• 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.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.9 no.4
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• pp.816-823
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• 2005

Non-Horn logic programs are extended from Horn logic programs to the level of 1st order predicate logic. Even though they are more expressive than Horn logic programs, They are not practically used because we do not have efficient implementations. Currently to execute non-Horn logic programs, we translate them to equivalent Horn logic programs using the proof procedure InH-Prolog and compile the Horn logic programs to WAM(Warren Abstract Machine) instructions. In this paper, we propose EWAM(Extended Warren Machine) that executes non-Horn logic programs more efficiently and a compilation scheme that compiles non-Horn logic programs to the EWAM instruction. We implement an EWAM emulator and a compiler and measured the performance of the EWAM emulator and the compiler and found that they are very efficient.

• East Asian mathematical journal
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• v.24 no.5
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• pp.527-545
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• 2008

One of the education purpose of the section "Figures" in the eighth grade is to develop students' deductive reasoning ability, which is basic and essential for living in a democratic society. However, most or middle school students feel much more difficulty or even frustration in the study of formal arguments for geometric situations than any other mathematical fields. It is owing to the big gap between inductive reasoning in elementary school education and deductive reasoning, which is not intuitive, in middle school education. Also, it is very burden for students to describe geometric statements exactly by using various appropriate symbols. Moreover, Usage of the same symbols for angle and angle measurement or segments and segments measurement makes students more confused. Since geometric relations is mainly determined by the measurements of geometric objects, students should be able to interpret the geometric properties to the algebraic properties, and vice verse. In this paper, we first compare and contrast inductive and deductive reasoning approaches to justify geometric facts and relations in school curricula. Convincing arguments are based on experiment and experience, then are developed from inductive reasoning to deductive proofs. We introduce teaching methods to help students's understanding for deductive reasoning in the textbook by using stepwise visualization materials. It is desirable that an effective proof instruction should be able to provide teaching methods and visual materials suitable for students' intellectual level and their own intuition.

• THE INTERNATIONAL COMMERCE & LAW REVIEW
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• v.51
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• pp.317-358
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• 2011

The purpose of this study aims to analyse the key differences of the sea waybill and electronic B/L in the international transport documents. Sea waybills look remarkably like ordinary bills of lading. Indeed, in two important ways, they are just like bills of lading: the front of the document will near a description of the quantity and apparent condition of the goods; and the back of the document provides evidence of the terms of the contract of carriage. They differ from bills of lading in that, far from indicating that the goods described are deliverable to the order of the shipper or of the consignee, they will make it explicit that the goods are deliverable only to the consignee. Again, different carries will do thai in a variety of ways. For example, the document may call itself non-negotiable, omitting the word order from the consignee box on the front of the document, and stating explicitly that the goods will be deliverable to the consignee or his authorised representative on proper proof of identity and authorisation. The Hague-Visby Rules and Hamburg Rules give no guidance as to any right to instruct the carrier in respect of goods while they are in transit. However, in applying Article 50 of the Rotterdam Rules, in particular when applying it in the context of seawaybills, straight bills of lading or ship's delivery orders, regard would need to be had to preserve the shipper's rights under any of those three documents even after the buyer of goods covered by them has acquired rights of its own. And, the right of control is defined at Article 1.12 of the Rotterdam Rules. The right to give instruction is further limited by the terms of Article 50.1 to three particular types of instruction in respect of the goods, relating broadly to the goods, their delivery en route, and the identity of the consignee. And, the CMI formulated the CMI Uniform Rules for Sea Waybills for voluntary incorporation into any contract of carriage covered by such a document. Recognising that neither the Hague nor the Hague-Visby Rules are applicable to sea waybills, the CMI Rules provide that a contract of carriage covered by a waybill shall be governed by whichever international or national law, if any, would have been compulsorily applicable if the contract had in fact been covered by a bill of lading or similar document of title.

• Journal of Educational Research in Mathematics
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• v.21 no.4
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• pp.379-398
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• 2011

In learning environment at mathematics education, prove and refute are essential abilities to demonstrate whether and why a statement is true or false. Learning proofs and counter examples within the domain of limit of sequence is important because preservice teacher encounter limit of sequence in many mathematics courses. Recently, a number of studies have showed evidence that pre service and students have problem with mathematical proofs but many research studies have focused on abilities to produce proofs and counter examples in domain of limit of sequence. The aim of this study is to contribute to research on preservice teachers' productions of proofs and counter examples, as participants showed difficulty in writing these proposition. More importantly, the analysis provides insight and understanding into the design of curriculum and instruction that may improve preservice teachers' learning in mathematics courses.

• The Mathematical Education
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• v.46 no.3
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• pp.331-349
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• 2007

This study aims to improve the teaching and learning method on the conic sections. To do that the researcher analyzed the impact of dynamic geometry software on students' problem solving of the conic sections. Students often say, "I have solved this kind of problem and remember hearing the problem solving process of it before." But they often are not able to resolve the question. Previous studies suggest that one of the reasons can be students' tendency to approach the conic sections only using algebra or analytic geometry without the geometric principle. So the researcher conducted instructions based on the geometric and historico-genetic principle on the conic sections using dynamic geometry software. The instructions were intended to find out if the experimental, intuitional, mathematic problem solving is necessary for the deductive process of solving geometric problems. To achieve the purpose of this study, the researcher video taped the instruction process and converted it to digital using the computer. What students' had said and discussed with the teacher during the classes was checked and their behavior was analyzed. That analysis was based on Branford's perspective, which included three different stage of proof; experimental, intuitive, and mathematical. The researcher got the following conclusions from this study. Firstly, students preferred their own manipulation or reconstruction to deductive mathematical explanation or proving of the problem. And they showed tendency to consider it as the mathematical truth when the problem is dealt with by their own manipulation. Secondly, the manipulation environment of dynamic geometry software help students correct their mathematical misconception, which result from their cognitive obstacles, and get correct ones. Thirdly, by using dynamic geometry software the teacher could help reduce the 'zone of proximal development' of Vigotsky.

• The Journal of Korean Association of Computer Education
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• v.12 no.5
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• pp.87-94
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• 2009

This paper presents a Web-based interactive and adaptive learning system that helps students learn the pumping lemma for the family of regular languages. Our system allows the students to proceed with their learning according to their individual differences through Web-Based Instruction and gives them opportunities for the interaction so that they can practice exercise related to the learning and gain feedbacks on the results of the exercises immediately. Especially, the system provides adaptive scaffolding that helps learners understand each step for the proof of the pumping lemma. Unlike existing systems that support learning the pumping lemma, the proposed system defines possible errors in advance and provides appropriate messages for corresponding errors. In addition, the system allows the learners to decompose a string into three parts so that they can understand the pumping lemma precisely.

• 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.

• Journal of Korean Home Economics Education Association
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• v.15 no.4
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• pp.115-132
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• 2003

The purpose of this study is aimed at giving proof that helps the elementary practical arts education system accomplish as the effects are turned out experimentally. Two classes of the sixth grade of J elementary school in Dae-gu have been selected in order to be experimented. One was chosen as an experimental group, the other was done as a comparative group. The creative-problem-solving learning-model was applied to the experimental group, and the traditional way of teaching was applied to the comparative group. For four classes of the sixth grades, ‘chapter 8: Making with recycled materials’ was proceeded as the content. Then. tests about the way of environmental awareness and creativity were carried out twice. After that, the results of pre and after-test in the comparative and experiment groups were compared using the t-test method. Following the analysis of the data collected in this study. the following major observations were obtained: First, children who were educated the creative problem-solving in a practical arts education achieved higher scores than before. Therefore, it turns out that the CPS method is an effective way to improve the environmental awareness in children. It showed that it included lots of daily habits connected with daily life and it made the intention to carry out the environment-preservation stronger and children´s attitude towards the environment improved. Moreover, making with recycled materials was used to solve an environmental problem, affecting in a positive way in our life. It also made the positive recognition about the environment. Second. the application of the creative problem-solving class of the practical arts education can make positive results to children. It helped children to have more interest in the environment around them. Children´s fluency, flexibility and originality in their ideas were improved as much as possible while they were solving problems. Consequently, the application of the creative problem-solving class model of elementary practical arts environmental education lets children expand environment consciousness and creativity.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1993.06a
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• pp.975-976
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• 1993

This talk presents the overview of the author's research and development activities on fuzzy inference hardware. We involved it with two distinct approaches. The first approach is to use application specific integrated circuits (ASIC) technology. The fuzzy inference method is directly implemented in silicon. The second approach, which is in its preliminary stage, is to use more conventional microprocessor architecture. Here, we use a quantitative technique used by designer of reduced instruction set computer (RISC) to modify an architecture of a microprocessor. In the ASIC approach, we implemented the most widely used fuzzy inference mechanism directly on silicon. The mechanism is beaded on a max-min compositional rule of inference, and Mandami's method of fuzzy implication. The two VLSI fuzzy inference chips are designed, fabricated, and fully tested. Both used a full-custom CMOS technology. The second and more claborate chip was designed at the University of North Carolina(U C) in cooperation with MCNC. Both VLSI chips had muliple datapaths for rule digital fuzzy inference chips had multiple datapaths for rule evaluation, and they executed multiple fuzzy if-then rules in parallel. The AT & T chip is the first digital fuzzy inference chip in the world. It ran with a 20 MHz clock cycle and achieved an approximately 80.000 Fuzzy Logical inferences Per Second (FLIPS). It stored and executed 16 fuzzy if-then rules. Since it was designed as a proof of concept prototype chip, it had minimal amount of peripheral logic for system integration. UNC/MCNC chip consists of 688,131 transistors of which 476,160 are used for RAM memory. It ran with a 10 MHz clock cycle. The chip has a 3-staged pipeline and initiates a computation of new inference every 64 cycle. This chip achieved an approximately 160,000 FLIPS. The new architecture have the following important improvements from the AT & T chip: Programmable rule set memory (RAM). On-chip fuzzification operation by a table lookup method. On-chip defuzzification operation by a centroid method. Reconfigurable architecture for processing two rule formats. RAM/datapath redundancy for higher yield It can store and execute 51 if-then rule of the following format: IF A and B and C and D Then Do E, and Then Do F. With this format, the chip takes four inputs and produces two outputs. By software reconfiguration, it can store and execute 102 if-then rules of the following simpler format using the same datapath: IF A and B Then Do E. With this format the chip takes two inputs and produces one outputs. We have built two VME-bus board systems based on this chip for Oak Ridge National Laboratory (ORNL). The board is now installed in a robot at ORNL. Researchers uses this board for experiment in autonomous robot navigation. The Fuzzy Logic system board places the Fuzzy chip into a VMEbus environment. High level C language functions hide the operational details of the board from the applications programme . The programmer treats rule memories and fuzzification function memories as local structures passed as parameters to the C functions. ASIC fuzzy inference hardware is extremely fast, but they are limited in generality. Many aspects of the design are limited or fixed. We have proposed to designing a are limited or fixed. We have proposed to designing a fuzzy information processor as an application specific processor using a quantitative approach. The quantitative approach was developed by RISC designers. In effect, we are interested in evaluating the effectiveness of a specialized RISC processor for fuzzy information processing. As the first step, we measured the possible speed-up of a fuzzy inference program based on if-then rules by an introduction of specialized instructions, i.e., min and max instructions. The minimum and maximum operations are heavily used in fuzzy logic applications as fuzzy intersection and union. We performed measurements using a MIPS R3000 as a base micropro essor. The initial result is encouraging. We can achieve as high as a 2.5 increase in inference speed if the R3000 had min and max instructions. Also, they are useful for speeding up other fuzzy operations such as bounded product and bounded sum. The embedded processor's main task is to control some device or process. It usually runs a single or a embedded processer to create an embedded processor for fuzzy control is very effective. Table I shows the measured speed of the inference by a MIPS R3000 microprocessor, a fictitious MIPS R3000 microprocessor with min and max instructions, and a UNC/MCNC ASIC fuzzy inference chip. The software that used on microprocessors is a simulator of the ASIC chip. The first row is the computation time in seconds of 6000 inferences using 51 rules where each fuzzy set is represented by an array of 64 elements. The second row is the time required to perform a single inference. The last row is the fuzzy logical inferences per second (FLIPS) measured for ach device. There is a large gap in run time between the ASIC and software approaches even if we resort to a specialized fuzzy microprocessor. As for design time and cost, these two approaches represent two extremes. An ASIC approach is extremely expensive. It is, therefore, an important research topic to design a specialized computing architecture for fuzzy applications that falls between these two extremes both in run time and design time/cost. TABLEI INFERENCE TIME BY 51 RULES {{{{Time }}{{MIPS R3000 }}{{ASIC }}{{Regular }}{{With min/mix }}{{6000 inference 1 inference FLIPS }}{{125s 20.8ms 48 }}{{49s 8.2ms 122 }}{{0.0038s 6.4㎲ 156,250 }} }}