• 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.

• Communications of Mathematical Education
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• v.27 no.4
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• pp.567-580
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• 2013

As integrated essay writing is performed in university entrance examinations, teachers and students recognize the importance of integrated essay, but teachers have still difficulties of teaching methods. The purpose of this study is to derive educational implications through case of mathematics instruction for integrated essay education to pre-service mathematics teachers. The content knowledge of this class is a definition of conic section in mathematics and properties of conic section in an antenna reflector. The students have to discover them using the history of math, manipulative material, paper-folding and computer simulation. In this teaching and learning process the students can realize mathematical knowledge invented by humans through history of mathematics. The students can evaluate the validity of that as create and justify a mathematical proposition. Also, the students can explain the relation between them logically and descript cause or basis convincingly in the process of justifying. We should keep our study to instructional materials and teaching methods in integrated essay education.

• School Mathematics
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• v.14 no.4
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• pp.469-493
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• 2012

The purpose of this study is find the relation between students' concept and types of proof construction. For this, four undergraduate students majored in mathematics education were evaluated to examine how they understand mathematical concepts and apply their concepts to their proving. Investigating students' proof with their concepts would be important to find implications for how students have to understand formal concepts to success in proving. The participants' proof productions were classified into syntactic proof productions and semantic proof productions. By comparing syntactic provers and semantic provers, we could reveal that the approaches to find idea for proof were different for two groups. The syntactic provers utilized procedural knowledges which had been accumulated from their proving experiences. On the other hand, the semantic provers made use of their concept images to understand why the given statements were true and to get a key idea for proof during this process. The distinctions of approaches to proving between two groups were related to students' concepts. Both two types of provers had accurate formal concepts. But the syntactic provers also knew how they applied formal concepts in proving. On the other hand, the semantic provers had concept images which contained the details and meaning of formal concept well. So they were able to use their concept images to get an idea of proving and to express their idea in formal mathematical language. This study leads us to two suggestions for helping students prove. First, undergraduate students should develop their concept images which contain meanings and details of formal concepts in order to produce a meaningful proof. Second, formal concepts with procedural knowledge could be essential to develop informal reasoning into mathematical proof.

• 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.

• Journal of Educational Research in Mathematics
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• v.14 no.3
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• pp.239-266
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• 2004

This article presents new perspectives for analysing and diagnosing students' mathematical errors on the basis of Pascaul-Leone's neo-Piagetian theory. Although Pascaul-Leone's theory is a cognitive developmental theory, its psychological mechanism gives us new insights on mathematical errors. We analyze mathematical errors in the domain of proof problem solving comparing Pascaul-Leone's psychological mechanism with mathematical errors and diagnose misleading factors using Schoenfeld's levels of analysis and structure and fuzzy cognitive map(FCM). FCM can present with cause and effect among preconceptions or misconceptions that students have about prerequisite proof knowledge and problem solving. Conclusions could be summarized as follows: 1) Students' mathematical errors on proof problem solving and LC learning structures have the same nature. 2) Structures in items of students' mathematical errors and misleading factor structures in cognitive tasks affect mental processes with the same activation mechanism. 3) LC learning structures were activated preferentially in knowledge structures by F operator. With the same activation mechanism, the process students' mathematical errors were activated firstly among conceptions could be explained.

• Journal of Educational Research in Mathematics
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• v.13 no.2
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• pp.159-178
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• 2003

This study analyzes on the reasoning in the process of justification and mathematical problem solving in our primary mathematics textbooks. In our analyses, we found that the inductive reasoning based on the paradima-tic example whose justification is founnded en a local deductive reasoning is the most important characteristics in our textbooks. We also found that some propositions on the properties of various quadrangles impose a deductive reasoning on primary students, which is very difficult to them. The inductive reasoning based on enumeration is used in a few cases, and analogies based on the similarity between the mathematical structures and the concrete materials are frequntly found. The exposition based en a paradigmatic example, which is the most important characteristics, have a problematic aspect that the level of reasoning is relatively low In Miyazaki's or Semadeni's respects. And some propositions on quadrangles is very difficult in Piagetian respects. As a result of our study, we propose that the level of reasoning in primary mathematics is leveled up by degrees, and the increasing levels are following: empirical justification on a paradigmatic example, construction of conjecture based on the example, examination on the various examples of the conjecture's validity, construction of schema on the generality, basic experiences for the relation of implication.