• Journal for History of Mathematics
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• v.21 no.3
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• pp.143-156
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• 2008

In this study, we divided the teaching and learning of proof into three steps in the demonstrative geometry of the middle school mathematics. And then we surveyed the student's difficulties in the teaching and learning of proof by using of questionnaire. Results of this survey suggest that students cannot only understand the meaning of proof in the teaching and learning of proof but also they cannot deduce simple mathematical reasoning as judgement for the truth of propositions. Moreover, they cannot follow the hypothesis to a conclusion of the proposition It results from the fact that students cannot understand clearly the meaning and the role of hypotheses and conclusions of propositions. So we need to focus more on teaching students about the meaning and role of hypotheses and conclusions of propositions.

• Research in Mathematical Education
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• v.2 no.2
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• pp.71-78
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• 1998

This paper investigates the practices of proof education in Korea by analyzing the teaching and learning of proofs in classes in the second year of middle school. With this purpose, this study examines the features and deficiencies of the ways of teaching proofs and investigates the difficulties which students have in learning them. Furthermore, it suggests methods for the improvement of teaching proofs.

• Journal of Educational Research in Mathematics
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• v.8 no.1
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• pp.291-298
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• 1998

This study deals with the problem of proof education in the middle school geometry bby examining van Hiele#s geometric thought level theory and Freudenthal#s mathematization teaching theory. The implications that have been revealed by examining the theory of van Hie이 and Freudenthal are as follows. First of all, the proof education at present that follows the order of #definition-theorem-proof#should be reconsidered. This order of proof-teaching may have the danger that fix the proof education poorly and formally by imposing the ready-made mathematics as the mere record of proof on students rather than suggesting the proof as the real thought activity. Hence we should encourage students in reinventing #proving#as the means of organization and mathematization. Second, proof-learning can not start by introducing the term of proof only. We should recognize proof-learning as a gradual process which forms with understanding the meaning of proof on the basic of the various activities, such as observation of geometric figures, analysis of the properties of geometric figures and construction of the relationship among those properties. Moreover students should be given this natural ground of proof.

• Journal of Educational Research in Mathematics
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• v.27 no.2
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• pp.191-205
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• 2017

The purpose of this study is to investigate the types and characteristics of the Seventh Graders' proofs. Harel, & Sowder's proof schemes were used to analyze the subjects' responses. As a result of the study, there was a difference in the type of proof schemes used by the students depending on the academic achievement level. While the proportion of students using a transformative proof scheme decreased from the top to the bottom, the proportion of students using inductive (measure) proof scheme increased. In addition, features of each type of proof schemes were shown, such as using informal codes in the proof process, and dividing a given picture into a specific ratio in the problem. Based on this, we extracted four meaningful conclusions and discussed implications for proof teaching and learning.

• 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.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.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.

• School Mathematics
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• v.4 no.3
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• pp.361-373
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• 2002

In this paper, a series of tasks related on polygonal numbers and pyramidal numbers are suggested for using them as teaching unit materials for teaching and learning of sequences in junior high school mathematics. Especially, finding n-th term in those seque-nces, relations among polygonal numbers, and relations among Pyramidal numbers are focused on. A series of tasks related on polygonal numbers and pyramidal numbers have three math-eucational values. First, they have a value as natural materials for teaching and teaming of finding nth term of original sequences using pro-gression of differences. Second, they have a value as materials for teaching and learning of mathematical thinking such as general-ization, analogy, etc. Third, they have a value as materials for teaching and learning of algebraic operation, proof, and connecting mathematical knowledges.

• Research in Mathematical Education
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• v.13 no.3
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• pp.217-233
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• 2009

The purpose of this study is to describe how dynamic geometry systems can be useful in proof activity; teaching sequences based on the use of dynamic geometry systems and to analyze the possible roles of dynamic geometry systems in both teaching and learning of proof. And also dynamic geometry environments can generate powerful interplay between empirical explorations and formal proofs. The point of this study was to show that how using dynamic geometry software can provide an opportunity to link between empirical and deductive reasoning, and how such software can be utilized to gain insight into a deductive argument.

• School Mathematics
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• v.13 no.3
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• pp.501-524
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• 2011

The purpose of this study was to develop teaching-learning materials for informal activities geared toward teaching the nature and structure of proof, to make a case analysis of the application of the developed instructional materials to students in an elementary gifted class, to discuss the feasibility of proof education for gifted elementary students and to give some suggestions on that proof education. It's ultimately meant to help improve the proof abilities of elementary gifted students. After the characteristics of the eight selected gifted elementary students were analyzed, instructional materials of nine sessions were developed to let them learn about the nature and structure of proof by utilizing informal activities. And then they took a lesson two times by using the instructional materials, and how they responded to that education was checked. An analysis framework was produced to assess how they solved the given proof problems, and another analysis framework was made to evaluate their understanding of the structure and nature of proof. In order to see whether they showed any improvement in proof abilities, their proof abilities and proof attitude were tested after they took lessons. And then they were asked to write how they felt, and there appeared seven kinds of significant responses when their writings were analyzed. Their responses proved the possibility of proof education for gifted elementary students, and seven suggestions were given on that education.