• Research in Mathematical Education
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• v.27 no.2
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• pp.195-210
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• 2024

Researchers and curricula continue to call for proof to serve a central role in learning of mathematics throughout kindergarten to grade 12 and beyond. Despite its prominence and recognition gained during past decades, proof is still a stumbling block for both teachers and students. Research efforts have been made to address issues related to teaching and learning of proof. An area in which such research efforts have been made is analysis of curriculum material (i.e. textbook analysis) with a focus on proof. This study is another research effort in this area of research through investigating the guidance offered in curriculum materials with the following research question: What is the nature (e.g., kinds of content knowledge, pedagogical content knowledge) of guidance is offered for teachers to implement proof tasks in grade 8 geometry textbooks? Results indicate that the guidance offered for proof tasks are concerned more with content knowledge about the content-specific instructional goals than with pedagogical content knowledge which supports teachers in preparing in-class interactions with students to teach proof.

• Journal of the Korean School Mathematics Society
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• v.11 no.1
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• pp.55-68
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• 2008

In this paper, we investigated difficulties that middle school students face in the teaming process of proof, and then inquired how does learning of proof using GSP ease students' difficulties. Throughout the inspection, we identified that students have difficulties in understanding process of premise and conclusion, use of notation, process of reasoning. And we identified, throughout learning process of proof using GSP, students can be feedbacked for their guess or reasoning, generalize the special case to general properties and have attitude checking ideas needed in proof by themselves.

• Communications of Mathematical Education
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• v.32 no.3
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• pp.275-296
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• 2018

The purpose of this study is to investigate the change of gaze and the change of the proof learning achievement after learning the analytic method for proof to mathematical gifted students using eye tracking technique. In order to complete the purpose of this study, a mixed method research was used, that is a combination of quantitative and qualitative research methods. Quantitative analysis was conducted based on the data obtained through the eye tracker, and qualitative analysis was also done using post interview data to make up for the quantitative analysis. The subjects of this study were 8 mathematical gifted 3rd grade middle school students in the gifted education center. The conclusions of this study are as follows. First, the learning of analysis leads to a change of gaze in the proof learning of students. The students, after learning the analysis, moved their gaze from the bottom to the top when solving the proof problem, and the occupancy rate of the gaze to the bottom of the proof was higher than the higher part. Second, the change of gaze caused by the learning of the analysis have a correlation with the achievement of the proof learning and it can be seen that the method learning improves the achievement of the proof learning of the students.

• 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.

• 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.

• The Journal of Bigdata
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• v.7 no.1
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• pp.1-14
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• 2022

The proof-of-work consensus algorithm used by most blockchains is causing a massive waste of computing resources in the form of mining. A useful proof-of-work consensus algorithm has been studied to reduce the waste of computing resources in proof-of-work, but there are still resource waste and mining centralization problems when creating blocks. In this paper, the problem of resource waste in block generation was solved by replacing the relatively inefficient computation process for block generation with distributed artificial intelligence model learning. In addition, by providing fair rewards to nodes participating in the learning process, nodes with weak computing power were motivated to participate, and performance similar to the existing centralized AI learning method was maintained. To show the validity of the proposed methodology, we implemented a blockchain network capable of distributed AI learning and experimented with reward distribution through resource verification, and compared the results of the existing centralized learning method and the blockchain distributed AI learning method. In addition, as a future study, the thesis was concluded by suggesting problems and development directions that may occur when expanding the blockchain main network and artificial intelligence model.

• 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 IEEK Conference
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• 2002.07c
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• pp.1483-1486
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• 2002

In this paper, a new scheme of iterative loaming control of a robot manipulator is presented. The proposed method uses a fuzzy sliding mode controller(FSMC), which is designed based on the similarity between the fuzzy logic control(FLC) and the sliding mode control(SMC), for the feedback. With this, the proposed method makes possible fDr fast iteration and has advantages that no linear approximation is used for the derivation of the learning law or in the stability proof Full proof of the convergence of the fuzzy sliding base learning scheme Is given.

• Journal of Educational Research in Mathematics
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• v.9 no.2
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• pp.419-438
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• 1999

This paper explored the impact of dynamic geometry software such as CabriII, GSP on student's understanding deductive justification, on the assumption that proof in school mathematics should be used in the broader, psychological sense of justification rather than in the narrow sense of deductive, formal proof. The following results have been drawn: Dynamic geometry provided positive impact on interacting between empirical justification and deductive justification, especially on understanding the necessity of deductive justification. And teacher in the computer environment played crucial role in reducing on difficulties in connecting empirical justification to deductive justification. At the beginning of the research, however, it was not the case. However, once students got intocul-de-sac in empirical justification and understood the need of deductive justification, they tried to justify deductively. Compared with current paper-and-pencil environment that many students fail to learn the basic knowledge on proof, dynamic geometry software will give more positive ffect for learning. Dynamic geometry software may promote interaction between empirical justification and edeductive justification and give a feedback to students about results of their own actions. At present, there is some very helpful computer software. However the presence of good dynamic geometry software can not be the solution in itself. Since learning on proof is a function of various factors such as curriculum organization, evaluation method, the role of teacher and student. Most of all, the meaning of proof need to be reconceptualized in the future research.