School Mathematics
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• v.14 no.3
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• pp.297-313
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• 2012

The purpose of this paper is to discuss the potential and to examine the effect of narrative, as an alternative approach to teach formal proof in more easier and comprehensible way. Identifying the key elements of narrative in proof, we constructed a narrative that derives the equation of function translation. We examined the effect of teaching through the narrative, in comparison with teaching the corresponding proof, on low-achieving students' instrumental understanding and relational understanding of function translation. Since we found no statistically significant differences between the experimental and the comparison group, this study could not conclude that teaching through the narrative was more effective than teaching the corresponding proof. But there were some qualitative differences in the relational understanding responses and the evaluation of the teaching between two groups. These findings suggested some potential of narratives that complement the formal proof.

• Communications of Mathematical Education
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• v.23 no.1
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• pp.85-108
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• 2009

The purposes of the study were to develop instructional materials based on Freudenthal's guided reinvention principle for teaching proofs and to investigate how the teaching method based on guided reinvention principle affects on 8th grade students' ability to write proofs and learning attitude toward proofs. Teaching based on guided reinvention principle placed emphasis on providing students opportunities to make a mathematical statement and prove the statement by themselves throughout various activities such as exploring, conjecturing, and testing the conjectures. The study found that students who studied proving with instructional materials developed by guided reinvention principle showed statistically higher mean scores on the posttest than students who studied by a traditional teaching method depending onteacher's explanation. Especially, on the posttest item which requested to prove a whole statement without presenting a picture corresponding to the statement, a big difference among students' responses was found. Many more students in the traditional group did not provide any response on the item. According to the results of the questionnaire regarding students' learning attitudes, the group who studied proving by guided reinvention principle indicated relatively more positive attitudes toward learning proofs than the counterparts.

• The Mathematical Education
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• v.43 no.4
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• pp.381-390
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• 2004

In this study, we conceptualized the ‘preformal proof’, which is a transitive level of proof from the experimental and inductive justification to the formalized mathematical proof. We investigated concrete features of the preformal proof in the historico-genetic and the didactical situations. The preformal proof can get the generality of the contents of proof, which makes a distinction from the experimental proof. And we can draw a distinction between the preformal and formal proof, in point that the preformal proof heads for the reality-oriented objects and does not use the formal language.

• School Mathematics
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• v.6 no.1
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• pp.59-90
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• 2004

This study has been established with two purposes. The first one is to development the learning-teaching model for enhancing students' creative proof capacities in the domain of demonstrative geometry as subject content. The second one is to aim at experimentally testing its effectiveness. First, we develop the learning-teaching model for enhancing students' proof capacities. This model is named the generative-convergent model based instruction. It consists of the following components: warming-up activities, generative activities, convergent activities, reflective discussion, other high quality resources etc. Second, to investigate the effects of the generative-convergent model based instruction, 160 8th-grade students are selected and are assigned to experimental and control groups. We focused that the generative-convergent model based instruction would be more effective than the traditional teaching method for improving middle school students' proof-writing capacities and error remediation. In conclusion, the generative-convergent model based instruction would be useful for improving middle grade students' proof-writing capacities. We suggest the following: first, it is required to refine the generative-convergent model for enhancing proof-problem solving capacities; second, it is also required to develop teaching materials in the generative-convergent model based instruction.

• Journal of Educational Research in Mathematics
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• v.21 no.2
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• pp.135-148
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• 2011

Middle school mathematics treats axiom as mere fact verified by experiment or observation and doesn't mention it axiom. But axiom is very important to understand the difference between empirical verification and mathematical proof, intuitive geometry and deductive geometry, proof and nonproof. This study analysed textbooks and surveyed gifted students' conception of axiom. The results showed the problem and limitation of middle school mathematics on the treatment of axiom and proof.

• Education of Primary School Mathematics
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• v.16 no.2
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• pp.123-145
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• 2013

This study was expected to yield the meaningful conclusions from the experimental group who took lessons based on inductive activities using GeoGebra at the beginning of proof learning and the comparison one who took traditional expository lessons based on deductive activities. The purpose of this study is to give some helpful suggestions for teaching proof to mathematically gifted elementary students. To attain the purpose, two research questions are established as follows. 1. Is there a significant difference in proof abilities between the experimental group who took inductive lessons using GeoGebra and comparison one who took traditional expository lessons? 2. Is there a significant difference in proof attitudes between the experimental group who took inductive lessons using GeoGebra and comparison one who took traditional expository lessons? To solve the above two research questions, they were divided into two groups, an experimental group of 10 students and a comparison group of 10 students, considering the results of gift and aptitude test, and the computer literacy among 20 elementary students that took lessons at some education institute for the gifted students located in K province after being selected in the mathematics. Special lesson based on the researcher's own lesson plan was treated to the experimental group while explanation-centered class based on the usual 8th grader's textbook was put into the comparison one. Four kinds of tests were used such as previous proof ability test, previous proof attitude test, subsequent proof ability test, and subsequent proof attitude test. One questionnaire survey was used only for experimental group. In the case of attitude toward proof test, the score of questions was calculated by 5-point Likert scale, and in the case of proof ability test was calculated by proper rating standard. The analysis of materials were performed with t-test using the SPSS V.18 statistical program. The following results have been drawn. First, experimental group who took proof lessons of inductive activities using GeoGebra as precedent activity before proving had better achievement in proof ability than the comparison group who took traditional proof lessons. Second, experimental group who took proof lessons of inductive activities using GeoGebra as precedent activity before proving had better achievement in the belief and attitude toward proof than the comparison group who took traditional proof lessons. Third, the survey about 'the effect of inductive activities using GeoGebra on the proof' shows that 100% of the students said that the activities were helpful for proof learning and that 60% of the reasons were 'because GeoGebra can help verify processes visually'. That means it gives positive effects on proof learning that students research constant character and make proposition by themselves justifying assumption and conclusion by changing figures through the function of estimation and drag in investigative software GeoGebra. In conclusion, this study may provide helpful suggestions in improving geometry education, through leading students to learn positive and active proof, connecting the learning processes such as induction based on activity using GeoGebra, simple deduction from induction(i.e. creating a proposition to distinguish between assumptions and conclusions), and formal deduction(i.e. proving).

• Proceedings of the Korean Information Science Society Conference
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• 2007.06c
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• pp.303-308
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• 2007

최근에 제안된 감독 지식을 융합하는 강화 학습 기법인 potential-based RL 기법의 효용성은 이론적 최적 정책으로의 수렴성 보장으로 증명되었고, policy-reuse RL 기법의 우수성은 감독지식을 융합하지 않는 기존의 강화학습과 실험적인 비교를 통하여 증명되었지만, policy-reuse RL 기법을 potential-based RL 기법과 비교한 연구는 아직까지 제시된 바가 없었다. 본 논문에서는 potential-based RL 기법과 policy-reuse RL 기법의 실험적인 성능 비교를 통하여 기법이 policy-reuse RL 기법이 policy-reuse RL 기법에 비하여 더 빠르게 수렴한다는 것을 보이며, 또한 policy-reuse RL 기법의 성능은 재사용하는 정책의 optimality에 영향을 받는다는 것을 보인다.