• School Mathematics
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• v.18 no.3
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• pp.647-666
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• 2016

This study investigates pre-service teacher's understanding of the concept and representations of irrational numbers. We classified the representations of irrational numbers into six categories; non-fraction, decimal, symbolic, geometric, point on a number line, approximation representation. The results of this study are as follows. First, pre-service teachers couldn't relate non-fractional definition and incommensurability of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations. Thirdly, pre-service teachers had more difficulty moving between symbolic representation and point on a number line representation of ${\pi}$ than that of $\sqrt{5}$ We suggested the concept of irrational numbers should be learned in relation to various representations of irrational numbers.

• Journal of the Korean Home Economics Association
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• v.46 no.9
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• pp.125-135
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• 2008

The purpose of this study was to examine the effects of young children's age and information processing style in understanding spatial coordinates. For sampling the subjects of this study, Korean version K-ABC Intelligence Test(Moon, Soo-Back, 1997)was conducted with 165 children aged 5-6 who were attending I and G kindergarten in D city. From this pool 30 children who possessed sequential processing style and 30 children who possessed simultaneous processing style were sampled. In order to analyze the understanding of spatial coordinates, a test tool was formulated according to methodology of Blades & Spencer(1989) which was modified. Acquired data was subjected to descriptive and comparative statistical analysis. The following conclusions were arrived at: Firstly, there was significant difference between 5-year-olds and 6-year-olds in understanding spatial coordinates. The 6-year-old group got statistically higher grades than the 5-year-old group in locating a point on the coordinate plane and reading the coordinate numbers. Secondly, there was significant difference between children's information processing style in understanding spatial coordinate. Children with high simultaneous-low sequential processing showed higher performance in locating a point on the coordinate plane and reading coordinate numbers than children with high sequential-low simultaneous processing. Thirdly, after verifying statistical significance of interactivity between young children's age and children's processing strength, there was significant interactive effects in both tasks.

• School Mathematics
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• v.10 no.4
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• pp.537-555
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• 2008

The study aims to analyze elementary school students' understanding of the concept of equality sign in contexts, to reflect the types of contexts for equality sign which mathematics textbook series for $1{\sim}4$ grades on natural numbers and its operation provide, and to invetigate the effects of teaching methods of the concept of equality sign suggested in this research. In order to achieve these purposes, the origin, concept, and contexts of equality sign were theoretically reviewed and organized. Also the error types in using equality sign were reflected. Modelling, discussing truth or falsity of equations, identifying relations between numbers and their operation, conjecturing basic properties of numbers and their operations, experiencing diverse contexts for equality sign, and creating contexts for equality sign are set up as teaching methods for better understanding the concept of equality sign. The conclusions are as follows. Firstly, elementary school students' under-standing of the concept of equality sign varied by context and was generally far from satisfactory. In particular, they had difficulties in understanding the concept of the equal sign in contexts with operations on both sides. The most frequently witnessed error was to recognize equality sign as a result of operations. Secondly, student' lack of understanding of the concept of equality sign came from the fact that elementary textbooks failed to provide diverse contexts for equality sign. According to the textbook analysis, contexts with operations on the left side of the equal sign in the form of $a{\pm}b=c$ were provided excessively, with the other contexts hardly seen. Thirdly, teaching methods provided in the study were found to be effective for enhancing understanding the concept of equality sign. In other words, these methods enabled students to focus on relational understanding of concept of equality sign rather than operational one.

• The Mathematical Education
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• v.47 no.4
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• pp.519-543
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• 2008

The study has been conducted with 29 children from 4th to 6th grades to realize how they understand addition, subtraction, multiplication, and division of whole numbers, and how their understanding influences solving of one-step word problems. Children's understanding of operations was categorized into "adding" and "combination" for additions, "taking away" and "comparison" for subtractions, "equal groups," "rectangular arrange," "ratio," and "Cartesian product" for multiplications, and "sharing," "measuring," "comparison," "ratio," "multiplicative inverse," and "repeated subtraction" for divisions. Overall, additions were mostly understood additions as "adding"(86.2%), subtractions as "taking away"(86.2%), multiplications as "equal groups"(100%), and divisions as "sharing"(82.8%). This result consisted with the Fischbein's intuitive models except for additions. Most children tended to solve the word problems based on their conceptual structure of the four arithmetic operations. Even though their conceptual structure of arithmetic operations helps to better solve problems, this tendency resulted in wrong solutions when problem situations were not related to their conceptual structure. Children in the same category of understanding for each operations showed some common features while solving the word problems. As children's understanding of operations significantly influences their solutions to word problems, they needs to be exposed to many different problem situations of the four arithmetic operations. Furthermore, the focus of teaching needs to be the meaning of each operations rather than computational algorithm.

• Journal of the Korea Institute of Military Science and Technology
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• v.10 no.1
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• pp.130-143
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• 2007

Using steady state aerodynamic theories, it has been claimed that insects and birds cannot fly. To make matters worse, insects and birds fly at low Reynolds numbers. Therefore, a recurring theme in the literature is the importance of understanding unsteady aerodynamic effect and how the vortices behave when they separate from the moving surface that created them. In flapping flight, birds and insects can modify wing beat amplitude, stroke angle, wing planform area, angle of attack, and to a lesser extent flapping frequency to optimize the generation of lift force. Some birds are thought to employ two different gaits(a vortex ring gait and a continuous vortex gait) and unsteady aerodynamic effect(Clap and fling, Delayed stall, Wake capture and Rotational Circulation) in flapping flight. Leading edge vortices may produce an increase in lift. The trailing edge vortex could be an important component in gliding flight. Tip vortices in hovering support the body weight of the hummingbirds. Thus, this study investigated how insects and birds generate lift at low Reynolds numbers. This research is written to further that as yet incomplete understanding.

• Journal for History of Mathematics
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• v.18 no.3
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• pp.55-66
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• 2005

In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.

• East Asian mathematical journal
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• v.25 no.3
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• pp.247-262
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• 2009

Permutation and combination are the part of mathematics which can be introduced the pliability and diversity of thought. In prior studies of permutation and combination, there treated difficulties of learning, strategy of problem solving, and errors that students might come up with. This paper provides the method so that meaningful teaching and learning might occur through the systematic approach of permutation and combination. But there were little prior studies treated counting numbers that basic of mathematics' action. Therefore this paper tries to help the understanding of permutation and combination with the process of changing from informal knowledge to formal knowledge.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.3 no.2
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• pp.173-177
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• 2003

In the first part an overview on fuzzy sets and fuzzy numbers is given. A detailed treatment of these notions is introduced in [1,2,3]. This sintetically presentation is useful in understanding and in developping the applications in context problems. In the second part, fuzzy context model is given as an application of fuzzy sets and the fuzzy equilibrium equation is solved [4,5].

• Journal of Educational Research in Mathematics
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• v.25 no.3
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• pp.263-279
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• 2015

The introduction of real numbers is one of the most difficult steps in the teaching of school mathematics since the mathematical justification of the extension from rational to real numbers requires the completeness property. The author elucidated what questions about real numbers can be unanswered as the "institutional didactic void" in school mathematics defining real numbers as the union of the rational and irrational numbers. The pre-service teachers' explanations on the extension from rational to real numbers and the raison d'$\hat{e}$tre of arbitrary non-recurring decimals showed the superficial and fragmentary understanding of real numbers. Connecting school mathematics to university mathematics via the didactic void, the author discussed how pre-service teachers could break the illusion of transparency about the real number.

• Journal of Educational Research in Mathematics
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• v.22 no.2
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• pp.163-178
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• 2012

The efforts to represent the numbers based on the decimal system give us fundamental understanding to construct and teach the concept network on the related knowledge of elementary and secondary school mathematics. In the process to represent natural numbers, integers, rational numbers, real numbers as decimal system, we will classify the extended decimal system. Moreover we will obtain the view to classify real numbers. In this paper, we will study the didactical significance of mathematical knowledge, which arise from process to represent real numbers as decimal system, starting from decimal system representation of natural numbers, and provide the theoretical base about the classification of real numbers. This study help math teachers to understand school mathematics in critical inside-measurement and provide the theore tical background of related knowledge. Furthermore, this study provide a clue to construct coherent curriculum and internal connections of related mathematical knowledge.