• Journal of Educational Research in Mathematics
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• v.9 no.1
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• pp.183-203
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• 1999

In this study, we analysed the constituents of proof and examined into the reasons why the students have trouble in learning the proof, and proposed directions for improving the teaming and teaching of proof. Through the reviews of the related literatures and the analyses of textbooks, the constituents of proof in the level of middle grades in our country are divided into two major categories 'Constituents related to the construction of reasoning' and 'Constituents related to the meaning of proof. 'The former includes the inference rules(simplification, conjunction, modus ponens, and hypothetical syllogism), symbolization, distinguishing between definition and property, use of the appropriate diagrams, application of the basic principles, variety and completeness in checking, reading and using the basic components of geometric figures to prove, translating symbols into literary compositions, disproof using counter example, and proof of equations. The latter includes the inferences, implication, separation of assumption and conclusion, distinguishing implication from equivalence, a theorem has no exceptions, necessity for proof of obvious propositions, and generality of proof. The results from three types of examinations; analysis of the textbooks, interview, writing test, are summarized as following. The hypothetical syllogism that builds the main structure of proofs is not taught in middle grades explicitly, so students have more difficulty in understanding other types of syllogisms than the AAA type of categorical syllogisms. Most of students do not distinguish definition from property well, so they find difficulty in symbolizing, separating assumption from conclusion, or use of the appropriate diagrams. The basic symbols and principles are taught in the first year of the middle school and students use them in proving theorems after about one year. That could be a cause that the students do not allow the exact names of the principles and can not apply correct principles. Textbooks do not describe clearly about counter example, but they contain some problems to solve only by using counter examples. Students have thought that one counter example is sufficient to disprove a false proposition, but in fact, they do not prefer to use it. Textbooks contain some problems to prove equations, A=B. Proving those equations, however, students do not perceive that writing equation A=B, the conclusion of the proof, in the first line and deforming the both sides of it are incorrect. Furthermore, students prefer it to developing A to B. Most of constituents related to the meaning of proof are mentioned very simply or never in textbooks, so many students do not know them. Especially, they accept the result of experiments or measurements as proof and prefer them to logical proof stated in textbooks.

• 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.

• Research in Mathematical Education
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• v.14 no.3
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• pp.219-231
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• 2010

Mathematically, a proof is to create a convincing argument through logical reasoning towards a given proposition or a given statement. Mathematics educators have been working diligently to create environments that will assist students to perform proofs. One of such environments is the use of dynamic-geometry-software in the classroom. This paper reports on a case study and intends to probe into students' own thinking, patterns they used in completing certain tasks, and the extent to which they have utilized technology. Their tasks were to explore the shape-to-shape, shape-to-part, and part-to-part interrelationships of geometric objects when dealing with certain geometric problem-solving situations utilizing dissection-motion-operation (DMO).

• Research in Mathematical Education
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• v.27 no.1
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• pp.75-92
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• 2024

Mathematical argument has been given much attention in the research literature as a mediating construct between reasoning and proof. However, there have been relatively less efforts made in the research that examined the nature of empirical arguments represented in textbooks and how students perceive them as proofs. Cases of point include Intermediate Value Theorem [IVT] and Maximum-Minimum theorem [MMT] in grade 11 in Korea. In this study, using Toulmin's framework (1958), the author analyzed the substance of the empirical arguments provided for both MMT and IVT to draw comparisons between the nature of datum, claims, and warrants among empirical arguments offered in textbooks. Also, an online survey was administered to learn about how students view as proofs the empirical arguments provided for MMT and IVT. Results indicate that nearly half of students tended to accept the empirical arguments as proofs. Implications are discussed to suggest alternative approaches for teaching MMT and IVT.

• School Mathematics
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• v.8 no.3
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• pp.291-300
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• 2006

Proofs in school mathematics are regarded as the procedures to examine a proposition's truth or falsehood. However, they are not based on an axiomatic system in general. This implies the possible existence of vicious circles in proofs in school mathematics. The concept of proof can be more completely acquired when accompanied with concept of circular reasoning and necessity of axiomatic system. Therefore, it is necessary to discuss on the axiomatic system in school mathematics curriculum. The vicious circle can be found in computing an area of a circle by using definite integral in differentiation/integration part of high school textbooks. This paper will first illustrate this in detail and then offer several comments on the axiomatic methods related to the dissolution of that circular reasoning. To further the discussion, Archimedes' derivation for the area of a circle will be considered next. Finally, several arguments on circular reasoning, local organization, and axiomatic system in school curriculum will be presented at the end of the paper.

• The Mathematical Education
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• v.63 no.1
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• pp.105-122
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• 2024

This study analyzed the research trends of 101 articles published in prominent international mathematics education journals over 24 years from 2000, when NCTM's recommendation emphasizing argumentation was released, until September 2023. We first examined the overall trend of argumentation research and then analyzed representative research topics. We found that students were the focus of the studies. However, several studies focused on teachers. More studies were examined in secondary school than in elementary school, and many were conducted in argumentation in classroom contexts. We also found that argumentation research is becoming increasingly popular in international journals. The representative research topics included 'teaching practice,' 'argumentation structure,' 'proof,' 'student understanding,' and 'student reasoning.' Based on our findings, we could categorize three perspectives on argumentation: formal, contextual, and purposeful. This paper concludes with implications on the meaning and role of argumentation in Korean mathematics education.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.401-420
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• 2010

The purpose of this research is to find out effectiveness of geometry learning through spatial reasoning activities on mathematical problem solving ability and mathematical attitude. In order to proof this research problem, the controlled experiment was done on two groups of 6th graders in N elementary school; one group went through the geometry learning style through spatial reasoning activities, and the other group went through the general geometry learning style. As a result, the experimental group and the comparing group on mathematical problem solving ability have statistically meaningful difference. However, the experimental group and the comparing group have not statistically meaningful difference on mathematical attitude. But the mathematical attitude in the experimental group has improved clearly after all the process of experiment. With these results we came up with this conclusion. First, the geometry learning through spatial reasoning activities enhances the ability of analyzing, spatial sensibility and logical ability, which is effective in increasing the mathematical problem solving ability. Second, the geometry learning through spatial reasoning activities enhances confidence in problem solving and an interest in mathematics, which has a positive influence on the mathematical attitude.

• Journal of Environmental Health Sciences
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• v.42 no.6
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• pp.375-384
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• 2016

Objectives: This paper analyzes the intersection of tort law and environmental health in a recent court decision. Methods: This paper analyzes Supreme Court Decision 2011Da7437, Decided on September 4, 2014 and related lower court decisions. Results: The plaintiffs sought financial compensation from the defendants, arguing that air pollutants in gases emitted by vehicles produced by the defendants had caused them to acquire respiratory diseases. The district court highlighted the need to mitigate the burden of proof for the plaintiffs, but proceeded to review whether the plaintiffs proved the actual toxicity levels of the air pollutants, whether the defendant's vehicles were the main source of the emissions, the plaintiff's level of exposure to the pollutants, and causation between the emissions and the injury. By doing so, the district court required the plaintiffs to prove both indirect and direct facts of causation, increasing burden of proof for plaintiffs. The appellate court upheld the district court's decision, adding that the defendant's conduct did not constitute an illegal act because it did not violate the emissions standards set by environmental law. The Supreme Court upheld the appellate court's decision, reasoning that the epidemiological evidence cannot establish a direct causation for diseases that lack specificity. Conclusion: This case demonstrates that discussions in environmental health have significance in tort lawsuits. For each fact that the plaintiffs and defendants attempted to prove, environmental health research studies were offered as evidence. In addition, the courts decided the legality of the defendant's conduct based on emission standards set by environmental law.

• IEMEK Journal of Embedded Systems and Applications
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• v.8 no.6
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• pp.347-357
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• 2013

Cyber-Physical system(CPS) is characterized by collaborating computational elements controlling physical entities. In CPS, human desire to acquire useful information and control devices anytime and anywhere automatically has increased the necessity of a high reliable system. However, the physical world where CPS is deployed has management complexity and maintenance cost of 'CPS', so that it is impossible to make reliable systems. Thus, this paper presents an 'Autonomic Control System towards High-reliable Cyber-Physical Systems' that comprise 8-steps including 'fault analysis', 'fault event analysis', 'fault modeling', 'fault state interpretation', 'fault strategy decision', 'fault detection', 'diagnosis&reasoning' and 'maneuver execution'. Through these activities, we fascinate to design and implement 'Autonomic control system' than before. As a proof of the approach, we used a ISR(Intelligent Service Robot) for case study. The experimental results show that it achieves to detect a fault event for autonomic control of 'CPS'.

• Research in Mathematical Education
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• v.14 no.3
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• pp.211-218
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• 2010

It is a truism that mathematics is about relations (cf. [Halford, G. S. (1999). The properties of representations used in higher cognitive processes: Developmental implications. In: Sigel, I. E. (Ed.), The Development of Mental Representation: Theories and Applications (pp. 147-168). Mahwah, New Jersey: Erlbaum]). In this article we are considering few problems related to the Viviani's and Routh's Theorems. All Problems are connected by the relation which exists between the distances of the point inside the triangle to it sides. We show how reasoning about the relations could lead the student's problem solving process and give easy to understand solutions of the problems. Among the problems being considered are the proof of the Converse to Viviani's Theorem, the formulas for areas of all figures formed by the sides of triangle and its cevians.